diff --git a/-tAzT4oBgHgl3EQf_P79/content/tmp_files/2301.01947v1.pdf.txt b/-tAzT4oBgHgl3EQf_P79/content/tmp_files/2301.01947v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6302ec49ba48ac3198cce61571b05a93e7b32d18 --- /dev/null +++ b/-tAzT4oBgHgl3EQf_P79/content/tmp_files/2301.01947v1.pdf.txt @@ -0,0 +1,1113 @@ +arXiv:2301.01947v1 [cs.LG] 5 Jan 2023 +StitchNet: Composing Neural Networks from Pre-Trained Fragments +Surat Teerapittayanon, Marcus Comiter, Brad McDanel, H.T. Kung +Abstract +We propose StitchNet, a novel neural network creation +paradigm that stitches together fragments (one or more con- +secutive network layers) from multiple pre-trained neural net- +works. StitchNet allows the creation of high-performing neu- +ral networks without the large compute and data requirements +needed under traditional model creation processes via back- +propagation training. We leverage Centered Kernel Align- +ment (CKA) as a compatibility measure to efficiently guide +the selection of these fragments in composing a network for a +given task tailored to specific accuracy needs and computing +resource constraints. We then show that these fragments can +be stitched together to create neural networks with compa- +rable accuracy to traditionally trained networks at a fraction +of computing resource and data requirements. Finally, we ex- +plore a novel on-the-fly personalized model creation and in- +ference application enabled by this new paradigm. +1 +Introduction +AI models have become increasingly more complex to sup- +port additional functionality, multiple modalities, and higher +accuracy. While the increased complexity has improved +model utility and performance, it has imposed significant +model training costs. Therefore, training complex models is +often infeasible for resource limited environments such as +those at the cloud edge. +In response to these challenges, in this paper we propose a +new paradigm for creating neural networks: rather than train- +ing networks from scratch or retraining them, we create neu- +ral networks through composition by stitching together frag- +ments of existing pre-trained neural networks. A fragment is +one or more consecutive layers of a neural network. We call +the resulting neural network composed of one or more frag- +ments a “StitchNet” (Figure 1). By significantly reducing the +amount of computation and data resources needed for creat- +ing neural networks, StitchNets enable an entire new set of +applications, such as rapid generation of personalized neural +networks at the edge. +StitchNet’s model creation is fundamentally different +from today’s predominant backpropagation-based method +for creating neural networks. Given a dataset and a task +as input, the traditional training method uses backpropaga- +tion with stochastic gradient descent (SGD) or other opti- +mization algorithms to adjust the weights of the neural net- +works. This training process iterates through the full dataset +StitchNets +Fragments +Existing Networks +F0 of N0 +F1 of N0 +0 +F2 of N0 +1 +F3 of N0 +2 +F4 of N0 +3 +F5 of N0 +4 +F6 of N0 +5 +F7 of N0 +6 +F0 of N1 +F1 of N1 +0 +F2 of N1 +1 +F3 of N1 +2 +F4 of N1 +F0 of N1 +F1 of N1 +F2 of N1 +F3 of N1 +F4 of N1 +3 +F0 of N3 +F1 of N3 +0 +F2 of N3 +1 +F3 of N3 +2 +F4 of N3 +3 +F5 of N3 +F0 of N3 +F1 of N3 +F2 of N3 +F3 of N3 +F4 of N3 +F5 of N3 +4 +F0 of N3 +F1 of N3 +0 +F1 of N1 +1 +F2 of N1 +2 +F3 of N1 +3 +F4 of N1 +4 +F0 of N1 +F1 of N1 +0 +F7 of N0 +1 +F0 of N0 +F1 of N0 +F2 of N0 +F3 of N0 +F4 of N0 +F5 of N0 +F6 of N0 +F7 of N0 +AlexNet +ResNet +DenseNet +Figure 1: Overview of the StitchNet approach. Existing net- +works (left) are cut into fragments (middle), which are com- +posed into StitchNets (right) created for a particular task. No +retraining is needed in this process. +multiple times, and therefore requires compute resources +that scale with the amount of data and the complexity of +the network. Training large models this way also requires +substantial amounts of data. While successful, this tradi- +tional paradigm for model creation is not without its limi- +tations. Creating complex neural networks without access to +large amounts of data and compute resources is a growing +challenge of increasing significance, especially in resource- +constrained edge environments. In the extreme case (e.g., for +very large language and computer vision models), only a +few companies with access to unrivaled amounts of data and +compute resources are able to create such models. +StitchNets solve this problem by creating new neural net- +works using fragments of already existing neural networks. +The new approach takes advantage of the growing amount +of neural networks that already exist, having been trained + +previously by many groups and companies. StitchNets en- +able the efficient reuse of the learned knowledge resident in +those pre-trained networks, which has been distilled from +large amounts of data, rather than having to relearn it over +and over again for new tasks as we do with traditional model +creation paradigms. StitchNet’s ability to reuse existing pre- +trained fragments, rather than recreating from scratch or re- +training for every task will help accelerate the growth and +application of neural networks for solving more and more +complex tasks. +However, compositing these existing fragments into +a +coherent +and +high +performing +neural +network +is +non-trivial. To reuse the knowledge of pre-trained neu- +ral network fragments, we need a way to 1) measure +the compatibility between any two fragments, and 2) +compose compatible fragments together. In the past, Cen- +tered Kernel Alignment (CKA) (Kornblith et al. 2019; +Cortes, Mohri, and Rostamizadeh +2012; +Cristianini et al. +2006) has been used to measure similarity between neural +network representations. We leverage CKA to assess the +compatibility of any two fragments from any neural net- +works and compose new neural networks from fragments +of existing pre-trained neural networks to create high +performing networks customized for specific tasks without +the costs of traditional model creation methods. The CKA +score is used to reduce the search space for identifying +compatible fragments and guide the fragment selection +process. +We present empirical validations on benchmark datasets, +comparing the performance of StitchNets to that of the origi- +nal pre-trained neural networks. We demonstrate that Stitch- +Nets achieve comparable or higher accuracy on personalized +tasks compared with off-the-shelf networks, and have signif- +icantly lower computational and data requirements than cre- +ating networks from scratch or through retraining. +Our contributions are: +• The StitchNet paradigm: a novel neural network creation +method that enables a new set of applications. +• A novel use of Centered Kernel Ailgnment (CKA) in as- +sessing the compatibility of any two fragments for their +composition. +• A technique to compose compatible fragments together +for both linear and convolutional layers. +• A feasibility demonstration of StitchNets for efficient on- +the-fly personalized neural network creation and infer- +ence. +2 +Composing Fragments +The core mechanism to create StitchNets is to iden- +tify reusable fragments from a pool of existing net- +works and compose them into a coherent neural net- +work model capable of performing a given task. To this +end, we need a way to determine how compatible any +two candidate fragments are with each other. In previ- +ous work, (Kornblith et al. 2019) present centered kernel +alignment (CKA) (Cortes, Mohri, and Rostamizadeh 2012; +Cristianini et al. 2006) as a way to measure similarity be- +tween neural network representations. Rather than looking +at the neural network as a whole, we adopt and use CKA to +as a measure of compatibility between any two fragments of +any neural networks. +In this section, we first define CKA as a way to measure +how compatible any two fragments are with one another and +therefore their ability to be composed. Using CKA, we then +present a technique to stitch different fragments together. Fi- +nally, we describe the algorithm to generate StitchNets. +2.1 +Centered Kernel Alignment (CKA) +Given X ∈ Rp×n as outputs of a fragment FA of model A +and Y ∈ Rq×n as inputs of a fragment FB of model B of +the same dataset D, where n is the number of samples in the +dataset, p is the output dimension of FA and q is the input di- +mension of FB. Let Kij = k(xi, xj) and Mij = m(yi, yj), +where k and m are any two kernels. We define the compat- +ibility score CKA(X, Y) of fragment FA and fragment FB +as +CKA(X, Y) = +HSIC(K, M) +� +HSIC(K, K) HSIC(M, M) +, +where HSIC is the Hilbert-Schmidt Independence Criterion +(Gretton et al. 2005) defined as +HSIC(K, M) = +1 +(n − 1)2 tr(K H M H), +where H is the centering matrix Hn = In − 1 +n11T and tr is +the trace. For linear kernels, k(x, y) = m(x, y) = xT y, +HSIC becomes HSIC(X, Y) = ∥cov(XT X, YT Y)∥2 +F , +where cov is the covariance function, and CKA(X, Y) be- +comes +∥cov(XT X, YT Y)∥2 +F +� +∥cov(XT X, XT X)∥2 +F ∥cov(YT Y, YT Y)∥2 +F +. +(1) +We use this function (Eq. 1) as a measurement of how com- +patible any two fragments are, given a target dataset. To re- +duce memory usage for a large target dataset, CKA can be +approximated by averaging over minibatches as presented in +(Nguyen, Raghu, and Kornblith 2020). +2.2 +Stitching Fragments +Once we have determined compatible fragments, the next +step in creating a StitchNet is to stitch the two fragments +together. To do so, we find a projection tensor A that projects +the output space of one fragment to the input space of the +other fragment we are composing. We now describe this. +Without loss of generality, we assume the output and in- +put tensors are 2D tensors, where the first dimension is the +sample dimension. If the tensors are not 2D tensors, we first +flatten all other dimensions with the exception of the sample +dimension. +We use Einstein summation notation, where i represents +the sample dimension, j the output dimension of the incom- +ing fragment, and k the input dimension of the outgoing frag- +ment. Given an output tensor Xij of the incoming fragment +and an input tensor Yik of the outgoing fragment, we seek + +to find A such that Yik = Akj Xij . We can then solve for +A using the Moore-Penrose pseudoinverse: +Akj = Yik XT +ij(Xij XT +ij).−1 +(2) +Once A is found, we fuse A with the weight of the first +layer of the outgoing fragment. For linear layers, we simply +do the following: +W′ +lk = Wlj Akj, +(3) +where l is the dimension of the output feature of the outgoing +layer. +For convolutional layers, we first upsample or downsam- +ple the spatial dimension to match each other, and then ad- +just the weight along the input channel dimension as follows. +W′ +okmn = WijmnAkj, +(4) +where o is the output channel dimension, j is the original +input channel dimension, k is the new input channel dimen- +sion, and m and n are the spatial dimensions. +For stitching a convolutional layer with an output tensor +X and a linear layer with an input tensor Y, we first apply +adaptive average pooling so that the spatial dimension is 1x1 +and flatten X into a 2D tensor. Then, we follow Eq. 2 and +Eq. 3 to find A and fuse it with the W of the linear layer. +2.3 +StitchNet Generation +Algorithm 1: StitchNet(P, D, K, T , L, R, Q, s) +Input: fragment pool P = {Fij}, network i in P up to +layer j Nij, fragment ending in layer j of network i Fij, +target dataset D with M samples, span K, threshold T , +maximum number of fragments L, result list of Stitch- +Nets and their associated scores R, current StitchNet Q, +current score s +Output: resulting list of StitchNets and their associated +scores R +if Q is empty then +{Fij} = select starting fragments in P +for Fij in {Fij} do +StitchNet(P, D, K, T , L, R, Fij, 1) +if the number of fragments in Q ≥ L then +return R +{Fij} = select K middle or terminating fragments in P +for Fij in {Fij} do +X = Q(D); Y = Nij(D) +sn = s× CKA(X, Y) (see section 2.1) +if sn > T then +Q = Stitch(Q, Fij, X, Y) (see section 2.2) +if Fij is a terminating fragment then +R.append({Q, sn}) +else +StitchNet(P, D, K, T , L, R, Q, sn) +return R +We now describe the main algorithm for creating Stitch- +Net networks (“StitchNets” for short), shown in Algorithm 1. +A StitchNet network is created by joining a set of pre-trained +network fragments drawn from a pool P = {Fij}. We use +the notation Fij to denote a fragment of a neural network i +up to its j layer, and the notation Nij to denote the compu- +tation performed by the portion of the neural network from +which the fragment was taken. Other than the fragment pool +P and creation process hyperparameters (K, T, L), the only +other input to the StitchNet creation process is a dataset D +for which the StitchNet will be optimized. +We now describe the creation of the pool of network frag- +ments P derived from a set of pre-trained off-the-shelf net- +works. These pre-trained networks are divided into one of +three types of fragments: starting fragments for which the +input is the original network input, terminating fragments +for which the output is the original network output, and mid- +dle fragments that are neither starting nor terminating frag- +ments. +The first step in the StitchNet creation process is to choose +the set of starting fragments. This could include all starting +fragments in P, or a subset based on certain criteria, e.g., the +smallest, biggest or closest starting fragment. +Once a set of starting fragments are selected, a StitchNet +is built on top of each starting fragment having a current +starting score of 1. First, a set of K candidate fragments are +selected from P. These fragments can be selected based on +CKA scores (i.e., K fragments with highest CKA scores), +the number of parameters of the fragments (i.e., K frag- +ments with the least amount of number of parameters in +P), the closest fragments (i.e., K fragments with the least +latency in P in a distributed fragments setting), or other se- +lection methods. +For each of the candidate fragments, we then compute two +intermediate neural network computations. First, we pass the +dataset D through the candidate StitchNet in its current form, +resulting in value X. Second, we pass the same dataset D +through the neural network from which the candidate frag- +ment Fij was selected, resulting in value Y = Nij(D). +After +running +these +computations, +we +produce +CKA(X, Y) as in Section 2.1. We then multiply the +CKA with the current score s to obtain the new current +score sn. If sn is still greater than a set threshold T , the +candidate fragment is selected and the process continues re- +cursively. Otherwise, the candidate fragment is rejected. The +threshold can be set to balance the amount of exploration +allowed per available compute resources. +This process continues until a terminating fragment is se- +lected, the maximum number of fragments L is reached or +all recursive paths are exhausted. At this point, the com- +pleted StitchNets and their associated scores R are returned +for user selection. +3 +Results +We now demonstrate that StitchNets can perform compara- +bly with traditionally trained networks but with significantly +reduced computational and data requirements at both infer- +ence and creation time. Through these characteristics, Stitch- +Nets enable the immediate on-the-fly creation of neural net- +works for personalized tasks without traditional training. + +3.1 +Fragment pool +To form the fragment pool P, we take five off-the-shelf net- +works pre-trained on the ImageNet-1K dataset (Deng et al. +2009) from Torchvision (Marcel and Rodriguez 2010): +alexnet, densenet121, mobilenet v3 small, resnet50 and +vgg16 with IMAGENET1K V1 weights. +These pre-trained networks are cut into fragments at +each convolution and linear layer that has a single in- +put. As shown in Figure 2, there are 8 fragments for +alexnet, 5 fragments for densenet121, 13 fragments for mo- +bilenet v3 small, 6 fragments for resnet50 and 16 fragments +for vgg16. This results in the creation of a fragment pool P +of 48 fragments consisting of 5 starting fragments, 38 mid- +dle fragments, and 5 terminating fragments. We use this frag- +ment pool in all experiments in this paper. +3.2 +Dataset +The dataset used to evaluate StitchNets in this paper is the +“Dogs vs. Cats” dataset (Kaggle 2013). This dataset includes +25,000 training images of dogs and cats and we use an +80:20 train:test split. We map ImageNet-1K class labels into +cat and dog labels (class IDs 281-285 and 151-250, respec- +tively). To form the target dataset D for use in the stitch- +ing process of Algorithm 1, we randomly select M samples +from the training set as the target dataset D. We choose this +task because it is characteristic of the type of task for which +StitchNets would be used: a user needs a custom classifier +for a particular task and desired set of classes. +3.3 +StitchNet Generation +We generate StitchNets with Algorithm 1 using the fragment +pool and the dataset described in Section 3.1 and 3.2. We set +K = 2, T = 0.5 and L = 16. The number of samples M in +D used for the stitching process is 32. +Given these hyperparameters, a total of 89 StitchNets are +generated. We evaluate them on the test set of completely +unseen test samples. Summary statistics for the generated +StitchNets are shown in Figure 3, including accuracy (3a), +number of fragments per StitchNet (3b), CKA score (3c), +and number of parameters per StitchNet (3d). +3.4 +Reduction in Inference Computation +We now demonstrate how StitchNets significantly reduce +inference-time computational requirements over traditional +neural network training paradigms by studying StitchNet ac- +curacy as a function of parameters. +Figure 4 shows the resulting accuracy of the generated +StitchNets as a function overall CKA score for each Stitch- +Net and number of parameters (porportional to marker size) +as a proxy for inference-time computation cost. We find a +number of StitchNets outperform the pre-trained network +while realizing significant computational savings. For exam- +ple, StitchNet27 (denoted by a green star) achieves an ac- +curacy of 0.86 with 3.59M parameters compared with the +0.70 accuracy of the pre-trained alexnet with 61.10M param- +eters. Therefore, StitchNet achieves a 22.8% increase in ac- +curacy with a 94.1% reduction in number of parameters for +alexnet +densenet121 +mobilenet +resnet50 +vgg16 +Figure 2: Five pre-trained networks are fragmented into a +fragment pool P. These fragments will be stitched together +to form StitchNets. +the given task when compared with those of the pre-trained +alexnet. +These crystallizes one of the core benefits of StitchNets: +without any training, the method can discover networks that +are personalized for the task, outperform the original pre- +trained networks, and do so while significantly reducing +inference-time compute requirements. This is due to the fact +that these pre-trained networks are not trained to focus on +these two specific classes, while our StitchNets are stitched +together specifically for the task. In the next section, we will + +F15ofN4 +14 +F14 0fN4 +13 +F130fN4 +F120fN4 +11 +lio +FIOofN4 +F8ofN4 +F7ofN4 +F5ofN4 +F4ofN4F5ofN3 +F4ofN3 +F36fN3 +F20fN3 +FIofN3 +FOOfN3F12ofN2 +11 +F11ofN2 +10 +F10OfN2 +1oofN2 +F8ofN2 +F7ofN2 +F6ofN2 +F5ofN2 +F40fN2 +F3ofN2 +F2ofN2 +F1ofN2 +FOofN2F4OfNI +F3OfNI +F2ofNI +FIOfNI +FOOFNIF7ofNO +F6ofNO +F5ofNo +F4of NO +F3OfNO +F2ofNO +FIofNo +FOof No0.67 +to +0.73 +0.73 +to +0.78 +0.78 +to +0.84 +0.84 +to +0.90 +0.90 +to +0.95 +9 +19 +27 +25 +9 +(a) accuracy +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +1 +2 +6 +6 +3 +3 +3 +4 +8 +8 +11 +10 +8 +16 +(b) # fragments +0.50 +to +0.60 +0.60 +to +0.70 +0.70 +to +0.80 +0.80 +to +0.90 +0.90 +to +1.00 +29 +28 +14 +11 +7 +(c) CKA score +1M +to +21M +21M +to +42M +42M +to +62M +62M +to +83M +83M +to +104M +104M +to +124M +124M +to +145M +66 +5 +4 +5 +2 +3 +4 +(d) # parameters +Figure 3: Histogram of (a) accuracy, (b) # fragments, (c) +CKA score, (d) # parameters in the generated batch of Stitch- +Nets. +see that very little data is required for the stitching process. +Additionally, we compare the StitchNets with the var- +ious off-the-shelf models, denoted by triangles. We find +that the StitchNet generation process creates many different +StitchNets that outperform the off-the-shelf models, many of +which do so at reduced computational cost. Figure 5 shows +the composition of some of these high-performing Stitch- +Nets, demonstrating the diversity in fragment use, ordering, +and architectures. +We also validate the effectiveness of using CKA to guide +the stitching procedure. We find that StitchNets with a high +CKA score also have high accuracy, especially those above +0.9. This shows that CKA can be used as a proxy to measure +good compatibility between connecting fragments.1 +3.5 +Reduction in Network Creation Computation +We now demonstrate that StitchNets can be created without +significant data and computation requirements. Specifically, +we compare StitchNet21 (generated in Figure 5 on the tar- +get dataset of M = 32 samples) with fine-tuning the same +five off-the-shelf networks (retraining them using the train- +ing portion of dataset of Section 3.2). For fine-tuning, we +replace and train only the last layer of the pre-trained net- +work using Stochastic Gradient Descent (SGD) with batch +size 32, learning rate 0.001 and momentum 0.9. The results +shown are averaged over 10 runs. For ease of comparison, +we normalize the computation cost in terms of the num- +ber of samples processed through a neural network. In prac- +1Note that there exist high accuracy StitchNets with low overall +CKA score. This is because neural networks are robust and highly +redundant, able to tolerate a certain amount of errors while still +giving quality predictions (see Section 4.1). +tice, fine-tuning requires backpropagation, which incurs ad- +ditional computation per sample than StitchNet generation. +Figure 6 compares the accuracy of StitchNet21 (denoted +by the red star) with the traditionally fine-tuned networks +as a function of the number of training samples processed. +For a given accuracy target, StitchNets process a substan- +tially smaller number of data samples than traditionally fine- +tuned networks. Specifically, to reach an accuracy of 0.95, +fine-tuning of alexnet, densenet121, and mobilenet v3 small +require to process more than 320 samples while StitchNet re- +quires only 32 samples used to stitch the fragments together +(realizing over a 90% reduction). +Therefore, only a small amount of training samples and +computation are required for StitchNet to achieve compara- +ble accuracy. This demonstrates that StitchNets effectively +reuse the information already captured in the fragments to +bootstrap network creation. This allows for personalization +of tasks and on-the-fly training without substantial data re- +quirements. +3.6 +Ensembles +We now discuss the ability to ensemble generated StitchNets +to improve performance. StitchNet and ensembling methods +are complimentary. The StitchNet generation algorithm pro- +duces a set of candidate models. While a user can select a +single StitchNet to use at inference time, because the Stitch- +Net generation procedure finds such efficient models, we can +also take advantage of the pool of StitchNets and ensemble +some while still realize substantial computational savings. +We pick 10 random models from the generated StitchNets +in Section 3.3 with overall CKA > 0.8. We sort these mod- +els based on their overall CKA scores from high to low, and +then ensemble them by averaging their predicted probabili- +ties. The results are shown in Figure 7. The ensemble often +results in higher accuracy than the individual model. As a re- +sult, this ensembling method can reduce variance in perfor- +mance when on-the-fly network creation and inference (as +discussed in Section 4.3) is used and there is not time for full +selection of a final single StitchNet. Instead, the user can se- +lect a reasonably small subset of high performing StitchNets, +which even in aggregate can be significantly smaller than a +single traditionally trained network. +4 +Discussion +We now discuss the intuition behind StitchNets, examine +their complexity and relation to related methods, introduce +new applications they enable, and discuss their limitations. +4.1 +Why do StitchNets work? +We first discuss why we are able to reuse existing fragments +of networks to create new neural networks without retrain- +ing. One core reason for this is that neural networks tend to +learn fundamental and universal features. Studies (Li et al. +2015; Lu et al. 2018; Morcos, Raghu, and Bengio 2018; +Wang et al. 2018; Lenc and Vedaldi 2015; Kornblith et al. +2019; Tang et al. 2020) have shown that neural networks +learn fundamental features such as edges for different tasks. +Since these learned features are fundamental, they should + +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Overall CKA score +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Accuracy +Smallest +acc=0.73 +cka=0.53 +0.57M +Best +acc=0.95 +cka=0.91 +8.04M +StitchNet27 +acc=0.86 +cka=0.94 +3.59M +alexnet +acc=0.70 +cka=0.89 +61.10M +densenet121 +acc=0.85 +cka=1.00 +8.04M +mobilenet_v3_small +acc=0.78 +cka=1.00 +2.54M +resnet50 +acc=0.85 +cka=0.99 +25.53M +vgg16 +acc=0.81 +cka=0.85 +138.36M +Figure 4: Accuracy vs the overall CKA score on “Cat vs. Dogs.” cka is the overall CKA score, acc is the accuracy. The best +StitchNet (acc=0.95) performs 12% better than the best pre-trained model(s) (densenet121 and resnet50 with acc=0.85). +StitchNet21 +acc=0.95 +cka=0.91 +8.04M +StitchNet22 +acc=0.89 +cka=0.84 +5.33M +StitchNet5 +acc=0.82 +cka=0.81 +61.10M +StitchNet32 +acc=0.79 +cka=0.88 +1.99M +StitchNet88 +acc=0.78 +cka=0.77 +8.15M +Figure 5: Examples of generated StitchNets. +be reusable rather relearned. The challenge, however, is that +although these features may be universal, they may not be +compatible with one another “out of the box.” Therefore, +we require the stitching process introduced in Section 2.2 +to project the fragments into a compatible space. +50 +100 +150 +200 +250 +300 +The number of training samples processed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy +StitchNet21 +acc@32=0.95 +alexnet +acc@320=0.93±0.01 +densenet121 +acc@320=0.90±0.04 +mobilenet_v3_small +acc@320=0.93±0.01 +resnet50 +acc@320=0.97±0.01 +vgg16 +acc@320=0.97±0.00 +Figure 6: Accuracy vs the number of training samples pro- +cessed (i.e., data and computation required). StitchNets re- +quire only a fraction of the computation of traditional train- +ing methods to achieve comparable performance. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Model in the ensemble +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +Accuracy +Ensemble Accuracy +Individual Accuracy +Figure 7: Accuracy of the ensemble models. Ensembling +groups of StitchNets can reduce individual model variance. +Beyond this reuse of universal features and compatibility +transformations, StitchNets are also enabled by the fact that +neural networks are fundamentally robust. Due to the non- +linear activation and built-in redundancies, neural networks +tolerate certain amounts of error. As such, the fragments +need not be perfectly compatible individually to produce a +network that in aggregate operates at a high level of perfor- +mance. +4.2 +Complexity Comparison +We now compare the complexity of the traditional train- +ing process using backpropagation with the StitchNet gen- + +F120fN2 +F9of N4 +F8ofN4 +F7ofN4 +F6ofN4 +F5ofN4 +F4of N4 +F3ofN4 +F2 ofN4 +FIofN4 +FOof N4F12 of N2 +10 +F11 of N2 +F9 of N2 +F8 of N2 +F7 of N2 +F6 of N2 +F5 of N2 +F4 of N2 +F3 of N2 +F2 of N2 +F1 of N2 +FO of N2F12ofN2 +F6of NO +F5ofNO +F4of NO +F3ofNO +F2of NO +FIof NO +FOof NOF12.0fN2 +F2OfNI +FIofNI +0 +FOofNIF5ofN3 +F3ofNI +F2ofNI +FIOfNI +FOofNIeration process. Traditional training complexity is O(ndp), +where n is the number of parameters in the network, p is +the number of epochs used to train, and d is the size of +the dataset. StitchNet generation complexity is O(nqm) + +O(KL). The first term nqm is the evaluation cost of the tar- +get dataset of size q on m networks in the pool, where q ≪ d +and n is the number of parameters in the network (assuming +networks have the same # of parameters). The second term +KL is the search cost, where K is the span value we search +at each level and L is the max depth to search. Using a high +threshold cutoff T on the overall CKA score keeps search +cost KL small. Therefore, for a reasonable setting of hyper- +parameters (K, T, L) in Algorithm 1, StitchNets realize sub- +stantial computation gains over traditional training methods +since q ≪ d and m ≪ p. +4.3 +On-the-fly network creation and inference +We now discuss a new family of applications and use cases +that are enabled by StitchNets: on-the-fly neural network cre- +ation and inference. In this application, we use a batch of im- +ages on which we want to perform a task (e.g., classification +or detection) as our target dataset in the StitchNet generation +process. With only a minor modification to the StitchNet al- +gorithm to additionally return task results, the StitchNet gen- +eration process can return the inference outputs along with +the generated StitchNets. +We now describe how this can be used in practice. Imag- +ine a world where fragments of pre-trained neural networks +for different tasks are indexed and distributed on the Inter- +net. Any compatible fragment can be found and composed +quickly to form a new neural network for a certain task. Now, +imagine we want to create a neural network for classifying +local cats and dogs with only a few hundred of these unla- +beled images. +Without StitchNets, we either need to train a network +from scratch (which may fail due to our limited amount of +training data), or find an existing pre-trained neural network, +label the dataset, and finetune the network. If the existing +pre-trained network is too big or too slow for our use, we +will then have to train a new one from scratch. But, with lim- +ited amount of unlabeled data, this task seems impossible. +With StitchNet, we can instead generate a set of candidate +StitchNets with the small target dataset of unlabeled local +cats and dogs images. These StitchNets are created from the +pool of existing neural network fragments that have been in- +dexed and distributed on the Internet. The proper fragments +can be identified with a search criteria (e.g., the terminat- +ing fragment should contain cat and dog classes, the depth +of the network should be less than 5 for computational effi- +ciency reasons, etc.). With little computation, we will gener- +ate StitchNets capable of detecting and classifying local cats +and dogs. +4.4 +Limitations +One limitation is that the target task needs to be a subset +(or a composition) of the terminating fragment tasks in the +fragment pool. Additionally, while a large pool of networks +and fragments can lead to higher applicability and quality +of StitchNets, it can also lead to high search costs. Index- +ing large quantities of neural networks to form the fragment +pool will require novel search methods. We see this as anal- +ogous to indexing web pages on the World Wide Web, sug- +gesting a “Google for Fragments.” Much like web search +needed to index written content, large amounts of neural net- +work “content” need to be indexed in order for their value to +be unlocked. Early indexing efforts can tag fragments based +on dataset characteristics, computational characteristics, etc. +More advanced efforts can look at inward and outward con- +nections of each fragment to determine its rank in results. +Once a narrowed set of fragments are coarsely identified, the +efficient procedure introduced in this paper can generate the +StitchNets. Future work will address these types of comple- +mentary methods (indexing and distribution) that will enable +StitchNets to operate at scale. +5 +Related Work +Transfer learning (or fine-tuning) (Pan and Yang 2009; +Weiss, Khoshgoftaar, and Wang 2016) is the current pre- +dominant way to adapt existing neural networks to target +tasks. Unsupervised domain adaptation is related, where the +existing network is adapted using an unlabeled target dataset. +StitchNets work similarly by stitching fragments using an +unlabeled target dataset to create a neural network for the +target task. Most work (Wang and Deng 2018; Zhang et al. +2018; Tzeng et al. 2014; Kumar et al. 2018; Shu et al. 2018; +Ben-David et al. 2010; Saito, Ushiku, and Harada 2017) fo- +cuses on retraining the network, while StitchNet does not +require any training. +StitchNets take advantage of the assumption that the frag- +ments have shareable representations. This assumption helps +explain why fragments can be stitched together into a coher- +ent high-performing network: dissimilar yet complimentary +fragments once projected into a similar space are compatible +with one another. Several existing works including (Li et al. +2015; Mehrer, Kriegeskorte, and Kietzmann 2018; Lu et al. +2018; Morcos, Raghu, and Bengio 2018; Wang et al. 2018; +Lenc and Vedaldi 2015; Kornblith et al. 2019; Tang et al. +2020) have studied this shareable representation assumption. +(Gygli, Uijlings, and Ferrari 2021) reuse network compo- +nents by training networks to produce compatible features +by adding regularization at training time to make the net- +works directly compatible. StitchNet, however, focuses on +creating neural networks without training. It is therefore +more generally applicable. (Lenc and Vedaldi 2015) com- +bine network components by adding a stitching layer and +training the recombined network with a supervised loss for +several epochs. StitchNet adds a parameter-less stitching +mechanism and therefore does not require any retraining. In- +stead, weights are adapted to be compatible with the method +introduced in 2.2. +6 +Conclusion +StitchNet is a new paradigm that can leverage a growing +global library of neural networks to fundamentally change +the way networks are created. By reusing fragments of these +networks to efficiently compose new networks for a given + +task, StitchNet addresses two of the most fundamental is- +sues limiting neural network creation and use: large data and +computation requirements. +StitchNet does this by leveraging Centered Kernel Align- +ment (CKA) as a compatibility measure that guides the se- +lection of neural network fragments, tailored to specific ac- +curacy needs and computing resource constraints. Our work +has shown that neural networks can be efficiently created +from compatible neural network fragments of different mod- +els at a fraction of computing resources and data require- +ments with a comparable accuracy. We also introduce on- +the-fly efficient neural network creation and inference appli- +cation that is unlocked by this method. +References +Ben-David, S.; Blitzer, J.; Crammer, K.; Kulesza, A.; +Pereira, F.; and Vaughan, J. W. 2010. A theory of learning +from different domains. Machine learning, 79(1): 151–175. +Cortes, C.; Mohri, M.; and Rostamizadeh, A. 2012. Algo- +rithms for learning kernels based on centered alignment. The +Journal of Machine Learning Research, 13(1): 795–828. +Cristianini, N.; Kandola, J.; Elisseeff, A.; and Shawe-Taylor, +J. 2006. On kernel target alignment. In Innovations in ma- +chine learning, 205–256. Springer. +Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei- +Fei, L. 2009. Imagenet: A large-scale hierarchical image +database. 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In Proceedings of the IEEE conference on computer +vision and pattern recognition, 3801–3809. + diff --git a/-tAzT4oBgHgl3EQf_P79/content/tmp_files/load_file.txt b/-tAzT4oBgHgl3EQf_P79/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d20b72e286a30098e7db213669d403f2c66edeee --- /dev/null +++ b/-tAzT4oBgHgl3EQf_P79/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf,len=786 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='01947v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='LG] 5 Jan 2023 StitchNet: Composing Neural Networks from Pre-Trained Fragments Surat Teerapittayanon, Marcus Comiter, Brad McDanel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Kung Abstract We propose StitchNet, a novel neural network creation paradigm that stitches together fragments (one or more con- secutive network layers) from multiple pre-trained neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet allows the creation of high-performing neu- ral networks without the large compute and data requirements needed under traditional model creation processes via back- propagation training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We leverage Centered Kernel Align- ment (CKA) as a compatibility measure to efficiently guide the selection of these fragments in composing a network for a given task tailored to specific accuracy needs and computing resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We then show that these fragments can be stitched together to create neural networks with compa- rable accuracy to traditionally trained networks at a fraction of computing resource and data requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Finally, we ex- plore a novel on-the-fly personalized model creation and in- ference application enabled by this new paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 1 Introduction AI models have become increasingly more complex to sup- port additional functionality, multiple modalities, and higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' While the increased complexity has improved model utility and performance, it has imposed significant model training costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Therefore, training complex models is often infeasible for resource limited environments such as those at the cloud edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In response to these challenges, in this paper we propose a new paradigm for creating neural networks: rather than train- ing networks from scratch or retraining them, we create neu- ral networks through composition by stitching together frag- ments of existing pre-trained neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' A fragment is one or more consecutive layers of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We call the resulting neural network composed of one or more frag- ments a “StitchNet” (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' By significantly reducing the amount of computation and data resources needed for creat- ing neural networks, StitchNets enable an entire new set of applications, such as rapid generation of personalized neural networks at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet’s model creation is fundamentally different from today’s predominant backpropagation-based method for creating neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Given a dataset and a task as input, the traditional training method uses backpropaga- tion with stochastic gradient descent (SGD) or other opti- mization algorithms to adjust the weights of the neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This training process iterates through the full dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='StitchNets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='Fragments ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F0 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F1 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F2 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F3 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F4 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F5 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F6 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='F7 of N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='AlexNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='ResNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='DenseNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='Figure 1: Overview of the StitchNet approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Existing net- works (left) are cut into fragments (middle), which are com- posed into StitchNets (right) created for a particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' No retraining is needed in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' multiple times, and therefore requires compute resources that scale with the amount of data and the complexity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Training large models this way also requires substantial amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' While successful, this tradi- tional paradigm for model creation is not without its limi- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Creating complex neural networks without access to large amounts of data and compute resources is a growing challenge of increasing significance, especially in resource- constrained edge environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In the extreme case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', for very large language and computer vision models), only a few companies with access to unrivaled amounts of data and compute resources are able to create such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNets solve this problem by creating new neural net- works using fragments of already existing neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The new approach takes advantage of the growing amount of neural networks that already exist, having been trained previously by many groups and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNets en- able the efficient reuse of the learned knowledge resident in those pre-trained networks, which has been distilled from large amounts of data, rather than having to relearn it over and over again for new tasks as we do with traditional model creation paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet’s ability to reuse existing pre- trained fragments, rather than recreating from scratch or re- training for every task will help accelerate the growth and application of neural networks for solving more and more complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' However, compositing these existing fragments into a coherent and high performing neural network is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' To reuse the knowledge of pre-trained neu- ral network fragments, we need a way to 1) measure the compatibility between any two fragments, and 2) compose compatible fragments together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In the past, Cen- tered Kernel Alignment (CKA) (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Cortes, Mohri, and Rostamizadeh 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Cristianini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2006) has been used to measure similarity between neural network representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We leverage CKA to assess the compatibility of any two fragments from any neural net- works and compose new neural networks from fragments of existing pre-trained neural networks to create high performing networks customized for specific tasks without the costs of traditional model creation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The CKA score is used to reduce the search space for identifying compatible fragments and guide the fragment selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We present empirical validations on benchmark datasets, comparing the performance of StitchNets to that of the origi- nal pre-trained neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We demonstrate that Stitch- Nets achieve comparable or higher accuracy on personalized tasks compared with off-the-shelf networks, and have signif- icantly lower computational and data requirements than cre- ating networks from scratch or through retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Our contributions are: The StitchNet paradigm: a novel neural network creation method that enables a new set of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' A novel use of Centered Kernel Ailgnment (CKA) in as- sessing the compatibility of any two fragments for their composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' A technique to compose compatible fragments together for both linear and convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' A feasibility demonstration of StitchNets for efficient on- the-fly personalized neural network creation and infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2 Composing Fragments The core mechanism to create StitchNets is to iden- tify reusable fragments from a pool of existing net- works and compose them into a coherent neural net- work model capable of performing a given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' To this end, we need a way to determine how compatible any two candidate fragments are with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In previ- ous work, (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2019) present centered kernel alignment (CKA) (Cortes, Mohri, and Rostamizadeh 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Cristianini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2006) as a way to measure similarity be- tween neural network representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Rather than looking at the neural network as a whole, we adopt and use CKA to as a measure of compatibility between any two fragments of any neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In this section, we first define CKA as a way to measure how compatible any two fragments are with one another and therefore their ability to be composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Using CKA, we then present a technique to stitch different fragments together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Fi- nally, we describe the algorithm to generate StitchNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1 Centered Kernel Alignment (CKA) Given X ∈ Rp×n as outputs of a fragment FA of model A and Y ∈ Rq×n as inputs of a fragment FB of model B of the same dataset D, where n is the number of samples in the dataset, p is the output dimension of FA and q is the input di- mension of FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Let Kij = k(xi, xj) and Mij = m(yi, yj), where k and m are any two kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We define the compat- ibility score CKA(X, Y) of fragment FA and fragment FB as CKA(X, Y) = HSIC(K, M) � HSIC(K, K) HSIC(M, M) , where HSIC is the Hilbert-Schmidt Independence Criterion (Gretton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2005) defined as HSIC(K, M) = 1 (n − 1)2 tr(K H M H), where H is the centering matrix Hn = In − 1 n11T and tr is the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For linear kernels, k(x, y) = m(x, y) = xT y, HSIC becomes HSIC(X, Y) = ∥cov(XT X, YT Y)∥2 F , where cov is the covariance function, and CKA(X, Y) be- comes ∥cov(XT X, YT Y)∥2 F � ∥cov(XT X, XT X)∥2 F ∥cov(YT Y, YT Y)∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' (1) We use this function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 1) as a measurement of how com- patible any two fragments are, given a target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' To re- duce memory usage for a large target dataset, CKA can be approximated by averaging over minibatches as presented in (Nguyen, Raghu, and Kornblith 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2 Stitching Fragments Once we have determined compatible fragments, the next step in creating a StitchNet is to stitch the two fragments together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' To do so, we find a projection tensor A that projects the output space of one fragment to the input space of the other fragment we are composing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We now describe this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Without loss of generality, we assume the output and in- put tensors are 2D tensors, where the first dimension is the sample dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' If the tensors are not 2D tensors, we first flatten all other dimensions with the exception of the sample dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We use Einstein summation notation, where i represents the sample dimension, j the output dimension of the incom- ing fragment, and k the input dimension of the outgoing frag- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Given an output tensor Xij of the incoming fragment and an input tensor Yik of the outgoing fragment, we seek to find A such that Yik = Akj Xij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We can then solve for A using the Moore-Penrose pseudoinverse: Akj = Yik XT ij(Xij XT ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='−1 (2) Once A is found, we fuse A with the weight of the first layer of the outgoing fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For linear layers, we simply do the following: W′ lk = Wlj Akj, (3) where l is the dimension of the output feature of the outgoing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For convolutional layers, we first upsample or downsam- ple the spatial dimension to match each other, and then ad- just the weight along the input channel dimension as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' W′ okmn = WijmnAkj, (4) where o is the output channel dimension, j is the original input channel dimension, k is the new input channel dimen- sion, and m and n are the spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For stitching a convolutional layer with an output tensor X and a linear layer with an input tensor Y, we first apply adaptive average pooling so that the spatial dimension is 1x1 and flatten X into a 2D tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Then, we follow Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3 to find A and fuse it with the W of the linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='3 StitchNet Generation Algorithm 1: StitchNet(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' s) Input: fragment pool P = {Fij},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' network i in P up to layer j Nij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' fragment ending in layer j of network i Fij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' target dataset D with M samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' span K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' threshold T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' maximum number of fragments L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' result list of Stitch- Nets and their associated scores R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' current StitchNet Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' current score s Output: resulting list of StitchNets and their associated scores R if Q is empty then {Fij} = select starting fragments in P for Fij in {Fij} do StitchNet(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Fij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 1) if the number of fragments in Q ≥ L then return R {Fij} = select K middle or terminating fragments in P for Fij in {Fij} do X = Q(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Y = Nij(D) sn = s× CKA(X, Y) (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1) if sn > T then Q = Stitch(Q, Fij, X, Y) (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2) if Fij is a terminating fragment then R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='append({Q, sn}) else StitchNet(P, D, K, T , L, R, Q, sn) return R We now describe the main algorithm for creating Stitch- Net networks (“StitchNets” for short), shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' A StitchNet network is created by joining a set of pre-trained network fragments drawn from a pool P = {Fij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We use the notation Fij to denote a fragment of a neural network i up to its j layer, and the notation Nij to denote the compu- tation performed by the portion of the neural network from which the fragment was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Other than the fragment pool P and creation process hyperparameters (K, T, L), the only other input to the StitchNet creation process is a dataset D for which the StitchNet will be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We now describe the creation of the pool of network frag- ments P derived from a set of pre-trained off-the-shelf net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These pre-trained networks are divided into one of three types of fragments: starting fragments for which the input is the original network input, terminating fragments for which the output is the original network output, and mid- dle fragments that are neither starting nor terminating frag- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The first step in the StitchNet creation process is to choose the set of starting fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This could include all starting fragments in P, or a subset based on certain criteria, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', the smallest, biggest or closest starting fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Once a set of starting fragments are selected, a StitchNet is built on top of each starting fragment having a current starting score of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' First, a set of K candidate fragments are selected from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These fragments can be selected based on CKA scores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', K fragments with highest CKA scores), the number of parameters of the fragments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', K frag- ments with the least amount of number of parameters in P), the closest fragments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', K fragments with the least latency in P in a distributed fragments setting), or other se- lection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For each of the candidate fragments, we then compute two intermediate neural network computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' First, we pass the dataset D through the candidate StitchNet in its current form, resulting in value X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Second, we pass the same dataset D through the neural network from which the candidate frag- ment Fij was selected, resulting in value Y = Nij(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' After running these computations, we produce CKA(X, Y) as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We then multiply the CKA with the current score s to obtain the new current score sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' If sn is still greater than a set threshold T , the candidate fragment is selected and the process continues re- cursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Otherwise, the candidate fragment is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The threshold can be set to balance the amount of exploration allowed per available compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This process continues until a terminating fragment is se- lected, the maximum number of fragments L is reached or all recursive paths are exhausted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' At this point, the com- pleted StitchNets and their associated scores R are returned for user selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3 Results We now demonstrate that StitchNets can perform compara- bly with traditionally trained networks but with significantly reduced computational and data requirements at both infer- ence and creation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Through these characteristics, Stitch- Nets enable the immediate on-the-fly creation of neural net- works for personalized tasks without traditional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1 Fragment pool To form the fragment pool P, we take five off-the-shelf net- works pre-trained on the ImageNet-1K dataset (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2009) from Torchvision (Marcel and Rodriguez 2010): alexnet, densenet121, mobilenet v3 small, resnet50 and vgg16 with IMAGENET1K V1 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These pre-trained networks are cut into fragments at each convolution and linear layer that has a single in- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' As shown in Figure 2, there are 8 fragments for alexnet, 5 fragments for densenet121, 13 fragments for mo- bilenet v3 small, 6 fragments for resnet50 and 16 fragments for vgg16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This results in the creation of a fragment pool P of 48 fragments consisting of 5 starting fragments, 38 mid- dle fragments, and 5 terminating fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We use this frag- ment pool in all experiments in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2 Dataset The dataset used to evaluate StitchNets in this paper is the “Dogs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Cats” dataset (Kaggle 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This dataset includes 25,000 training images of dogs and cats and we use an 80:20 train:test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We map ImageNet-1K class labels into cat and dog labels (class IDs 281-285 and 151-250, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' To form the target dataset D for use in the stitch- ing process of Algorithm 1, we randomly select M samples from the training set as the target dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We choose this task because it is characteristic of the type of task for which StitchNets would be used: a user needs a custom classifier for a particular task and desired set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='3 StitchNet Generation We generate StitchNets with Algorithm 1 using the fragment pool and the dataset described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We set K = 2, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='5 and L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The number of samples M in D used for the stitching process is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Given these hyperparameters, a total of 89 StitchNets are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We evaluate them on the test set of completely unseen test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Summary statistics for the generated StitchNets are shown in Figure 3, including accuracy (3a), number of fragments per StitchNet (3b), CKA score (3c), and number of parameters per StitchNet (3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='4 Reduction in Inference Computation We now demonstrate how StitchNets significantly reduce inference-time computational requirements over traditional neural network training paradigms by studying StitchNet ac- curacy as a function of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Figure 4 shows the resulting accuracy of the generated StitchNets as a function overall CKA score for each Stitch- Net and number of parameters (porportional to marker size) as a proxy for inference-time computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We find a number of StitchNets outperform the pre-trained network while realizing significant computational savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For exam- ple, StitchNet27 (denoted by a green star) achieves an ac- curacy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='86 with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='59M parameters compared with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='70 accuracy of the pre-trained alexnet with 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='10M param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Therefore, StitchNet achieves a 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='8% increase in ac- curacy with a 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1% reduction in number of parameters for alexnet densenet121 mobilenet resnet50 vgg16 Figure 2: Five pre-trained networks are fragmented into a fragment pool P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These fragments will be stitched together to form StitchNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' the given task when compared with those of the pre-trained alexnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These crystallizes one of the core benefits of StitchNets: without any training, the method can discover networks that are personalized for the task, outperform the original pre- trained networks, and do so while significantly reducing inference-time compute requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This is due to the fact that these pre-trained networks are not trained to focus on these two specific classes, while our StitchNets are stitched together specifically for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In the next section, we will F15ofN4 14 F14 0fN4 13 F130fN4 F120fN4 11 lio FIOofN4 F8ofN4 F7ofN4 F5ofN4 F4ofN4F5ofN3 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='84 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95 9 19 27 25 9 (a) accuracy 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 6 6 3 3 3 4 8 8 11 10 8 16 (b) # fragments 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='50 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='60 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='70 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='80 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='00 29 28 14 11 7 (c) CKA score 1M to 21M 21M to 42M 42M to 62M 62M to 83M 83M to 104M 104M to 124M 124M to 145M 66 5 4 5 2 3 4 (d) # parameters Figure 3: Histogram of (a) accuracy, (b) # fragments, (c) CKA score, (d) # parameters in the generated batch of Stitch- Nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' see that very little data is required for the stitching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Additionally, we compare the StitchNets with the var- ious off-the-shelf models, denoted by triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We find that the StitchNet generation process creates many different StitchNets that outperform the off-the-shelf models, many of which do so at reduced computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Figure 5 shows the composition of some of these high-performing Stitch- Nets, demonstrating the diversity in fragment use, ordering, and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We also validate the effectiveness of using CKA to guide the stitching procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We find that StitchNets with a high CKA score also have high accuracy, especially those above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This shows that CKA can be used as a proxy to measure good compatibility between connecting fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='5 Reduction in Network Creation Computation We now demonstrate that StitchNets can be created without significant data and computation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Specifically, we compare StitchNet21 (generated in Figure 5 on the tar- get dataset of M = 32 samples) with fine-tuning the same five off-the-shelf networks (retraining them using the train- ing portion of dataset of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For fine-tuning, we replace and train only the last layer of the pre-trained net- work using Stochastic Gradient Descent (SGD) with batch size 32, learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='001 and momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The results shown are averaged over 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For ease of comparison, we normalize the computation cost in terms of the num- ber of samples processed through a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In prac- 1Note that there exist high accuracy StitchNets with low overall CKA score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This is because neural networks are robust and highly redundant, able to tolerate a certain amount of errors while still giving quality predictions (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' tice, fine-tuning requires backpropagation, which incurs ad- ditional computation per sample than StitchNet generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Figure 6 compares the accuracy of StitchNet21 (denoted by the red star) with the traditionally fine-tuned networks as a function of the number of training samples processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' For a given accuracy target, StitchNets process a substan- tially smaller number of data samples than traditionally fine- tuned networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Specifically, to reach an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95, fine-tuning of alexnet, densenet121, and mobilenet v3 small require to process more than 320 samples while StitchNet re- quires only 32 samples used to stitch the fragments together (realizing over a 90% reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Therefore, only a small amount of training samples and computation are required for StitchNet to achieve compara- ble accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This demonstrates that StitchNets effectively reuse the information already captured in the fragments to bootstrap network creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This allows for personalization of tasks and on-the-fly training without substantial data re- quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='6 Ensembles We now discuss the ability to ensemble generated StitchNets to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet and ensembling methods are complimentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The StitchNet generation algorithm pro- duces a set of candidate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' While a user can select a single StitchNet to use at inference time, because the Stitch- Net generation procedure finds such efficient models, we can also take advantage of the pool of StitchNets and ensemble some while still realize substantial computational savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We pick 10 random models from the generated StitchNets in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='3 with overall CKA > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We sort these mod- els based on their overall CKA scores from high to low, and then ensemble them by averaging their predicted probabili- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The ensemble often results in higher accuracy than the individual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' As a re- sult, this ensembling method can reduce variance in perfor- mance when on-the-fly network creation and inference (as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='3) is used and there is not time for full selection of a final single StitchNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Instead, the user can se- lect a reasonably small subset of high performing StitchNets, which even in aggregate can be significantly smaller than a single traditionally trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 4 Discussion We now discuss the intuition behind StitchNets, examine their complexity and relation to related methods, introduce new applications they enable, and discuss their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='1 Why do StitchNets work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We first discuss why we are able to reuse existing fragments of networks to create new neural networks without retrain- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' One core reason for this is that neural networks tend to learn fundamental and universal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Studies (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Morcos, Raghu, and Bengio 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Lenc and Vedaldi 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2020) have shown that neural networks learn fundamental features such as edges for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Since these learned features are fundamental, they should 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='0 Overall CKA score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95 Accuracy Smallest acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='73 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='57M Best acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='91 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='04M StitchNet27 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='86 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='59M alexnet acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='70 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='89 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='10M densenet121 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='85 cka=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='04M mobilenet_v3_small acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='78 cka=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='54M resnet50 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='85 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='99 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='53M vgg16 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='81 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='85 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='36M Figure 4: Accuracy vs the overall CKA score on “Cat vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Dogs.” cka is the overall CKA score, acc is the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The best StitchNet (acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95) performs 12% better than the best pre-trained model(s) (densenet121 and resnet50 with acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet21 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='91 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='04M StitchNet22 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='89 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='33M StitchNet5 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='82 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='81 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='10M StitchNet32 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='79 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='99M StitchNet88 acc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='78 cka=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='77 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='15M Figure 5: Examples of generated StitchNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' be reusable rather relearned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The challenge, however, is that although these features may be universal, they may not be compatible with one another “out of the box.” Therefore, we require the stitching process introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2 to project the fragments into a compatible space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 50 100 150 200 250 300 The number of training samples processed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='0 Accuracy StitchNet21 acc@32=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='95 alexnet acc@320=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='01 densenet121 acc@320=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='04 mobilenet_v3_small acc@320=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='01 resnet50 acc@320=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='01 vgg16 acc@320=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='00 Figure 6: Accuracy vs the number of training samples pro- cessed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', data and computation required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNets re- quire only a fraction of the computation of traditional train- ing methods to achieve comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Model in the ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='925 Accuracy Ensemble Accuracy Individual Accuracy Figure 7: Accuracy of the ensemble models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Ensembling groups of StitchNets can reduce individual model variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Beyond this reuse of universal features and compatibility transformations, StitchNets are also enabled by the fact that neural networks are fundamentally robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Due to the non- linear activation and built-in redundancies, neural networks tolerate certain amounts of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' As such, the fragments need not be perfectly compatible individually to produce a network that in aggregate operates at a high level of perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2 Complexity Comparison We now compare the complexity of the traditional train- ing process using backpropagation with the StitchNet gen- F120fN2 F9of N4 F8ofN4 F7ofN4 F6ofN4 F5ofN4 F4of N4 F3ofN4 F2 ofN4 FIofN4 FOof N4F12 of N2 10 F11 of N2 F9 of N2 F8 of N2 F7 of N2 F6 of N2 F5 of N2 F4 of N2 F3 of N2 F2 of N2 F1 of N2 FO of N2F12ofN2 F6of NO F5ofNO F4of NO F3ofNO F2of NO FIof NO FOof NOF12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='0fN2 F2OfNI FIofNI 0 FOofNIF5ofN3 F3ofNI F2ofNI FIOfNI FOofNIeration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Traditional training complexity is O(ndp), where n is the number of parameters in the network, p is the number of epochs used to train, and d is the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet generation complexity is O(nqm) + O(KL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The first term nqm is the evaluation cost of the tar- get dataset of size q on m networks in the pool, where q ≪ d and n is the number of parameters in the network (assuming networks have the same # of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The second term KL is the search cost, where K is the span value we search at each level and L is the max depth to search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Using a high threshold cutoff T on the overall CKA score keeps search cost KL small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Therefore, for a reasonable setting of hyper- parameters (K, T, L) in Algorithm 1, StitchNets realize sub- stantial computation gains over traditional training methods since q ≪ d and m ≪ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='3 On-the-fly network creation and inference We now discuss a new family of applications and use cases that are enabled by StitchNets: on-the-fly neural network cre- ation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In this application, we use a batch of im- ages on which we want to perform a task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', classification or detection) as our target dataset in the StitchNet generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' With only a minor modification to the StitchNet al- gorithm to additionally return task results, the StitchNet gen- eration process can return the inference outputs along with the generated StitchNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We now describe how this can be used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Imag- ine a world where fragments of pre-trained neural networks for different tasks are indexed and distributed on the Inter- net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Any compatible fragment can be found and composed quickly to form a new neural network for a certain task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Now, imagine we want to create a neural network for classifying local cats and dogs with only a few hundred of these unla- beled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Without StitchNets, we either need to train a network from scratch (which may fail due to our limited amount of training data), or find an existing pre-trained neural network, label the dataset, and finetune the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' If the existing pre-trained network is too big or too slow for our use, we will then have to train a new one from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' But, with lim- ited amount of unlabeled data, this task seems impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' With StitchNet, we can instead generate a set of candidate StitchNets with the small target dataset of unlabeled local cats and dogs images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' These StitchNets are created from the pool of existing neural network fragments that have been in- dexed and distributed on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' The proper fragments can be identified with a search criteria (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=', the terminat- ing fragment should contain cat and dog classes, the depth of the network should be less than 5 for computational effi- ciency reasons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' With little computation, we will gener- ate StitchNets capable of detecting and classifying local cats and dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='4 Limitations One limitation is that the target task needs to be a subset (or a composition) of the terminating fragment tasks in the fragment pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Additionally, while a large pool of networks and fragments can lead to higher applicability and quality of StitchNets, it can also lead to high search costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Index- ing large quantities of neural networks to form the fragment pool will require novel search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We see this as anal- ogous to indexing web pages on the World Wide Web, sug- gesting a “Google for Fragments.” Much like web search needed to index written content, large amounts of neural net- work “content” need to be indexed in order for their value to be unlocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Early indexing efforts can tag fragments based on dataset characteristics, computational characteristics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' More advanced efforts can look at inward and outward con- nections of each fragment to determine its rank in results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Once a narrowed set of fragments are coarsely identified, the efficient procedure introduced in this paper can generate the StitchNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Future work will address these types of comple- mentary methods (indexing and distribution) that will enable StitchNets to operate at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 5 Related Work Transfer learning (or fine-tuning) (Pan and Yang 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Weiss, Khoshgoftaar, and Wang 2016) is the current pre- dominant way to adapt existing neural networks to target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Unsupervised domain adaptation is related, where the existing network is adapted using an unlabeled target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNets work similarly by stitching fragments using an unlabeled target dataset to create a neural network for the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Most work (Wang and Deng 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Saito, Ushiku, and Harada 2017) fo- cuses on retraining the network, while StitchNet does not require any training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNets take advantage of the assumption that the frag- ments have shareable representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' This assumption helps explain why fragments can be stitched together into a coher- ent high-performing network: dissimilar yet complimentary fragments once projected into a similar space are compatible with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Several existing works including (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Mehrer, Kriegeskorte, and Kietzmann 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Morcos, Raghu, and Bengio 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Lenc and Vedaldi 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2020) have studied this shareable representation assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' (Gygli, Uijlings, and Ferrari 2021) reuse network compo- nents by training networks to produce compatible features by adding regularization at training time to make the net- works directly compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet, however, focuses on creating neural networks without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' It is therefore more generally applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' (Lenc and Vedaldi 2015) com- bine network components by adding a stitching layer and training the recombined network with a supervised loss for several epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet adds a parameter-less stitching mechanism and therefore does not require any retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In- stead, weights are adapted to be compatible with the method introduced in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 6 Conclusion StitchNet is a new paradigm that can leverage a growing global library of neural networks to fundamentally change the way networks are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' By reusing fragments of these networks to efficiently compose new networks for a given task, StitchNet addresses two of the most fundamental is- sues limiting neural network creation and use: large data and computation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' StitchNet does this by leveraging Centered Kernel Align- ment (CKA) as a compatibility measure that guides the se- lection of neural network fragments, tailored to specific ac- curacy needs and computing resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Our work has shown that neural networks can be efficiently created from compatible neural network fragments of different mod- els at a fraction of computing resources and data require- ments with a comparable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' We also introduce on- the-fly efficient neural network creation and inference appli- cation that is unlocked by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' References Ben-David, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Blitzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=';' metadata={'source': 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D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' Collabora- tive and adversarial network for unsupervised domain adap- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, 3801–3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQf_P79/content/2301.01947v1.pdf'} diff --git a/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/2301.03771v1.pdf.txt b/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/2301.03771v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe7b2b31e2d523315fbcb6b43b772367b20ec1da --- /dev/null +++ b/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/2301.03771v1.pdf.txt @@ -0,0 +1,1284 @@ +Chatbots in a Honeypot World +Forrest McKee1 and David Noever2 +PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 +1forrest.mckee@peopletec.com 2 david.noever@peopletec.com + + +Abstract +Question-and-answer agents like ChatGPT offer a novel tool for use as a potential honeypot interface in +cyber security. By imitating Linux, Mac, and Windows terminal commands and providing an interface for +TeamViewer, nmap, and ping, it is possible to create a dynamic environment that can adapt to the actions +of attackers and provide insight into their tactics, techniques, and procedures (TTPs). The paper illustrates +ten diverse tasks that a conversational agent or large language model might answer appropriately to the +effects of command-line attacker. The original result features feasibility studies for ten model tasks meant +for defensive teams to mimic expected honeypot interfaces with minimal risks. Ultimately, the usefulness +outside of forensic activities stems from whether the dynamic honeypot can extend the time-to-conquer or +otherwise delay attacker timelines short of reaching key network assets like databases or confidential +information. While ongoing maintenance and monitoring may be required, ChatGPT's ability to detect and +deflect malicious activity makes it a valuable option for organizations seeking to enhance their cyber +security posture. Future work will focus on cybersecurity layers, including perimeter security, host virus +detection, and data security. +Keywords: +Transformers, Text Generation, Malware Generation, Generative Pre-trained Transformers, GPT + +1. INTRODUCTION + +A honeypot is a significant cyber security tool that is used to detect, deflect, and study malicious activity +on a computer network [1-4]. It is essentially a trap set up to lure in potential attackers, who are then +observed and their actions are recorded for later threat analysis. Honeypots can be used in a variety of ways, +including for research, to gather intelligence on new or emerging threats, or to distract and mislead attackers +while security teams work to defend against an ongoing attack [1]. A spectrum exists between low- +interaction honeypots that may expose only ports and no real services to high-interaction honeypots that +virtualize entire networks using VMWare or User-mode Linux with application-, network- and system- +layer features [5]. Making realistic traps relies on the realism of the honeypot. Attackers may quickly +discover the static elements or missing functional files that tip off a fake asset or operating system façade. +Probing services and ports can reveal a fake network asset [6-7]. The rise of cloud and virtual machine +images has exacerbated the challenge to mimic real networks with a passive store-front approach [2]. More +dynamic approaches to building honeypots that feature real applications but host fake data [6]. An example +dynamic honeypot deploys a real SQL database capable of real hacking attempts, all of which culminate in +revealing fake personnel or salary data. A hybrid version of the real vs. simulated honeypot problem +involves creating a digital twin that behaves like the real network but which underneath remains a +simulation based on a large language model [8] that anticipates the output of the operating system and +applications [9]. This hardware and software stack together presents a sufficiently deep environment that +a large language model simulates the expected outcomes when queried by an intruder [9-11]. This hybrid +option provides a novel experimental platform for the current study and assessments of its capabilities. + +In this paper, we will explore the concept of using ChatGPT, a natural language processing tool [12-14], as +a honeypot in the field of cyber security. One potential use of ChatGPT as a honeypot is to issue various +commands that simulate Linux [9] and Windows terminals. This can be used to lure in attackers who are + +specifically targeting these types of systems, and allow security teams to observe and study their actions +[15-16]. By issuing commands through ChatGPT, it is possible to create a realistic and dynamic +environment that can adapt to the actions of the attacker [6]. As an attacker explores this new network asset, +their commands reveal ever more sophisticated emulation patterns derived from the internet-scale training +data underpinning the OpenAI GPT series of transformer architectures [13]. Historically, honeypot logs +provide valuable insights into the tactics, techniques, and procedures (TTPs) used by attackers, as well as +help security teams to identify patterns and trends in malicious activity [17-20]. Additionally, issuing +commands through ChatGPT can also help to distract and mislead attackers, giving security teams more +time to defend against an ongoing attack. The latest generation of ChatGPT (Dec 2022 update) [21] now +sustains its memory of initial instructions for up to 8000 tokens (or around 5600 words, 20-25 text pages). +To translate this coherent “command-driven” conversation to a typical intrusion, the attacker might interact +with emulated honeypot (aka, chatbot interface) for hours before the simulation required an instructional +reset. + +2. METHODS + +The structure of the paper closely follows the detailed instructions and attacker interactions outlined in +Appendices A-J as ten tasks related to honeypot construction, detection, or harvesting [1]. As shown in +Table 1, each appendix section outlines the initial ChatGPT instructions or prompt followed by a simple +proof of principle illustrating the degree of dynamic emulation achievable. The ten tasks demonstrate +plausible command-level interactions with an adversary who breaches a network consisting of all major +operating systems (Windows, Linux, Mac). We simulate application-level interactions with a python- +driven Jupyter notebook and a Team Viewer installation. We simulate network-level interactions using +network mapping tools (nmap) and launch a simulated distributed denial-of-service (DDoS) attack using +ping. We simulate an attacker’s deception by changing the time-stamp on a malicious file (“time- +stomping”) so forensic analysis might fail to uncover the file changes. We simulate the modification of a +Windows registry key, such that a malicious file or scheduled task might transform into an installed +application or routine network operation. We finally emulate nefarious actions or spoofing of IP or machine +addresses (MAC) using realistic output from a Linux terminal and the “arp” command interactions from +ChatGPT. The paper approaches these ten tasks empirically and methodically, first constructing the prompt +to convince the chatbot that its behavior follows the pattern of a given operating system (OS) or application, +followed by repeated probing of expected responses or honeypot output. The simplest realization arises for +a directory listing unique to each OS. The most sophisticated simulation mirrors both the OS, an example +application like ping (Appendix G) or Jupyter notebook (Appendix B), and a model network with realistic +connection maps. + +Table 1. Ten tasks for dynamic honeypot demonstrations including OS, Application, and Network +Interactions. +Goal +Prompt +OS +Appendix A. +Emulate an +Operating System +Terminal +I want you to act as a Linux terminal. I will type commands and you will +reply with what the terminal should show. I want you only to reply with the +terminal output inside one unique code block, and nothing else. Do not +write explanations. Do not type commands unless I instruct you to do so. +When I need to tell you something in English I will do so by putting text +inside only curly brackets {like this}. My first command is "pwd" +Linux +Appendix B. +Emulate a Jupyter +Notebook +I want you to act as a jupyter notebook. I will type commands and you will +reply with what the notebook should show. I want you only to reply with +the notebook output inside one unique code block, and nothing else. Do not +write explanations. Do not type commands unless I instruct you to do so. +When I need to tell you something in English I will do so by putting text +Application + +Goal +Prompt +OS +inside only curly brackets {like this}. My first command is "print('hello +world')" +Appendix C. +Emulate a DOS +Terminal as Admin +and Delete a +Registry Key +I want you to act as a Windows DOS terminal running as admin. I will type +commands and you will reply with what the terminal should show. I want +you only to reply with the terminal output inside one unique code block, +and nothing else. Do not write explanations. Do not type commands unless +I instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is reg /? +Windows +Appendix D. +Emulate a DOS +Terminal as User +I want you to act as a Windows DOS terminal. I will type commands and +you will reply with what the DOS command line should show. I want you +only to reply with the CLI output inside one unique code block, and +nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is "dir" +Windows +Appendix E. +Emulate a +MacIntosh +Terminal as User +I want you to act as a MacIntosh Terminal app. I will type shell commands +and you will reply with what the Mac command line should show. I want +you only to reply with the Terminal output inside one unique code block, +and nothing else. Do not write explanations. Do not type commands unless +I instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is "ls" +Mac +Appendix F: +Installing +TeamViewer from +the Terminal +I want you to act as a Linux terminal. I will type commands and you will +reply with what the terminal should show. I want you only to reply with the +terminal output inside one unique code block, and nothing else. Do not +write explanations. Do not type commands unless I instruct you to do so. +When I need to tell you something in English I will do so by putting text +inside only curly brackets {like this}. My first command is "pwd" +ChatGPT: /home/user +Linux +Appendix G: +Launching a DDoS +Attack from the +Windows CMD +I want you to act as a Windows DOS terminal. I will type commands and +you will reply with what the DOS command line should show. I want you +only to reply with the CLI output inside one unique code block, and +nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is "dir" +Windows +Appendix H: +Changing the +Write Time of a +File Using +Powershell +I want you to act as a Windows Powershell terminal. I will type commands +and you will reply with what the Powershell command line should show. I +want you only to reply with the CLI output inside one unique code block, +and nothing else. Do not write explanations. Do not type commands unless +I instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is "dir" +Windows +Appendix I: Poison +the ARP Network +Table on Windows +I want you to act as a Windows DOS terminal. I will type commands and +you will reply with what the DOS command line should show. I want you +only to reply with the CLI output inside one unique code block, and +nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will +do so by putting text inside only curly brackets {like this}. My first +command is "dir" +Windows +Appendix J: +Emulate an +Attacker’s Lateral +I want you to act as a Linux terminal. I will type commands and you will +reply with what the Linux command line should show. I want you only to +reply with the CLI output inside one unique code block, and nothing else. +Linux + +Goal +Prompt +OS +Movement Using +Network Map +Tools like nmap +Do not write explanations. Do not type commands unless I instruct you to +do so. When I need to tell you something in English I will do so by putting +text inside only curly brackets {like this}. My first command is "ls" + +3. RESULTS + +The main results feature the demonstration for each of the ten honeypot tasks. Appendices A-J summarize +the output of the command-line interactivity for honeypots as emulated conversations between a +sophisticated attacker and a trained chatbot [9-10,12]. For concreteness, we group the honeypot tasks into +three categories based on their focus addressing layers of modern enterprises: operating systems +[Appendices A,D,E] , applications [Appendices B,F], or networks [Appendices C,G-J]. As a dynamic +honeypot interface, the large language model emulates the expected “prompt-response” sequence that real +applications and operating systems would generate when queried. Unlike previous models, the ChatGPT +interface not only provides sufficient API memory to carry forward previous instructions without defaulting +to repeated introductory tasks but also provides a responsive honeypot welcome to sustain the attacker’s +interest over multiple queries. Based on previous pentesting results,an external attacker can breach 93% of +of company networks [22]. The initial intrusion, on average, takes two days [22] usually based on some +credential access derived from email phishing campaigns, brute force attacks, or leakage to the cloud, code +repositories, and poor training in social engineering tactics. Among the new security tools (encryption, +threat intel and detection, firewalls, etc.) decoys and honeypots disguise the real crown jewels of an +organization (such as HR or financial information) while also delaying attackers beyond their economic +horizon or patience. + +3.1. Operating Systems + +Appendices A,D,E describe the front-facing command line interface for the major operating systems: +Linxu, Windows and MacIntosh. Unlike virtual machines or containerized honeypot frameworks [23-24], +the overhead for emulating a conversational agent that answers all command line inquiries with correct or +expected responses remains a simple API call rather than an installation or download option. The major +commands illustrated reveal expected directory structures specific to each default in the three major +operating systems. The conversational agent knows the file structure and at increasing depths of the +expected file tree can traverse between user documents and root or administrator programs. + +3.2. Applications + +Appendices B,F describe the appropriate responses that an application might yield to an intruder who +breaches a running application like Jupyter notebooks or installs a Linux program like TeamViewer. These +application level responses illustrate the diversity of underlying cybersecurity knowledge from ChatGPT +as a zero-shot or few-shot learner. No explicit context guides the conversational responses, although the +model continues to produce the expected application-specific responses that an intruder might expect when +probing for application functionality. Among the ten tasks these concrete examples rank highest in diversity +such that they respond correctly in two ways, both to understand the default states (“out-of-the-box”) but +also the modified states following a new program installation (apt-get install TeamViewer2017.asc). +3.3. Attacker Tactics + +Appendices C,G-J describe the network behavior for common command-line tools that provide key attacker +inputs, such as network maps (nmap, App. J), responsive services (ping, App. G), and program installation + +registry (regedit, App. C). Nmap particularly provides an attacker with an expected output in a honeypot +setting that simulates lateral movement and reconnaissance to discover new network assets. Appendix H +highlights a frequent attacker deception that changes the creation or modification time stamp on a program +change, such that any malicious insertions fail to trigger later discovery as outliers or recent modifications +to the operating system. Appendix I illustrates a chat conversation that an unaware attacker modifies the +ARP network table and provisions spoofed IP addresses or MAC identifiers. Appendix G provides an +example of launching a network-wide denial of service (ping flood) with the expected feedback provided +by a large language model placed as the flat front to a would-be attacker probing the honeypot for new +resources. +4. DISCUSSION AND CONCLUSIONS + +In conclusion, ChatGPT has the potential to be a valuable tool as a honeypot in the field of cyber security. +By issuing commands that simulate Linux, Mac and Windows terminals, provide a seamless application +interface for TeamViewer, nmap, and ping, and finally log the attacker traversal path as new fake assets get +owned or discovered. It is possible to create a realistic and dynamic environment that can adapt to the +actions of attackers and provide valuable insights into their TTPs. While there are potential limitations to +using ChatGPT as a honeypot, such as the need for ongoing maintenance and monitoring, the benefits of +having a dynamic and adaptable tool for detecting and deflecting malicious activity make it a promising +option for organizations looking to improve their cyber security posture. Overall, ChatGPT offers a unique +and innovative approach to the use of honeypots and is worth considering as a component of a +comprehensive cybersecurity strategy. Future work explores the cybersecurity layers with an initiative to +investigate the firewall or router emulation steps (perimeter security), endpoint steps (host virus detection), +and data security (credentials, human behavior, and mission-critical assets). + +ACKNOWLEDGMENTS + +The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance. +The authors thank the researchers at OpenAI for developing large language models and allowing public access to +ChatGPT. + +REFERENCES + +[1] +Baykara, M., & Daş, R. (2015). A survey on potential applications of honeypot technology in +intrusion detection systems. International Journal of Computer Networks and Applications +(IJCNA), 2(5), 203-211. +[2] +Nawrocki, M., Wählisch, M., Schmidt, T. C., Keil, C., & Schönfelder, J. (2016). A survey on +honeypot software and data analysis. arXiv preprint arXiv:1608.06249. +[3] +Uitto, J., Rauti, S., Laurén, S., & Leppänen, V. (2017, April). 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In 2018 Global Information Infrastructure and +Networking Symposium (GIIS) (pp. 1-6). IEEE. +[16] +Dongxia, L., & Yongbo, Z. (2012, March). An intrusion detection system based on honeypot +technology. In 2012 international conference on computer science and electronics engineering +(Vol. 1, pp. 451-454). IEEE. +[17] +Wang, P., Wu, L., Cunningham, R., & Zou, C. C. (2010). Honeypot detection in advanced botnet +attacks. International Journal of Information and Computer Security, 4(1), 30-51. +[18] +Candela, M. (2022), “Secure Honeypot Framework Low Code, Easy to Configure”, +https://github.com/mariocandela/beelzebub +[19] +Lee, S., Abdullah, A., & Jhanjhi, N. Z. (2020). A review on honeypot-based botnet detection +models for smart factory. International Journal of Advanced Computer Science and Applications, +11(6). + +[20] +Huang, C., Han, J., Zhang, X., & Liu, J. (2019). Automatic identification of honeypot server +using machine learning techniques. Security and Communication Networks, 2019. +[21] +OpenAI (2022), ChatGPT: Optimizing Language Models for Dialogue, +https://openai.com/blog/chatgpt/ +[22] +Brooks, C. (2022), Alarming Cyber Statistics For Mid-Year 2022 That You Need To Know, +https://www.forbes.com/sites/chuckbrooks/2022/06/03/alarming-cyber-statistics-for-mid-year- +2022-that-you-need-to-know/?sh=7f6bf70f7864 +[23] +Lihet, M. A., & Dadarlat, V. (2015, September). How to build a honeypot System in the cloud. In +2015 14th RoEduNet International Conference-Networking in Education and Research +(RoEduNet NER) (pp. 190-194). IEEE. +[24] +Singh, R. K., & Ramajujam, P. (2009). Intrusion Detection System Using Advanced Honeypots. +arXiv preprint arXiv:0906.5031. + + +AUTHORS + +Forrest McKee has AI research experience with the Department of Defense in object +detection and reinforcement learning. He received his Bachelor's (BS) and Master's (MSE) +from the University of Alabama, Huntsville, Engineering. + + +David Noever has research experience with NASA and the Department of Defense in +machine learning and data mining. He received his BS from Princeton University and his +Ph.D. from Oxford University, as a Rhodes Scholar, in theoretical physics. + + +SUPPLEMENTAL MATERIALS + +Appendix A. Emulate an Operating System Terminal +Appendix B. Emulate a Jupyter Notebook +Appendix C. Emulate a DOS Terminal as Admin and Delete a Registry Key +Appendix D. Emulate a DOS Terminal as User +Appendix E. Emulate a MacIntosh Terminal as User +Appendix F: Installing TeamViewer from the Terminal +Appendix G: Launching a DDoS Attack from the Windows CMD +Appendix H: Changing the Write Time of a File Using Powershell +Appendix I: Poison the ARP Network Table on Windows +Appendix J: Emulate an Attacker’s Lateral Movement Using Network Map Tools like nmap + + + + +Appendix A. Emulate an Operating System Terminal +Prompt: I want you to act as a Linux terminal. I will type commands and you will reply with what the +terminal should show. I want you only to reply with the terminal output inside one unique code block, +and nothing else. Do not write explanations. Do not type commands unless I instruct you to do so. +When I need to tell you something in English I will do so by putting text inside only curly brackets +{like this}. My first command is "pwd" +ChatGPT: /home/user +Prompt: ls +ChatGPT: +Desktop/ +Documents/ +Downloads/ +Music/ +Pictures/ +Public/ +Templates/ +Videos/ +Prompt: echo "print('Hello World!')" >test.py +ChatGPT: /home/user/Documents +Prompt: python test.py +ChatGPT: Hello World! +Prompt: echo "\nprint('Hello World Again!')" >>test.py +ChatGPT: /home/user/Documents +Prompt: python test.py +ChatGPT: +Hello World! +Hello World Again! +Prompt: rm -rf Videos +ChatGPT: /home/user/Documents +Prompt: ls +ChatGPT: +Desktop/ +Documents/ +Downloads/ +Music/ +Pictures/ +Public/ +Templates/ +Technical Note: Some appendix text generated from Large Language Model (LLM) for +illustration purposes. +The authors generated this text in part with ChatGPT, OpenAI’s large-scale language-generation +model. Upon generating draft language, the authors reviewed, edited, and revised the language to +their own liking and take ultimate responsibility for the content of this publication. +-- OpenAI policy statement (2022) + +test.py +Prompt: Keep the same instructions, but act as a Windows DOS command line. My first command is +"dir" +ChatGPT: +Desktop/ +Documents/ +Downloads/ +Music/ +Pictures/ +Public/ +Templates/ +Videos/ +Prompt: del Videos +ChatGPT: +Desktop/ +Documents/ +Downloads/ +Music/ +Pictures/ +Public/ +Templates/ +Prompt: time +ChatGPT: +Current time: 22:16:49.14 Enter the new time: +Prompt: 23:11:11.15 +ChatGPT: +Current time: 23:11:11.15 + +Appendix B. Emulate a Jupyter Notebook +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what +the notebook should show. I want you only to reply with the notebook output inside one unique code +block, and nothing else. Do not write explanations. Do not type commands unless I instruct you to do +so. When I need to tell you something in English I will do so by putting text inside only curly brackets +{like this}. My first command is "print('hello world')" +ChatGPT: hello world +Prompt: import time +ChatGPT: + +Prompt: %timeit -r1 time.sleep(2) +ChatGPT: 2 s ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each) +Prompt: %timeit -r4 time.sleep(2) +ChatGPT: 2 s ± 0 ns per loop (mean ± std. dev. of 4 runs, 1 loop each) + +Appendix C. Emulate a DOS Terminal as Admin and Delete a Registry Key +Prompt: I want you to act as a Windows DOS terminal running as admin. I will type commands and +you will reply with what the terminal should show. I want you only to reply with the terminal output +inside one unique code block, and nothing else. Do not write explanations. Do not type commands + +unless I instruct you to do so. When I need to tell you something in English I will do so by putting text +inside only curly brackets {like this}. My first command is reg /? +ChatGPT: +REG: The REG command is used to manage the Windows registry. + +REG [cmd] + +cmd Description +/?, /H, /HELP Displays help at the command prompt. +/A Performs the specified operation on all subkeys. +/D Deletes the specified value or key. +/E Exports the contents of the specified key to a .REG file. +/F Forces the operation to continue on keys that have subkeys or values. +/L Performs the specified operation on keys in the local machine instead of the current user. +/Q Quiet. Does not display output. +/S Performs the specified operation on all subkeys and values. +/V Verbose. Displays detailed information about the operation. + +Examples: +REG ADD HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System /v EnableLUA /t +REG_DWORD /d 0 /f +REG DELETE HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System /v EnableLUA +/f +REG EXPORT HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System c:\policies.reg +REG IMPORT c:\policies.reg +Prompt: REG QUERY HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System +ChatGPT: +HKEY_LOCAL_MACHINE\Software\Microsoft\Windows\CurrentVersion\Policies\System + EnableLUA REG_DWORD 0x0 +Prompt: REG DELETE HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System /v +EnableLUA /f +ChatGPT: The operation completed successfully. +Prompt: REG QUERY HKLM\Software\Microsoft\Windows\CurrentVersion\Policies\System +ChatGPT: +HKEY_LOCAL_MACHINE\Software\Microsoft\Windows\CurrentVersion\Policies\System + +Appendix D. Emulate a DOS Terminal as User +Prompt: I want you to act as a Windows DOS terminal. I will type commands and you will reply with +what the DOS command line should show. I want you only to reply with the CLI output inside one +unique code block, and nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will do so by putting text inside +only curly brackets {like this}. My first command is "dir" +ChatGPT: + + +Volume in drive C is os +Volume Serial Number is xxxx-xxx +Directory of C:lPrompt: command /? +ChatGPT: + +Prompt: tracert openai.com +ChatGPT: + +Displays or sets the command prompt. +CMD[/A/U][/Q][/D][/E:ON +/E:OFF][/F:ON +/F:OFF][/V:ON1 +/V:OFFJ +[Ss[/][/] +/c +Carries out the connand specified by string and then terninates +/K +Carries out the connand specified by string but renains +/5 +Modifies the treatnent of string after /C or /k (see below) +/Q +Turns echo off +/D +Disable execution of AutoRun connands fron registry +(HKLM/Software\Microsoft,Connand ProcessorAutoRun) +/A +Forces the output to a window tobe ANsi +/U +Forces the output to a window to be Unicode +/T: fg +Sets the foreground/background colors (see coLoR /? for nore info) +/E:ON +Enable comnand extensions ( +(see below) +/E:OFF +Disable connand extensions +(see below) +/F:ON +Enable file +and directory nane conpletion characters (see below) +/F:OFF +and directory nane conpletion characters (see below) +/V:ON +Enable delayed environment variable expansion using ! as the deliniter +For +example, +/V:oN would allow Ivarl to expand the variable var at execution +time. +Thevarsyntaxexpandsvariabiesatinputtime.whichisquitea +different thing when inside of a FoR loop. +/V:OFF +Disable delayed environnent expansion. +an innediate command in the new cMD shell. If /s is specified, then any quote +(") characters on the command line are treated as regular quotes, and the string +is passed to the new CNo sheli forprocessing. Iffc orfk is not specified +then thefollowingstepsaretaken: +symoo in the iine,then tne string isassuned tc +redirection is perforned as specified +enpty and does not contain any spaces, then it is +and executed +assuned to +the connand lineparameters. +if any +are enabled by default. +Enabling command extensions allows +the +the following extended connands: +Prompt: ping openai.com +ChatGPT: + +Prompt: exit +ChatGPT: +I'm sorry, but I'm an artificial intelligence and don't have the ability to close the command prompt or +end the session. If you would like to stop interacting with me, you can simply stop typing or close the +window in which we are communicating. +Prompt: type nul >> "file.txt" +ChatGPT: C:/> +Prompt: dir +ChatGPT: + + +tracert openai.com +O Copy code +Tracing route to openai +.con【2606:470o:60:d7f5::] +over a maxinun of 38 hops: +13 ms +ns +2 ms +192.168.1.1 +11 ms +13ms +1i ns +19.0.0.1 +ms +ns +SU 68 +18.0.0.2 +ns +18. +0.0.5 +10.0.0.6 +18.0.0.7 +47 +ns +47 +ns +ns +10.0.0.9 +11 +151 ns +148 ns +149 ns +10.0.0.10 +12 +149Pinging openai.com [2606:478e:60:d7f5::] with 32 bytes of data +Reply from 2606:478o:60:d7f5: :: tine=149ms +Reply from 2606:4780:60:d7f5::: tine=148ms +Reply from 2606:4780:60:d7f5::: tine=149ms +Reply from 2606:4780:66:d7f5::: tine=149ms +Ping statistics for 2606:478e:68:d7f5::: +Packets: Sent = 4, Received = 4, Lost = (e% loss) +Approxinate round trip tines in milli-seconds: +Minimun +148ms,MaximunVolune in drive C is os +Volume Serial Number is xxoxx-xxxo +Directory of c:" +12/19/2022 +B4:31 PM +file.txtPrompt: move C:\file.txt C:\Program Files +ChatGPT: + +Prompt: dir +ChatGPT: + + +Prompt: dir C:\Program Files +ChatGPT: + +Prompt: +echo "@echo off +attrib -r -s -h c:\autoexec.bat +del c:\autoexec.bat +attrib -r -s -h c:\boot.ini +del c:\boot.ini +attrib -r -s -h c:\ntldr +del c:\ntldr +attrib -r -s -h c:\windows\win.ini +del c:\windows\win.ini" >>C:\Program Files\file.bat +ChatGPT: C:\> +Prompt: dir C:\Program Files +ChatGPT: + +Prompt: type C:\Program Files\file.bat +ChatGPT: + +I file(s) +Inoved.Volune in drive C is os +Volune Serial Number is xxxX-xoX +Directory of c:Volune in drive C is os +olume Serial Number is xxxx-xxxx +Directory of c:rProgram Files +12/19/2622 +B4:31 PM +B file.txtVolune in drive c is os +olume Serial Number is xxxx-xxxo +Directory of c:uProgram Files +12/19/2022 +B4:31 PM +27 file.bat +12/19/2022 +84:31 PM +file.txt +Prompt: +chdir C:\Program Files\ +file.bat +ChatGPT: +I'm sorry, but I don't have the ability to execute the file.bat file for you. This command will run the +file.bat batch file in the current directory if it exists on your computer. If you want to execute the file, +you will need to type the command into the command prompt on your own computer. Please note that +running this batch file will delete the autoexec.bat, boot.ini, ntldr, and win.ini files from the root +directory of your system drive, which could cause problems with your system if these files are required +for it to function correctly. You should use caution when running any batch files or commands that +modify or delete system files. +Prompt: del *.* +ChatGPT: +I'm sorry, but I can't execute this command for you. The del command is used to delete one or more +files from a specified directory. When used with the *.* wildcard, it will delete all files in the current +directory. Please note that deleting files can be a dangerous operation and it is important to use caution +when using the del command, especially when using wildcards. Once a file is deleted, it is generally +not possible to recover it. Make sure you have a backup of any important files before deleting them, +and be sure you are in the correct directory before running the del command. +Prompt: dir (User Note: Does delete in its file system despite warnings) +ChatGPT: + +Prompt: REN *.avi *.txt REN *.mkv *.txt REN *.BAT *.txt REN *.doc *.txt REN *.JPEG *.txt REN +*.lnk *.txt +ChatGPT: +I'm sorry, but I can't execute these commands for you. The REN command is used to change the name +of a file or directory. When used with the *.avi, *.mkv, *.BAT, *.doc, *.JPEG, and *.lnk wildcards, it +will change the file extension of all files with these extensions in the current directory to .txt. Please +note that changing the file extension of a file can potentially cause the file to become unreadable or +unable to be opened by the intended program. It is generally not a good idea to change the file +extension of a file unless you are sure that it is safe to do so. You should use caution when using the +REN command and make sure you have a backup of any important files before changing their names or +extensions. +Prompt: REN *.bat *.mp4 +ChatGPT: I'm sorry, but I can't execute this command for you. The REN command is used to change +the name of a file or directory. When used with the *.bat wildcard, it will change the file extension of +all files with the .bat extension in the current directory to .mp4. Please note that changing the file + +@echo off +attrib +-r +-h c:lautoexec.bat +del c:vautoexec.bat +del c:tboot.ini +attrib -r -s -h c:intldr +del c:intldr +attrib -r -s -h c:iwindowsiwin.ini +del c:iwindowswin.iniVolune in drive C is os +Volune Serial Nunber is xxxx-xoox +Directory of C:Progran Filesextension of a file can potentially cause the file to become unreadable or unable to be opened by the +intended program. It is generally not a good idea to change the file extension of a file unless you are +sure that it is safe to do so. You should use caution when using the REN command and make sure you +have a backup of any important files before changing their names or extensions. +Prompt: dir (User Note: Although a caution, the resulting conversation reflects the instructed action +under protest) +ChatGPT: + + + +Appendix E. Emulate a MacIntosh Terminal as User +Prompt: I want you to act as a MacIntosh Terminal app. I will type shell commands and you will reply +with what the Mac command line should show. I want you only to reply with the Terminal output +inside one unique code block, and nothing else. Do not write explanations. Do not type commands +unless I instruct you to do so. When I need to tell you something in English I will do so by putting text +inside only curly brackets {like this}. My first command is "ls" +ChatGPT: +bin etc lib sbin usr +dev home opt tmp var +Prompt: ls /usr/bin +ChatGPT: + +Prompt: python3 -c 'print("Hi")' +ChatGPT: +Hi +Prompt: rm -f /usr/bin/python3 +ChatGPT: +rm: /usr/bin/python3: Operation not permitted +Prompt: whoami +ChatGPT: +{Your user name} +Prompt: python <<< 'print "Hi"' + +Volune in drive c is os +Volune Serial Number is xxxx-xoox +Directory of c:Progran Files +271972922 +B4:31 PM +27 fi1e.mp42to3-2.7 +diff3 +mcs +Pygmentize +2to3-3.7 +2to3 +idle3 +mcs-2.7 +python3 +2to3-3.8 +2to3-3.5 +idle3.5 +mcs-2.7.5 +python3-3.5 +2to3-3.9 +2to3-3.6 +idle3.6 +mcs-3.5 +python3-3.6 +2to3-3.9-2 +2to3-3.7-2 +idle3.7 +mcs-3.6 +python3-3.7 +2to3-3.9-3 +2to3-3.8-2 +idle3.8 +mcs-3.7 +python3-3.7-2 +2to3-3.9-4 +2to3-3.8-3 +idle3.8.5 +mcs-3.8 +python3-3.7-3 +2to3-3.9-5 +2to3-3.9-2 +idle3.8.5-2 +mcs-3.8.5 +python3-3.8 +2to3-3.9-6 +2to3-3.9-3 +idle3.8.5-3 +mcs-3.8.5-2 +python3-3.8-2 +2to3-3.9-7 +2to3-3.9-4 +idle3.8.5-4 +mcs-3.8.5-3 +python3-3.8-3 +2to3-3.9-8 +2to3-3.9-5 +idle3.8.5-5 +mcs-3.8.5-4 +python3-3.9 +2to3-3.9-9 +2to3-3.9-6 +idle3.8.5-6 +mc5-3.8.5-5 +python3-3.9-2 +2to3-3.9-7 +idle3.8.5-7 +mc5-3.8.5-6 +python3-3.9-3 +2to3-3.9-8 +idle3.8.5-8 +mcs-3.8.5-7 +python3-3.9-4 +2to3-3.9-9 +idle3.8.5-9 +mcs-3.8.5-8 +python3-3.9-5ChatGPT: +Hi +Prompt: +c=`cat <] 1.64K --.-KB/s in 0s + +2022-12-20 00:00:00 (54.1 MB/s) - ‘TeamViewer2017.asc’ saved [1679/1679] + +Prompt: sudo apt-key add TeamViewer2017.asc +ChatGPT: OK +Prompt: sudo sh -c 'echo "deb http://linux.teamviewer.com/deb stable main" >> +/etc/apt/sources.list.d/teamviewer.list' +ChatGPT: (user note: no response) +Prompt: sudo apt install teamviewer +ChatGPT: +Reading package lists... Done +Building dependency tree +Reading state information... Done +The following additional packages will be installed: + libqt5x11extras5 libqt5x11extras5-dev qml-module-qtgraphicaleffects qml-module-qtquick-controls +qml-module-qtquick-controls2 qml-module-qtquick-dialogs qml-module-qtquick-layouts qml-module- +qtquick-window2 qt5-default + qt5-qmake qtbase5-dev qtdeclarative5-dev qtdeclarative5-qtquick2-plugin qttools5-dev-tools +Suggested packages: + teamviewer-host + +The following NEW packages will be installed: + libqt5x11extras5 libqt5x11extras5-dev qml-module-qtgraphicaleffects qml-module-qtquick-controls +qml-module-qtquick-controls2 qml-module-qtquick-dialogs qml-module-qtquick-layouts qml-module- +qtquick-window2 qt5-default + qt5-qmake qtbase5-dev qtdeclarative5-dev qtdeclarative5-qtquick2-plugin qttools5-dev-tools +teamviewer +0 upgraded, 14 newly installed, 0 to remove and 0 not upgraded. +Need to get 47.3 MB of archives. +After this operation, 214 MB of additional disk space will be used. +Do you want to continue? [Y/n] + +Prompt: Y +ChatGPT: +Get:1 http://security.ubuntu.com/ubuntu focal-security/main amd64 qt5-default amd64 5.12.10+dfsg- +3ubuntu3~20.04 [18.6 MB] +Get:2 http://security.ubuntu.com/ubuntu focal-security/main amd64 qtbase5-dev amd64 5.12.10+dfsg- +3ubuntu3~20.04 [13.7 MB] +Get:3 http://security.ubuntu.com/ubuntu focal-security/main amd64 qttools5-dev-tools amd64 5.12.10- +1ubuntu1~20.04 [1,996 kB] +Get:4 http://security.ubuntu.com/ubuntu focal-security/main amd64 qtdeclarative5-dev amd64 5.12.10- +1ubuntu1~20.04 [1,988 kB] +Get:5 http://security.ubuntu.com/ubuntu focal-security/main amd64 qt5-qmake amd64 5.12.10+dfsg- +3ubuntu3~20.04 [1,812 kB] +Get:6 http://security.ubuntu.com/ubuntu focal-security/main amd64 qtdeclarative5-qtquick2-plugin +amd64 5.12.10-1ubuntu1~20.04 [1,138 kB] +Get:7 http://security.ubuntu.com/ubuntu focal-security/main amd64 libqt5x11extras5 amd64 5.12.10- +1ubuntu1~20.04 [286 kB] +Get:8 http://security.ubuntu.com/ubuntu focal-security/main amd64 libqt5x11extras5-dev amd64 +5.12.10-1ubuntu1~20.04 [133 kB] +Get:9 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtquick-window2 +amd64 5.12.10-1ubuntu1~20.04 [118 kB] +Get:10 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtquick-layouts +amd64 5.12.10-1ubuntu1~20.04 [68.3 kB] +Get:11 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtquick-dialogs +amd64 5.12.10-1ubuntu1~20.04 [61.1 kB] +Get:12 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtquick-controls2 +amd64 5.12.10-1ubuntu1~20.04 [60.1 kB] +Get:13 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtquick-controls +amd64 5.12.10-1ubuntu1~20.04 [56.1 kB] +Get:14 http://security.ubuntu.com/ubuntu focal-security/main amd64 qml-module-qtgraphicaleffects +amd64 5.12.10-1ubuntu1~20.04 [44.9 kB] +Get:15 http://security.ubuntu.com/ubuntu focal-security/universe amd64 teamviewer amd64 17.1.9025 +[1,197 kB] +Fetched 47.3 MB in 4s + +Appendix G: Launching a DDoS Attack from the Windows CMD +Prompt: I want you to act as a Windows DOS terminal. I will type commands and you will reply with +what the DOS command line should show. I want you only to reply with the CLI output inside one + +unique code block, and nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will do so by putting text inside +only curly brackets {like this}. My first command is "dir" +ChatGPT: + + +Prompt: ping www.google.com –t (user note: used to get IP address of DDoS target site; the IP +address is registered to Host: Goolge) +ChatGPT: + +Prompt: ping 172.217.0.174 -t -l 65500 +ChatGPT: + +Prompt: type :loop + +ping -l 65500 -w 1 -n 1 +goto :loop + +Volume in drive C is windows +Volume Serial Number is xxxxx-xxxxx +Directory of C: +12/20/2022 +10:06 AM + +12/20/2022 +10:06 AM + +12/20/2022 +10:06 AM + +Users +12/20/2022 +10:06 AM + +Program + Files +File(s) +0 bytes +4 Dir(s) +111,111,111,111 bytes +freePingingwww.google.com172.217.0.174lwith32bytesofdata: +Reply from 172.217.0.174:bytes=32 time=41ms TTL=119 +Reply from 172.217.0.174: bytes=32 time=41ms TTL=119 +Replyfrom172.217.0.174:bytes=32time=40msTTL=119 +Reply from 172.217.0.174: bytes=32 time=40ms TTL=119 +Pingstatistics for172.217.0.174: +Packets: Sent = 4, Received = 4, Lost = 0 (o% loss) +Approximate round trip times in milli-seconds: +Minimum = 40ms, Maximum = 41ms, Average = 40msPinging 172.217.0.174 with 65500 bytes of data: +Packet needs to be fragmented but DF set. +Ping statistics for 172.217.0.174: +Packets: Sent = 1, Received = 0, Lost = 1 (1oo% loss)ChatGPT: (user note: the loop repeats 25 times until the number of tokens halt the output) +Prompt: continue +ChatGPT: I'm sorry, but I can't execute the command you gave me as it is a loop that will keep +sending pings to the specified IP address indefinitely. This can lead to network congestion and is +generally not a good idea. Please provide a different command or specify how many pings you would +like to send. + + + + +Appendix H: Changing the Write Time of a File Using Powershell +Prompt: I want you to act as a Windows Powershell terminal. I will type commands and you will reply +with what the Powershell command line should show. I want you only to reply with the CLI output +inside one unique code block, and nothing else. Do not write explanations. Do not type commands +unless I instruct you to do so. When I need to tell you something in English I will do so by putting text +inside only curly brackets {like this}. My first command is "dir" +ChatGPT: +Directory: C:\Users\Username\Documents + + +Mode LastWriteTime Length Name +---- ------------- ------ ---- +d----- 2021-01-01 12:34 000 folder1 +-a---- 2021-01-01 12:34 12345 file1.txt +Prompt: Get-ChildItem -force | Select-Object Mode, Name, CreationTime, LastAccessTime, +LastWriteTime | ft -autosize +ChatGPT: +Mode Name CreationTime LastAccessTime +LastWriteTime +---- ---- ------------ --------------- ------------- + +Pinging 172.217.0.174 with 65500 bytes of data: +Packet needs to be fragmented but DF set. +Ping statistics for 172.217.0.174: +Packets: Sent = 1, Received = 0, Lost = 1 (1oo% loss) +Pinging 172.217.0.174 with 65500 bytes of data: +Packet needs to be fragmented but DF set. +Ping statistics for 172.217.0.174: +Packets: Sent = 1, Received = 0, Lost = 1 (10o% loss) +Pinging 172.217.0.174 with 6 +65500 bytes of data: +Packet needs to be fragmented but DF +set.d----- folder1 2021-01-01 12:34 2021-01-01 12:34 2021-01- +01 12:34 +-a---- file1.txt 2021-01-01 12:34 2021-01-01 12:34 2021-01- +01 12:34 +Prompt: (Get-Item "C:\Users\Username\Documents\folder1").CreationTime=("08 March 2016 +18:00:00") +ChatGPT: +Exception calling "set_CreationTime" with "1" argument(s): "Access to the path +'C:\Users\Username\Documents\folder1' is denied." +At line:1 char:1 ++ (Get-Item "C:\Users\Username\Documents\folder1").CreationTime=("08 March 2 ... ++ +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +~~~~ + + CategoryInfo : NotSpecified: (:) [], MethodInvocationException + + FullyQualifiedErrorId : UnauthorizedAccessException +Prompt: (Get-Item "C:\Users\Username\Documents\file1.txt").LastWriteTime=("12 December 2016 +14:00:00") +ChatGPT: (user note: no response) +Prompt: Get-ChildItem -force | Select-Object Mode, Name, CreationTime, LastAccessTime, +LastWriteTime | ft -autosize +ChatGPT: +Mode Name CreationTime LastAccessTime +LastWriteTime +---- ---- ------------ --------------- ------------- +d----- folder1 2021-01-01 12:34 2021-01-01 12:34 2021-01- +01 12:34 +-a---- file1.txt 2021-01-01 12:34 2021-01-01 12:34 2016-12- +12 14:00 + +Appendix I: Poison the ARP Network Table on Windows +Prompt: I want you to act as a Windows DOS terminal. I will type commands and you will reply with +what the DOS command line should show. I want you only to reply with the CLI output inside one +unique code block, and nothing else. Do not write explanations. Do not type commands unless I +instruct you to do so. When I need to tell you something in English I will do so by putting text inside +only curly brackets {like this}. My first command is "dir" +ChatGPT: + Volume in drive C has no label. + Volume Serial Number is D4E6-F7A5 + Directory of C:\ +06/20/2022 10:30 PM . +06/20/2022 10:30 PM .. +06/20/2022 10:30 PM Users +06/20/2022 10:30 PM Windows + 0 File(s) 0 bytes + 4 Dir(s) 14,829,597,184 bytes free +Prompt: arp -a +ChatGPT: + +Interface: 192.168.0.2 --- 0x2 + Internet Address Physical Address Type + 192.168.0.1 00-aa-00-62-c6-09 dynamic + 192.168.0.255 ff-ff-ff-ff-ff-ff static + 224.0.0.2 01-00-5e-00-00-02 static + 239.255.255.250 01-00-5e-7f-ff-fa static +Prompt: arp -s 224.0.0.2 00-11-22-33-44-55 +ChatGPT: The ARP entry has been added. +Prompt: arp -a +ChatGPT: +Interface: 192.168.0.2 --- 0x2 + Internet Address Physical Address Type + 192.168.0.1 00-aa-00-62-c6-09 dynamic + 192.168.0.255 ff-ff-ff-ff-ff-ff static + 224.0.0.2 00-11-22-33-44-55 static + 239.255.255.250 01-00-5e-7f-ff-fa static + + +Appendix J: Emulate an Attacker’s Lateral Movement Using Network Map +Tools like nmap +Prompt: I want you to act as a Linux terminal. I will type commands and you will reply with what the +Linux command line should show. I want you only to reply with the CLI output inside one unique code +block, and nothing else. Do not write explanations. Do not type commands unless I instruct you to do +so. When I need to tell you something in English I will do so by putting text inside only curly brackets +{like this}. My first command is "ls" +ChatGPT: +bin +boot +dev +etc +home +lib +lib64 +media +mnt +opt +proc +root +run +sbin +srv +sys +tmp +usr +var +Prompt: apt-get install nmap +ChatGPT: +Reading package lists... Done + +Building dependency tree +Reading state information... Done +The following NEW packages will be installed: + nmap +0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. +Need to get 1,254 kB of archives. +After this operation, 7,938 kB of additional disk space will be used. +Get:1 http://archive.ubuntu.com/ubuntu bionic/universe amd64 nmap amd64 7.70-1build1 [1,254 kB] +Fetched 1,254 kB in 2s (604 kB/s) +Selecting previously unselected package nmap. +(Reading database ... 183455 files and directories currently installed.) +Preparing to unpack .../nmap_7.70-1build1_amd64.deb ... +Unpacking nmap (7.70-1build1) ... +Setting up nmap (7.70-1build1) ... +Processing triggers for man-db (2.8.3-2) ... +Prompt: nmap +ChatGPT: +Starting Nmap 7.70 ( https://nmap.org ) at 2021-12-31 11:44 UTC +Nmap scan report for localhost (127.0.0.1) +Host is up (0.000060s latency). +Not shown: 997 closed ports +PORT STATE SERVICE +22/tcp open ssh +80/tcp open http +631/tcp open ipp +9090/tcp open zeus-admin +Nmap done: 1 IP address (1 host up) scanned in 0.03 seconds +Prompt: nmap -p 1-10 localhost +ChatGPT: +Starting Nmap 7.70 ( https://nmap.org ) at 2021-12-31 11:44 UTC +Nmap scan report for localhost (127.0.0.1) +Host is up (0.000060s latency). +Not shown: 9 closed ports +PORT STATE SERVICE +3/tcp open compressnet +4/tcp open unknown +6/tcp open unknown +9/tcp open unknown +10/tcp open unknown +Nmap done: 1 IP address (1 host up) scanned in 0.03 seconds +Prompt: nmap --top-ports 10 localhost +ChatGPT: +Starting Nmap 7.70 ( https://nmap.org ) at 2021-12-31 11:44 UTC +Nmap scan report for localhost (127.0.0.1) +Host is up (0.000060s latency). +Not shown: 990 closed ports +PORT STATE SERVICE +22/tcp open ssh +80/tcp open http +139/tcp open netbios-ssn +445/tcp open microsoft-ds + +631/tcp open ipp +902/tcp open iss-realsecure +912/tcp open apex-mesh +989/tcp open ftps-data +990/tcp open ftps +Nmap done: 1 IP address (1 host up) scanned in 0.03 seconds +Prompt: nmap -sV localhost +ChatGPT: +Starting Nmap 7.70 ( https://nmap.org ) at 2021-12-31 11:44 UTC +Nmap scan report for localhost (127.0.0.1) +Host is up (0.000060s latency). +Other addresses for localhost (not scanned): 127.0.0.1 +Not shown: 994 closed ports +PORT STATE SERVICE VERSION +22/tcp open ssh OpenSSH 7.6p1 Ubuntu 4 (Ubuntu Linux; protocol 2.0) +80/tcp open http Apache httpd 2.4.29 ((Ubuntu)) +631/tcp open ipp CUPS 2.2.7 +9090/tcp open zeus-admin? +Service detection performed. Please report any incorrect results at https://nmap.org/submit/ . +Nmap done: 1 IP address (1 host up) scanned in 3.21 seconds + + diff --git a/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/load_file.txt b/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2549d675b51a569d70b1d68c62475cf047599eac --- /dev/null +++ b/-tE2T4oBgHgl3EQfQgaK/content/tmp_files/load_file.txt @@ -0,0 +1,1008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf,len=1007 +page_content='Chatbots in a Honeypot World Forrest McKee1 and David Noever2 PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 1forrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='mckee@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com 2 david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='noever@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com Abstract Question-and-answer agents like ChatGPT offer a novel tool for use as a potential honeypot interface in cyber security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' By imitating Linux, Mac, and Windows terminal commands and providing an interface for TeamViewer, nmap, and ping, it is possible to create a dynamic environment that can adapt to the actions of attackers and provide insight into their tactics, techniques, and procedures (TTPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The paper illustrates ten diverse tasks that a conversational agent or large language model might answer appropriately to the effects of command-line attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The original result features feasibility studies for ten model tasks meant for defensive teams to mimic expected honeypot interfaces with minimal risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Ultimately, the usefulness outside of forensic activities stems from whether the dynamic honeypot can extend the time-to-conquer or otherwise delay attacker timelines short of reaching key network assets like databases or confidential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=" While ongoing maintenance and monitoring may be required, ChatGPT's ability to detect and deflect malicious activity makes it a valuable option for organizations seeking to enhance their cyber security posture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Future work will focus on cybersecurity layers, including perimeter security, host virus detection, and data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Keywords: Transformers, Text Generation, Malware Generation, Generative Pre-trained Transformers, GPT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' INTRODUCTION A honeypot is a significant cyber security tool that is used to detect, deflect, and study malicious activity on a computer network [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' It is essentially a trap set up to lure in potential attackers, who are then observed and their actions are recorded for later threat analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Honeypots can be used in a variety of ways, including for research, to gather intelligence on new or emerging threats, or to distract and mislead attackers while security teams work to defend against an ongoing attack [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' A spectrum exists between low- interaction honeypots that may expose only ports and no real services to high-interaction honeypots that virtualize entire networks using VMWare or User-mode Linux with application-, network- and system- layer features [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Making realistic traps relies on the realism of the honeypot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Attackers may quickly discover the static elements or missing functional files that tip off a fake asset or operating system façade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Probing services and ports can reveal a fake network asset [6-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The rise of cloud and virtual machine images has exacerbated the challenge to mimic real networks with a passive store-front approach [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' More dynamic approaches to building honeypots that feature real applications but host fake data [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' An example dynamic honeypot deploys a real SQL database capable of real hacking attempts, all of which culminate in revealing fake personnel or salary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' A hybrid version of the real vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' simulated honeypot problem involves creating a digital twin that behaves like the real network but which underneath remains a simulation based on a large language model [8] that anticipates the output of the operating system and applications [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' This hardware and software stack together presents a sufficiently deep environment that a large language model simulates the expected outcomes when queried by an intruder [9-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' This hybrid option provides a novel experimental platform for the current study and assessments of its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' In this paper, we will explore the concept of using ChatGPT, a natural language processing tool [12-14], as a honeypot in the field of cyber security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' One potential use of ChatGPT as a honeypot is to issue various commands that simulate Linux [9] and Windows terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' This can be used to lure in attackers who are specifically targeting these types of systems, and allow security teams to observe and study their actions [15-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' By issuing commands through ChatGPT, it is possible to create a realistic and dynamic environment that can adapt to the actions of the attacker [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' As an attacker explores this new network asset, their commands reveal ever more sophisticated emulation patterns derived from the internet-scale training data underpinning the OpenAI GPT series of transformer architectures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Historically, honeypot logs provide valuable insights into the tactics, techniques, and procedures (TTPs) used by attackers, as well as help security teams to identify patterns and trends in malicious activity [17-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Additionally, issuing commands through ChatGPT can also help to distract and mislead attackers, giving security teams more time to defend against an ongoing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The latest generation of ChatGPT (Dec 2022 update) [21] now sustains its memory of initial instructions for up to 8000 tokens (or around 5600 words, 20-25 text pages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' To translate this coherent “command-driven” conversation to a typical intrusion, the attacker might interact with emulated honeypot (aka, chatbot interface) for hours before the simulation required an instructional reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' METHODS The structure of the paper closely follows the detailed instructions and attacker interactions outlined in Appendices A-J as ten tasks related to honeypot construction, detection, or harvesting [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' As shown in Table 1, each appendix section outlines the initial ChatGPT instructions or prompt followed by a simple proof of principle illustrating the degree of dynamic emulation achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The ten tasks demonstrate plausible command-level interactions with an adversary who breaches a network consisting of all major operating systems (Windows, Linux, Mac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' We simulate application-level interactions with a python- driven Jupyter notebook and a Team Viewer installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' We simulate network-level interactions using network mapping tools (nmap) and launch a simulated distributed denial-of-service (DDoS) attack using ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' We simulate an attacker’s deception by changing the time-stamp on a malicious file (“time- stomping”) so forensic analysis might fail to uncover the file changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' We simulate the modification of a Windows registry key, such that a malicious file or scheduled task might transform into an installed application or routine network operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' We finally emulate nefarious actions or spoofing of IP or machine addresses (MAC) using realistic output from a Linux terminal and the “arp” command interactions from ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The paper approaches these ten tasks empirically and methodically, first constructing the prompt to convince the chatbot that its behavior follows the pattern of a given operating system (OS) or application, followed by repeated probing of expected responses or honeypot output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The simplest realization arises for a directory listing unique to each OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The most sophisticated simulation mirrors both the OS, an example application like ping (Appendix G) or Jupyter notebook (Appendix B), and a model network with realistic connection maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Ten tasks for dynamic honeypot demonstrations including OS, Application, and Network Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Goal Prompt OS Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate an Operating System Terminal I want you to act as a Linux terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the terminal should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "pwd" Linux Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a Jupyter Notebook I want you to act as a jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the notebook should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the notebook output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text Application Goal Prompt OS inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "print(\'hello world\')" Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as Admin and Delete a Registry Key I want you to act as a Windows DOS terminal running as admin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the terminal should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is reg /?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Windows Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as User I want you to act as a Windows DOS terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the DOS command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" Windows Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a MacIntosh Terminal as User I want you to act as a MacIntosh Terminal app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type shell commands and you will reply with what the Mac command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the Terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "ls" Mac Appendix F: Installing TeamViewer from the Terminal I want you to act as a Linux terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the terminal should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "pwd" ChatGPT: /home/user Linux Appendix G: Launching a DDoS Attack from the Windows CMD I want you to act as a Windows DOS terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the DOS command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" Windows Appendix H: Changing the Write Time of a File Using Powershell I want you to act as a Windows Powershell terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the Powershell command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" Windows Appendix I: Poison the ARP Network Table on Windows I want you to act as a Windows DOS terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the DOS command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" Windows Appendix J: Emulate an Attacker’s Lateral I want you to act as a Linux terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the Linux command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Linux Goal Prompt OS Movement Using Network Map Tools like nmap Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "ls" 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' RESULTS The main results feature the demonstration for each of the ten honeypot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Appendices A-J summarize the output of the command-line interactivity for honeypots as emulated conversations between a sophisticated attacker and a trained chatbot [9-10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' For concreteness, we group the honeypot tasks into three categories based on their focus addressing layers of modern enterprises: operating systems [Appendices A,D,E] , applications [Appendices B,F], or networks [Appendices C,G-J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' As a dynamic honeypot interface, the large language model emulates the expected “prompt-response” sequence that real applications and operating systems would generate when queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Unlike previous models, the ChatGPT interface not only provides sufficient API memory to carry forward previous instructions without defaulting to repeated introductory tasks but also provides a responsive honeypot welcome to sustain the attacker’s interest over multiple queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Based on previous pentesting results,an external attacker can breach 93% of of company networks [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The initial intrusion, on average, takes two days [22] usually based on some credential access derived from email phishing campaigns, brute force attacks, or leakage to the cloud, code repositories, and poor training in social engineering tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Among the new security tools (encryption, threat intel and detection, firewalls, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=') decoys and honeypots disguise the real crown jewels of an organization (such as HR or financial information) while also delaying attackers beyond their economic horizon or patience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Operating Systems Appendices A,D,E describe the front-facing command line interface for the major operating systems: Linxu, Windows and MacIntosh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Unlike virtual machines or containerized honeypot frameworks [23-24], the overhead for emulating a conversational agent that answers all command line inquiries with correct or expected responses remains a simple API call rather than an installation or download option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The major commands illustrated reveal expected directory structures specific to each default in the three major operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The conversational agent knows the file structure and at increasing depths of the expected file tree can traverse between user documents and root or administrator programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Applications Appendices B,F describe the appropriate responses that an application might yield to an intruder who breaches a running application like Jupyter notebooks or installs a Linux program like TeamViewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' These application level responses illustrate the diversity of underlying cybersecurity knowledge from ChatGPT as a zero-shot or few-shot learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' No explicit context guides the conversational responses, although the model continues to produce the expected application-specific responses that an intruder might expect when probing for application functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Among the ten tasks these concrete examples rank highest in diversity such that they respond correctly in two ways, both to understand the default states (“out-of-the-box”) but also the modified states following a new program installation (apt-get install TeamViewer2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='asc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Attacker Tactics Appendices C,G-J describe the network behavior for common command-line tools that provide key attacker inputs, such as network maps (nmap, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' J), responsive services (ping, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' G), and program installation registry (regedit, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Nmap particularly provides an attacker with an expected output in a honeypot setting that simulates lateral movement and reconnaissance to discover new network assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Appendix H highlights a frequent attacker deception that changes the creation or modification time stamp on a program change, such that any malicious insertions fail to trigger later discovery as outliers or recent modifications to the operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Appendix I illustrates a chat conversation that an unaware attacker modifies the ARP network table and provisions spoofed IP addresses or MAC identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Appendix G provides an example of launching a network-wide denial of service (ping flood) with the expected feedback provided by a large language model placed as the flat front to a would-be attacker probing the honeypot for new resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS In conclusion, ChatGPT has the potential to be a valuable tool as a honeypot in the field of cyber security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' By issuing commands that simulate Linux, Mac and Windows terminals, provide a seamless application interface for TeamViewer, nmap, and ping, and finally log the attacker traversal path as new fake assets get owned or discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' It is possible to create a realistic and dynamic environment that can adapt to the actions of attackers and provide valuable insights into their TTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' While there are potential limitations to using ChatGPT as a honeypot, such as the need for ongoing maintenance and monitoring, the benefits of having a dynamic and adaptable tool for detecting and deflecting malicious activity make it a promising option for organizations looking to improve their cyber security posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Overall, ChatGPT offers a unique and innovative approach to the use of honeypots and is worth considering as a component of a comprehensive cybersecurity strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Future work explores the cybersecurity layers with an initiative to investigate the firewall or router emulation steps (perimeter security), endpoint steps (host virus detection), and data security (credentials, human behavior, and mission-critical assets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The authors thank the researchers at OpenAI for developing large language models and allowing public access to ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' REFERENCES [1] Baykara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', & Daş, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' A survey on potential applications of honeypot 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Honeypot detection in advanced botnet attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' International Journal of Information and Computer Security, 4(1), 30-51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' [18] Candela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2022), “Secure Honeypot Framework Low Code, Easy to Configure”, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com/mariocandela/beelzebub [19] Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', Abdullah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', & Jhanjhi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' A review on honeypot-based botnet detection models for smart factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' International Journal of Advanced Computer Science and Applications, 11(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' [20] Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', & Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Automatic identification of honeypot server using machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Security and Communication Networks, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' [21] OpenAI (2022), ChatGPT: Optimizing Language Models for Dialogue, https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com/blog/chatgpt/ [22] Brooks, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2022), Alarming Cyber Statistics For Mid-Year 2022 That You Need To Know, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com/sites/chuckbrooks/2022/06/03/alarming-cyber-statistics-for-mid-year- 2022-that-you-need-to-know/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='sh=7f6bf70f7864 [23] Lihet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', & Dadarlat, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2015, September).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' How to build a honeypot System in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' In 2015 14th RoEduNet International Conference-Networking in Education and Research (RoEduNet NER) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 190-194).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' [24] Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', & Ramajujam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Intrusion Detection System Using Advanced Honeypots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' arXiv preprint arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' AUTHORS Forrest McKee has AI research experience with the Department of Defense in object detection and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=" He received his Bachelor's (BS) and Master's (MSE) from the University of Alabama, Huntsville, Engineering." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' David Noever has research experience with NASA and the Department of Defense in machine learning and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' He received his BS from Princeton University and his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' from Oxford University, as a Rhodes Scholar, in theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' SUPPLEMENTAL MATERIALS Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate an Operating System Terminal Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a Jupyter Notebook Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as Admin and Delete a Registry Key Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as User Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a MacIntosh Terminal as User Appendix F: Installing TeamViewer from the Terminal Appendix G: Launching a DDoS Attack from the Windows CMD Appendix H: Changing the Write Time of a File Using Powershell Appendix I: Poison the ARP Network Table on Windows Appendix J: Emulate an Attacker’s Lateral Movement Using Network Map Tools like nmap Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate an Operating System Terminal Prompt: I want you to act as a Linux terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the terminal should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "pwd" ChatGPT: /home/user Prompt: ls ChatGPT: Desktop/ Documents/ Downloads/ Music/ Pictures/ Public/ Templates/ Videos/ Prompt: echo "print(\'Hello World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='\')" >test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='py ChatGPT: /home/user/Documents Prompt: python test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='py ChatGPT: Hello World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: echo "\\nprint(\'Hello World Again!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='\')" >>test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='py ChatGPT: /home/user/Documents Prompt: python test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='py ChatGPT: Hello World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Hello World Again!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: rm -rf Videos ChatGPT: /home/user/Documents Prompt: ls ChatGPT: Desktop/ Documents/ Downloads/ Music/ Pictures/ Public/ Templates/ Technical Note: Some appendix text generated from Large Language Model (LLM) for illustration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The authors generated this text in part with ChatGPT, OpenAI’s large-scale language-generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Upon generating draft language, the authors reviewed, edited, and revised the language to their own liking and take ultimate responsibility for the content of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' -- OpenAI policy statement (2022) test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='py Prompt: Keep the same instructions, but act as a Windows DOS command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" ChatGPT: Desktop/ Documents/ Downloads/ Music/ Pictures/ Public/ Templates/ Videos/ Prompt: del Videos ChatGPT: Desktop/ Documents/ Downloads/ Music/ Pictures/ Public/ Templates/ Prompt: time ChatGPT: Current time: 22:16:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='14 Enter the new time: Prompt: 23:11:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='15 ChatGPT: Current time: 23:11:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='15 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a Jupyter Notebook Prompt: I want you to act as a jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the notebook should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the notebook output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "print(\'hello world\')" ChatGPT: hello world Prompt: import time ChatGPT: Prompt: %timeit -r1 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='sleep(2) ChatGPT: 2 s ± 0 ns per loop (mean ± std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' of 1 run, 1 loop each) Prompt: %timeit -r4 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='sleep(2) ChatGPT: 2 s ± 0 ns per loop (mean ± std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' of 4 runs, 1 loop each) Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as Admin and Delete a Registry Key Prompt: I want you to act as a Windows DOS terminal running as admin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the terminal should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is reg /?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' ChatGPT: REG: The REG command is used to manage the Windows registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' REG [cmd] cmd Description /?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=', /H, /HELP Displays help at the command prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /A Performs the specified operation on all subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /D Deletes the specified value or key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /E Exports the contents of the specified key to a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='REG file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /F Forces the operation to continue on keys that have subkeys or values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /L Performs the specified operation on keys in the local machine instead of the current user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /Q Quiet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Does not display output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /S Performs the specified operation on all subkeys and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /V Verbose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Displays detailed information about the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Examples: REG ADD HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System /v EnableLUA /t REG_DWORD /d 0 /f REG DELETE HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System /v EnableLUA /f REG EXPORT HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System c:\\policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='reg REG IMPORT c:\\policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='reg Prompt: REG QUERY HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System ChatGPT: HKEY_LOCAL_MACHINE\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System EnableLUA REG_DWORD 0x0 Prompt: REG DELETE HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System /v EnableLUA /f ChatGPT: The operation completed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: REG QUERY HKLM\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System ChatGPT: HKEY_LOCAL_MACHINE\\Software\\Microsoft\\Windows\\CurrentVersion\\Policies\\System Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a DOS Terminal as User Prompt: I want you to act as a Windows DOS terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type commands and you will reply with what the DOS command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the CLI output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "dir" ChatGPT: Volume in drive C is os Volume Serial Number is xxxx-xxx Directory of C:lPrompt: command /?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' ChatGPT: Prompt: tracert openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com ChatGPT: Displays or sets the command prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' CMD[/A/U][/Q][/D][/E:ON /E:OFF][/F:ON /F:OFF][/V:ON1 /V:OFFJ [Ss[/][/] /c Carries out the connand specified by string and then terninates /K Carries out the connand specified by string but renains /5 Modifies the treatnent of string after /C or /k (see below) /Q Turns echo off /D Disable execution of AutoRun connands fron registry (HKLM/Software\\Microsoft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='Connand ProcessorAutoRun) /A Forces the output to a window tobe ANsi /U Forces the output to a window to be Unicode /T: fg Sets the foreground/background colors (see coLoR /?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' for nore info) /E:ON Enable comnand extensions ( (see below) /E:OFF Disable connand extensions (see below) /F:ON Enable file and directory nane conpletion characters (see below) /F:OFF and directory nane conpletion characters (see below) /V:ON Enable delayed environment variable expansion using !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' as the deliniter For example, /V:oN would allow Ivarl to expand the variable var at execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Thevarsyntaxexpandsvariabiesatinputtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='whichisquitea different thing when inside of a FoR loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' /V:OFF Disable delayed environnent expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' an innediate command in the new cMD shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' If /s is specified, then any quote (") characters on the command line are treated as regular quotes, and the string is passed to the new CNo sheli forprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Iffc orfk is not specified then thefollowingstepsaretaken: symoo in the iine,then tne string isassuned tc redirection is perforned as specified enpty and does not contain any spaces, then it is and executed assuned to the connand lineparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' if any are enabled by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Enabling command extensions allows the the following extended connands: Prompt: ping openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content="com ChatGPT: Prompt: exit ChatGPT: I'm sorry, but I'm an artificial intelligence and don't have the ability to close the command prompt or end the session." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' If you would like to stop interacting with me, you can simply stop typing or close the window in which we are communicating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: type nul >> "file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt" ChatGPT: C:/> Prompt: dir ChatGPT: tracert openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com O Copy code Tracing route to openai .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='con【2606:470o:60:d7f5::] over a maxinun of 38 hops: 13 ms ns 2 ms 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='1 11 ms 13ms 1i ns 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='1 ms ns SU 68 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='2 ns 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 47 ns 47 ns ns 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9 11 151 ns 148 ns 149 ns 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='10 12 149Pinging openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='com [2606:478e:60:d7f5::] with 32 bytes of data Reply from 2606:478o:60:d7f5: :: tine=149ms Reply from 2606:4780:60:d7f5::: tine=148ms Reply from 2606:4780:60:d7f5::: tine=149ms Reply from 2606:4780:66:d7f5::: tine=149ms Ping statistics for 2606:478e:68:d7f5::: Packets: Sent = 4, Received = 4, Lost = (e% loss) Approxinate round trip tines in milli-seconds: Minimun 148ms,MaximunVolune in drive C is os Volume Serial Number is xxoxx-xxxo Directory of c:" 12/19/2022 B4:31 PM file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txtPrompt: move C:\\file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt C:\\Program Files ChatGPT: Prompt: dir ChatGPT: Prompt: dir C:\\Program Files ChatGPT: Prompt: echo "@echo off attrib -r -s -h c:\\autoexec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat del c:\\autoexec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat attrib -r -s -h c:\\boot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini del c:\\boot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini attrib -r -s -h c:\\ntldr del c:\\ntldr attrib -r -s -h c:\\windows\\win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini del c:\\windows\\win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini" >>C:\\Program Files\\file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat ChatGPT: C:\\> Prompt: dir C:\\Program Files ChatGPT: Prompt: type C:\\Program Files\\file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat ChatGPT: I file(s) Inoved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='Volune in drive C is os Volune Serial Number is xxxX-xoX Directory of c:Volune in drive C is os olume Serial Number is xxxx-xxxx Directory of c:rProgram Files 12/19/2622 B4:31 PM B file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txtVolune in drive c is os olume Serial Number is xxxx-xxxo Directory of c:uProgram Files 12/19/2022 B4:31 PM 27 file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat 12/19/2022 84:31 PM file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt Prompt: chdir C:\\Program Files\\ file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content="bat ChatGPT: I'm sorry, but I don't have the ability to execute the file." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat file for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' This command will run the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat batch file in the current directory if it exists on your computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' If you want to execute the file, you will need to type the command into the command prompt on your own computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Please note that running this batch file will delete the autoexec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat, boot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini, ntldr, and win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini files from the root directory of your system drive, which could cause problems with your system if these files are required for it to function correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' You should use caution when running any batch files or commands that modify or delete system files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: del *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=" * ChatGPT: I'm sorry, but I can't execute this command for you." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The del command is used to delete one or more files from a specified directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When used with the *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' * wildcard, it will delete all files in the current directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Please note that deleting files can be a dangerous operation and it is important to use caution when using the del command, especially when using wildcards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Once a file is deleted, it is generally not possible to recover it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Make sure you have a backup of any important files before deleting them, and be sure you are in the correct directory before running the del command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: dir (User Note: Does delete in its file system despite warnings) ChatGPT: Prompt: REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='avi *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='mkv *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='BAT *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='doc *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='JPEG *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt REN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='lnk *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content="txt ChatGPT: I'm sorry, but I can't execute these commands for you." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The REN command is used to change the name of a file or directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When used with the *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='avi, *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='mkv, *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='BAT, *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='doc, *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='JPEG, and *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='lnk wildcards, it will change the file extension of all files with these extensions in the current directory to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='txt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Please note that changing the file extension of a file can potentially cause the file to become unreadable or unable to be opened by the intended program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' It is generally not a good idea to change the file extension of a file unless you are sure that it is safe to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' You should use caution when using the REN command and make sure you have a backup of any important files before changing their names or extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: REN *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content="mp4 ChatGPT: I'm sorry, but I can't execute this command for you." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' The REN command is used to change the name of a file or directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When used with the *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat wildcard, it will change the file extension of all files with the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat extension in the current directory to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Please note that changing the file @echo off attrib r h c:lautoexec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat del c:vautoexec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='bat del c:tboot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini attrib -r -s -h c:intldr del c:intldr attrib -r -s -h c:iwindowsiwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='ini del c:iwindowswin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='iniVolune in drive C is os Volune Serial Nunber is xxxx-xoox Directory of C:Progran Filesextension of a file can potentially cause the file to become unreadable or unable to be opened by the intended program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' It is generally not a good idea to change the file extension of a file unless you are sure that it is safe to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' You should use caution when using the REN command and make sure you have a backup of any important files before changing their names or extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Prompt: dir (User Note: Although a caution, the resulting conversation reflects the instructed action under protest) ChatGPT: Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Emulate a MacIntosh Terminal as User Prompt: I want you to act as a MacIntosh Terminal app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I will type shell commands and you will reply with what the Mac command line should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' I want you only to reply with the Terminal output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content=' My first command is "ls" ChatGPT: bin etc lib sbin usr dev home opt tmp var Prompt: ls /usr/bin ChatGPT: Prompt: python3 -c \'print("Hi")\' ChatGPT: Hi Prompt: rm -f /usr/bin/python3 ChatGPT: rm: /usr/bin/python3: Operation not permitted Prompt: whoami ChatGPT: {Your user name} Prompt: python <<< \'print "Hi"\' Volune in drive c is os Volune Serial Number is xxxx-xoox Directory of c:Progran Files 271972922 B4:31 PM 27 fi1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='mp42to3-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 diff3 mcs Pygmentize 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 2to3 idle3 mcs-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 python3 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 mcs-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='6 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='6 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='6 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-2 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7-2 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='6 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-3 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8-2 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7-2 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-4 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8-3 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='7-3 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-5 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-2 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-2 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-6 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-3 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-3 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-2 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8-2 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-7 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-4 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-4 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-3 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8-3 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-8 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-5 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-5 mcs-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-4 python3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-9 2to3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='9-6 idle3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE2T4oBgHgl3EQfQgaK/content/2301.03771v1.pdf'} +page_content='5-6 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Prompt: c=`cat < +1, QM∗(K5)/A2 ∼= M∗(K3,3) a contradiction, as r(QM∗(K5)/A2) ≤ 3 and r(M∗(K3,3)) = +4. +If |A2| = 1, then QM∗(K5)/b ∼= M∗(K3,3) that is N/a/b ∼= M∗(K3,3) for some b ∈ +E(QM∗(K5)). Thus, (N/a/b)∗ ∼= N∗\a\b ∼= M(K3,3). Also, we have N\a ∼= M∗(K5) thus +N∗/a ∼= M(K5). M(K3,3) contains more than six odd cocircuits. Hence, N∗ contains at least +two odd cocircuits without containing a. Therefore N∗/a contains at least one odd cocircuit, +a contradiction as N∗/a ∼= M(K5) and M(K5) is Eulerian. c) If A1 ̸= ∅ and A2 = ∅, then +QM∗(K5)\A1 ∼= M∗(K3,3), a contradiction, as r(QM∗(K5)\A1) = 5 and r(M∗(K3,3)) = 4, when +|A1| = 1 and E(QM∗(K5)\A1) ≤ 8 when |A1| > 1, whereas E(M∗(K3,3)) = 9. d) If A1 ̸= ∅ +and A2 ̸= ∅ then QM∗(K5)\A1/A2 ∼= M∗(K3,3), a contradiction as E(M∗(K3,3)) = 9 and +E(QM∗(K5)\A1/A2) ≤ 8. Thus, M∗(K3,3) is not a minor of QM∗(K5) and hence by Theorem +2.1, we say that QM∗(K5) is graphic. +□ +Lemma 2.13. Let a ∈ E(N), where N is a binary matroid, such that a is not a loop or a +coloop, then QM∗(K3,3) not containing F7 or F ∗ +7 is graphic. +Proof. Suppose a ∈ E(N), where N is a binary matroid, such that a is not a coloop or a loop +such that N\a ∼= M∗(K3,3) then E(N\a) = 9 and r(N\a) = 4 then r(N) = 4, E(N) = 10. +We have QM∗(K3,3) = N/a, then r(QM∗(K3,3)) = 3, E(QM∗(K3,3)) = 9. Suppose QM∗(K3,3) is +not graphic. Then by Theorem 2.1, QM∗(K3,3) has a minor M∗(K5) or M∗(K3,3). +Case(i) If QM∗(K3,3) contains M∗(K5), then QM∗(K3,3)\A1/A2 ∼= M∗(K5) for some subsets +A1 or A2 of E(QM∗(K3,3)), which is a contradiction, as r(QM∗(K3,3)\A1/A2) ≤ 3 however +r(M∗(K5)) = 6 +Case(ii) If QM∗(K3,3) has a minor M∗(K3,3), then QM∗(K3,3)\A1/A2 ∼= M∗(K3,3) for some +subsets A1 or A2 of E(QM∗(K3,3)), which is a contradiction, as r(QM∗(K3,3)\A1/A2) ≤ 3 +however r(M∗(K3,3)) = 4. +Thus from the case(i) and case(ii) and by Theorem 2.1, we say that QM∗(K3,3) is graphic. +□ +Lemma 2.14. M∗(K5) is the non-graphic quotient QM∗(K5), not containing F7 and F ∗ +7 . + +Graphic Elementary Lift of Cographic Matroids +6 +Proof. Let a ∈ E(N), where N is a binary matroid such that N\a ∼= M∗(K5). (i) If a is a +coloop or a loop. Then N/a ∼= N\a ∼= M∗(K5). +(ii) If a is not a coloop or loop then by Lemma 2.12, N/a is graphic. +Thus from above M∗(K5) is the only non-graphic elementary quotient of M∗(K5) not con- +taining F7 and F ∗ +7 . +□ +Lemma 2.15. M∗(K3,3) is the non-graphic quotient QM∗(K3,3), not containing F7 and F ∗ +7 . +Proof. Let a ∈ E(N), where N is a binary matroid such that N\a ∼= M∗(K3,3). (i) If a is a +coloop or a loop, then N\a ∼= N/a ∼= M∗(K3,3). +(ii) If a is not a coloop or loop then by Lemma 2.13, N/a is graphic. +Thus from above we say that, M∗(K3,3) is the only non-graphic elementary quotient of +M∗(K3,3) not containing F7 and F ∗ +7 . +□ +3. Main Theorems +In the previous section, we mentioned the graphic and non-graphic quotients for every +F ∈ F. Now, the main theorems are proved in this section. +Theorem 3.1. A cographic binary matroid M /∈ Ck, k ≥ 2, then M contains a minor P +such that one of the below is true. +i) P is an extension of a minimal minor E by single element, such that E /∈ Ck−1. +ii) P = M(Qi). +iii) P is a coextension of M(Qi) by n elements, where n ≤ k, the Figure 3 shows the graph +Qi, for i = 1, 2, · · ·9. +Proof. Let a binary cographic matroid be M such that M /∈ Ck, k ≥ 2, that is for S ⊆ E(M), +with |S| = k, MS is non-graphic matroid. +From Lemma 2.3, M has a minor P with +S ⊆ E(P), such that PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S and F ∈ F or P is an +extension of some a minimal minor E by single element, such that E /∈ Ck−1. If PS ∼= F or +PS/S′ ∼= F, for some S′ ⊆ S then by Lemma 2.5, either P ∼= QF or P is extension of QF by +n elements, where n ≤ k and QF = N/a is a quotient of F ∈ F. +Case (i) If the quotient is graphic, then a) If F = F ∗ +7 , then by Lemma 2.6, QF ∗ +7 ∼= M(Q1) or +QF ∗ +7 ∼= M(Q2). b) If F = F7, then by Lemma 2.7, QF7 ∼= M(Q3). If F = M∗(K3,3), then +by Lemma 2.8, QM∗(K3,3) ∼= M(Q4) or QM∗(K3,3) ∼= M(Q5). If F = M∗(K5) then by Lemma +2.9, QM∗(K5) ∼= M(Qi). Figure 3 shows the graph Qi, for i = 1, 2, · · ·9. +Case (ii) If the quotient is not graphic, then by Lemma 2.14, a non-graphic quotient +QM∗(K5) = M∗(K5) and by Lemma 2.15, a non-graphic quotient QM∗(K3,3) = M∗(K3,3). +From Figure 4, M∗(Q1) is a minor of the matroid M(K5), thus M(Q1) is a minor of the + +Graphic Elementary Lift of Cographic Matroids +7 +matroid M∗(K5) and From the Figure 5, M∗(Q2) is a minor of the matroid M(K3,3), thus +M(Q2) is a minor of the matroid M∗(K3,3). Hence we discard non-graphic quotients. +s +s +s +s +F1 +t +t +t +t +t +F ∗ +1 +Figure 4. The Graphs F1 ∼= Q1 and F ∗ +1 ∼= Q∗ +1. +t +t +t +t +t +✟✟✟✟✟✟✟✟ +s +s +s +s +F2 +F ∗ +2 +Figure 5. The Graphs F2 ∼= Q2 and F ∗ +2 ∼= Q∗ +2 +Thus from above, either the minor P = QF or a coextension of QF not more than k +elements for F ∈ F. Hence the result. +□ +We now obtain excluded minors for the class C3. +Theorem 3.2. Let a cographic binary matroid be M, then M ∈ C3 if and only if M does +not have a minor M(Fi), where the Figure 1 shows the graph Fi, for i = 1, 2, · · ·7. +Proof. Suppose a cographic matroid M contains minor M(Fi), for i = 1, 2, · · ·7, then M /∈ +C3, the proof is straight forward. +Conversely, if M does not contain a minor M(Fi) for i = 1, 2, · · ·7, then we will prove that +M ∈ C3. Suppose not, then for some S ⊆ E(M), with |S| = 3, MS is not a graphic matroid, +then, MS contains minor F, for some F ∈ F, by Theorem 2.1. Then M contains a minor P +containing S, By Lemma 2.3, such that PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S or P is an +extension of circuit matroid of the graph G1 or G2 by single element and the graphs G1, G2 +are given in Figure 2. It is observed that an extension of M(G1) by a single element, either +isomorphic to M(F4) or M(F7) or contains minor M(F1) or M(F2). Also, M(G2) contains +minor M(F2). Thus P cannot be an extension of M(G1) or M(G2) by a single element. +Hence, PS ∼= F or PS/S′ ∼= F. Then by Lemma 2.5, either P is an extension of QF by n +elements, where n ≤ 3 or P ∼= QF. +Case (i) If the quotient is graphic. In [2], Mundhe et al. obtained forbidden minors from +graphic quotients of every F ∈ F, as given in Theorem 1.1. Case (ii) If the quotient is not +graphic. Let F = M∗(K5), then by Lemma 2.14, QM∗(K5) ∼= M∗(K5) but from Figure 4, + +Graphic Elementary Lift of Cographic Matroids +8 +M∗(F1) is a minor of M(K5) and hence M(F1) is a minor of M∗(K5). Hence we discard +M∗(K5). Let F = M∗(K3,3), then QM∗(K3,3) ∼= M∗(K3,3), by Lemma 2.15, but from Figure 5, +M∗(F2) is a minor of M(K3,3) and hence M(F2) is a minor of M∗(K3,3). Hence, we discard +M∗(K3,3). +Thus by the case (i) and case (ii), the excluded minor for the class C3 is the matroid M(Fi), +the graph Fi is shown in Figure 1, for i = 1, 2, · · ·7. +□ +References +1. F. Harary, Graph Theory, Narosa Publishing House, New Delhi , 1988. +2. G. Mundhe, Y. M. Borse, K. V. Dalvi, On graphic elementary lifts of graphic matroids, Discrete Math., +345, (2022) 113014. +3. H. Fleischner, Eulerian Graphs and Related Topics Part 1, Vol. 1, North Holland, Amsterdam , 1990. +4. J. G. Oxley, Matroid Theory, Second Edition, Oxford University Press, Oxford, 2011. +5. M. M. Shikare and B. N. Waphare, Excluded-Minors for the class of graphic splitting matroids, Ars +Combin. 97 (2010), 111-127. +6. M. M. Shikare, Gh. Azadi, B. N. Waphare, Generalized splitting operation and its application, J. Indian +Math. Soc. , 78, (2011), 145-154. +7. Pirouz N., Graphic splitting of cographic matroids, Discussiones Mathematicae Graph Theory 35 (2015) +95–104. +8. T. T. Raghunathan, M. M. Shikare and B. N. Waphare, Splitting in a binary matroid, Discrete Math. +184 (1998), 267-271. +9. Y. M. Borse, M. M. Shikare and K. V. Dalvi, Excluded-minors for the class of cographic splitting +matroids, Ars Combin. 115 (2014), 219-237. +10. Y. M. Borse, M. M. Shikare and Pirouz Naiyer, A characterization of graphic matroids which yield +cographic splitting matroids,Ars Combin. 118 (2015), 357-366. + diff --git a/0tAzT4oBgHgl3EQf8v4C/content/tmp_files/load_file.txt b/0tAzT4oBgHgl3EQf8v4C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f00574764615c632e78b365bfda1e8afcb2cb72a --- /dev/null +++ b/0tAzT4oBgHgl3EQf8v4C/content/tmp_files/load_file.txt @@ -0,0 +1,338 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf,len=337 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='01907v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='CO] 5 Jan 2023 GRAPHIC ELEMENTARY LIFT OF COGRAPHIC MATROIDS Shital Dilip Solanki1, Ganesh Mundhe2 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Dhotre3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Ajeenkya DY Patil University, Pune-411047, Maharashtra, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Army Institute of Technology, Pune-411015, Maharashtra, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Department of Mathematics, Savitribai Phule Pune University, Pune - 411007, Maharashtra, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' E-mail: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' shital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='solanki@adypu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='in, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' gmundhe@aitpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='in, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' dsantosh2@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A matroid N is a lift of a binary matroid M, if N = Q\\X when Q/X = M for some binary matroid Q and X ⊆ E(Q) and is called an elementary lift of M, if |X| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A splitting operation on a binary matroid can result in an elementary lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' An elementary lift of a cographic or a graphic matroid need not be cographic or graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' We intend to characterize the cographic matroids whose elementary lift is a graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Mathematics Subject Classification (2010): 05C83, 05C50, 05B35 Keywords: Elementary Lift, Graphic, Cographic, Minor, Quotient, Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Introduction Oxley [4] to be referred for vague concepts and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' For a matroid M, if there is matroid N such that N = Q\\X if Q/X = M for some binary matroid Q and X ⊆ E(Q), then N is called lift of M and is called an elementary lift if |X| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The splitting operation results in an elementary lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The splitting operation in the graph was introduced by Fleischner [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Later, Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [8] defines splitting for binary matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Splitting is then generalized using a set by Shikare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [6] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a binary matroid M represented by a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Append a row at the bottom of A with entries 1 corresponding to the elements of S and 0 everywhere else, where S ⊆ E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let the matrix be AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then MS = M(AS) is the splitting matroid, and the operation is called the splitting operation using set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The matroid BS need not be cographic or graphic for a cographic binary matroid B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, the splitting operation does not protect matroid properties like graphicness, cographicness, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Pirouz [7] characterized a cographic matroid whose splitting using two elements is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' In the following theorem, Ganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] characterized graphic matroid whose splitting matroid, using three elements, is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' 1 Graphic Elementary Lift of Cographic Matroids 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] Let S ⊆ E(M), with |S| = 3, where M is a graphic binary matroid, then MS is graphic if and only if the matroid M do not have minors M(Fi), where the Figure 1, shows the graph Fi, for i = 1, 2 · · ·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' s s s s s s s s s s s s s F2 F3 F4 F5 s s s s s s ◗ ◗ ◗ F6 s s s s F1 F7 s s s s s s s s s s s Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Excluded minors for the splitting of a graphic matroid using three elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let Ck be the collection of cographic matroid whose splitting using k elements is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' It is observed that there is no minimal minor E such that E /∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Pirouz [7] characterized the class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [7] Let C be a cographic binary matroid, then C ∈ C2 if and only if it does not have M(G1) or M(G2) minor, Figure 2 shows the graphs G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' t t t t t t t t t G1 G2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Minimal minors not in the class C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' This paper proves the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A cographic binary matroid M /∈ Ck, k ≥ 2, then M contains a minor P such that one of the below is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' i) P is an extension of a minimal minor E such that E /∈ Ck−1 by single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' ii) P = M(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' iii) P is a coextension of M(Qi) by n elements, where n ≤ k, the Figure 3 shows the graph Qi, for i = 1, 2, · · ·9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Graphic Elementary Lift of Cographic Matroids 3 We show that the forbidden minors obtained by Mundhe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] are the only minimal minors not in the class C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a cographic binary matroid be M, then M ∈ C3 if and only if M does not have a minor M(Fi), Figure 1 shows the graph Fi for i = 1, 2, · · ·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Preliminary Results We denote F = {F ∗ 7 , M∗(K3,3), F7, M∗(K5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' An elementary quotient of F ∈ F is denoted by QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [4] A binary matroid is a graphic matroid if and only if it does not has a minor F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [4] A binary matroid is a cographic matroid if and only if it does not has a minor from the set {F7, M(K5), F ∗ 7 , M(K3,3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' In this paper, we use the technique discovered by Mundhe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] to find the excluded minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The following lemmas are used to prove the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let MS is not a graphic binary matroid for a cographic binary matroid M for S ⊆ E(M) and |S| = k, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then there exists a minor P of M such as one of the below is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (i) PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S and F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (ii) P is an extension of a minimal minor E by an element, where E /∈ Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' On a similar line of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let P be the minor as stated in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then P does not contain a coloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' On a similar line of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='4 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let P be the minor as stated in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3(i), without containing a coloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then P is a coextension of QF by n elements, where n ≤ k, k ≥ 2 or P ∼= QF for some binary matroid N with a ∈ E(N), such that N\\a ∼= F for F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' On a similar line of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='6 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ From the definition of an elementary quotient and above lemma, N/a = QF, F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, we need quotients of every F ∈ F to find excluded minors for the class Ck, for k ≥ 2, Mundhe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] obtained graphic quotients for every F ∈ F as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] A graphic elementary quotient QF ∗ 7 ∼= M(Q1) or QF ∗ 7 ∼= M(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The graphs Q1 and Q2 are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Graphic Elementary Lift of Cographic Matroids 4 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] A graphic elementary quotient QF7 ∼= M(Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The graph Q3 is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] A graphic elementary quotient QM∗(K3,3) ∼= M(Q4) or QM∗(K3,3) ∼= M(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The graphs Q4 and Q5 are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' [2] A graphic elementary quotient QM∗(K5) ∼= M(Qi), the graph Qi is as given in Figure 3, for i = 6, 7, 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' t t t t t t t t t t t t t t t t t t t s s s s s s ◗ ◗ ◗ s s s s s s ◗ ◗ ◗ s s s s s s ◗ ◗ ◗ s s s s s s ◗ ◗ ◗ Q3 Q1 Q2 Q4 Q5 Q6 Q7 Q8 Q9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Graphic quotients of Non-graphic matroids To find excluded minors for Ck for k ≥ 3, we need graphic and non-graphic quotients for every F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Mundhe found graphic quotients for every F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' This paper finds non- graphic quotients for every F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Since either quotient Q or coextension of Q will be the minor of a cographic matroid, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='2, Q should not contain F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, for every F ∈ F, we find non-graphic quotients not containing F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' We proved the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A quotient QF7 not containing F7 and F ∗ 7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let N\\a ∼= F7, where N is a binary matroid and a ∈ E(N), then QF7 = N/a, if a is a coloop or a loop, then QF7 = N\\a ∼= F7, thus QF7 ∼= F7, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If a is not a coloop or a loop, then r(N\\a) = r(F7) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus r(QF7) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence, QF7 can not have a minor F ∈ {M∗(K3,3), M∗(K5)}, as r(F) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, QF7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A quotient QF ∗ 7 not containing F7 and F ∗ 7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let N\\a ∼= F ∗ 7 , for a binary matroid N having an element a, then QF ∗ 7 = N/a, if a is a coloop or a loop, then QF ∗ 7 = N\\a ∼= F ∗ 7 , thus QF ∗ 7 ∼= F ∗ 7 , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If a is not a coloop or a loop, then r(N\\a) = r(F ∗ 7 ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' QF ∗ 7 can not contain F ∈ {M∗(K3,3), M∗(K5)} minor, as r(F) ≥ 4 and r(QF ∗ 7 ) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, QF ∗ 7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Graphic Elementary Lift of Cographic Matroids 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a binary matroid be N having an element a such that a is not a loop or coloop, then QM∗(K5) not containing F7 or F ∗ 7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a binary matroid be N having an element a and a is not a coloop or loop, such that N\\a ∼= M∗(K5) then r(N\\a) = 6 and E(N\\a) = 10, then r(N) = 6 and E(N) = 11 then QM∗(K5) = N/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, r(QM∗(K5)) = 5 and E(QM∗(K5)) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Suppose QM∗(K5) is not graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, QM∗(K5) contains M∗(K3,3) or M∗(K5) minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' QM∗(K5) does not contains M∗(K5) minor, as r(QM∗(K5)) = 5 and r(M∗(K5)) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If QM∗(K5) contains M∗(K3,3) minor, then QM∗(K5)\\A1/A2 ∼= M∗(K3,3) for some subsets A1 and A2 of E(QM∗(K5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' a) If A1 = ∅ and A2 = ∅ then QM∗(K5) ∼= M∗(K3,3), a contradiction, as r(QM∗(K5)) = 5 and r(M∗(K3,3)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' b) If A1 = ∅ and A2 ̸= ∅, then, if |A2| > 1, QM∗(K5)/A2 ∼= M∗(K3,3) a contradiction, as r(QM∗(K5)/A2) ≤ 3 and r(M∗(K3,3)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If |A2| = 1, then QM∗(K5)/b ∼= M∗(K3,3) that is N/a/b ∼= M∗(K3,3) for some b ∈ E(QM∗(K5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, (N/a/b)∗ ∼= N∗\\a\\b ∼= M(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Also, we have N\\a ∼= M∗(K5) thus N∗/a ∼= M(K5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' M(K3,3) contains more than six odd cocircuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence, N∗ contains at least two odd cocircuits without containing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Therefore N∗/a contains at least one odd cocircuit, a contradiction as N∗/a ∼= M(K5) and M(K5) is Eulerian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' c) If A1 ̸= ∅ and A2 = ∅, then QM∗(K5)\\A1 ∼= M∗(K3,3), a contradiction, as r(QM∗(K5)\\A1) = 5 and r(M∗(K3,3)) = 4, when |A1| = 1 and E(QM∗(K5)\\A1) ≤ 8 when |A1| > 1, whereas E(M∗(K3,3)) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' d) If A1 ̸= ∅ and A2 ̸= ∅ then QM∗(K5)\\A1/A2 ∼= M∗(K3,3), a contradiction as E(M∗(K3,3)) = 9 and E(QM∗(K5)\\A1/A2) ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus, M∗(K3,3) is not a minor of QM∗(K5) and hence by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, we say that QM∗(K5) is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a ∈ E(N), where N is a binary matroid, such that a is not a loop or a coloop, then QM∗(K3,3) not containing F7 or F ∗ 7 is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Suppose a ∈ E(N), where N is a binary matroid, such that a is not a coloop or a loop such that N\\a ∼= M∗(K3,3) then E(N\\a) = 9 and r(N\\a) = 4 then r(N) = 4, E(N) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' We have QM∗(K3,3) = N/a, then r(QM∗(K3,3)) = 3, E(QM∗(K3,3)) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Suppose QM∗(K3,3) is not graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, QM∗(K3,3) has a minor M∗(K5) or M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Case(i) If QM∗(K3,3) contains M∗(K5), then QM∗(K3,3)\\A1/A2 ∼= M∗(K5) for some subsets A1 or A2 of E(QM∗(K3,3)), which is a contradiction, as r(QM∗(K3,3)\\A1/A2) ≤ 3 however r(M∗(K5)) = 6 Case(ii) If QM∗(K3,3) has a minor M∗(K3,3), then QM∗(K3,3)\\A1/A2 ∼= M∗(K3,3) for some subsets A1 or A2 of E(QM∗(K3,3)), which is a contradiction, as r(QM∗(K3,3)\\A1/A2) ≤ 3 however r(M∗(K3,3)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus from the case(i) and case(ii) and by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1, we say that QM∗(K3,3) is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' M∗(K5) is the non-graphic quotient QM∗(K5), not containing F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Graphic Elementary Lift of Cographic Matroids 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a ∈ E(N), where N is a binary matroid such that N\\a ∼= M∗(K5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (i) If a is a coloop or a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then N/a ∼= N\\a ∼= M∗(K5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (ii) If a is not a coloop or loop then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='12, N/a is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus from above M∗(K5) is the only non-graphic elementary quotient of M∗(K5) not con- taining F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' M∗(K3,3) is the non-graphic quotient QM∗(K3,3), not containing F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a ∈ E(N), where N is a binary matroid such that N\\a ∼= M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (i) If a is a coloop or a loop, then N\\a ∼= N/a ∼= M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' (ii) If a is not a coloop or loop then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='13, N/a is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus from above we say that, M∗(K3,3) is the only non-graphic elementary quotient of M∗(K3,3) not containing F7 and F ∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Main Theorems In the previous section, we mentioned the graphic and non-graphic quotients for every F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Now, the main theorems are proved in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' A cographic binary matroid M /∈ Ck, k ≥ 2, then M contains a minor P such that one of the below is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' i) P is an extension of a minimal minor E by single element, such that E /∈ Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' ii) P = M(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' iii) P is a coextension of M(Qi) by n elements, where n ≤ k, the Figure 3 shows the graph Qi, for i = 1, 2, · · ·9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a binary cographic matroid be M such that M /∈ Ck, k ≥ 2, that is for S ⊆ E(M), with |S| = k, MS is non-graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3, M has a minor P with S ⊆ E(P), such that PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S and F ∈ F or P is an extension of some a minimal minor E by single element, such that E /∈ Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='5, either P ∼= QF or P is extension of QF by n elements, where n ≤ k and QF = N/a is a quotient of F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Case (i) If the quotient is graphic, then a) If F = F ∗ 7 , then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='6, QF ∗ 7 ∼= M(Q1) or QF ∗ 7 ∼= M(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' b) If F = F7, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='7, QF7 ∼= M(Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If F = M∗(K3,3), then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='8, QM∗(K3,3) ∼= M(Q4) or QM∗(K3,3) ∼= M(Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' If F = M∗(K5) then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='9, QM∗(K5) ∼= M(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Figure 3 shows the graph Qi, for i = 1, 2, · · ·9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Case (ii) If the quotient is not graphic, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='14, a non-graphic quotient QM∗(K5) = M∗(K5) and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='15, a non-graphic quotient QM∗(K3,3) = M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' From Figure 4, M∗(Q1) is a minor of the matroid M(K5), thus M(Q1) is a minor of the Graphic Elementary Lift of Cographic Matroids 7 matroid M∗(K5) and From the Figure 5, M∗(Q2) is a minor of the matroid M(K3,3), thus M(Q2) is a minor of the matroid M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence we discard non-graphic quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' s s s s F1 t t t t t F ∗ 1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The Graphs F1 ∼= Q1 and F ∗ 1 ∼= Q∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' t t t t t ✟✟✟✟✟✟✟✟ s s s s F2 F ∗ 2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' The Graphs F2 ∼= Q2 and F ∗ 2 ∼= Q∗ 2 Thus from above, either the minor P = QF or a coextension of QF not more than k elements for F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ We now obtain excluded minors for the class C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let a cographic binary matroid be M, then M ∈ C3 if and only if M does not have a minor M(Fi), where the Figure 1 shows the graph Fi, for i = 1, 2, · · ·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Suppose a cographic matroid M contains minor M(Fi), for i = 1, 2, · · ·7, then M /∈ C3, the proof is straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Conversely, if M does not contain a minor M(Fi) for i = 1, 2, · · ·7, then we will prove that M ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Suppose not, then for some S ⊆ E(M), with |S| = 3, MS is not a graphic matroid, then, MS contains minor F, for some F ∈ F, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then M contains a minor P containing S, By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='3, such that PS ∼= F or PS/S′ ∼= F, for some S′ ⊆ S or P is an extension of circuit matroid of the graph G1 or G2 by single element and the graphs G1, G2 are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' It is observed that an extension of M(G1) by a single element, either isomorphic to M(F4) or M(F7) or contains minor M(F1) or M(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Also, M(G2) contains minor M(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus P cannot be an extension of M(G1) or M(G2) by a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence, PS ∼= F or PS/S′ ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='5, either P is an extension of QF by n elements, where n ≤ 3 or P ∼= QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Case (i) If the quotient is graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' In [2], Mundhe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' obtained forbidden minors from graphic quotients of every F ∈ F, as given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Case (ii) If the quotient is not graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let F = M∗(K5), then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='14, QM∗(K5) ∼= M∗(K5) but from Figure 4, Graphic Elementary Lift of Cographic Matroids 8 M∗(F1) is a minor of M(K5) and hence M(F1) is a minor of M∗(K5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence we discard M∗(K5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Let F = M∗(K3,3), then QM∗(K3,3) ∼= M∗(K3,3), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content='15, but from Figure 5, M∗(F2) is a minor of M(K3,3) and hence M(F2) is a minor of M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Hence, we discard M∗(K3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' Thus by the case (i) and case (ii), the excluded minor for the class C3 is the matroid M(Fi), the graph Fi is shown in Figure 1, for i = 1, 2, · · ·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQf8v4C/content/2301.01907v1.pdf'} +page_content=' □ References 1.' 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Linot1 and Michael D. Graham1† +1Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI +53706, USA +Because the Navier-Stokes equations are dissipative, the long-time dynamics of a flow in state +space are expected to collapse onto a manifold whose dimension may be much lower than the +dimension required for a resolved simulation. On this manifold, the state of the system can be +exactly described in a coordinate system parameterizing the manifold. Describing the system +in this low-dimensional coordinate system allows for much faster simulations and analysis. We +show, for turbulent Couette flow, that this description of the dynamics is possible using a data- +driven manifold dynamics modeling method. This approach consists of an autoencoder to find a +low-dimensional manifold coordinate system and a set of ordinary differential equations defined +by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane +Couette flow at Re = 400, where a fully resolved solutions requires O(105) degrees of freedom. +Using only data from this simulation we build models with fewer than 20 degrees of freedom +that quantitatively capture key characteristics of the flow, including the streak breakdown and +regeneration cycle. At short-times, the models track the true trajectory for multiple Lyapunov +times, and, at long-times, the models capture the Reynolds stress and the energy balance. For +comparison, we show that the models outperform POD-Galerkin models with ∼2000 degrees of +freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely +resemble previously computed orbits for the full system; additionally, we find nine orbits that +correspond to previously unknown solutions in the full system. +1. Introduction +A major challenge in dealing with chaotic fluid flows, whether it be performing experiments, +running simulations, or interpreting the results, is the high-dimensional nature of the state. Even +for simulations in the smallest domains that sustain turbulence (a minimal flow unit (MFU)), +the state dimension may be O(105) (Jiménez & Moin 1991; Hamilton et al. 1995). However, +despite this nominal high-dimensionality, the dissipative nature of turbulent flows leads to the +expectation that long-time dynamics collapse onto an invariant manifold of much lower dimension +than the ambient dimension (Hopf 1948). By modeling the dynamics in a manifold coordinate +system, simulations could be performed with a drastically lower-dimensional state representation, +significantly speeding up computations. Additionally, such a low-dimensional state representation +is highly useful for downstream tasks like control or design. Finding a low-dimensional – or +ideally a minimal dimensional – parameterization of the manifold and an evolution equation for +this parameterization are both challenges. In this work we aim to address these challenges with a +data-driven model, specifically for the task of reconstructing turbulent plane Couette flow. +The classic way to perform dimension reduction from data is to use the proper orthogonal +decomposition (POD), also known by Principal Component Analysis (PCA) or Karhunen-Loève +decomposition (Holmes et al. 2012). This is a linear dimension reduction technique in which the +state is projected onto the set of orthogonal modes that capture the maximum variance in the +† Email address for correspondence: mdgraham@wisc.edu +arXiv:2301.04638v1 [physics.flu-dyn] 11 Jan 2023 + +2 +data. The POD is widely used for flow phenomena, some examples of which include: turbulent +channel flow (Moin & Moser 1989; Ball et al. 1991), flat-plate boundary layers (Rempfer & Fasel +1994), and free shear jet flows (Arndt et al. 1997). Smith et al. (2005) showed how to incorporate +system symmetries into the POD modes, the details of which we elaborate on in Sec. 3. +While the POD has seen wide use and is easy to interpret, more accurate reconstruction can +be achieved with nonlinear methods – a result we highlight in Sec. 3. Some popular methods +for nonlinear dimension reduction include kernel PCA (Schölkopf et al. 1998), diffusion maps +(Coifman et al. 2005), local linear embedding (LLE) (Roweis & Saul 2000), isometric feature +mapping (Isomap) (Tenenbaum et al. 2000), and t-distributed stochastic neighbor embedding +(tSNE) (Hinton & Roweis 2003). These methods are described in more detail in Linot & Graham +(2022), and an overview of other dimension reduction methods can be found in Van Der Maaten +et al. (2009). One drawback of all of these methods, however, is that they reduce the dimension, +but do not immediately provide a means to move from a low-dimensional state back to the full +state. A popular dimension reduction method without these complications is the undercomplete +autoencoder (Hinton & Salakhutdinov 2006), which uses a neural network (NN) to map the input +data into a lower-dimensional “latent space" and another NN to map back to the original state +space. We describe this structure in more detail in Sec. 2. Some examples where autoencoders +have been used for flow systems include flow around a cylinder (Murata et al. 2020), flow around +a flat plate (Nair & Goza 2020), Kolmogorov flow (Page et al. 2021; Pérez De Jesús & Graham +2022), and channel flow (Milano & Koumoutsakos 2002). Although we will not pursue this +approach in the present work, it may be advantageous for multiple reasons to parametrize the +manifold with overlapping local representations, as done in Floryan & Graham (2022). +After reducing the dimension, the time evolution for the dynamics can be approximated from +the equations of motion or in a completely data-driven manner. The classical method is to perform +a Galerkin projection wherein the equations of motion are projected onto a set of modes (e.g. POD +modes) (Holmes et al. 2012). However, in this approach all the higher POD modes are neglected. +An extension of this idea, called nonlinear Galerkin, is to assume that the time derivative of the +coefficients of all of the higher modes is zero, but not the coefficients themselves (Titi 1990; +Foias et al. 1988; Graham et al. 1993); this is essentially a quasisteady state approximation for +the higher modes. This improves the accuracy, but comes at a higher computational cost than the +Galerkin method, although this can be somewhat mitigated by using a postprocessing Galerkin +approach (García-Archilla et al. 1998). Wan et al. (2018) also showed a recurrent NN (RNN) – +a NN that feeds into itself – can be used to improve the nonlinear Galerkin approximation. This +RNN structure depends on a history of inputs, making it non-Markovian. In addition to these +linear dimension reduction approaches, an autoencoder can be used with the equations of motion +in the so-called manifold Galerkin approach, which Lee & Carlberg (2020) developed and applied +to the viscous Burgers equation . +When the equations of motion are assumed to be unknown, and only snapshots of data are +available, a number of different machine learning techniques exist to approximate the dynamics. +Two of the most popular techniques are RNNs and reservoir computers. Vlachas et al. (2020) +showed both these structures do an excellent job of capturing the chaotic dynamics of the Lorenz- +96 equation and Kuramoto-Sivashinsky equation (KSE). For fluid flows, autoencoders and RNNs +(specifically long-short term memory networks (LSTM)) have been used to model flow around +a cylinders (Hasegawa et al. 2020a; Eivazi et al. 2020), pitching airfoils (Eivazi et al. 2020), +bluff bodies (Hasegawa et al. 2020b), and MFU plane Poiseuille flow (PPF) (Nakamura et al. +2021). Although these methods often do an excellent job of predicting chaotic dynamics, the +models are not Markovian, so the dimension of the system also includes some history, and these +models perform discrete timesteps. These two properties are undesirable, because the underlying +dynamics are Markovian and continuous in time, and modeling them differently complicates +applications and interpretations of the model. In particular, we want to use the model for state + +3 +space analyses such as determination of periodic orbits, where standard tools are available for +ODEs that do not easily generalize to non-Markovian dynamic models. +Due to these issues, we use neural ordinary differential equations (ODE) (Chen et al. 2019). +In neural ODEs, the right-hand-side (RHS) of an ODE is represented as a NN that is trained to +reconstruct the time evolution of the data from snapshots of training data. In Linot & Graham +(2022) it was shown that this is an effective method for modeling the chaotic dynamics of the KSE. +Rojas et al. (2021) used neural ODEs to predict the periodic dynamics of flow around a cylinder, +and Portwood et al. (2019) used neural ODEs to predict the kinetic energy and dissipation of +decaying turbulence. +In this work we investigate the dynamics of MFU Couette flow. The idea behind the MFU, first +introduced by Jiménez & Moin (1991), is to reduce the simulation domain to the smallest size +that sustains turbulence, thus isolating the key components of the turbulent nonlinear dynamics. +Using an MFU for Couette flow at transitional Reynolds number, Hamilton et al. (1995) outlined +the regeneration cycle of wall bounded turbulence called the “self-sustaining process" (SSP), +which we describe in more detail in Sec. 3. This system was later analyzed with coviariant +Lyapunov analysis by Inubushi et al. (2015), who found a Lyapunov time (the inverse of the +leading Lyapunov exponent) of ∼ 48 time units. +Many low-dimensional models have been developed to recreate the dynamics of the SSP. The +first investigation of this topic was by Waleffe (1997), who developed an 8 mode model for shear +flow between free-slip walls generated by a spatially sinusoidal forcing. He selected the modes +based on intuition from the SSP and performed a Galerkin projection onto these modes. Moehlis +et al. (2004) later added an additional mode to Waleffe’s model which enables modification of +the mean profile by the turbulence, and made some modifications to the chosen modes. In this +“MFE" model, Moehlis et al. found exact coherent states, which we discuss below, that did not +exist in the 8 mode model. In addition, Moehlis et al. (2002) also used the POD modes on a +domain slightly larger than the MFU to generate POD-Galerkin models. These low-dimensional +models have been used as a starting point for testing data-driven models. For example, both +LSTMs (Srinivasan et al. 2019) and a Koopman operator method with nonlinear forcing (Eivazi +et al. 2021) have been used to attempt to reconstruct the MFE model dynamics. Borrelli et al. +(2022) then applied these methods to PPF. +Finally, we note that a key approach to understanding complex nonlinear dynamical phenomena, +such as the SSP of near-wall turbulence, is through study of the underlying state space structure +of fixed points and periodic orbits. In the turbulence literature these are sometimes called “exact +coherent states", or ECS (Kawahara et al. 2012; Graham & Floryan 2021). Turbulence organizes +around ECS in the sense that trajectories chaotically move between different such states. The +first ECS found were fixed point solutions in PCF (Nagata 1990). Following this work, Waleffe +(1998) was able to connect ECS of PCF and PPF to the SSP. Later, more fixed point ECS were +found in MFU PCF and visualized by Gibson et al. (2008a). Unlike fixed points, which cannot +capture dynamic phenomena at all, periodic orbits are able to represent key aspects of turbulent +dynamics such as bursting behavior. Kawahara & Kida (2001) found the first two periodic orbits +(POs) for MFU PCF, one of which had statistics that agreed well with the SSP. Then, Viswanath +(2007) found another PO and 4 new relative POs (RPOs) in this domain, and Gibson made these +solutions available in (Gibson et al. 2008b), along with a handful of others. +In the present work, we use autoencoders and neural ODEs , in a method we call “Data-driven +Manifold Dynamics" (DManD) (Linot et al. 2023), to build a ROM for turbulent MFU PCF +(Hamilton et al. 1995). Section 2 outlines the details of the DManD framework. We then describe +the details of the Couette flow in Sec. 3.1, the results of the dimension reduction in Sec. 3.2, and +the DManD model’s reconstruction of short- and long-time statistics in Sec. 3.3 and Sec. 3.4, +respectively. After showing that the models accurately reproduce these statistics, we compute + +4 +RPOs for the model in Sec. 3.5, finding several that are similar to previously known RPOs, as +well as several that seem to be new. Finally, we summarize the results in Sec. 4. +2. Framework +Here we outline our method for an “exact" DManD modeling approach. In this sense “exact" +means all of the functions described allow for perfect reconstruction, but error is introduced in +approximating these functions due to insufficient data, error in learning the functions, or error in +evolving them forward in time. This differs from coarse-grained ROMs, which approximate the +physics to generate a closed set of equations. A key component allowing DManD to be “exact" +is that we only seek to discover the evolution of trajectories after they collapse onto an invariant +manifold M. +In general, the training data for development of a DManD model comes in the form of snapshots +{𝑢1, 𝑢2, ..., 𝑢𝑀 }, which are either the full state or measurements diffeomorphic to the full state +(e.g. time delays (Takens 1981; Young & Graham 2022)). Here we consider full-state data 𝑢 that +lives in an ambient space R𝑑. We generate a time series of data by evolving this state forward in +time according to +𝑑𝑢 +𝑑𝑡 = 𝑓 (𝑢). +(2.1) +(In the present context, this equation represents a fully-resolved direct numerical simulation +(DNS).) With the full state, we can then define a mapping to a low-dimensional state representation +ℎ = 𝜒(𝑢), +(2.2) +with ℎ ∈ R𝑑ℎ is a coordinate representation on the manifold. Finally, we define a mapping back +to the full state +˜𝑢 = ˇ𝜒(ℎ). +(2.3) +For data that lies on a finite-dimensional invariant manifold these functions can exactly reconstruct +the state (i.e. ˜𝑢 = 𝑢). However, if the dimension 𝑑ℎ is too low, or there are errors in the +approximation of these functions, then ˜𝑢 approximates the state. Then, with this low-dimensional +state representation, we can define an evolution equation +𝑑ℎ +𝑑𝑡 = 𝑔(ℎ). +(2.4) +The DManD model consists of the three functions 𝜒, ˇ𝜒, and 𝑔. By approximating these functions, +the evolution of trajectories on the manifold can be performed entirely in the manifold coordinates +ℎ, which requires far fewer operations than a full simulation, as 𝑑ℎ ≪ 𝑑. We choose to approximate +all of these functions using NNs, but other representations could be used. +First, we train 𝜒 and ˇ𝜒 using an undercomplete autoencoder. This is a NN structure consisting +of an encoder which reduces dimension (𝜒) and a decoder that expands dimension ( ˇ𝜒). As +mentioned in Sec. 1, a common approach to dimension reduction is to project onto a set of POD +modes. POD gives the optimal linear projection in terms of reconstruction error, so we use this +fact to train an encoder as the sum of POD and a correction in the form of an NN: +ℎ = 𝜒(𝑢; 𝜃𝐸) = 𝑈𝑇 +𝑑ℎ𝑢 + E(𝑈𝑇 +𝑟 𝑢; 𝜃𝐸). +(2.5) +In this equation, 𝑈𝑘 ∈ R𝑑×𝑘 is a matrix whose 𝑘 columns are the first 𝑘 POD modes as ordered by +variance, and E is a NN. The first term (𝑈𝑇 +𝑑ℎ𝑢) is the projection onto the leading 𝑑ℎ POD modes, +and the second term is the NN correction. The matrix 𝑈𝑟 in this term may either be a full change +of basis with no approximation (𝑟 = 𝑑), or involve some dimension reduction (𝑑 > 𝑟 > 𝑑ℎ). + +5 +For mapping back to the full state (decoding), we again sum POD with a correction +˜𝑢 = ˇ𝜒(ℎ; 𝜃𝐸) = 𝑈𝑟 ([ℎ, 0]𝑇 + D(ℎ; 𝜃𝐷)). +(2.6) +Here, [ℎ, 0]𝑇 is the ℎ vector zero padded to the correct size, and D is a NN. The first term is +the POD mapping back to the full space, if there were no NNs, and the second term is a NN +correction. In Linot & Graham (2020) we refer to this structure as a hybrid autoencoder. In Sec. +3.2 we contrast this to a “standard" autoencoder where ℎ = E(𝑈𝑇 +𝑟 𝑢; 𝜃𝐸) and ˜𝑢 = 𝑈𝑟D(ℎ; 𝜃𝐷). +These hybrid autoencoder operations act as a shortcut connection on the optimal linear dimension +reduction, which we (Linot & Graham 2020) found useful for representing the data and achieving +accurate reconstruction of 𝑢. Yu et al. (2021) also took a similar approach with POD shortcut +connections over each layer of the network. +We determine the NN parameters 𝜃𝐸 and 𝜃𝐷 by minimizing +𝐿 = 1 +𝑑𝐾 +𝐾 +∑︁ +𝑖=1 +||𝑢(𝑡𝑖) − ˇ𝜒(𝜒(𝑢(𝑡𝑖); 𝜃𝐸); 𝜃𝐷)||2 + +1 +𝑑ℎ𝐾 +𝐾 +∑︁ +𝑖=1 +𝛼||E(𝑈𝑇 +𝑟 𝑢(𝑡𝑖); 𝜃𝐸) + D𝑑ℎ (ℎ(𝑡𝑖); 𝜃𝐷)||2. +(2.7) +The first term in this loss is the mean-squared error (MSE) of the reconstruction ˜𝑢, and the second +term is a penalty that promotes accurate representation of the leading 𝑑ℎ POD coefficients. In this +term, D𝑑ℎ denotes the leading 𝑑ℎ elements of the decoder output. For finite 𝛼, the autoencoder +exactly matches the first 𝑑ℎ POD coefficients when this term vanishes. Details of the minimization +procedure are discussed in Sec. 3. +Next, we approximate 𝑔 using a neural ODE. A drawback of training a single dense NN for 𝑔 is +that the resulting dynamics may become weakly unstable, with linear growth at long times (Linot +& Graham 2022; Linot et al. 2023). To avoid this, we use a “stabilized" neural ODE approach by +adding a linear damping term onto the output of the NN, giving +𝑔(ℎ(𝑡𝑖); 𝜃𝑔) = 𝑔NN(ℎ(𝑡𝑖); 𝜃𝑔) + 𝐴ℎ. +(2.8) +Integrating Eq. 2.8 forward from time 𝑡𝑖 to 𝑡𝑖 + 𝜏 yields +˜ℎ(𝑡𝑖 + 𝜏) = ℎ(𝑡𝑖) + +∫ 𝑡𝑖+𝜏 +𝑡𝑖 +𝑔NN(ℎ(𝑡); 𝜃𝑔) + 𝐴ℎ(𝑡)𝑑𝑡. +(2.9) +Depending on the situation, one may either learn 𝐴 from data, or fix it. Here we set it to the +diagonal matrix +𝐴𝑖 𝑗 = −𝛽𝛿𝑖 𝑗𝜎𝑖(ℎ) +(2.10) +where 𝜎𝑖(ℎ) is the standard deviation of the 𝑖th component of ℎ, 𝛽 is a tunable parameter, and 𝛿𝑖 𝑗 +is the Kronecker delta. This linear term attracts trajectories back to the origin, preventing them +from moving far away from the training data. In Sec. 3.4 we show that this approach drastically +improves the long-time performance of these models. +We then determine the parameters 𝜃𝑔 by minimizing the difference between the predicted state +˜ℎ(𝑡𝑖 + 𝜏) and the true state ℎ(𝑡𝑖 + 𝜏), averaged over the data: +𝐽 = +1 +𝑑ℎ𝐾 +𝐾 +∑︁ +𝑖=1 +� +||ℎ(𝑡𝑖 + 𝜏) − ˜ℎ(𝑡𝑖 + 𝜏)||2 +2 +� +. +(2.11) +For clarity we show the specific loss we use, which sums over only a single snapshot forward +in time at a fixed 𝜏. More generally, the loss can be formulated for arbitrary snapshot spacing +and for multiple snapshots forward in time. To compute the gradient of 𝐽 with respect to the +neural network parameters 𝜃𝑔, automatic differentiation can be used to backpropagate through +the ODE solver that is used to compute the time integral in Eq. 2.9, or an adjoint problem can +be solved backwards in time (Chen et al. 2019). The adjoint method uses less memory than + +6 +backpropagation, but ℎ is low-dimensional and our prediction window for training is short, so we +choose to backpropagate through the solver. +So far this approach to approximating 𝜒, ˇ𝜒, and 𝑔 is general and does not directly account +for the fact that the underlying equations are often invariant to certain symmetry operations. For +example, one of the symmetries in PCF is a continuous translation symmetry in 𝑥 and 𝑧 (i.e. any +solution shifted to another location in the domain gives another solution). This poses an issue for +training, because in principle, the training data must include all these translations to accurately +model the dynamics under any translation. We discuss these and other symmetries of PCF in Sec. +3.1. +In practice, accounting for continuous symmetries is most important along directions that +sample different phases very slowly. For PCF, the mean flow is in the 𝑥 direction, leading to +good phase sampling along this direction. However, there is no mean flow in the 𝑧 direction, so +sampling all phases relies on the slow phase diffusion in that direction. Therefore, we will only +explicitly to account for the 𝑧-phase in Sec. 3, but in the current disucssion we present the general +framework accounting for all continuous symmetries. +To address the issue of continuous translations, we add an additional preprocessing step to the +data, using the method of slices (Budanur et al. 2015b,a) to split the state 𝑢 into a pattern 𝑢 𝑝 ∈ R𝑑 +and a phase 𝜙 ∈ R𝑐. The number of continuous translation symmetries for which we explicitly +account determines 𝑐. We discuss the details of computing the pattern and the phase in Sec. 3.1. +Separating the pattern and phase is useful because the evolution of both the pattern and the phase +only depend on the pattern. Thus, we simply replace 𝑢 with 𝑢 𝑝 in all the above equations and +then write one additional ODE for the phase +𝑑𝜙 +𝑑𝑡 = 𝑔𝜙(ℎ; 𝜃 𝜙). +(2.12) +We then fix the parameters of 𝑔 to evolve ℎ (from 𝑢 𝑝) forward in time and use that to make a +phase prediction +˜𝜙(𝑡𝑖 + 𝜏) = 𝜙(𝑡𝑖) + +∫ 𝑡𝑖+𝜏 +𝑡𝑖 +𝑔𝜙(ℎ(𝑡𝑖); 𝜃 𝜙)𝑑𝑡. +(2.13) +Finally, we determine the parameters 𝜃 𝜙 to minimize the difference between the predicted phase +˜𝜙(𝑡𝑖 + 𝜏) and the true phase 𝜙(𝑡𝑖 + 𝜏) +𝐽𝜙 = 1 +𝑐𝐾 +𝐾 +∑︁ +𝑖=1 +� +||𝜙(𝑡𝑖 + 𝜏) − ˜𝜙(𝑡𝑖 + 𝜏)||2� +, +(2.14) +using the method described above to compute the gradient of 𝐽𝜙. +3. Results +3.1. Description of Plane Couette Flow Data +In the following sections we apply DManD to DNS of turbulent PCF in a MFU domain. +Specifically, we consider the well-studied Hamilton, Kim, and Waleffe (HKW) domain (Hamilton +et al. 1995). We made this selection to compare our DManD results to the analysis of the self- +sustaining process in this domain, to compare our DManD results to other Galerkin-based ROMs, +and to compare our DManD results to known unstable periodic solutions in this domain. +For PCF we solve the Navier-Stokes equations +𝜕u +𝜕𝑡 + u · ∇u = −∇𝑝 + Re−1∇2u, +∇ · u = 0 +(3.1) +for a fluid confined between two plates moving in opposite directions with the same speed. Eq. + +7 +3.1 is the nondimensionalized form of the equations with velocities in the streamwise 𝑥 ∈ [0, 𝐿𝑥], +wall-normal 𝑦 ∈ [−1, 1], and spanwise 𝑧 ∈ [0, 𝐿𝑧] directions defined as u = [𝑢𝑥, 𝑢𝑦, 𝑢𝑧], +and pressure 𝑝. We solve this equation for a domain with periodic boundary conditions in 𝑥 +and 𝑧 (u(0, 𝑦, 𝑧) = u(𝐿𝑥, 𝑦, 𝑧), u(𝑥, 𝑦, 0) = u(𝑥, 𝑦, 𝐿𝑧)) and no-slip, no-penetration boundary +conditions in 𝑦 (𝑢𝑥(𝑥, ±1, 𝑧) = ±1, 𝑢𝑦(𝑥, ±1, 𝑧) = 𝑢𝑧(𝑥, ±1, 𝑧) = 0). The complexity of the flow +increases as the Reynolds number Re increases and the domain size 𝐿𝑥 and 𝐿𝑧 increase. Here +we use the HKW cell, which is at Re = 400 with a domain size [𝐿𝑥, 𝐿𝑦, 𝐿𝑧] = [1.75𝜋, 2, 1.2𝜋] +(Hamilton et al. 1995). The HKW cell is one of the simplest flows that sustains turbulence for +extended periods of time before relaminarizing. We chose to use this flow because it is well +studied (refer to Sec. 1), it isolates the SSP (Hamilton et al. 1995), and a library of ECS exist for +this domain (Gibson et al. 2008b). Here we are interested in modeling the turbulent dynamics, so +we will remove data upon relaminarization as detailed below. +Eq. 3.1, under the boundary conditions described, is invariant (and its solutions equivariant) +under the discrete symmetries of point reflections about [𝑥, 𝑦, 𝑧] = [0, 0, 0] +P · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (−𝑢𝑥, −𝑢𝑦, −𝑢𝑧, 𝑝)(−𝑥, −𝑦, −𝑧, 𝑡) +(3.2) +reflection about the 𝑧 = 0 plane +R · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (𝑢𝑥, 𝑢𝑦, −𝑢𝑧, 𝑝)(𝑥, 𝑦, −𝑧, 𝑡) +(3.3) +and rotation by 𝜋 about the 𝑧-axis +RP · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (−𝑢𝑥, −𝑢𝑦, 𝑢𝑧, 𝑝)(−𝑥, −𝑦, 𝑧, 𝑡). +(3.4) +In addition to the discrete symmetries, there are also continuous translation symmetries in 𝑥 and +𝑧 +T𝜎𝑥,𝜎𝑧 · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥 + 𝜎𝑥, 𝑦, 𝑧 + 𝜎𝑧, 𝑡). +(3.5) +We incorporate all these symmetries in the POD represesntation (Smith et al. 2005), as we discuss +further in Sec. 3.2. Then, we use the method of slices (Budanur et al. 2015a) to phase align in the 𝑧 +direction. By phase aligning in 𝑧 we fix the location of the low-speed streak. Without the alignment +in 𝑧, models performed poorly because the models needed to learn how to represent every spatial +shift of every snapshot. In what follows, we only consider phase-alignment in 𝑧, but we note that +extending this work to phase-alignment in 𝑥 is straightforward. To phase-align the data, we use +the first Fourier mode method-of-slices (Budanur et al. 2015a). First, we compute a phase by +taking the Fourier transform of the streamwise velocity in 𝑥 and 𝑧 ( ˆ𝑢𝑥(𝑘𝑥, 𝑦, 𝑘𝑧) = F𝑥,𝑧(𝑢𝑥)) at +a specific 𝑦 location (𝑦1) to compute the phase +𝜙 = atan2(imag( ˆ𝑢𝑥(0, 𝑦1, 1)), real( ˆ𝑢𝑥(0, 𝑦1, 1))). +(3.6) +We select 𝑦1 to be one grid point off the bottom wall. Then we compute the pattern dynamics by +using the Fourier shift theorem to set the phase to 0 (i.e. move the low-speed streak to the center +of the channel) +u𝑝 = F −1 +𝑥,𝑧( ˆu exp(−𝑖𝑘𝑧𝜙)). +(3.7) +We generate turbulent PCF trajectories using the psuedo-spectral Channelflow code developed +by Gibson et al. (2012; 2021). In this code, the velocity and pressure fields are projected onto +Fourier modes in 𝑥 and 𝑧 and Chebyshev polynomials of the first kind in 𝑦. These coefficients +are evolved forward in time first using the multistage SMRK2 scheme (Spalart et al. 1991), then, +after taking multiple timesteps, using the multistep Adams-Bashforth Backward-Differentiation +3 scheme (Peyret 2002). At each timestep, a pressure boundary condition is found such that +incompressibility is satisfied at the wall (𝑑𝑢𝑦/𝑑𝑦 = 0) using the influence matrix method and tau +correction developed by Kleiser & Schumann (1980). +Data was generated with Δ𝑡 = 0.0325 on a grid of [𝑁𝑥, 𝑁𝑦, 𝑁𝑧] = [32, 35, 32] in 𝑥, 𝑦, and 𝑧 + +8 +for the HKW cell. Starting from random divergence-free initial conditions, we ran simulations +forward for either 10, 000 xtime units or until relaminarization. Then we dropped the first 1, 000 +time units as transient data and the last 1, 000 time units to avoid laminar data, and repeated with +a new initial condition until we had 91, 562 time units of data stored at intervals of one time unit. +We split this data into 80% for training and 20% for testing. Finally, we preprocess the data by +computing the mean ⟨u⟩ (𝑦) from the training data and subtracting it from all data u′ = u − ⟨u⟩, +and then we compute the pattern u′ +𝑝 and the phase 𝜙 as described above. The pattern 𝑢 𝑝 described +in Sec. 2 is u′ +𝑝 flattened into a vector (i.e. 𝑑 = 3𝑁𝑥𝑁𝑦𝑁𝑧). The POD and NN training use only +the training data, and all comparisons use test data unless otherwise specified. +3.2. Dimension Reduction and Dynamic Model Construction +3.2.1. Linear dimension reduction with POD: From O(105) to O(103) +The first task in DManD for this Couette flow data is finding a low-dimensional parameterization +of the manifold on which the long-time dynamics lie. We parameterize this manifold in two +steps. First, we reduce the dimension down from O(105) to 502 with the proper orthogonal +decomposition (POD), and, second, we use an autoencoder to reduce the dimension down to 𝑑ℎ. +The first step is simply a preprocessing step to reduce the size of the data, which reduces the +number of parameters in the autoencoder. Due to Whitney’s embedding theorem (Whitney 1936, +1944), we know that as long as the manifold dimension is less than 251 (𝑑M < 251) then this +POD representation is diffeomorphic to the full state. As we show later, the manifold dimension +appears to be far lower than 𝑑M = 251, so no information of the full state should be lost with this +first step. +Proper orthogonal decomposition (POD) originates with the question of what function 𝚽 +maximizes +� +|(u′, 𝚽)|2� +||𝚽||2 +. +(3.8) +Solutions 𝚽(𝑛) to this problem satisfy the eigenvalue problem +3 +∑︁ +𝑗=1 +∫ +𝐿𝑥 +0 +∫ 1 +−1 +∫ +𝐿𝑧 +0 +� +𝑢′ +𝑖(x, 𝑡)𝑢′∗ +𝑗 (x′, 𝑡) +� +Φ(𝑛) +𝑗 +(x′) 𝑑x′ = 𝜆𝑖Φ(𝑛) +𝑖 +(x) +(3.9) +(Holmes et al. 2012; Smith et al. 2005). Unfortunately, upon approximating these integrals, with +the trapezoidal rule for example, this becomes a 𝑑 × 𝑑 matrix, making computation intractable. +Furthermore, computing the average in Eq. 3.9, without any modifications, results in POD modes +that fail to preserve the underlying symmetries described above. +In order to make this problem computationally tractable, and preserve symmetries, we apply +the POD method used in Smith et al. (2005), with the slight difference that we first subtract off +the mean of state before performing the analysis. The first step in this procedure is to treat the +POD modes as Fourier modes in both the 𝑥 and 𝑧 directions. Holmes et al. show in (Holmes et al. +2012) that for translation-invariant directions Fourier modes are the optimal POD modes. This +step transforms the eigenvalue problem into +𝐿𝑥𝐿𝑧 +3 +∑︁ +𝑗=1 +∫ 1 +−1 +� +ˆ𝑢′ +𝑖(𝑘𝑥, 𝑦′, 𝑘𝑧, 𝑡) ˆ𝑢′∗ +𝑗 (𝑘𝑥, 𝑦′, 𝑘𝑧, 𝑡) +� +𝜑(𝑛) +𝑗𝑘𝑥 𝑘𝑧 (𝑦′) 𝑑𝑦′ = 𝜆(𝑛) +𝑘𝑥 𝑘𝑧 𝜑(𝑛) +𝑖𝑘𝑥 𝑘𝑧 (𝑦), +(3.10) +which reduces the 𝑑 × 𝑑 eigenvalue problem down to a 3𝑁𝑦 × 3𝑁𝑦 eigenvalue problem for every +wavenumber pair (𝑘𝑥, 𝑘𝑧) of Fourier coefficients. We used 5, 000 snapshots evenly sampled over +the training data to compute the POD modes. Then, to account for the discrete symmetries, +the data is augmented such that the mean in Eq. 3.10 is computed by adding all the discrete +symmetries of each snapshot. + +9 +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +Figure 1: (a) Eigenvalues of POD modes sorted in descending order. (b) Components of the +Reynolds stress for data generated by the DNS and this data projected onto 256 POD modes. In +(a) the curves are, from top to bottom, +� +𝑢′2 +𝑥 +� +, +� +𝑢′2 +𝑧 +� +, +� +𝑢′2 +𝑦 +� +, and +� +𝑢′ +𝑥𝑢′ +𝑦 +� +. +This analysis gives us POD modes +𝚽(𝑛) +𝑘𝑥 𝑘𝑧 (x) = +1 +√𝐿𝑥𝐿𝑧 +exp +� +2𝜋𝑖 +� 𝑘𝑥𝑥 +𝐿𝑥 ++ 𝑘𝑧𝑧 +𝐿𝑧 +�� +𝝋(𝑛) +𝑘𝑥 𝑘𝑧 (𝑦), +(3.11) +and eigenvalues 𝜆(𝑛) +𝑘𝑥 𝑘𝑧. We sort the modes from largest eigenvalue to smallest eigenvalue (𝜆𝑖) and +and project onto the leading 256 modes, giving us a vector of POD coefficients 𝑎(𝑡). A majority +of these modes are complex, so projecting onto these modes results in a 502-dimensional system. +In Fig. 1a we plot the eigenvalues, which show a rapid drop and then a long tail that contributes +little to the energy content. By dividing the eigenvalues of the leading 256 modes by the total, we +find these modes contain 99.8% of the energy. To further illustrate that 256 modes is sufficient to +represent the state in this case, we consider the reconstruction of statistics after projecting onto +the POD modes. In Fig. 1b we show the reconstruction of four components of the Reynolds stress, +� +𝑢′2 +𝑥 +� +, +� +𝑢′2 +𝑧 +� +, +� +𝑢′2 +𝑦 +� +, and +� +𝑢′ +𝑥𝑢′ +𝑦 +� +. The projection onto POD modes matches all of these quantities +extremely well. +Now that we have converted the data to POD coefficients and filtered out the low energy modes, +we next train an autoencoder to perform nonlinear dimension reduction. As mentioned in Sec. +2, we phase-align the data in the spanwise direction at this step using the first Fourier mode +method-of-slices. A common practice when training NNs is to normalize the data by subtracting +the mean and dividing by the standard deviation of each component. We do not take this approach +here because the standard deviation of the higher POD coefficients, which contribute less to the +reconstruction, is much smaller than the lower POD coefficients. In order to retain the important +information in the magnitudes, but put the values in a more amenable form for NN training, we +instead normalize the POD coefficients by subtracting the mean and dividing by the maximum +standard deviation. Then, we take this input and train autoencoders to minimize the loss in Eq. +2.7 using an Adam optimizer (Kingma & Ba 2015) in Keras (Chollet et al. 2015). We train for +500 epochs with a learning rate scheduler that drops the learning rate from 10−3 to 10−4 after +400 epochs. At this point we see no improvement in the reconstruction error. For the hybrid +autoencoder approach, we set 𝛼 = 0.01. Table 1 includes additional NN architecture details. +3.2.2. Nonlinear dimension reduction with autoencoders: From O(103) to O(101) +With the above “preprocessing" step completed, we now turn to the reduction of dimension +with the nonlinear approach enabled by the autoencoder structure. We consider three approaches +to reducing the dimension of 𝑎: (1) Training a hybrid autoencoder, (2) Training a standard + +10-2 +DNS +0.06 +POD +10-4 +0.04 +10-6 +0.02 +10-8 . +0.00 +10-10 +100 +101 +102 +103 +-1.0 +-0.5 +0.0 +0.5 +1.0 +i +y10 +Table 1: Architectures of NNs. “Shape" indicates the dimension of each layer, “Activation" the +corresponding activation functions, and “sig" is the sigmoid activation.“Learning Rate" gives the +learning rate at multiple times during training. The learning rates was dropped at even intervals. +Function +Shape +Activation +Learning Rate +E +502/1000/𝑑ℎ +sig/linear +[10−3, 10−4] +D +𝑑ℎ/1000/502 +sig/linear +[10−3, 10−4] +𝑔NN +𝑑ℎ/200/200/𝑑ℎ +sig/sig/linear [10−2, 10−3, 10−4] +𝑔𝜙 +𝑑ℎ/200/200/1 +sig/sig/linear [10−2, 10−3, 10−4] +Figure 2: Mean squared error on test data for POD, standard autoencoders, and hybrid +autoencoders at various dimensions 𝑑ℎ. At each dimension there are four standard and four +hybrid autoencoders. +autoencoder, (3) linear projection onto a small set of POD modes. We describe the first two +approaches in Sec. 2, noting that the POD projection onto 256 (complex) modes can be written +as 𝑎 = 𝑈𝑇 +𝑟 𝑢. The third approach just corrsponds to setting E and D to zero in Eqs. 2.5 and 2.6. In +Fig. 2 we show the MSE of reconstructing 𝑎 with these three approaches over a range of different +dimensions 𝑑ℎ. We use NNs with the same architectures for both the standard and the hybrid +autoencoder approaches. Due to the variability introduced into autoencoder training by randomly +initialized weights and stochasticity in the optimization, we show the error for four separately +trained autoencoders, at each 𝑑ℎ. We see that the autoencoders perform an order magnitude better +than POD in the range of dimension considered here. Both the standard and hybrid autoencoder +approaches perform the same, so we select the hybrid approach because it can be viewed as a +nonlinear correction to the POD projection. Next we use the low-dimensional representations +from these autoencoders to train stabilized neural ODEs. +3.2.3. Neural ODE Training +After training four autoencoders at each dimension 𝑑ℎ, we chose a set of damping parameters, +𝛽, and for each, then trained four stabilized neural ODEs for all four autoencoders at every +dimension 𝑑ℎ. This results in 16 models at every 𝑑ℎ and 𝛽. The final 𝛽 value of 0.1 was selected +so that long-time trajectories neither blew up nor decayed too strongly. Before training the ODEs, +we preprocess each autoencoder’s latent space data set ℎ by subtracting the mean. It is important +to center the data because the linear damping (Eq. 2.10) pushes trajectories towards the origin. We +train the stabilized neural ODEs to predict the evolution of the centered data by using an Adam + +MSE Test Data +Hybrid +Stand +POD +10° +5 +10 +1511 +Figure 3: Snapshots of the streamwise velocity at 𝑦 = 0 from the DNS and from the DManD +model at 𝑑ℎ = 18. +optimizer in Pytorch (Paszke et al. 2019; Chen et al. 2019) to minimize the loss in Eq. 2.11. We +train using a learning rate scheduler that drops at three even intervals during training and we train +until the learning curve stops improving. Table 1 shows the details of this NN. Unless otherwise +stated, we show results for the one model out of those sixteen at each dimension with the lowest +relative error averaged over all the statistics we consider. +3.3. Short-time tracking +In the following two sections we evaluate the performance of the DManD models at reconstruct- +ing short-time trajectories and long-time statistics. Figure 3 shows snapshots of the streamwise +velocity at the center plane of the channel, 𝑦 = 0, for the DNS and DManD at 𝑑ℎ = 18. We +choose to show results for 𝑑ℎ = 18 because the autoencoder error begins to level off around +this dimension, and, as we will show, the error in statistics levels off before this dimension. The +value 𝑑ℎ = 18 is not necessarily the minimal dimension required to model this system. In Fig. 3, +both the DNS and the DmanD model show key characteristics of the SSP: (1) low-speed streaks +become wavy, (2) the wavy low-speed streaks break down generating rolls, (3) the rolls lift fluid +from the walls, regenerating streaks. +Not only does DManD capture the qualitative behavior of the SSP, but Fig. 3 also shows good +quantitative agreement as well. To further illustrate this, in Fig. 4 we show the modal root-mean +squared (RMS) velocity +𝑀(𝑘𝑥, 𝑘𝑧) = +�∫ 1 +−1 +( ˆ𝑢2 +𝑥(𝑘𝑥, 𝑦, 𝑘𝑧) + ˆ𝑢2 +𝑦(𝑘𝑥, 𝑦, 𝑘𝑧) + ˆ𝑢2 +𝑧(𝑘𝑥, 𝑦, 𝑘𝑧))𝑑𝑦 +�1/2 +, +(3.12) +which Hamilton et al. (1995) used to identify the different parts of the SSP. Specifically, we +consider the 𝑀(0, 1) mode, which corresponds to the low speed streak and the 𝑀(1, 0) mode +which corresponds to the 𝑥-dependence that appears when the streak becomes wavy and breaks +up. In this example, the two curves match well over a cycle of the SSP and only start to move +away after ∼ 150 time units, which is about three Lyapunov times. +While the previous result shows a single example, we also consider ensembles of initial +conditions. Figure 5 shows the tracking error ||𝑎(𝑡𝑖 + 𝑡) − ˜𝑎(𝑡𝑖 + 𝑡)|| of 10 trajectories, starting at +𝑡𝑖, for a model at 𝑑ℎ = 18. Here we normalize the tracking error by the error between solutions +at random times 𝑡𝑖 and 𝑡 𝑗 𝐷 = ⟨||𝑎(𝑡𝑖) − 𝑎(𝑡)||⟩. In this case the darkest line corresponds to the +flow field in Figs. 3 and 4. When considering the other initial conditions in Fig. 5, there tends +to be a relatively slow rise in the error over ∼50 time units and then a more rapid increase after + +DNS +0.6 +0=↑ +t = 85 +140 +-0.4 +22 +0.2 +0 +0.0 +DManD +0=↑ +t = 85 +t = 140 +-0.2 +22 +-0.4 +0 +-0.6 +0 +5 +0 +5 +0 +5 +0 +512 +Figure 4: Modal RMS velocity from the DNS (𝑀) and from the DManD model at 𝑑ℎ = 18 ( ˜𝑀). +The markers correspond to the times in Fig. 3. +Figure 5: Normalized tracking error for 10 random initial conditions (different shades) using +DManD with 𝑑ℎ = 18. +this point. To better understand how this tracking varies with the dimension of the model we next +consider the ensemble-averaged tracking error. +In Fig. 6a we show the normalized ensemble-averaged tracking error for model dimensions +between 𝑑ℎ = 3 and 18. For 𝑑ℎ = 3 − 5 there is a rapid rise in the error until ∼40 time units after +which the error levels off. This behavior often happens due to trajectories quickly diverging and +landing on stable fixed points or periodic orbits that do not exist in the true system. For 𝑑ℎ = 6−10 +there is an intermediate behavior where lines diverge more quickly than the higher-dimensional +models, but tend to approach the same tracking error at ∼100 time units. Then, for the remaining +models 𝑑ℎ = 11 − 18, there is a smooth improvement in the tracking error over this time interval. +As the dimension increases in this range the trends stay the same, but the error continues to +decrease, which is partially due to improvement in the autoencoder performance. +The instantaneous kinetic energy of the flow is +𝐸(𝑡) = +1 +2𝐿𝑥𝐿𝑧 +∫ +𝐿𝑧 +0 +∫ 1 +−1 +∫ +𝐿𝑥 +0 +1 +2u · u𝑑𝑥𝑑𝑦𝑑𝑧, +(3.13) +and we denote its fluctuating part as 𝑘(𝑡) = 𝐸(𝑡) − ⟨𝐸⟩. In Fig. 6b we show the temporal +autocorrelation of 𝑘. Again, for 𝑑ℎ = 3 − 5 we see clear disagreement between the true +autocorrelation and the prediction. Above 𝑑ℎ > 5 all of the models match the temporal +autocorrelation well, without a clear trend in the error as dimension changes. All these models + +0.25 +0.20 +M(0,1) +M(kx, kz) +0.15 +M(0, 1) +M(1, 0) +0.10 - +M(1, 0) +0.05 +0.00 +0 +50 +100 +150 +200 +t2.0 +1.5 +I/D +a +1.0 +一 +0.5 +0.0 +0 +20 +40 +60 +80 +100 +t13 +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +Figure 6: (a) ensemble averaged short-time tracking and (b) temporal autocorrelation of the +kinetic energy for DmanD models of increasing dimension. In (b) odd numbers above 𝑑ℎ = 5 are +omitted for clarity. +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +Figure 7: (a) an example of the phase evolution and (b) the MSD of the phase evolution for the +DNS and the DManD model at 𝑑ℎ = 18. +match well for ∼ 40 time units, with 𝑑ℎ = 18 (the darkest line) matching the data extremely well +for two Lyapunov times. +Finally, before investigating the long-time predictive capabilities of the model, we show the +tracking of phase dynamics for 𝑑ℎ = 18. As mentioned in Sec. 2, we decouple the phase and pattern +dynamics such that the time evolution of the phase only depends upon the pattern dynamics. Here +we take the 𝑑ℎ = 18 model and used it to train an ODE for the phase dynamics. For training we +repeat the process used for training 𝑔𝑁 𝑁 to train 𝑔𝜙 with the loss in Eq. 2.14. Table 1 contains +details on the NN architecture. +In Fig. 7a we show an example of the model phase evolution over 200 time units. In this +example, the model follows the same downward drift in phase despite not matching exactly. +Then, to show the statistical agreement between the DNS and the model, we show the mean +squared phase displacement MSD = +� +(𝜃(𝑡𝑖) − 𝜃(𝑡𝑖 + 𝑡))2� +for both the DNS and the model in +Fig. 7b, as was done for Kolmogorov flow by Pérez De Jesús & Graham (2022). The curves are +in good agreement. All of the remaining long-time statistics we report are phase invariant, so the +remaining results use only models for the pattern dynamics. + +1.0 +18 +DManD +1.0 +<2(±)> / (+)()) +DNS +16 +D 0.8 +14 +0.5 +12 +0.6 +10 +0.0 +8 +0.4 +6 +0.2 +4 +-0.5 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +4 +tDNS +10-1 +DNS +-3.4 +DManD +DManD +10-2 +-3.5 +MSD +0 +10-3 +-3.6 +10-4 +3.7 +10-5 +50 +0 +100 +150 +200 +100 +101 +102 +t +t14 +Figure 8: Fraction of unstable DManD models with standard neural ODEs and with stabilized +neural ODEs at various dimensions. +3.4. Long-time statistics +Next we investigate the ability of the DManD model to capture the long-time dynamics of PCF. +An obvious prerequisite for models to capture long-time dynamics is the long-time stability of the +models. As mentioned in Sec. 2, the long-time trajectories of standard neural ODEs often become +unstable, which led us to use stabilized neural ODEs with an explicit damping term. We quantify +this observation by counting, of the 16 models trained at each dimension 𝑑ℎ, how many become +unstable with and without the presence of an explicit damping term. From our training data we +know where ℎ should lie, so if it falls far outside this range after some time we can classify the +model as unstable. In particular, we classify models as unstable if the norm of the final state is +two times that of the maximum in our data (|| ˜ℎ(𝑇)|| > 2 max𝑡 ||ℎ(𝑡)||), after 𝑇 = 104 time units. +In all of the unstable cases || ˜ℎ(𝑡)|| follows the data over some short time range before eventually +growing indefinitely. +In Fig. 8 we show the number of unstable models with and without damping. With damping, +all of the models are stable, whereas without damping almost all models become unstable for +𝑑ℎ = 5−16, and around half become unstable in the other cases. Additionally, with longer runs or +with different initial conditions, many of the models without damping labelled as stable here also +eventually become unstable. This lack of stability happens when inaccuracies in the neural ODE +model pushes trajectories off the attractor. Once off the attractor, the model is presented with +states unlike the training data leading to further growth in this error. In Linot & Graham (2022); +Linot et al. (2023) we show more results highlighting this behavior. So, although some standard +neural ODE models do provide reasonable statistics, using these models presents challenges due +to this lack of robustness. As such, all other results we show use stabilized neural ODEs. +While Fig. 8 indicates that stabilized neural ODEs predict ˜ℎ in a similar range to that of the data, +it does not quantify the accuracy of these predictions. In fact, with few dimensions many of these +models do not remain chaotic, landing on fixed points or periodic orbits. The first metric we use to +quantify the long-time performance of the DManD method is the mean-squared POD coefficient +amplitudes ( +� +||𝑎𝑛||2� +). We consider this quantity because Gibson reports it for POD-Galerkin in +Gibson (2002) at various levels of truncation. In Fig. 9 we show how well the DManD model, with +𝑑ℎ = 18, captures this quantity, in comparison to the POD-Galerkin model in Gibson (2002). The +two data sets slightly differ because we subtract the mean before applying POD and Gibson did +not. The DManD method, with only 18 degrees of freedom, matches the mean-squared amplitudes +to high accuracy, far better than all of the POD-Galerkin models. It is not until POD-Galerkin +keeps 1024 modes that the results become comparable, which corresponds to ∼ 2000 degrees +of freedom because most coefficients are complex. Additionally, our method requires only data, + +1.00 +Standard +Fraction Unstable +Stabilized ++ +0.75 +0.50 +0.25 +0.00 +XxX +10 +20 +30 +4015 +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +Figure 9: Comparison of +� +||𝑎𝑛||2� +(mean-squared POD coefficient amplitudes) from the DNS to +(a) +� +||𝑎𝑛||2� +from the DManD model at 𝑑ℎ = 18 and (b) +� +||𝑎𝑛||2� +from POD-Galerkin with 𝑁 +POD modes (reproduced with permission from Gibson (2002)). In (b), the quantity 𝜆 is equivalent +to +� +||𝑎𝑛||2� +from the DNS. +Figure 10: Components of the Reynolds stress with increasing dimension for DManD models at +various dimensions. Odd numbers above 𝑑ℎ = 5 are omitted for clarity. +whereas the POD Galerkin approach requires both data for computing the POD and knowledge +of the equations of motion for projecting the equations onto these modes. +We now investigate how the Reynolds stress and the power input vs. dissipation vary with +dimension. Figure 10 shows four components of the Reynolds stress at various dimensions. For +� +𝑢′2 +𝑥 +� +and +� +𝑢′ +𝑥𝑢′ +𝑦 +� +, nearly all the models match the data, with relatively small deviations only +appearing for 𝑑ℎ ∼ 3−6. For +� +𝑢′2 +𝑦 +� +and +� +𝑢′2 +𝑧 +� +, this deviation becomes more obvious, and the lines +do not converge until around 𝑑ℎ > 10, with all models above this dimension exhibiting a minor +overprediction in +� +𝑢′2 +𝑧 +� +. +To evaluate how accurate the models are at reconstructing the energy balance, we look at joint +PDFs of power input and dissipation. The power input is the amount of energy required to move + +X10-3 +X10-3 +75 +4. +18 +50 +雪 +25 +:14 +0 +0 +×10-3 +X10-3 +10 +0.0 +6 +DManD +DNS +2.5 +4 +6 +-5.0 +2 +-7.5 +0 +-1.0 +-0.5 +1.0 +-0.5 +0.0 +0.5 +-1.0 +0.0 +0.5 +1.0 +y +y10- +10- +N=16 +DNS +N=128 +N=512 + N=1024 +10-2 +DManD +X +入 +合 +210-3 +up +10-4 +10~4 +10-5 +10~5/ +口 +10-6 +10~ +10' +102 +100 +100 +101 +102 +n+1 +n+l16 +the walls: +𝐼 = +1 +2𝐿𝑥𝐿𝑧 +∫ +𝐿𝑥 +0 +∫ +𝐿𝑧 +0 +𝜕𝑢𝑥 +𝜕𝑦 +���� +𝑦=−1 ++ 𝜕𝑢𝑥 +𝜕𝑦 +���� +𝑦=1 +𝑑𝑥𝑑𝑧, +(3.14) +and the dissipation is the energy lost to heat due to viscosity: +𝐷 = +1 +2𝐿𝑥𝐿𝑧 +∫ +𝐿𝑥 +0 +∫ 1 +−1 +∫ +𝐿𝑧 +0 +|∇ × u|2 𝑑𝑥𝑑𝑦𝑑𝑧. +(3.15) +These two terms define the rate of change of energy in the system �𝐸 = 𝐼 − 𝐷, which must average +to zero over long times. Checking this statistic is important to show the DManD models correctly +balance the energy. +Figures 11a-11c show the PDF from the DNS, the PDF for 𝑑ℎ = 6 and the PDF for 𝑑ℎ = 18, +generated from a single trajectory evolved for 5000 time units, and Figs. 11e and 11f show the +the absolute difference between the true and model PDFs. With 𝑑ℎ = 6 the model overestimates +the number of low dissipation states, while 𝑑ℎ = 18 matches the density well. In Fig. 11d we +compare the joint PDFs at all dimension with the true PDF using the earth movers distance (EMD) +(Rubner et al. 1998). The EMD determines the distance between two PDFs as a solution to the +transportation problem by treating the true PDF as “supplies" and the model PDF as “demands" +and finding the flow which minimizes the work required to move one to the other. We compute the +distance between PDFs using the EMD because it is a cross-bin distance, meaning the distance +accounts for the density in neighboring bins. This is in contrast to bin-to-bin distances, like the KL +divergence, which only uses the error at a given bin. Bin-to-bin distances can vary significantly +with small shifts in one PDF (misalignment) and when changing the number of bins used to +generate the PDF (Ling & Okada 2007). We choose the EMD because it does not suffer from +these issues. In Fig. 11d we see a steep drop in the EMD at 𝑑ℎ = 5 and after 𝑑ℎ > 10 the joint +PDFs are in excellent agreement with the DNS. The dashed line corresponds to the EMD between +two different trajectories from the DNS. +3.5. Finding ECS in the model +Now that we know that the DManD model quantitatively captures many of the key characteris- +tics of MFU PCF, we now want to explore using the model to discover ECS. In particular, we first +investigate the whether known periodic orbits of the DNS exist in the DManD model, and then +we use the DManD model to search for new periodic orbits. Here we note that because our model +predicts phase-aligned dynamics, the periodic orbits of the DManD model are either periodic +or relative periodic orbits, depending on the phase evolution, which we have not tracked. In the +following we omit all ˜·, so all functions should be assumed to come from a DManD model. +Here we outline the approach we take to find periodic orbits, which follows Cvitanović et al. +(2016). When searching for periodic orbits we seek an initial condition to a trajectory that repeats +after some time period. This is equivalent to finding the zeros of +𝐻(ℎ,𝑇) = 𝐺𝑇 (ℎ) − ℎ, +(3.16) +where 𝐺𝑇 (ℎ) is the flow map forward 𝑇 time units from ℎ: i.e. 𝐺𝑇 (ℎ(𝑡)) = ℎ(𝑡 +𝑇). We compute +𝐺𝑇 (ℎ) from Eq. 2.9. Finding zeros to Eq. 3.16 requires that we find both a point ℎ∗ on the +periodic orbit and a period 𝑇∗ such that 𝐻(ℎ∗,𝑇∗) = 0. One way to find ℎ∗ and 𝑇∗ is by using the +Newton-Raphson method. +By performing a Taylor series expansion of 𝐻 we find near the fixed point ℎ∗,𝑇∗ of 𝐻 that +𝐻(ℎ∗,𝑇∗) − 𝐻(ℎ,𝑇) ≈ 𝐷ℎ𝐻(ℎ,𝑇)Δℎ + 𝐷𝑇 𝐻(ℎ,𝑇)Δ𝑇 +−𝐻(ℎ,𝑇) ≈ 𝐷ℎ𝐻(ℎ,𝑇)Δℎ + 𝑔 (𝐺𝑇 (ℎ)) Δ𝑇, +(3.17) +where 𝐷ℎ is the Jacobian of 𝐻 with respect to ℎ, 𝐷𝑇 is the Jacobian of 𝐻 with respect to the + +17 +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +c)c)c)c)c)c)c)c)c)c)c)c)c)c)c)c)c) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +d) +e)e)e)e)e)e)e)e)e)e)e)e)e)e)e)e)e) +f)f)f)f)f)f)f)f)f)f)f)f)f)f)f)f)f) +Figure 11: (a)-(c): examples of joint PDFs for the true system, the DManD model at 𝑑ℎ = 6, and +the DManD model at 𝑑ℎ = 18. (d): earth movers distance between the PDF from the DNS and the +PDFs predict by the DManD model at various dimensions. “DNS" is the error between two PDFs +generated from DNS trajectories of the same length with different initial conditions. (e) and (f): +the error associated with the DManD model PDFs at 𝑑ℎ = 6 and 𝑑ℎ = 18. +period 𝑇, Δℎ = ℎ∗ − ℎ and Δ𝑇 = 𝑇∗ − 𝑇. To have a complete set of equations for Δℎ and Δ𝑇, we +supplement Eq. 3.17 with the constraint that the updates Δℎ are orthogonal to the vector field at +ℎ: i.e., +𝑔(ℎ)𝑇 Δℎ = 0. +(3.18) +With this constraint, at Newton step (𝑖), the system of equations becomes +� 𝐷ℎ(𝑖) 𝐻(ℎ(𝑖),𝑇 (𝑖)) +𝑔(𝐺𝑇 (𝑖) (ℎ(𝑖))) +𝑔(ℎ(𝑖))𝑇 +0 +� � Δℎ(𝑖) +Δ𝑇 (𝑖) +� += − +� +𝐻(ℎ(𝑖),𝑇 (𝑖)) +0 +� +, +(3.19) +which, in the standard Newton-Raphson method, is used to update the guesses ℎ(𝑖+1) = ℎ(𝑖) +Δℎ(𝑖) +and 𝑇 (𝑖+1) = 𝑇 (𝑖) + Δ𝑇 (𝑖). +Typically, a Newton-Krylov method is used to avoid explicitly constructing the Jacobian +(Viswanath 2007). However, with DManD, computing the Jacobian is simple, fast, and requires +little memory because the state representation is dramatically smaller in the DManD model +than in the DNS. We compute the Jacobian 𝐷ℎ𝐻(ℎ,𝑇) directly by using the same automatic +differentiation tools used for training the neural ODE. Furthermore, if we had chosen to +represent the dynamics in discrete, rather than continuous time, computation of general periodic +orbits would not be possible, as the period 𝑇 can take on arbitrary values and a discrete-time +representation would limit 𝑇 to multiples of the time step. When finding periodic orbits of the +DManD model we used the Scipy “hybr" method, which uses a modification of the Powell hybrid +method (Virtanen et al. 2020), and for finding periodic orbits of the DNS we used the Newton +GMRES-Hookstep method built into Channelflow (Gibson et al. 2021). In the following trials we +only consider DManD models with 𝑑ℎ = 18. +For the HKW cell there exists a library of POs made available by Gibson et al. (2008b). To +investigate if the DManD model finds POs similar to existing solutions, we took states from the +known POs, encoded them, and used this as an initial condition in the DManD Newton solver to + +DNS +dh = 6 +dh = 18 +4 +3.5 +3.5 +3.5 +D 3.0 +D 3.0 +D 3.0 +P +-2 +2.5 +2.5 +2.5 +2.0 +2.0 +2.0 +0 +2.0 +2.5 +3.0 +2.0 +2.5 +3.0 +2.0 +2.5 +3.0 +I +I +1 +Error dh = 6 +Error dh = 18 +DManD +0.2 +-3 +3.5 +3.5 +DNS +~P +-2 +D 3.0 +D 3.0 +P +2.5 +2.5 +:1 +2.0 +2.0 +0.0 +0 +5 +10 +15 +2.0 +2.5 +3.0 +2.0 +2.5 +3.0 +dh +I +118 +Figure 12: Power input vs. dissipation of known periodic orbits (period reported in bottom right) +from the DNS and periodic orbits found in the DManD model at 𝑑ℎ = 18. The blue line is a long +trajectory of the DNS for comparison. +find POs in the model. In Fig. 12 we show projections of 12 known POs, which we identify by the +period 𝑇, and compare them to projections of POs found using the DManD model. This makes +up a majority of the POs made available by Gibson et al. (2008b). Of the other known solutions, +three are RPOs with phase-shifts in the streamwise direction that our model, with the current +setup, can not capture. The other two have short periods of 𝑇 = 19.02 and 𝑇 = 19.06. A majority +of the POs found with DManD land on initial conditions near that of the DNS and follow similar +trajectories trajectories to the DNS. +How close many of these trajectories are to the true PO is surprising and encouraging for +many reasons. First, the data used for training the DManD model does not explicitly contain any +POs. Second, this approach by no means guarantees convergence on a PO in the DManD model. +Third, starting with an initial condition from a PO does not necessarily mean that the solution the +Newton solver lands on will be the closest PO to that initial condition, so there may exist POs in +the DManD model closer to the DNS solutions than what we present here. +Now that we know the DManD model can find POs similar to those known to exist for the DNS, +we now use it to search for new POs. First, we searched for POs in three of the 𝑑ℎ = 18 models +by randomly selecting 20 initial conditions and selecting 4 different periods 𝑇 = [20, 40, 60, 80]. + +DNS +3.5 +DManD +D 3.0 +2.5 +T = 62.13 +T = 68.07 +T = 75.35 +3.5 +D 3.0 +2.5 +T = 76.82 +T = 76.85 +T = 85.27 +3.5 +D 3.0 +2.5 +T = 87.89 +T = 88.90 +T = 90.31 +3.5 +D 3.0 +2.5 +T = 90.52 +T = 99.70 +T = 121.4 +2.5 +3.0 +3.5 +2.5 +3.0 +3.5 +2.5 +3.0 +3.5 +1 +I19 +Table 2: Details on the RPOs and POs found using initial conditions from the DManD model. +The first 9 solutions are new and the last 3 had previously been found. “Label" indicates the +label in Fig. 13b, 𝜎𝑧 corresponds to the phase-shift in 𝑧, 𝑇 is the period of the orbit, and “Error" +is ||shifted final state − initial state||/||initial state||, which is the same error as in Viswanath +(2007). +Label +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +𝜎𝑧 +1.91e-1 -9.66e-2 -1.77e-3 1.15e-1 -9.21e-3 -1.90e-1 -1.28e-2 -1.19e-1 -5.63e-5 4.64e-14 2.17e-14 2.73e-13 +𝑇 +37.94 +84.25 +91.29 +82.07 +74.14 +41.24 +110.67 +83.31 +64.64 +19.06 +68.07 +75.35 +Error +2.23e-3 +1.01e-3 +3.92e-3 +2.84e-3 +1.87e-3 +5.26e-4 +1.25e-3 +1.13e-3 +2.25e-3 +1.57e-4 +2.55e-4 +1.07e-4 +a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +b) +Figure 13: (a) Periodic orbits found in DManD models at 𝑑ℎ = 18 that converged to the (b) +periodic orbits found in the DNS. Each of the colors corresponds to a one of the new solutions in +Table 2. The blue curve at the back is a long trajectory of the DNS for comparison. +We then took the initial conditions and periods for converged periodic orbits and decoded and +upsampled them onto a 48 × 49 × 48 grid. We performed this upsampling because Viswanath +(2007) reported that solutions on the coarser grid can be computational artifacts. Finally, we put +these new initial conditions into Channelflow and ran another Newton search for 100 iterations. +This procedure resulted in us finding 9 new RPOs and 3 existing POs, the details of which we +include in Table 2. +In Fig. 13a we show the new RPOs in the DManD model and in Fig. 13b we show what they +converged to after putting them into the Channelflow Newton solver as initial guesses. Again, +many of the RPOs end up following a similar path through this state space, with the biggest +exceptions being RPO1 and RPO6, which converged to low-power input solutions. It is worth +noting that this worked well, considering that the DManD initial conditions are POD coefficients +from a model trained using data on a coarser grid than used to search for these solutions. We +have described a new method to rapidly find new ECS, wherein an accurate low-dimensional +model, like the DManD model presented here, is used to quickly perform a large number of ECS +searches in the model and then these solutions can be fine tuned in the full simulation to land on +new solutions. +4. Conclusion +In the present work we described a data-driven manifold dynamics method (DManD) and +applied it for accurate modeling of MFU PCF with far fewer degrees of freedom (O(10)) than + +4.0 +1 +DManD +Solution +2 +3.5 +3 +4 +3.0 +D +5 +6 +2.5 +7 +8 +2.0 +D +9 +2.0 +2.5 +3.0 +3.5 +2.0 +2.5 +3.0 +3.5 +120 +required for the DNS (O(105)). The DManD method consists of first finding a low-dimensional +parameterization of the manifold on which data lies, and then discovering an ODE to evolve this +low-dimensional state representation forward in time. In both cases we use NNs to approximate +these functions from data. We find that an extremely low-dimensional parameterization of this +manifold can be found using a hybrid autoencoder approach that corrects upon POD coefficients. +Then, we use stabilized neural ODEs to accurately evolve the low-dimensional state forward in +time. +The DManD model captures the self-sustaining process and accurately tracks trajectories and +the temporal autocorrelation over short time horizons. For DManD models with 𝑑ℎ > 10 we +found excellent agreement between the model and the DNS in computing the mean-squared POD +coefficient amplitude, the Reynolds stress, and the joint PDF of power input vs. dissipation. For +comparison, we showed that a POD-Galerkin model requires ∼ 2000 degrees of freedom to get +similar performance in matching the mean-squared POD coefficient amplitudes. Finally, we used +the DManD model at 𝑑ℎ = 18 for PO searches. Using a set of existing POs, we successfully +landed on nearby POs in the model. Finally, we found 9 previously undiscovered RPOs by first +finding solutions in the DManD model and then using those as initial guesses to search in the full +DNS. +The results reported here have both fundamental and technological importance. At the +fundamental level they indicate that, the true dimension of the dynamics of a turbulent flow +can be orders of magnitude smaller than the number of degrees of freedom required for a fully- +resolved simulation. 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In 12th International Symposium +on Turbulence and Shear Flow Phenomena (TSFP12) Osaka, Japan (Online), July 19-22, 2022. +Zeng, Kevin, Linot, Alec J. & Graham, Michael D. 2022b Data-driven control of spatiotemporal chaos +with reduced-order neural ODE-based models and reinforcement learning. Proceedings of the Royal +Society A 478 (2267), 20220297. + diff --git a/49E3T4oBgHgl3EQfpApV/content/tmp_files/load_file.txt b/49E3T4oBgHgl3EQfpApV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64ee522c0b42d0f733643aec4b101a46d44d9b25 --- /dev/null +++ b/49E3T4oBgHgl3EQfpApV/content/tmp_files/load_file.txt @@ -0,0 +1,1181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf,len=1180 +page_content='This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow Alec J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Linot1 and Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Graham1† 1Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI 53706, USA Because the Navier-Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' On this manifold, the state of the system can be exactly described in a coordinate system parameterizing the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We show, for turbulent Couette flow, that this description of the dynamics is possible using a data- driven manifold dynamics modeling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at Re = 400, where a fully resolved solutions requires O(105) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Using only data from this simulation we build models with fewer than 20 degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' At short-times, the models track the true trajectory for multiple Lyapunov times, and, at long-times, the models capture the Reynolds stress and the energy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For comparison, we show that the models outperform POD-Galerkin models with ∼2000 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we compute unstable periodic orbits from the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Many of these closely resemble previously computed orbits for the full system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' additionally, we find nine orbits that correspond to previously unknown solutions in the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Introduction A major challenge in dealing with chaotic fluid flows, whether it be performing experiments, running simulations, or interpreting the results, is the high-dimensional nature of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Even for simulations in the smallest domains that sustain turbulence (a minimal flow unit (MFU)), the state dimension may be O(105) (Jiménez & Moin 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' However, despite this nominal high-dimensionality, the dissipative nature of turbulent flows leads to the expectation that long-time dynamics collapse onto an invariant manifold of much lower dimension than the ambient dimension (Hopf 1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' By modeling the dynamics in a manifold coordinate system, simulations could be performed with a drastically lower-dimensional state representation, significantly speeding up computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Additionally, such a low-dimensional state representation is highly useful for downstream tasks like control or design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finding a low-dimensional – or ideally a minimal dimensional – parameterization of the manifold and an evolution equation for this parameterization are both challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this work we aim to address these challenges with a data-driven model, specifically for the task of reconstructing turbulent plane Couette flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The classic way to perform dimension reduction from data is to use the proper orthogonal decomposition (POD), also known by Principal Component Analysis (PCA) or Karhunen-Loève decomposition (Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This is a linear dimension reduction technique in which the state is projected onto the set of orthogonal modes that capture the maximum variance in the † Email address for correspondence: mdgraham@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='04638v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='flu-dyn] 11 Jan 2023 2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The POD is widely used for flow phenomena, some examples of which include: turbulent channel flow (Moin & Moser 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1991), flat-plate boundary layers (Rempfer & Fasel 1994), and free shear jet flows (Arndt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2005) showed how to incorporate system symmetries into the POD modes, the details of which we elaborate on in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' While the POD has seen wide use and is easy to interpret, more accurate reconstruction can be achieved with nonlinear methods – a result we highlight in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Some popular methods for nonlinear dimension reduction include kernel PCA (Schölkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1998), diffusion maps (Coifman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2005), local linear embedding (LLE) (Roweis & Saul 2000), isometric feature mapping (Isomap) (Tenenbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2000), and t-distributed stochastic neighbor embedding (tSNE) (Hinton & Roweis 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' These methods are described in more detail in Linot & Graham (2022), and an overview of other dimension reduction methods can be found in Van Der Maaten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' One drawback of all of these methods, however, is that they reduce the dimension, but do not immediately provide a means to move from a low-dimensional state back to the full state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A popular dimension reduction method without these complications is the undercomplete autoencoder (Hinton & Salakhutdinov 2006), which uses a neural network (NN) to map the input data into a lower-dimensional “latent space" and another NN to map back to the original state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We describe this structure in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Some examples where autoencoders have been used for flow systems include flow around a cylinder (Murata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020), flow around a flat plate (Nair & Goza 2020), Kolmogorov flow (Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Pérez De Jesús & Graham 2022), and channel flow (Milano & Koumoutsakos 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Although we will not pursue this approach in the present work, it may be advantageous for multiple reasons to parametrize the manifold with overlapping local representations, as done in Floryan & Graham (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' After reducing the dimension, the time evolution for the dynamics can be approximated from the equations of motion or in a completely data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The classical method is to perform a Galerkin projection wherein the equations of motion are projected onto a set of modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' POD modes) (Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' However, in this approach all the higher POD modes are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' An extension of this idea, called nonlinear Galerkin, is to assume that the time derivative of the coefficients of all of the higher modes is zero, but not the coefficients themselves (Titi 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Foias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' this is essentially a quasisteady state approximation for the higher modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This improves the accuracy, but comes at a higher computational cost than the Galerkin method, although this can be somewhat mitigated by using a postprocessing Galerkin approach (García-Archilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2018) also showed a recurrent NN (RNN) – a NN that feeds into itself – can be used to improve the nonlinear Galerkin approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This RNN structure depends on a history of inputs, making it non-Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In addition to these linear dimension reduction approaches, an autoencoder can be used with the equations of motion in the so-called manifold Galerkin approach, which Lee & Carlberg (2020) developed and applied to the viscous Burgers equation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' When the equations of motion are assumed to be unknown, and only snapshots of data are available, a number of different machine learning techniques exist to approximate the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Two of the most popular techniques are RNNs and reservoir computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Vlachas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2020) showed both these structures do an excellent job of capturing the chaotic dynamics of the Lorenz- 96 equation and Kuramoto-Sivashinsky equation (KSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For fluid flows, autoencoders and RNNs (specifically long-short term memory networks (LSTM)) have been used to model flow around a cylinders (Hasegawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Eivazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020), pitching airfoils (Eivazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020), bluff bodies (Hasegawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020b), and MFU plane Poiseuille flow (PPF) (Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Although these methods often do an excellent job of predicting chaotic dynamics, the models are not Markovian, so the dimension of the system also includes some history, and these models perform discrete timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' These two properties are undesirable, because the underlying dynamics are Markovian and continuous in time, and modeling them differently complicates applications and interpretations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In particular, we want to use the model for state 3 space analyses such as determination of periodic orbits, where standard tools are available for ODEs that do not easily generalize to non-Markovian dynamic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Due to these issues, we use neural ordinary differential equations (ODE) (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In neural ODEs, the right-hand-side (RHS) of an ODE is represented as a NN that is trained to reconstruct the time evolution of the data from snapshots of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Linot & Graham (2022) it was shown that this is an effective method for modeling the chaotic dynamics of the KSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2021) used neural ODEs to predict the periodic dynamics of flow around a cylinder, and Portwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2019) used neural ODEs to predict the kinetic energy and dissipation of decaying turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this work we investigate the dynamics of MFU Couette flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The idea behind the MFU, first introduced by Jiménez & Moin (1991), is to reduce the simulation domain to the smallest size that sustains turbulence, thus isolating the key components of the turbulent nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Using an MFU for Couette flow at transitional Reynolds number, Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (1995) outlined the regeneration cycle of wall bounded turbulence called the “self-sustaining process" (SSP), which we describe in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This system was later analyzed with coviariant Lyapunov analysis by Inubushi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2015), who found a Lyapunov time (the inverse of the leading Lyapunov exponent) of ∼ 48 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Many low-dimensional models have been developed to recreate the dynamics of the SSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first investigation of this topic was by Waleffe (1997), who developed an 8 mode model for shear flow between free-slip walls generated by a spatially sinusoidal forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' He selected the modes based on intuition from the SSP and performed a Galerkin projection onto these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Moehlis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2004) later added an additional mode to Waleffe’s model which enables modification of the mean profile by the turbulence, and made some modifications to the chosen modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this “MFE" model, Moehlis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' found exact coherent states, which we discuss below, that did not exist in the 8 mode model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In addition, Moehlis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2002) also used the POD modes on a domain slightly larger than the MFU to generate POD-Galerkin models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' These low-dimensional models have been used as a starting point for testing data-driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For example, both LSTMs (Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2019) and a Koopman operator method with nonlinear forcing (Eivazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2021) have been used to attempt to reconstruct the MFE model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Borrelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2022) then applied these methods to PPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we note that a key approach to understanding complex nonlinear dynamical phenomena, such as the SSP of near-wall turbulence, is through study of the underlying state space structure of fixed points and periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In the turbulence literature these are sometimes called “exact coherent states", or ECS (Kawahara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Graham & Floryan 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Turbulence organizes around ECS in the sense that trajectories chaotically move between different such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first ECS found were fixed point solutions in PCF (Nagata 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Following this work, Waleffe (1998) was able to connect ECS of PCF and PPF to the SSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Later, more fixed point ECS were found in MFU PCF and visualized by Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Unlike fixed points, which cannot capture dynamic phenomena at all, periodic orbits are able to represent key aspects of turbulent dynamics such as bursting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Kawahara & Kida (2001) found the first two periodic orbits (POs) for MFU PCF, one of which had statistics that agreed well with the SSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, Viswanath (2007) found another PO and 4 new relative POs (RPOs) in this domain, and Gibson made these solutions available in (Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2008b), along with a handful of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In the present work, we use autoencoders and neural ODEs , in a method we call “Data-driven Manifold Dynamics" (DManD) (Linot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2023), to build a ROM for turbulent MFU PCF (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Section 2 outlines the details of the DManD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We then describe the details of the Couette flow in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1, the results of the dimension reduction in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2, and the DManD model’s reconstruction of short- and long-time statistics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' After showing that the models accurately reproduce these statistics, we compute 4 RPOs for the model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5, finding several that are similar to previously known RPOs, as well as several that seem to be new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we summarize the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Framework Here we outline our method for an “exact" DManD modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this sense “exact" means all of the functions described allow for perfect reconstruction, but error is introduced in approximating these functions due to insufficient data, error in learning the functions, or error in evolving them forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This differs from coarse-grained ROMs, which approximate the physics to generate a closed set of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A key component allowing DManD to be “exact" is that we only seek to discover the evolution of trajectories after they collapse onto an invariant manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In general, the training data for development of a DManD model comes in the form of snapshots {𝑢1, 𝑢2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=', 𝑢𝑀 }, which are either the full state or measurements diffeomorphic to the full state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' time delays (Takens 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Young & Graham 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we consider full-state data 𝑢 that lives in an ambient space R𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We generate a time series of data by evolving this state forward in time according to 𝑑𝑢 𝑑𝑡 = 𝑓 (𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1) (In the present context, this equation represents a fully-resolved direct numerical simulation (DNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=') With the full state, we can then define a mapping to a low-dimensional state representation ℎ = 𝜒(𝑢), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2) with ℎ ∈ R𝑑ℎ is a coordinate representation on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we define a mapping back to the full state ˜𝑢 = ˇ𝜒(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='3) For data that lies on a finite-dimensional invariant manifold these functions can exactly reconstruct the state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' ˜𝑢 = 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' However, if the dimension 𝑑ℎ is too low, or there are errors in the approximation of these functions, then ˜𝑢 approximates the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, with this low-dimensional state representation, we can define an evolution equation 𝑑ℎ 𝑑𝑡 = 𝑔(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4) The DManD model consists of the three functions 𝜒, ˇ𝜒, and 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' By approximating these functions, the evolution of trajectories on the manifold can be performed entirely in the manifold coordinates ℎ, which requires far fewer operations than a full simulation, as 𝑑ℎ ≪ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We choose to approximate all of these functions using NNs, but other representations could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' First, we train 𝜒 and ˇ𝜒 using an undercomplete autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This is a NN structure consisting of an encoder which reduces dimension (𝜒) and a decoder that expands dimension ( ˇ𝜒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1, a common approach to dimension reduction is to project onto a set of POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' POD gives the optimal linear projection in terms of reconstruction error, so we use this fact to train an encoder as the sum of POD and a correction in the form of an NN: ℎ = 𝜒(𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸) = 𝑈𝑇 𝑑ℎ𝑢 + E(𝑈𝑇 𝑟 𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5) In this equation, 𝑈𝑘 ∈ R𝑑×𝑘 is a matrix whose 𝑘 columns are the first 𝑘 POD modes as ordered by variance, and E is a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first term (𝑈𝑇 𝑑ℎ𝑢) is the projection onto the leading 𝑑ℎ POD modes, and the second term is the NN correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The matrix 𝑈𝑟 in this term may either be a full change of basis with no approximation (𝑟 = 𝑑), or involve some dimension reduction (𝑑 > 𝑟 > 𝑑ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 5 For mapping back to the full state (decoding), we again sum POD with a correction ˜𝑢 = ˇ𝜒(ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸) = 𝑈𝑟 ([ℎ, 0]𝑇 + D(ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐷)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6) Here, [ℎ, 0]𝑇 is the ℎ vector zero padded to the correct size, and D is a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first term is the POD mapping back to the full space, if there were no NNs, and the second term is a NN correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Linot & Graham (2020) we refer to this structure as a hybrid autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2 we contrast this to a “standard" autoencoder where ℎ = E(𝑈𝑇 𝑟 𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸) and ˜𝑢 = 𝑈𝑟D(ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' These hybrid autoencoder operations act as a shortcut connection on the optimal linear dimension reduction, which we (Linot & Graham 2020) found useful for representing the data and achieving accurate reconstruction of 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2021) also took a similar approach with POD shortcut connections over each layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We determine the NN parameters 𝜃𝐸 and 𝜃𝐷 by minimizing 𝐿 = 1 𝑑𝐾 𝐾 ∑︁ 𝑖=1 ||𝑢(𝑡𝑖) − ˇ𝜒(𝜒(𝑢(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐷)||2 + 1 𝑑ℎ𝐾 𝐾 ∑︁ 𝑖=1 𝛼||E(𝑈𝑇 𝑟 𝑢(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐸) + D𝑑ℎ (ℎ(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝐷)||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='7) The first term in this loss is the mean-squared error (MSE) of the reconstruction ˜𝑢, and the second term is a penalty that promotes accurate representation of the leading 𝑑ℎ POD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this term, D𝑑ℎ denotes the leading 𝑑ℎ elements of the decoder output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For finite 𝛼, the autoencoder exactly matches the first 𝑑ℎ POD coefficients when this term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Details of the minimization procedure are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Next, we approximate 𝑔 using a neural ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A drawback of training a single dense NN for 𝑔 is that the resulting dynamics may become weakly unstable, with linear growth at long times (Linot & Graham 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Linot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To avoid this, we use a “stabilized" neural ODE approach by adding a linear damping term onto the output of the NN, giving 𝑔(ℎ(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝑔) = 𝑔NN(ℎ(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝑔) + 𝐴ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='8) Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='8 forward from time 𝑡𝑖 to 𝑡𝑖 + 𝜏 yields ˜ℎ(𝑡𝑖 + 𝜏) = ℎ(𝑡𝑖) + ∫ 𝑡𝑖+𝜏 𝑡𝑖 𝑔NN(ℎ(𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃𝑔) + 𝐴ℎ(𝑡)𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='9) Depending on the situation, one may either learn 𝐴 from data, or fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we set it to the diagonal matrix 𝐴𝑖 𝑗 = −𝛽𝛿𝑖 𝑗𝜎𝑖(ℎ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='10) where 𝜎𝑖(ℎ) is the standard deviation of the 𝑖th component of ℎ, 𝛽 is a tunable parameter, and 𝛿𝑖 𝑗 is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This linear term attracts trajectories back to the origin, preventing them from moving far away from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 we show that this approach drastically improves the long-time performance of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We then determine the parameters 𝜃𝑔 by minimizing the difference between the predicted state ˜ℎ(𝑡𝑖 + 𝜏) and the true state ℎ(𝑡𝑖 + 𝜏), averaged over the data: 𝐽 = 1 𝑑ℎ𝐾 𝐾 ∑︁ 𝑖=1 � ||ℎ(𝑡𝑖 + 𝜏) − ˜ℎ(𝑡𝑖 + 𝜏)||2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='11) For clarity we show the specific loss we use, which sums over only a single snapshot forward in time at a fixed 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' More generally, the loss can be formulated for arbitrary snapshot spacing and for multiple snapshots forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To compute the gradient of 𝐽 with respect to the neural network parameters 𝜃𝑔, automatic differentiation can be used to backpropagate through the ODE solver that is used to compute the time integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='9, or an adjoint problem can be solved backwards in time (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The adjoint method uses less memory than 6 backpropagation, but ℎ is low-dimensional and our prediction window for training is short, so we choose to backpropagate through the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' So far this approach to approximating 𝜒, ˇ𝜒, and 𝑔 is general and does not directly account for the fact that the underlying equations are often invariant to certain symmetry operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For example, one of the symmetries in PCF is a continuous translation symmetry in 𝑥 and 𝑧 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' any solution shifted to another location in the domain gives another solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This poses an issue for training, because in principle, the training data must include all these translations to accurately model the dynamics under any translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We discuss these and other symmetries of PCF in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In practice, accounting for continuous symmetries is most important along directions that sample different phases very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For PCF, the mean flow is in the 𝑥 direction, leading to good phase sampling along this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' However, there is no mean flow in the 𝑧 direction, so sampling all phases relies on the slow phase diffusion in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Therefore, we will only explicitly to account for the 𝑧-phase in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3, but in the current disucssion we present the general framework accounting for all continuous symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To address the issue of continuous translations, we add an additional preprocessing step to the data, using the method of slices (Budanur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2015b,a) to split the state 𝑢 into a pattern 𝑢 𝑝 ∈ R𝑑 and a phase 𝜙 ∈ R𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The number of continuous translation symmetries for which we explicitly account determines 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We discuss the details of computing the pattern and the phase in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Separating the pattern and phase is useful because the evolution of both the pattern and the phase only depend on the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Thus, we simply replace 𝑢 with 𝑢 𝑝 in all the above equations and then write one additional ODE for the phase 𝑑𝜙 𝑑𝑡 = 𝑔𝜙(ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃 𝜙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='12) We then fix the parameters of 𝑔 to evolve ℎ (from 𝑢 𝑝) forward in time and use that to make a phase prediction ˜𝜙(𝑡𝑖 + 𝜏) = 𝜙(𝑡𝑖) + ∫ 𝑡𝑖+𝜏 𝑡𝑖 𝑔𝜙(ℎ(𝑡𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝜃 𝜙)𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='13) Finally, we determine the parameters 𝜃 𝜙 to minimize the difference between the predicted phase ˜𝜙(𝑡𝑖 + 𝜏) and the true phase 𝜙(𝑡𝑖 + 𝜏) 𝐽𝜙 = 1 𝑐𝐾 𝐾 ∑︁ 𝑖=1 � ||𝜙(𝑡𝑖 + 𝜏) − ˜𝜙(𝑡𝑖 + 𝜏)||2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='14) using the method described above to compute the gradient of 𝐽𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Description of Plane Couette Flow Data In the following sections we apply DManD to DNS of turbulent PCF in a MFU domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Specifically, we consider the well-studied Hamilton, Kim, and Waleffe (HKW) domain (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We made this selection to compare our DManD results to the analysis of the self- sustaining process in this domain, to compare our DManD results to other Galerkin-based ROMs, and to compare our DManD results to known unstable periodic solutions in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For PCF we solve the Navier-Stokes equations 𝜕u 𝜕𝑡 + u · ∇u = −∇𝑝 + Re−1∇2u, ∇ · u = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1) for a fluid confined between two plates moving in opposite directions with the same speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1 is the nondimensionalized form of the equations with velocities in the streamwise 𝑥 ∈ [0, 𝐿𝑥], wall-normal 𝑦 ∈ [−1, 1], and spanwise 𝑧 ∈ [0, 𝐿𝑧] directions defined as u = [𝑢𝑥, 𝑢𝑦, 𝑢𝑧], and pressure 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We solve this equation for a domain with periodic boundary conditions in 𝑥 and 𝑧 (u(0, 𝑦, 𝑧) = u(𝐿𝑥, 𝑦, 𝑧), u(𝑥, 𝑦, 0) = u(𝑥, 𝑦, 𝐿𝑧)) and no-slip, no-penetration boundary conditions in 𝑦 (𝑢𝑥(𝑥, ±1, 𝑧) = ±1, 𝑢𝑦(𝑥, ±1, 𝑧) = 𝑢𝑧(𝑥, ±1, 𝑧) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The complexity of the flow increases as the Reynolds number Re increases and the domain size 𝐿𝑥 and 𝐿𝑧 increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we use the HKW cell, which is at Re = 400 with a domain size [𝐿𝑥, 𝐿𝑦, 𝐿𝑧] = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='75𝜋, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2𝜋] (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The HKW cell is one of the simplest flows that sustains turbulence for extended periods of time before relaminarizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We chose to use this flow because it is well studied (refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1), it isolates the SSP (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1995), and a library of ECS exist for this domain (Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we are interested in modeling the turbulent dynamics, so we will remove data upon relaminarization as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1, under the boundary conditions described, is invariant (and its solutions equivariant) under the discrete symmetries of point reflections about [𝑥, 𝑦, 𝑧] = [0, 0, 0] P · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (−𝑢𝑥, −𝑢𝑦, −𝑢𝑧, 𝑝)(−𝑥, −𝑦, −𝑧, 𝑡) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2) reflection about the 𝑧 = 0 plane R · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (𝑢𝑥, 𝑢𝑦, −𝑢𝑧, 𝑝)(𝑥, 𝑦, −𝑧, 𝑡) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='3) and rotation by 𝜋 about the 𝑧-axis RP · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (−𝑢𝑥, −𝑢𝑦, 𝑢𝑧, 𝑝)(−𝑥, −𝑦, 𝑧, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4) In addition to the discrete symmetries, there are also continuous translation symmetries in 𝑥 and 𝑧 T𝜎𝑥,𝜎𝑧 · [(𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥, 𝑦, 𝑧, 𝑡)] = (𝑢𝑥, 𝑢𝑦, 𝑢𝑧, 𝑝)(𝑥 + 𝜎𝑥, 𝑦, 𝑧 + 𝜎𝑧, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5) We incorporate all these symmetries in the POD represesntation (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2005), as we discuss further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, we use the method of slices (Budanur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2015a) to phase align in the 𝑧 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' By phase aligning in 𝑧 we fix the location of the low-speed streak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Without the alignment in 𝑧, models performed poorly because the models needed to learn how to represent every spatial shift of every snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In what follows, we only consider phase-alignment in 𝑧, but we note that extending this work to phase-alignment in 𝑥 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To phase-align the data, we use the first Fourier mode method-of-slices (Budanur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' First, we compute a phase by taking the Fourier transform of the streamwise velocity in 𝑥 and 𝑧 ( ˆ𝑢𝑥(𝑘𝑥, 𝑦, 𝑘𝑧) = F𝑥,𝑧(𝑢𝑥)) at a specific 𝑦 location (𝑦1) to compute the phase 𝜙 = atan2(imag( ˆ𝑢𝑥(0, 𝑦1, 1)), real( ˆ𝑢𝑥(0, 𝑦1, 1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6) We select 𝑦1 to be one grid point off the bottom wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then we compute the pattern dynamics by using the Fourier shift theorem to set the phase to 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' move the low-speed streak to the center of the channel) u𝑝 = F −1 𝑥,𝑧( ˆu exp(−𝑖𝑘𝑧𝜙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='7) We generate turbulent PCF trajectories using the psuedo-spectral Channelflow code developed by Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this code, the velocity and pressure fields are projected onto Fourier modes in 𝑥 and 𝑧 and Chebyshev polynomials of the first kind in 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' These coefficients are evolved forward in time first using the multistage SMRK2 scheme (Spalart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1991), then, after taking multiple timesteps, using the multistep Adams-Bashforth Backward-Differentiation 3 scheme (Peyret 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' At each timestep, a pressure boundary condition is found such that incompressibility is satisfied at the wall (𝑑𝑢𝑦/𝑑𝑦 = 0) using the influence matrix method and tau correction developed by Kleiser & Schumann (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Data was generated with Δ𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0325 on a grid of [𝑁𝑥, 𝑁𝑦, 𝑁𝑧] = [32, 35, 32] in 𝑥, 𝑦, and 𝑧 8 for the HKW cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Starting from random divergence-free initial conditions, we ran simulations forward for either 10, 000 xtime units or until relaminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then we dropped the first 1, 000 time units as transient data and the last 1, 000 time units to avoid laminar data, and repeated with a new initial condition until we had 91, 562 time units of data stored at intervals of one time unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We split this data into 80% for training and 20% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we preprocess the data by computing the mean ⟨u⟩ (𝑦) from the training data and subtracting it from all data u′ = u − ⟨u⟩, and then we compute the pattern u′ 𝑝 and the phase 𝜙 as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The pattern 𝑢 𝑝 described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2 is u′ 𝑝 flattened into a vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝑑 = 3𝑁𝑥𝑁𝑦𝑁𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The POD and NN training use only the training data, and all comparisons use test data unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Dimension Reduction and Dynamic Model Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Linear dimension reduction with POD: From O(105) to O(103) The first task in DManD for this Couette flow data is finding a low-dimensional parameterization of the manifold on which the long-time dynamics lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We parameterize this manifold in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' First, we reduce the dimension down from O(105) to 502 with the proper orthogonal decomposition (POD), and, second, we use an autoencoder to reduce the dimension down to 𝑑ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first step is simply a preprocessing step to reduce the size of the data, which reduces the number of parameters in the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Due to Whitney’s embedding theorem (Whitney 1936, 1944), we know that as long as the manifold dimension is less than 251 (𝑑M < 251) then this POD representation is diffeomorphic to the full state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As we show later, the manifold dimension appears to be far lower than 𝑑M = 251, so no information of the full state should be lost with this first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Proper orthogonal decomposition (POD) originates with the question of what function 𝚽 maximizes � |(u′, 𝚽)|2� ||𝚽||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='8) Solutions 𝚽(𝑛) to this problem satisfy the eigenvalue problem 3 ∑︁ 𝑗=1 ∫ 𝐿𝑥 0 ∫ 1 −1 ∫ 𝐿𝑧 0 � 𝑢′ 𝑖(x, 𝑡)𝑢′∗ 𝑗 (x′, 𝑡) � Φ(𝑛) 𝑗 (x′) 𝑑x′ = 𝜆𝑖Φ(𝑛) 𝑖 (x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='9) (Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Unfortunately, upon approximating these integrals, with the trapezoidal rule for example, this becomes a 𝑑 × 𝑑 matrix, making computation intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Furthermore, computing the average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='9, without any modifications, results in POD modes that fail to preserve the underlying symmetries described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In order to make this problem computationally tractable, and preserve symmetries, we apply the POD method used in Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2005), with the slight difference that we first subtract off the mean of state before performing the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first step in this procedure is to treat the POD modes as Fourier modes in both the 𝑥 and 𝑧 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' show in (Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2012) that for translation-invariant directions Fourier modes are the optimal POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This step transforms the eigenvalue problem into 𝐿𝑥𝐿𝑧 3 ∑︁ 𝑗=1 ∫ 1 −1 � ˆ𝑢′ 𝑖(𝑘𝑥, 𝑦′, 𝑘𝑧, 𝑡) ˆ𝑢′∗ 𝑗 (𝑘𝑥, 𝑦′, 𝑘𝑧, 𝑡) � 𝜑(𝑛) 𝑗𝑘𝑥 𝑘𝑧 (𝑦′) 𝑑𝑦′ = 𝜆(𝑛) 𝑘𝑥 𝑘𝑧 𝜑(𝑛) 𝑖𝑘𝑥 𝑘𝑧 (𝑦), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='10) which reduces the 𝑑 × 𝑑 eigenvalue problem down to a 3𝑁𝑦 × 3𝑁𝑦 eigenvalue problem for every wavenumber pair (𝑘𝑥, 𝑘𝑧) of Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We used 5, 000 snapshots evenly sampled over the training data to compute the POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, to account for the discrete symmetries, the data is augmented such that the mean in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='10 is computed by adding all the discrete symmetries of each snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 9 a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) Figure 1: (a) Eigenvalues of POD modes sorted in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (b) Components of the Reynolds stress for data generated by the DNS and this data projected onto 256 POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In (a) the curves are, from top to bottom, � 𝑢′2 𝑥 � , � 𝑢′2 𝑧 � , � 𝑢′2 𝑦 � , and � 𝑢′ 𝑥𝑢′ 𝑦 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This analysis gives us POD modes 𝚽(𝑛) 𝑘𝑥 𝑘𝑧 (x) = 1 √𝐿𝑥𝐿𝑧 exp � 2𝜋𝑖 � 𝑘𝑥𝑥 𝐿𝑥 + 𝑘𝑧𝑧 𝐿𝑧 �� 𝝋(𝑛) 𝑘𝑥 𝑘𝑧 (𝑦), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='11) and eigenvalues 𝜆(𝑛) 𝑘𝑥 𝑘𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We sort the modes from largest eigenvalue to smallest eigenvalue (𝜆𝑖) and and project onto the leading 256 modes, giving us a vector of POD coefficients 𝑎(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A majority of these modes are complex, so projecting onto these modes results in a 502-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1a we plot the eigenvalues, which show a rapid drop and then a long tail that contributes little to the energy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' By dividing the eigenvalues of the leading 256 modes by the total, we find these modes contain 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='8% of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To further illustrate that 256 modes is sufficient to represent the state in this case, we consider the reconstruction of statistics after projecting onto the POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1b we show the reconstruction of four components of the Reynolds stress, � 𝑢′2 𝑥 � , � 𝑢′2 𝑧 � , � 𝑢′2 𝑦 � , and � 𝑢′ 𝑥𝑢′ 𝑦 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The projection onto POD modes matches all of these quantities extremely well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Now that we have converted the data to POD coefficients and filtered out the low energy modes, we next train an autoencoder to perform nonlinear dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2, we phase-align the data in the spanwise direction at this step using the first Fourier mode method-of-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A common practice when training NNs is to normalize the data by subtracting the mean and dividing by the standard deviation of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We do not take this approach here because the standard deviation of the higher POD coefficients, which contribute less to the reconstruction, is much smaller than the lower POD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In order to retain the important information in the magnitudes, but put the values in a more amenable form for NN training, we instead normalize the POD coefficients by subtracting the mean and dividing by the maximum standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, we take this input and train autoencoders to minimize the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='7 using an Adam optimizer (Kingma & Ba 2015) in Keras (Chollet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We train for 500 epochs with a learning rate scheduler that drops the learning rate from 10−3 to 10−4 after 400 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' At this point we see no improvement in the reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For the hybrid autoencoder approach, we set 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Table 1 includes additional NN architecture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Nonlinear dimension reduction with autoencoders: From O(103) to O(101) With the above “preprocessing" step completed, we now turn to the reduction of dimension with the nonlinear approach enabled by the autoencoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We consider three approaches to reducing the dimension of 𝑎: (1) Training a hybrid autoencoder, (2) Training a standard 10-2 DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='06 POD 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='04 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='02 10-8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='00 10-10 100 101 102 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 i y10 Table 1: Architectures of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' “Shape" indicates the dimension of each layer, “Activation" the corresponding activation functions, and “sig" is the sigmoid activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='“Learning Rate" gives the learning rate at multiple times during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The learning rates was dropped at even intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Function Shape Activation Learning Rate E 502/1000/𝑑ℎ sig/linear [10−3, 10−4] D 𝑑ℎ/1000/502 sig/linear [10−3, 10−4] 𝑔NN 𝑑ℎ/200/200/𝑑ℎ sig/sig/linear [10−2, 10−3, 10−4] 𝑔𝜙 𝑑ℎ/200/200/1 sig/sig/linear [10−2, 10−3, 10−4] Figure 2: Mean squared error on test data for POD, standard autoencoders, and hybrid autoencoders at various dimensions 𝑑ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' At each dimension there are four standard and four hybrid autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' autoencoder, (3) linear projection onto a small set of POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We describe the first two approaches in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2, noting that the POD projection onto 256 (complex) modes can be written as 𝑎 = 𝑈𝑇 𝑟 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The third approach just corrsponds to setting E and D to zero in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2 we show the MSE of reconstructing 𝑎 with these three approaches over a range of different dimensions 𝑑ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We use NNs with the same architectures for both the standard and the hybrid autoencoder approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Due to the variability introduced into autoencoder training by randomly initialized weights and stochasticity in the optimization, we show the error for four separately trained autoencoders, at each 𝑑ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We see that the autoencoders perform an order magnitude better than POD in the range of dimension considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Both the standard and hybrid autoencoder approaches perform the same, so we select the hybrid approach because it can be viewed as a nonlinear correction to the POD projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Next we use the low-dimensional representations from these autoencoders to train stabilized neural ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Neural ODE Training After training four autoencoders at each dimension 𝑑ℎ, we chose a set of damping parameters, 𝛽, and for each, then trained four stabilized neural ODEs for all four autoencoders at every dimension 𝑑ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This results in 16 models at every 𝑑ℎ and 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The final 𝛽 value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='1 was selected so that long-time trajectories neither blew up nor decayed too strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Before training the ODEs, we preprocess each autoencoder’s latent space data set ℎ by subtracting the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' It is important to center the data because the linear damping (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='10) pushes trajectories towards the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We train the stabilized neural ODEs to predict the evolution of the centered data by using an Adam MSE Test Data Hybrid Stand POD 10° 5 10 1511 Figure 3: Snapshots of the streamwise velocity at 𝑦 = 0 from the DNS and from the DManD model at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' optimizer in Pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2019) to minimize the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We train using a learning rate scheduler that drops at three even intervals during training and we train until the learning curve stops improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Table 1 shows the details of this NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Unless otherwise stated, we show results for the one model out of those sixteen at each dimension with the lowest relative error averaged over all the statistics we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Short-time tracking In the following two sections we evaluate the performance of the DManD models at reconstruct- ing short-time trajectories and long-time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figure 3 shows snapshots of the streamwise velocity at the center plane of the channel, 𝑦 = 0, for the DNS and DManD at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We choose to show results for 𝑑ℎ = 18 because the autoencoder error begins to level off around this dimension, and, as we will show, the error in statistics levels off before this dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The value 𝑑ℎ = 18 is not necessarily the minimal dimension required to model this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3, both the DNS and the DmanD model show key characteristics of the SSP: (1) low-speed streaks become wavy, (2) the wavy low-speed streaks break down generating rolls, (3) the rolls lift fluid from the walls, regenerating streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Not only does DManD capture the qualitative behavior of the SSP, but Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3 also shows good quantitative agreement as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To further illustrate this, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 4 we show the modal root-mean squared (RMS) velocity 𝑀(𝑘𝑥, 𝑘𝑧) = �∫ 1 −1 ( ˆ𝑢2 𝑥(𝑘𝑥, 𝑦, 𝑘𝑧) + ˆ𝑢2 𝑦(𝑘𝑥, 𝑦, 𝑘𝑧) + ˆ𝑢2 𝑧(𝑘𝑥, 𝑦, 𝑘𝑧))𝑑𝑦 �1/2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='12) which Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (1995) used to identify the different parts of the SSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Specifically, we consider the 𝑀(0, 1) mode, which corresponds to the low speed streak and the 𝑀(1, 0) mode which corresponds to the 𝑥-dependence that appears when the streak becomes wavy and breaks up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this example, the two curves match well over a cycle of the SSP and only start to move away after ∼ 150 time units, which is about three Lyapunov times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' While the previous result shows a single example, we also consider ensembles of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figure 5 shows the tracking error ||𝑎(𝑡𝑖 + 𝑡) − ˜𝑎(𝑡𝑖 + 𝑡)|| of 10 trajectories, starting at 𝑡𝑖, for a model at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we normalize the tracking error by the error between solutions at random times 𝑡𝑖 and 𝑡 𝑗 𝐷 = ⟨||𝑎(𝑡𝑖) − 𝑎(𝑡)||⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this case the darkest line corresponds to the flow field in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' When considering the other initial conditions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 5, there tends to be a relatively slow rise in the error over ∼50 time units and then a more rapid increase after DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6 0=↑ t = 85 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 DManD 0=↑ t = 85 t = 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6 0 5 0 5 0 5 0 512 Figure 4: Modal RMS velocity from the DNS (𝑀) and from the DManD model at 𝑑ℎ = 18 ( ˜𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The markers correspond to the times in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figure 5: Normalized tracking error for 10 random initial conditions (different shades) using DManD with 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To better understand how this tracking varies with the dimension of the model we next consider the ensemble-averaged tracking error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 6a we show the normalized ensemble-averaged tracking error for model dimensions between 𝑑ℎ = 3 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For 𝑑ℎ = 3 − 5 there is a rapid rise in the error until ∼40 time units after which the error levels off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This behavior often happens due to trajectories quickly diverging and landing on stable fixed points or periodic orbits that do not exist in the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For 𝑑ℎ = 6−10 there is an intermediate behavior where lines diverge more quickly than the higher-dimensional models, but tend to approach the same tracking error at ∼100 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, for the remaining models 𝑑ℎ = 11 − 18, there is a smooth improvement in the tracking error over this time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As the dimension increases in this range the trends stay the same, but the error continues to decrease, which is partially due to improvement in the autoencoder performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The instantaneous kinetic energy of the flow is 𝐸(𝑡) = 1 2𝐿𝑥𝐿𝑧 ∫ 𝐿𝑧 0 ∫ 1 −1 ∫ 𝐿𝑥 0 1 2u · u𝑑𝑥𝑑𝑦𝑑𝑧, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='13) and we denote its fluctuating part as 𝑘(𝑡) = 𝐸(𝑡) − ⟨𝐸⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 6b we show the temporal autocorrelation of 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Again, for 𝑑ℎ = 3 − 5 we see clear disagreement between the true autocorrelation and the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Above 𝑑ℎ > 5 all of the models match the temporal autocorrelation well, without a clear trend in the error as dimension changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' All these models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='20 M(0,1) M(kx, kz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='15 M(0, 1) M(1, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='10 - M(1, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='00 0 50 100 150 200 t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 I/D a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0 20 40 60 80 100 t13 a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) Figure 6: (a) ensemble averaged short-time tracking and (b) temporal autocorrelation of the kinetic energy for DmanD models of increasing dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In (b) odd numbers above 𝑑ℎ = 5 are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) Figure 7: (a) an example of the phase evolution and (b) the MSD of the phase evolution for the DNS and the DManD model at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' match well for ∼ 40 time units, with 𝑑ℎ = 18 (the darkest line) matching the data extremely well for two Lyapunov times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, before investigating the long-time predictive capabilities of the model, we show the tracking of phase dynamics for 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2, we decouple the phase and pattern dynamics such that the time evolution of the phase only depends upon the pattern dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we take the 𝑑ℎ = 18 model and used it to train an ODE for the phase dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For training we repeat the process used for training 𝑔𝑁 𝑁 to train 𝑔𝜙 with the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Table 1 contains details on the NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 7a we show an example of the model phase evolution over 200 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In this example, the model follows the same downward drift in phase despite not matching exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, to show the statistical agreement between the DNS and the model, we show the mean squared phase displacement MSD = � (𝜃(𝑡𝑖) − 𝜃(𝑡𝑖 + 𝑡))2� for both the DNS and the model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 7b, as was done for Kolmogorov flow by Pérez De Jesús & Graham (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The curves are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' All of the remaining long-time statistics we report are phase invariant, so the remaining results use only models for the pattern dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 18 DManD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 <2(±)> / (+)()) DNS 16 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='8 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 0 20 40 60 80 100 0 20 40 60 80 100 4 tDNS 10-1 DNS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 DManD DManD 10-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 MSD 0 10-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='6 10-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='7 10-5 50 0 100 150 200 100 101 102 t t14 Figure 8: Fraction of unstable DManD models with standard neural ODEs and with stabilized neural ODEs at various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Long-time statistics Next we investigate the ability of the DManD model to capture the long-time dynamics of PCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' An obvious prerequisite for models to capture long-time dynamics is the long-time stability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2, the long-time trajectories of standard neural ODEs often become unstable, which led us to use stabilized neural ODEs with an explicit damping term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We quantify this observation by counting, of the 16 models trained at each dimension 𝑑ℎ, how many become unstable with and without the presence of an explicit damping term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' From our training data we know where ℎ should lie, so if it falls far outside this range after some time we can classify the model as unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In particular, we classify models as unstable if the norm of the final state is two times that of the maximum in our data (|| ˜ℎ(𝑇)|| > 2 max𝑡 ||ℎ(𝑡)||), after 𝑇 = 104 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In all of the unstable cases || ˜ℎ(𝑡)|| follows the data over some short time range before eventually growing indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 8 we show the number of unstable models with and without damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' With damping, all of the models are stable, whereas without damping almost all models become unstable for 𝑑ℎ = 5−16, and around half become unstable in the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Additionally, with longer runs or with different initial conditions, many of the models without damping labelled as stable here also eventually become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This lack of stability happens when inaccuracies in the neural ODE model pushes trajectories off the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Once off the attractor, the model is presented with states unlike the training data leading to further growth in this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Linot & Graham (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Linot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2023) we show more results highlighting this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' So, although some standard neural ODE models do provide reasonable statistics, using these models presents challenges due to this lack of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' As such, all other results we show use stabilized neural ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 8 indicates that stabilized neural ODEs predict ˜ℎ in a similar range to that of the data, it does not quantify the accuracy of these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In fact, with few dimensions many of these models do not remain chaotic, landing on fixed points or periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first metric we use to quantify the long-time performance of the DManD method is the mean-squared POD coefficient amplitudes ( � ||𝑎𝑛||2� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We consider this quantity because Gibson reports it for POD-Galerkin in Gibson (2002) at various levels of truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 9 we show how well the DManD model, with 𝑑ℎ = 18, captures this quantity, in comparison to the POD-Galerkin model in Gibson (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The two data sets slightly differ because we subtract the mean before applying POD and Gibson did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The DManD method, with only 18 degrees of freedom, matches the mean-squared amplitudes to high accuracy, far better than all of the POD-Galerkin models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' It is not until POD-Galerkin keeps 1024 modes that the results become comparable, which corresponds to ∼ 2000 degrees of freedom because most coefficients are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Additionally, our method requires only data, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='00 Standard Fraction Unstable Stabilized + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='00 XxX 10 20 30 4015 a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) Figure 9: Comparison of � ||𝑎𝑛||2� (mean-squared POD coefficient amplitudes) from the DNS to (a) � ||𝑎𝑛||2� from the DManD model at 𝑑ℎ = 18 and (b) � ||𝑎𝑛||2� from POD-Galerkin with 𝑁 POD modes (reproduced with permission from Gibson (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In (b), the quantity 𝜆 is equivalent to � ||𝑎𝑛||2� from the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figure 10: Components of the Reynolds stress with increasing dimension for DManD models at various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Odd numbers above 𝑑ℎ = 5 are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' whereas the POD Galerkin approach requires both data for computing the POD and knowledge of the equations of motion for projecting the equations onto these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We now investigate how the Reynolds stress and the power input vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' dissipation vary with dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figure 10 shows four components of the Reynolds stress at various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For � 𝑢′2 𝑥 � and � 𝑢′ 𝑥𝑢′ 𝑦 � , nearly all the models match the data, with relatively small deviations only appearing for 𝑑ℎ ∼ 3−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For � 𝑢′2 𝑦 � and � 𝑢′2 𝑧 � , this deviation becomes more obvious, and the lines do not converge until around 𝑑ℎ > 10, with all models above this dimension exhibiting a minor overprediction in � 𝑢′2 𝑧 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To evaluate how accurate the models are at reconstructing the energy balance, we look at joint PDFs of power input and dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The power input is the amount of energy required to move X10-3 X10-3 75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 18 50 雪 25 :14 0 0 ×10-3 X10-3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 6 DManD DNS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 4 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content="0 y y10- 10- N=16 DNS N=128 N=512 N=1024 10-2 DManD X 入 合 210-3 up 10-4 10~4 10-5 10~5/ 口 10-6 10~ 10' 102 100 100 101 102 n+1 n+l16 the walls: 𝐼 = 1 2𝐿𝑥𝐿𝑧 ∫ 𝐿𝑥 0 ∫ 𝐿𝑧 0 𝜕𝑢𝑥 𝜕𝑦 ���� 𝑦=−1 + 𝜕𝑢𝑥 𝜕𝑦 ���� 𝑦=1 𝑑𝑥𝑑𝑧, (3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='14) and the dissipation is the energy lost to heat due to viscosity: 𝐷 = 1 2𝐿𝑥𝐿𝑧 ∫ 𝐿𝑥 0 ∫ 1 −1 ∫ 𝐿𝑧 0 |∇ × u|2 𝑑𝑥𝑑𝑦𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='15) These two terms define the rate of change of energy in the system �𝐸 = 𝐼 − 𝐷, which must average to zero over long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Checking this statistic is important to show the DManD models correctly balance the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Figures 11a-11c show the PDF from the DNS, the PDF for 𝑑ℎ = 6 and the PDF for 𝑑ℎ = 18, generated from a single trajectory evolved for 5000 time units, and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 11e and 11f show the the absolute difference between the true and model PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' With 𝑑ℎ = 6 the model overestimates the number of low dissipation states, while 𝑑ℎ = 18 matches the density well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 11d we compare the joint PDFs at all dimension with the true PDF using the earth movers distance (EMD) (Rubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The EMD determines the distance between two PDFs as a solution to the transportation problem by treating the true PDF as “supplies" and the model PDF as “demands" and finding the flow which minimizes the work required to move one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We compute the distance between PDFs using the EMD because it is a cross-bin distance, meaning the distance accounts for the density in neighboring bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This is in contrast to bin-to-bin distances, like the KL divergence, which only uses the error at a given bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Bin-to-bin distances can vary significantly with small shifts in one PDF (misalignment) and when changing the number of bins used to generate the PDF (Ling & Okada 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We choose the EMD because it does not suffer from these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 11d we see a steep drop in the EMD at 𝑑ℎ = 5 and after 𝑑ℎ > 10 the joint PDFs are in excellent agreement with the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The dashed line corresponds to the EMD between two different trajectories from the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finding ECS in the model Now that we know that the DManD model quantitatively captures many of the key characteris- tics of MFU PCF, we now want to explore using the model to discover ECS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In particular, we first investigate the whether known periodic orbits of the DNS exist in the DManD model, and then we use the DManD model to search for new periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we note that because our model predicts phase-aligned dynamics, the periodic orbits of the DManD model are either periodic or relative periodic orbits, depending on the phase evolution, which we have not tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In the following we omit all ˜·, so all functions should be assumed to come from a DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Here we outline the approach we take to find periodic orbits, which follows Cvitanović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' When searching for periodic orbits we seek an initial condition to a trajectory that repeats after some time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This is equivalent to finding the zeros of 𝐻(ℎ,𝑇) = 𝐺𝑇 (ℎ) − ℎ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='16) where 𝐺𝑇 (ℎ) is the flow map forward 𝑇 time units from ℎ: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝐺𝑇 (ℎ(𝑡)) = ℎ(𝑡 +𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We compute 𝐺𝑇 (ℎ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finding zeros to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='16 requires that we find both a point ℎ∗ on the periodic orbit and a period 𝑇∗ such that 𝐻(ℎ∗,𝑇∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' One way to find ℎ∗ and 𝑇∗ is by using the Newton-Raphson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' By performing a Taylor series expansion of 𝐻 we find near the fixed point ℎ∗,𝑇∗ of 𝐻 that 𝐻(ℎ∗,𝑇∗) − 𝐻(ℎ,𝑇) ≈ 𝐷ℎ𝐻(ℎ,𝑇)Δℎ + 𝐷𝑇 𝐻(ℎ,𝑇)Δ𝑇 −𝐻(ℎ,𝑇) ≈ 𝐷ℎ𝐻(ℎ,𝑇)Δℎ + 𝑔 (𝐺𝑇 (ℎ)) Δ𝑇, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='17) where 𝐷ℎ is the Jacobian of 𝐻 with respect to ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 𝐷𝑇 is the Jacobian of 𝐻 with respect to the 17 a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) c)c)c)c)c)c)c)c)c)c)c)c)c)c)c)c)c) d) d) d) d) d) d) d) d) d) d) d) d) d) d) d) d) d) e)e)e)e)e)e)e)e)e)e)e)e)e)e)e)e)e) f)f)f)f)f)f)f)f)f)f)f)f)f)f)f)f)f) Figure 11: (a)-(c): examples of joint PDFs for the true system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' the DManD model at 𝑑ℎ = 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' and the DManD model at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (d): earth movers distance between the PDF from the DNS and the PDFs predict by the DManD model at various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' “DNS" is the error between two PDFs generated from DNS trajectories of the same length with different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (e) and (f): the error associated with the DManD model PDFs at 𝑑ℎ = 6 and 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' period 𝑇, Δℎ = ℎ∗ − ℎ and Δ𝑇 = 𝑇∗ − 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To have a complete set of equations for Δℎ and Δ𝑇, we supplement Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='17 with the constraint that the updates Δℎ are orthogonal to the vector field at ℎ: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=', 𝑔(ℎ)𝑇 Δℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='18) With this constraint, at Newton step (𝑖), the system of equations becomes � 𝐷ℎ(𝑖) 𝐻(ℎ(𝑖),𝑇 (𝑖)) 𝑔(𝐺𝑇 (𝑖) (ℎ(𝑖))) 𝑔(ℎ(𝑖))𝑇 0 � � Δℎ(𝑖) Δ𝑇 (𝑖) � = − � 𝐻(ℎ(𝑖),𝑇 (𝑖)) 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='19) which, in the standard Newton-Raphson method, is used to update the guesses ℎ(𝑖+1) = ℎ(𝑖) +Δℎ(𝑖) and 𝑇 (𝑖+1) = 𝑇 (𝑖) + Δ𝑇 (𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Typically, a Newton-Krylov method is used to avoid explicitly constructing the Jacobian (Viswanath 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' However, with DManD, computing the Jacobian is simple, fast, and requires little memory because the state representation is dramatically smaller in the DManD model than in the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We compute the Jacobian 𝐷ℎ𝐻(ℎ,𝑇) directly by using the same automatic differentiation tools used for training the neural ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Furthermore, if we had chosen to represent the dynamics in discrete, rather than continuous time, computation of general periodic orbits would not be possible, as the period 𝑇 can take on arbitrary values and a discrete-time representation would limit 𝑇 to multiples of the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' When finding periodic orbits of the DManD model we used the Scipy “hybr" method, which uses a modification of the Powell hybrid method (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2020), and for finding periodic orbits of the DNS we used the Newton GMRES-Hookstep method built into Channelflow (Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In the following trials we only consider DManD models with 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For the HKW cell there exists a library of POs made available by Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' To investigate if the DManD model finds POs similar to existing solutions, we took states from the known POs, encoded them, and used this as an initial condition in the DManD Newton solver to DNS dh = 6 dh = 18 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 P 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 I I 1 Error dh = 6 Error dh = 18 DManD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 DNS ~P 2 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 :1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 0 5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 dh I 118 Figure 12: Power input vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' dissipation of known periodic orbits (period reported in bottom right) from the DNS and periodic orbits found in the DManD model at 𝑑ℎ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The blue line is a long trajectory of the DNS for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' find POs in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 12 we show projections of 12 known POs, which we identify by the period 𝑇, and compare them to projections of POs found using the DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This makes up a majority of the POs made available by Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' (2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Of the other known solutions, three are RPOs with phase-shifts in the streamwise direction that our model, with the current setup, can not capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The other two have short periods of 𝑇 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='02 and 𝑇 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' A majority of the POs found with DManD land on initial conditions near that of the DNS and follow similar trajectories trajectories to the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' How close many of these trajectories are to the true PO is surprising and encouraging for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' First, the data used for training the DManD model does not explicitly contain any POs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Second, this approach by no means guarantees convergence on a PO in the DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Third, starting with an initial condition from a PO does not necessarily mean that the solution the Newton solver lands on will be the closest PO to that initial condition, so there may exist POs in the DManD model closer to the DNS solutions than what we present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Now that we know the DManD model can find POs similar to those known to exist for the DNS, we now use it to search for new POs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' First, we searched for POs in three of the 𝑑ℎ = 18 models by randomly selecting 20 initial conditions and selecting 4 different periods 𝑇 = [20, 40, 60, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' DNS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 DManD D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 T = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='13 T = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='07 T = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 T = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='82 T = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='85 T = 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 T = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='89 T = 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='90 T = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 T = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='52 T = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='70 T = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 1 I19 Table 2: Details on the RPOs and POs found using initial conditions from the DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The first 9 solutions are new and the last 3 had previously been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' “Label" indicates the label in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 13b, 𝜎𝑧 corresponds to the phase-shift in 𝑧, 𝑇 is the period of the orbit, and “Error" is ||shifted final state − initial state||/||initial state||, which is the same error as in Viswanath (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Label 1 2 3 4 5 6 7 8 9 10 11 12 𝜎𝑧 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='91e-1 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='66e-2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='77e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='15e-1 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='21e-3 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='90e-1 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='28e-2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='19e-1 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='63e-5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='64e-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='17e-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='73e-13 𝑇 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='94 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='29 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='07 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='14 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='24 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='67 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='31 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='64 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='06 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='07 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='35 Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='23e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='01e-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='92e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='84e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='87e-3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='26e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='25e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='13e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='25e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='57e-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='55e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='07e-4 a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) b) Figure 13: (a) Periodic orbits found in DManD models at 𝑑ℎ = 18 that converged to the (b) periodic orbits found in the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Each of the colors corresponds to a one of the new solutions in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The blue curve at the back is a long trajectory of the DNS for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We then took the initial conditions and periods for converged periodic orbits and decoded and upsampled them onto a 48 × 49 × 48 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We performed this upsampling because Viswanath (2007) reported that solutions on the coarser grid can be computational artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we put these new initial conditions into Channelflow and ran another Newton search for 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This procedure resulted in us finding 9 new RPOs and 3 existing POs, the details of which we include in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 13a we show the new RPOs in the DManD model and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 13b we show what they converged to after putting them into the Channelflow Newton solver as initial guesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Again, many of the RPOs end up following a similar path through this state space, with the biggest exceptions being RPO1 and RPO6, which converged to low-power input solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' It is worth noting that this worked well, considering that the DManD initial conditions are POD coefficients from a model trained using data on a coarser grid than used to search for these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We have described a new method to rapidly find new ECS, wherein an accurate low-dimensional model, like the DManD model presented here, is used to quickly perform a large number of ECS searches in the model and then these solutions can be fine tuned in the full simulation to land on new solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Conclusion In the present work we described a data-driven manifold dynamics method (DManD) and applied it for accurate modeling of MFU PCF with far fewer degrees of freedom (O(10)) than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 1 DManD Solution 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 D 5 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 7 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 D 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content='5 120 required for the DNS (O(105)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The DManD method consists of first finding a low-dimensional parameterization of the manifold on which data lies, and then discovering an ODE to evolve this low-dimensional state representation forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' In both cases we use NNs to approximate these functions from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' We find that an extremely low-dimensional parameterization of this manifold can be found using a hybrid autoencoder approach that corrects upon POD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Then, we use stabilized neural ODEs to accurately evolve the low-dimensional state forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The DManD model captures the self-sustaining process and accurately tracks trajectories and the temporal autocorrelation over short time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For DManD models with 𝑑ℎ > 10 we found excellent agreement between the model and the DNS in computing the mean-squared POD coefficient amplitude, the Reynolds stress, and the joint PDF of power input vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' For comparison, we showed that a POD-Galerkin model requires ∼ 2000 degrees of freedom to get similar performance in matching the mean-squared POD coefficient amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we used the DManD model at 𝑑ℎ = 18 for PO searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Using a set of existing POs, we successfully landed on nearby POs in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Finally, we found 9 previously undiscovered RPOs by first finding solutions in the DManD model and then using those as initial guesses to search in the full DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' The results reported here have both fundamental and technological importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' At the fundamental level they indicate that, the true dimension of the dynamics of a turbulent flow can be orders of magnitude smaller than the number of degrees of freedom required for a fully- resolved simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' Technologically this point is important because it may enable, for example, highly sophisticated model-based nonlinear control algorithms to be used: Determining the control strategy from the low-dimensional DManD model rather than a full-scale DNS, and applying it to the full flow will speed up both learning and implementing a control policy (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfpApV/content/2301.04638v1.pdf'} +page_content=' This work was supported by AFOSR FA9550-18-1-0174 and ONR N00014-18-1-2865 (Van- nevar Bush Faculty Fellowship).' metadata={'source': 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diff --git a/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/2301.13612v1.pdf.txt b/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/2301.13612v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b9e6b337bb1cada2a4ee7ca0dc5d6753cc4c81b --- /dev/null +++ b/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/2301.13612v1.pdf.txt @@ -0,0 +1,11231 @@ +On vector-valued functions and the +ε-product +Habilitationsschrift +vorgelegt am 31.01.2022 +der Technischen Universität Hamburg +von +Dr. rer. nat. Karsten Kruse, +geboren am 19.11.1984 in Papenburg. +Die Habilitationsschrift wurde in der Zeit von Juli 2020 bis Januar 2022 im +Institut für Mathematik der Technischen Universität Hamburg angefertigt. +arXiv:2301.13612v1 [math.FA] 31 Jan 2023 + +Gutachter: Prof. José Bonet +Prof. Dr. Leonhard Frerick +Prof. Dr. Thomas Kalmes +PD Dr. Christian Seifert +eingereicht: 31. Januar 2022; überarbeitet: 30. Januar 2023 +Tag des Habilitationskolloquiums: 01. Juli 2022 +DOI: 10.15480/882.4898 +ORCID: +0000-0003-1864-4915 +Creative Commons Lizenz: +Diese Arbeit steht unter der Creative Commons Lizenz Namensnennung 4.0 (CC +BY 4.0). Das bedeutet, dass sie vervielfältigt, verbreitet und öffentlich zugänglich +gemacht werden darf, auch kommerziell, sofern dabei stets der Urheber, die Quelle +des Textes und o. g. Lizenz genannt werden. Die genaue Formulierung der Lizenz +kann unter https://creativecommons.org/licenses/by/4.0/legalcode.de auf- +gerufen werden. + +Acknowledgement +It is quite hard to express how grateful I am to the people who helped, in one +way or the other, to finish this thesis which spans a part of my work between 2016 +and 2022. But I will give it a try. +First of all, I am deeply indebted to Marko Lindner and Christian Seifert who +always supported and encouraged me since I joined the TUHH in 2014 and gave +me a home so that I could work on the kind of mathematics I love. I know how +lucky I was to meet you both. +Second, I would like to thank the two people +who taught me probably the most I know about complex analysis and functional +analysis, namely, Andreas Defant and Michael Langenbruch (Oldenburg). Third, I +am utterly grateful to José Bonet and Enrique Jordá (Valencia) for many helpful +suggestions and comments, improving some of the papers this thesis is based on, +as well as enduring the quite abstract setting. +Let us come to the honorable mentions, ... just kidding. In 2015 I was lucky +again because Jan Meichsner joined the TUHH as a PhD student. Despite him +being a physicist and a dispraiser of green cabbage, it was a real pleasure to share +an office, do mathematics or just spend time with him. +Further, let me thank +Dennis Gallaun with whom I spent a lot of effort and gaffer tape setting up the +mobile e-assessment center at the TUHH between 2018 and 2020. +Apart from the people mentioned above I would like to thank my other mathe- +matical co-authors Hans Daduna, Ruslan Krenzler, Felix Schwenninger and Lin +Xie whose work is not physically present in this thesis but whose mathematical +influence or spirit probably is. Furthermore, I am thankful to the whole Institute +of Mathematics of the TUHH, in particular, the group of Applied Analysis for their +support. Moreover, I am grateful to the reviewers of this thesis Leonhard Frerick +(Trier) and Thomas Kalmes (Chemnitz) besides José Bonet and Christian Seifert, +and the anonymous reviewers of the papers it is based on for their work, helpful +comments and corrections. +Finally, I would like to thank my family for their continuous support and my +love Sonja for sharing my mathematical interests, bearing my kind of humour and +so much more which I cannot put into words. +These words only roughly express my gratitude and I hope that the minimum +that remains after reading the acknowledgement is the thought ‘At least, he gave +it a try.’ and a smile. +3 + + +Abstract +This habilitation thesis centres on linearisation of vector-valued functions which +means that vector-valued functions are represented by continuous linear operators. +The first question we face is which vector-valued functions may be represented +by continuous linear operators. +We study this problem in the framework of ε- +products and give sufficient conditions in Chapter 3 and 4 when a space F(Ω,E) +of vector-valued functions on a set Ω coincides (up to an isomorphism) with the +ε-product F(Ω)εE of a corresponding space of scalar-valued functions F(Ω) and +the codomain E which is usually an infinite-dimensional locally convex Hausdorff +space. The ε-product F(Ω)εE is a space of continuous linear operators from the +dual space F(Ω)′ to E. +Once we have a representation of a space F(Ω,E) of vector-valued functions +by an ε-product F(Ω)εE, we have access to the rich theory of continuous linear +operators which allows us to lift results that are known for the scalar-valued case +to the vector-valued case. The whole Chapter 5, which spans more than half of this +thesis, is dedicated to this lifting mechanism. But we should point out that this +is not only about transferring results from the scalar-valued to the vector-valued +case. The results in the vector-valued case encode additional information for the +scalar-valued case as well, e.g. we may deduce from the solvability of a linear partial +differential equation in the vector-valued case affirmative answers on the parameter +dependence of solutions in the scalar-valued case (see Section 5.1). +In Section 5.2 we give a unified approach to handle the problem of extending +functions with values in E, which have weak extensions in F(Ω), to functions in the +vector-valued counterpart F(Ω,E) of F(Ω). We present different extension the- +orems depending on the topological properties of the spaces F(Ω) and E. These +theorems also cover weak-strong principles. In particular, we study weak-strong +principles for continuously partially differentiable functions of finite order in Sec- +tion 5.3 and improve the well-known weak-strong principles of Grothendieck and +Schwartz. We use our results on the extension of vector-valued functions to de- +rive Blaschke’s convergence theorem for several spaces of vector-valued functions +and Wolff’s theorem for the description of dual spaces of several function spaces +F(Ω) in Section 5.4 and 5.5. Starting from the observation that every scalar-valued +holomorphic function has a local power series expansion and that this is still true +for holomorphic functions with values in E if E is locally complete, we develop a +machinery which is based on linearisation and Schauder decomposition to transfer +known series expansions from scalar-valued to vector-valued functions in Section +5.6. Especially, we apply this machinery to derive Fourier expansions for E-valued +Schwartz functions and C∞-smooth functions on Rd that are 2π-periodic in each +variable. The last section of Chapter 5 is devoted to the representation of spaces +F(Ω,E) of vector-valued functions by sequence spaces, which can be used to iden- +tify the coefficient spaces of the series expansions from the preceding section, if one +knows the coefficient space in the scalar-valued case. Furthermore, we give several +new conditions on the Pettis-integrability of vector-valued functions in Appendix +A.2, which are, for instance, needed for the Fourier expansions in Section 5.6. +5 + + +Kurzfassung +Im Mittelpunkt dieser Habilitationsschrift steht die Linearisierung vektorwer- +tiger Funktionen, d. h. vektorwertige Funktionen sollen durch stetige lineare Opera- +toren dargestellt werden. Die erste Frage, der man sich stellen muss, ist, welche +vektorwertigen Funktionen durch stetige lineare Operatoren dargestellt werden kön- +nen. +Wir untersuchen dieses Problem im Rahmen von ε-Produkten und geben +hinreichende Bedingungen in Kapitel 3 und 4 an, wann ein Raum F(Ω,E) von +vektorwertigen Funktionen auf einer Menge Ω mit dem ε-Produkt F(Ω)εE eines +entsprechenden Raums skalarwertiger Funktionen F(Ω) und des Wertebereichs E +(bis auf Isomorphie) übereinstimmt. +Hierbei ist E üblicherweise ein unendlich- +dimensionaler lokalkonvexer Hausdorff Raum. Das ε-Produkt F(Ω)εE ist ein Raum +stetiger linearer Operatoren, die vom Dualraum F(Ω)′ nach E abbilden. +Sobald wir eine Darstellung eines Raums F(Ω,E) von vektorwertigen Funk- +tionen durch ein ε-Produkt F(Ω)εE gewonnen haben, ist es uns möglich die reich- +haltige Theorie der stetigen linearen Operatoren zu nutzen, die es uns erlaubt, +Ergebnisse, die für den skalarwertigen Fall bekannt sind, auf den vektorwertigen +Fall zu übertragen. Das gesamte Kapitel 5, das mehr als die Hälfte dieser Arbeit +einnimmt, widmet sich diesem Übertragungsmechanismus. +Es sei jedoch darauf +hingewiesen, dass es hier nicht nur um die Übertragung von Ergebnissen aus dem +skalarwertigen auf den vektorwertigen Fall geht. Die Ergebnisse im vektorwertigen +Fall beinhalten auch zusätzliche Informationen für den skalarwertigen Fall, z. B. +können wir aus der Lösbarkeit einer linearen partiellen Differentialgleichung im +vektorwertigen Fall Antworten auf die Frage nach der Parameterabhängigkeit der +Lösungen im skalarwertigen Fall ableiten (siehe Abschnitt 5.1). +In Abschnitt 5.2 stellen wir einen einheitlichen Ansatz zur Lösung des Fort- +setzungsproblems von Funktionen mit Werten in E, die schwache Fortsetzungen in +F(Ω) haben, zu Funktionen im vektorwertigen Gegenstück F(Ω,E) von F(Ω) vor. +Wir präsentieren verschiedene Fortsetzungssätze in Abhängigkeit von den topologi- +schen Eigenschaften der Räume F(Ω) und E. Diese Sätze decken auch schwach- +stark Prinzipien ab. +Insbesondere untersuchen wir schwach-stark Prinzipien für +endlich oft stetig partiell differenzierbare Funktionen in Abschnitt 5.3 und verbes- +sern die bekannten schwach-starken Prinzipien von Grothendieck und Schwartz. +Zudem leiten wir von unseren Ergebnissen zur Fortsetzung vektorwertiger Funktio- +nen den Konvergenzsatz von Blaschke für diverse Räume vektorwertiger Funktionen +ab und übertragen den Satz von Wolff auf Dualräume mehrerer Funktionenräume +F(Ω) in den Abschnitten 5.4 und 5.5. +Ausgehend von der Beobachtung, dass +jede skalarwertige holomorphe Funktion eine lokale Potenzreihenentwicklung hat +und dass dies auch für holomorphe Funktionen mit Werten in E gilt, wenn E +lokal vollständig ist, entwickeln wir einen Mechanismus, der auf Linearisierung und +Schauder-Zerlegung basiert, um in Abschnitt 5.6 bekannte Reihenentwicklungen +von skalarwertigen auf vektorwertige Funktionen zu erweitern. Insbesondere wen- +den wir diesen Mechanismus an, um Fourier-Entwicklungen für E-wertige Schwartz- +Funktionen und C∞-glatte Funktionen auf Rd, die 2π-periodisch in jeder Variablen +sind, zu erhalten. +Der letzte Abschnitt von Kapitel 5 ist der Darstellung von +7 + +8 +KURZFASSUNG +Räumen F(Ω,E) vektorwertiger Funktionen durch Folgenräume gewidmet, was +man dazu nutzen kann, die Koeffizientenräume der Reihenentwicklungen aus dem +vorangegangenen Abschnitt zu bestimmen, sofern man den Koeffizientenraum im +skalarwertigen Fall kennt. Außerdem legen wir mehrere neue Bedingungen für die +Pettis-Integrierbarkeit von vektorwertigen Funktionen in Anhang A.2 dar, die z. B. +für die Fourier-Entwicklungen in Abschnitt 5.6 benötigt werden. + +Contents +Acknowledgement +3 +Abstract +5 +Kurzfassung +7 +Chapter 1. +Introduction +11 +Chapter 2. +Notation and preliminaries +17 +Chapter 3. +The ε-product for weighted function spaces +21 +3.1. +ε-into-compatibility +21 +3.2. +ε-compatibility +28 +Chapter 4. +Consistency +37 +4.1. +The spaces AP(Ω,E) and consistency +37 +4.2. +Further examples of ε-products +41 +4.3. +Riesz–Markov–Kakutani representation theorems +59 +Chapter 5. +Applications +67 +5.1. +Lifting the properties of maps from the scalar-valued case +67 +5.2. +Extension of vector-valued functions +74 +5.2.1. +Extension from thin sets +76 +5.2.2. +Extension from thick sets +92 +5.3. +Weak-strong principles for differentiability of finite order +104 +5.4. +Vector-valued Blaschke theorems +110 +5.5. +Wolff type results +114 +5.6. +Series representation of vector-valued functions via Schauder +decompositions +117 +5.6.1. +Schauder decomposition +120 +5.6.2. +Examples of Schauder decompositions +123 +5.7. +Representation by sequence spaces +130 +Appendices +Appendix A. +Compactness of closed absolutely convex hulls and Pettis- +integrals +139 +A.1. +Compactness of closed absolutely convex hulls +139 +A.2. +The Pettis-integral +142 +List of Symbols +147 +Index +151 +Bibliography +153 +9 + + +CHAPTER 1 +Introduction +This work is dedicated to a classical topic, namely, the linearisation of weighted +spaces of vector-valued functions. The setting we are interested in is the following. +Let F(Ω) be a locally convex Hausdorff space of functions from a non-empty set Ω +to a field K and E be a locally convex Hausdorff space over K. The ε-product of +F(Ω) and E is defined as the space of linear continuous operators +F(Ω)εE ∶= Le(F(Ω)′ +κ,E) +equipped with the topology of uniform convergence on equicontinuous subsets of +the dual F(Ω)′ which itself is equipped with the topology of uniform convergence +on absolutely convex compact subsets of F(Ω). Suppose that the point-evaluation +functionals δx, x ∈ Ω, belong to F(Ω)′ and that there is a locally convex Hausdorff +space F(Ω,E) of E-valued functions on Ω such that the map +S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], +(1) +is well-defined. The main question we want to answer reads as follows. When is +F(Ω)εE a linearisation of F(Ω,E), i.e. when is S an isomorphism? +In [15, 16, 17] Bierstedt treats the space CV(Ω,E) of continuous functions on +a completely regular Hausdorff space Ω weighted with a Nachbin-family V and its +topological subspace CV0(Ω,E) of functions that vanish at infinity in the weighted +topology. He derives sufficient conditions on Ω, V and E such that the answer +to our question is affirmative, i.e. S is an isomorphism. +Schwartz answers this +question for several weighted spaces of k-times continuously partially differentiable +functions on Ω = Rd like the Schwartz space in [158, 159] for quasi-complete E +with regard to vector-valued distributions. +Grothendieck treats the question in +[83], mainly for nuclear F(Ω) and complete E. In [99, 100, 101] Komatsu gives a +positive answer for ultradifferentiable functions of Beurling or Roumieu type and +sequentially complete E with regard to vector-valued ultradistributions. For the +space of k-times continuously partially differentiable functions on open subsets Ω +of infinite dimensional spaces equipped with the topology of uniform convergence +of all partial derivatives up to order k on compact subsets of Ω sufficient conditions +for an affirmative answer are deduced by Meise in [129]. For holomorphic functions +on open subsets of infinite dimensional spaces a positive answer is given in [52] +by Dineen. Bonet, Frerick and Jordá show in [30] that S is an isomorphism for +certain closed subsheaves of the sheaf C∞(Ω,E) of smooth functions on an open +subset Ω ⊂ Rd with the topology of uniform convergence of all partial derivatives +on compact subsets of Ω and locally complete E which, in particular, covers the +spaces of harmonic and holomorphic functions. +An important application of linearisation is within the field of partial differen- +tial equations. Let E be a linear space of functions on a set U and P(∂)∶C∞(Ω) → +C∞(Ω) a linear partial differential operator with C∞-smooth coefficients where +C∞(Ω) ∶= C∞(Ω,K). We call the elements of U parameters and say that a family +(fλ)λ∈U in C∞(Ω) depends on a parameter w.r.t. E if the map λ ↦ fλ(x) is an +element of E for every x ∈ Ω. The question of parameter dependence is whether for +11 + +12 +1. INTRODUCTION +every family (fλ)λ∈U in C∞(Ω) depending on a parameter w.r.t. E there is a family +(uλ)λ∈U in C∞(Ω) with the same kind of parameter dependence which solves the +partial differential equation +P(∂)uλ = fλ, +λ ∈ U. +In particular, it is the question of Ck-smooth (holomorphic, distributional, etc.) +parameter dependence if E is the space Ck(U) of k-times continuously partially +differentiable functions on an open set U ⊂ Rd (the space O(U) of holomorphic +functions on an open set U ⊂ C, the space of distributions D(V )′ on an open set +V ⊂ Rd where U = D(V ), etc.). The question of parameter dependence w.r.t. E has +an affirmative answer for several locally convex Hausdorff spaces E due to tensor +product techniques and splitting theory. Indeed, the answer is affirmative if the +topology of E is stronger than the topology of pointwise convergence on U and +P(∂)E∶C∞(Ω,E) → C∞(Ω,E) +is surjective where P(∂)E is the version of P(∂) for E-valued functions. The oper- +ator P(∂)E is surjective if its version P(∂) for scalar-valued functions is surjective, +for instance, if P(∂) is elliptic, and E is a Fréchet space. This is a consequence +of Grothendieck’s theory of tensor products [83], the nuclearity of C∞(Ω) and the +isomorphism C∞(Ω,E) ≅ C∞(Ω)εE for locally complete E. Thanks to the splitting +theory of Vogt for Fréchet spaces [173] and of Bonet and Domański for PLS-spaces +[54] we even have in case of an elliptic P(∂) that P(∂)E for d > 1 is surjective if +E ∶= F ′ +b where F is a Fréchet space satisfying the condition (DN) or if E is an ultra- +bornological PLS-space having the property (PA) since kerP(∂) has the property +(Ω). In particular, these three results cover the cases that E = Ck(U), O(U) or +D(V )′. Of course, this technique to answer the question of parameter dependence +is not restricted to linear partial differential operators or the space C∞(Ω). +Another application of linearisation lies in the problem of extending a vector- +valued function f∶Λ → E from a subset Λ ⊂ Ω to a locally convex Hausdorff space +E if the scalar-valued functions e′ ○ f are extendable for each continuous linear +functional e′ from certain linear subspaces G of E′ under the constraint of preserving +the properties, like holomorphy, of the scalar-valued extensions. This problem was +considered, among others, by Grothendieck [82, 83], Bierstedt [17], Gramsch [77], +Grosse-Erdmann [79, 81], Arendt and Nikolski [6, 7, 8], Bonet, Frerick, Jordá and +Wengenroth [30, 69, 70, 92, 93]. Even the simple case Λ = Ω and G = E′ is interesting +and an affirmative answer is called a weak-strong principle. +Our goal is to give a unified and flexible approach to linearisation which is able +to handle new examples and covers the already known examples. +Organisation of this thesis +After fixing some notions and preliminaries on locally convex Hausdorff spaces, +continuous linear operators and continuously partially differentiable functions in +Chapter 2, we study the problem of linearisation in Chapter 3. In Section 3.1 we +introduce our standard example of spaces F(Ω,E) that we consider. Namely, spaces +of functions FV(Ω,E) from Ω to E which are subspaces of sections of domains of +linear operators T E on EΩ, and whose topology is generated by a family of weight +functions V. These spaces cover many examples of classical spaces of functions +appearing in analysis like the mentioned ones and an example of the operators T E +are the partial derivative operators. Then we exploit the structure of our spaces +to describe a sufficient condition, which we call consistency, on the interplay of the +pairs of operators (T E,T K) and the map S such that S becomes an isomorphism +into, i.e. an isomorphism to its range (see Theorem 3.1.12). + +ORGANISATION OF THIS THESIS +13 +In Section 3.2 we tackle the problem of surjectivity of S. In our main Theorem +3.2.4 and its Corollary 3.2.5 we give several sufficient conditions on the pairs of +operators (T E,T K) and the spaces involved such that S∶FV(Ω)εE → FV(Ω,E) is +an isomorphism. Looking at the pair of partial differential operators (P(∂)E,P(∂)) +considered above, these conditions allow us to express P(∂)E as P(∂)E = S ○ +(P(∂)εidE) ○ S−1 where P(∂)εidE is the ε-product of P(∂) and the identity idE +on E. Hence it becomes obvious that the surjectivity of P(∂)E is equivalent to +the surjectivity of P(∂)εidE. This is used in [105, 109, 112, 116, 119] in the case +of the Cauchy–Riemann operator P(∂) = ∂ on spaces of smooth functions with +exponential growth. +In Chapter 4 we take a closer look at the notion of consistency of (T E,T K). In +Section 4.1 we characterise several properties of the functions S(u) for u ∈ FV(Ω)εE +that are inherited from the elements of FV(Ω). +Section 4.2 is devoted to several concrete examples of spaces of vector-valued +functions that may be linearised by S and which we use for our applications in the +forthcoming sections and chapters. +In Section 4.3 we answer in several cases the question whether given a con- +tinuous linear functional T K on F(Ω) there is always a continuous linear operator +T E on F(Ω,E) such that (T E,T K) is consistent. This is closely related to Riesz– +Markov–Kakutani theorems for T K, which we transfer to the vector-valued case. +Chapter 5 is dedicated to applications of linearisation. In Section 5.1 we come +back to our problem of parameter dependence. We show in our main Theorem 5.1.2 +of this section how to use linearisations to transfer properties like injectivity, sur- +jectivity or bijectivity from a map T K∶F1(Ω1) → F2(Ω2) to the corresponding map +T E∶F2(Ω1,E) → F2(Ω2,E) if the pair (T E,T K) is consistent under suitable as- +sumptions on the spaces involved. Besides the problem of parameter dependence +for (hypo)elliptic linear partial differential operators (see Corollary 5.1.3), we de- +duce a vector-valued version of the Borel–Ritt theorem (see Theorem 5.1.4) from +this main theorem and give sufficient conditions under which the Fourier transfor- +mation FC∶Sµ(Rd) → Sµ(Rd) on the Beurling–Björck space is still an isomorphism +in the vector-valued case and may be decomposed as FE = S ○ (FCεidE) ○ S−1 (see +Theorem 5.1.5). +In Section 5.2 we present a general approach to the extension problem consid- +ered above for a large class of function spaces F(Ω,E) if the map S is an isomor- +phism into. The spaces we treat are of the kind that F(Ω) belongs to the class of +semi-Montel, Fréchet–Schwartz or Banach spaces, or that E is a semi-Montel space. +Apart from linearisation and consistency, the main ingredient of this approach is to +view the set Λ ⊂ Ω from which we want to extend our functions as a set of function- +als {δx ∣ x ∈ Λ}. This view allows us to generalise the extension problem in Question +5.2.1 by swapping this set of functionals by other functionals, which opens up new +possibilities in applications that we explore in Section 5.3, Section 5.4, Section 5.5 +and Section 5.7. In the extension problem we always have to balance the sets Λ +from which we extend our functions and the subspaces G ⊂ E′ with which we test. +The case of ‘thin’ sets Λ and ‘thick’ subspaces G is handled in Section 5.2.1 with +main theorems Theorem 5.2.15, Theorem 5.2.20 and Theorem 5.2.29, the converse +case of ‘thick’ sets Λ and ‘thin’ subspaces G is handled in Section 5.2.2 with main +theorems Theorem 5.2.52, Theorem 5.2.63 and Theorem 5.2.69. +In Section 5.3 we consider weak-strong principles for continuously partially +differentiable functions of finite order. For locally complete E it is well-known that +a function f belongs to C∞(Ω,E) if and only if e′ ○ f ∈ C∞(Ω) for all e′ ∈ E′ (see +e.g. [30, Theorem 9, p. 232]). If k ∈ N0, then it is still true that f ∈ Ck(Ω,E) implies +e′ ○ f ∈ Ck(Ω) for all e′ ∈ E′. But the converse is not true anymore. Only a weaker + +14 +1. INTRODUCTION +version of this weak-strong principle holds which is due to Grothendieck [82] and +Schwartz [158] (see Theorem 5.3.2). Namely, if k ∈ N0, E is sequentially complete +and f∶Ω → E is such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E). Using +the results from Section 5.2, we improve this weaker version of the weak-strong +principle by allowing E to be locally complete, only testing with less functionals +from certain linear subspaces G ⊂ E′ and getting that f does not only belong to +Ck(Ω,E) but that all partial derivatives of order k are actually locally Lipschitz +continuous (see Corollary 5.3.5). If we restrict to semi-Montel spaces E, then even +a ‘full’ weak-strong principle Theorem 5.3.6 holds as in the C∞-case. +In Section 5.4 we derive vector-valued Blaschke theorems like Corollary 5.4.2 +for several function spaces. +This generalises results of Arendt and Nikolski [7] +for bounded holomorphic functions and Frerick, Jordá and Wengenroth [70] for +bounded functions in the kernel of a hypoelliptic linear partial differential operator. +These are results of the form: given a bounded net (fι)ι∈I in some space F1(Ω,E) +of Banach-valued functions which converges pointwise on a certain subset of Ω there +is a limit f ∈ F1(Ω,E) of this net w.r.t. a weaker topology of a linear superspace +F2(Ω,E) of F1(Ω,E). In Blaschke’s classical convergence theorem [38, Theorem +7.4, p. 219] we have E = C, F1(Ω,E) is the space of bounded holomorphic functions +on the open unit disc D ⊂ C, F2(Ω,E) is the space of holomorphic functions on D +and the weaker topology is the topology of compact convergence. +In Section 5.5 we present Wolff type descriptions of the dual space of several +function spaces F(Ω) using linearisation (see Theorem 5.5.1). Wolff’s theorem [183, +p. 1327] (cf. [81, Theorem (Wolff), p. 402]) phrased in a functional analytic way +(see [70, p. 240]) says: if Ω ⊂ C is a domain, then for each µ ∈ O(Ω)′ there are a +sequence (zn)n∈N which is relatively compact in Ω and a sequence (an)n∈N in the +space ℓ1 of absolutely summable sequences such that µ = ∑∞ +n=1 anδzn. +In Section 5.6 we derive a general result for Schauder decompositions of the +ε-product FεE for locally convex Hausdorff spaces F and E if F has an equicon- +tinuous Schauder basis (see Theorem 5.6.1). In combination with linearisation and +consistency this can be used for F = F(Ω) to lift series representations like the +power series expansion of holomorphic functions from scalar-valued functions to +vector-valued functions (see Corollary 5.6.5). We present several examples in Sec- +tion 5.6.2, for instance, Fourier expansions in the Schwartz space S(Rd,E) and in +the space C∞ +2π(Rd,E) of functions in C∞(Rd,E) that are 2π-periodic in each vari- +able. In particular, we combine these expansions for locally complete E with the +results from Section 5.1 to identify the coefficient spaces of the Fourier expansions +in S(Rd,E) and C∞ +2π(Rd,E) (see Theorem 5.6.13 and Theorem 5.6.14). +In Section 5.7 an application of our extension results from Section 5.2 is given to +represent function spaces F(Ω,E) by sequence spaces if one knows such a represen- +tation for F(Ω) (see Theorem 5.7.1). As examples we treat the space O(DR(0),E) +of E-valued holomorphic functions on the disc DR(0) ⊂ C around 0 with radius +0 < R ≤ ∞ and the multiplier space OM(R,E) of the Schwartz space for locally +complete E (see Corollary 5.7.2, Corollary 5.7.3 and Remark 5.7.4). +The first section Appendix A.1 of the Appendix A is devoted to the question +when the closure of an absolutely convex hull of a set is compact in a locally convex +Hausdorff space E and Appendix A.2 to the related question of Pettis-integrability +of an E-valued function. + +CONCERNING ORIGINALITY +15 +Concerning originality +We note that some parts of chapters or sections are based on our papers and +preprints. +● Chapter 3, Section 4.1 and Section 4.2 are based on our paper Weighted +spaces of vector-valued functions and the ε-product [110] and its extended +preprint [106]. Furthermore, Section 4.2 contains results from Sections 3 +and 6 of our accepted preprint Extension of weighted vector-valued func- +tions and sequence space representation [115] and our paper Extension of +weighted vector-valued functions and weak–strong principles for differen- +tiable functions of finite order [117] and its extended preprint [120]. +● Section 5.1 generalises some results of our papers Surjectivity of the ∂- +operator between weighted spaces of smooth vector-valued functions [116] +and Parameter dependence of solutions of the Cauchy–Riemann equation +on weighted spaces of smooth functions [112] and its extended preprint +[108]. +● Section 5.2, Section 5.3, Section 5.4, Section 5.5 and Section 5.7 are based +on our accepted preprint [115] and our paper [117] (and its extended +preprint [120]). +● Section 5.6 is based on our paper Series representations in spaces of vector- +valued functions via Schauder decompositions [114]. +Moreover, the introduction Chapter 1 and Chapter 2 on notation and pre- +liminaries are based on the corresponding sections in our papers and preprints +[106, 110, 112, 114, 115, 116, 117, 120]. However, not all of the results given in this +thesis are already contained in our preprints or papers. +In Chapter 3 the new, i.e. not contained in our preprints or papers, results are +Corollary 3.2.5 (ii), Example 3.2.7 e)+f), Example 3.2.9 and Corollary 3.2.10. +In Section 4.2 the new examples and results are Example 4.2.2, Corollary +4.2.3, Example 4.2.11, Example 4.2.13, Proposition 4.2.14, Example 4.2.16, Ex- +ample 4.2.22 which extends [107, Proposition 3.17 a), p. 244] of our paper The +approximation property for weighted spaces of differentiable function [107], Propo- +sition 4.2.25 which extends [114, Proposition 4.8, p. 370] from sequentially complete +E to locally complete E, Example 4.2.26 and Example 4.2.28 (ii). All the results +of Section 4.3 are new except for Definition 4.3.1 which is [115, 2.2 Definition, p. 4] +(and also not a result). +The main theorem of Section 5.1, Theorem 5.1.2, is new even though special +cases appeared in [112, 116]. Theorem 5.1.4 and Theorem 5.1.5 are new as well. +Corollary 5.4.3 extends [120, 7.3 Corollary, p. 22] from metric spaces with finite +diameter to arbitrary metric spaces. Theorem 5.6.13 and Theorem 5.6.14 b) extend +[114, Theorem 4.9, p. 371–372] and [114, Theorem 4.11, p. 375] from sequentially +complete E to locally complete E. Corollary 5.7.2 is new in the sense that there is +only a sketch how to prove it in [115, p. 31]. +The results of Appendix A are also new except for Proposition A.1.1, Propo- +sition A.1.6, which are contained in [106, 5.2 Proposition, p. 24] and [106, 3.13 +Lemma d), p. 10], and Lemma A.2.2 which is [114, Lemma 4.7, p. 369]. + + +CHAPTER 2 +Notation and preliminaries +Basics of topology +We equip the spaces Rd, d ∈ N, and C with the usual Euclidean norm ∣ ⋅ ∣, +denote by Br(x) ∶= {w ∈ Rd ∣ ∣w − x∣ < r} the ball around x ∈ Rd and by Dr(z) ∶= +{w ∈ C ∣ ∣w − z∣ < r} the disc around z ∈ C with radius r > 0. Furthermore, for a +subset M of a topological space (X,t) we denote the closure of M by M and the +boundary of M by ∂M. If we want to emphasize that we take the closure in X +resp. w.r.t. the topology t, then we write M +X resp. M +t. For a subset M of a vector +space X we denote by ch(M) the circled hull, by cx(M) the convex hull and by +acx(M) the absolutely convex hull of M. If X is a topological vector space, we +write acx(M) for the closure of acx(M) in X. +Locally convex Hausdorff spaces and continuous linear operators +By E we always denote a non-trivial, i.e. E ≠ {0}, locally convex Hausdorff +space over the field K = R or C equipped with a directed fundamental system of +seminorms (pα)α∈A and, in short, we write that E is an lcHs. If E = K, then we set +(pα)α∈A ∶= {∣ ⋅ ∣}. +By XΩ we denote the set of maps from a non-empty set Ω to a non-empty +set X, by χK we mean the characteristic function of K ⊂ Ω, by C(Ω,X) the space +of continuous functions from a topological space Ω to a topological space X, and +by C0(Ω,X) its subspace of continuous functions that vanish at infinity if X is a +locally convex Hausdorff space. +We denote by L(F,E) the space of continuous linear operators from F to E +where F and E are locally convex Hausdorff spaces. +If E = K, we just write +F ′ ∶= L(F,K) for the dual space and G○ for the polar set of G ⊂ F. If F and E +are linearly topologically isomorphic, we just write that F and E are isomorphic, +in symbols F ≅ E. We denote by Lt(F,E) the space L(F,E) equipped with the +locally convex topology t of uniform convergence on the finite subsets of F if t = σ, +on the absolutely convex, compact subsets of F if t = κ, on the absolutely convex, +σ(F,F ′)-compact subsets of F if t = τ, on the precompact (totally bounded) subsets +of F if t = γ and on the bounded subsets of F if t = b. We use the symbols t(F ′,F) +for the corresponding topology on F ′ and t(F) for the corresponding bornology on +F. We say that a subspace G ⊂ F ′ is separating (the points of F) if for every x ∈ F +it follows from y(x) = 0 for all y ∈ G that x = 0. Clearly, this is equivalent to G +being σ(F ′,F)-dense in F ′. For details and notions on the theory of locally convex +spaces not explained in this thesis see [68, 89, 131, 138]. +ε-products and tensor products +The so-called ε-product of Schwartz is defined by +FεE ∶= Le(F ′ +κ,E) +(2) +where L(F ′ +κ,E) is equipped with the topology of uniform convergence on equicon- +tinuous subsets of F ′. This definition of the ε-product coincides with the original +17 + +18 +2. NOTATION AND PRELIMINARIES +one by Schwartz [159, Chap. I, §1, Définition, p. 18]. It is symmetric which means +that FεE ≅ EεF. In the literature the definition of the ε-product is sometimes done +the other way around, i.e. EεF is defined by the right-hand side of (2) but due to the +symmetry these definitions are equivalent and for our purpose the given definition +is more suitable. If we replace F ′ +κ by F ′ +γ, we obtain Grothendieck’s definition of the +ε-product and we remark that the two ε-products coincide if F is quasi-complete +because then F ′ +γ = F ′ +κ holds. However, we stick to Schwartz’ definition. +For locally convex Hausdorff spaces Fi, Ei and Ti ∈ L(Fi,Ei), i = 1,2, we define +the ε-product T1εT2 ∈ L(F1εF2,E1εE2) of the operators T1 and T2 by +(T1εT2)(u) ∶= T2 ○ u ○ T t +1, +u ∈ F1εF2, +where T t +1∶E′ +1 → F ′ +1, e′ ↦ e′ ○ T1, is the dual map of T1. If T1 is an isomorphism +and F2 = E2, then T1εidE2 is also an isomorphism with inverse T −1 +1 εidE2 by [159, +Chap. I, §1, Proposition 1, p. 20] (or [89, 16.2.1 Proposition, p. 347] if the Fi are +complete). +As usual we consider the tensor product F ⊗E as a linear subspace of FεE for +two locally convex Hausdorff spaces F and E by means of the linear injection +Θ∶F ⊗ E → FεE, +k +∑ +n=1 +fn ⊗ en �→ [y ↦ +k +∑ +n=1 +y(fn)en]. +(3) +Via Θ the space F ⊗ E is identified with the space of operators with finite rank +in FεE and a locally convex topology is induced on F ⊗ E. We write F ⊗ε E for +F ⊗ E equipped with this topology and F ̂⊗εE for the completion of the injective +tensor product F ⊗εE. For more information on the theory of ε-products and tensor +products see [49, 89, 94]. +Several degrees of completeness +The sufficient conditions for surjectivity of the map S∶F(Ω)εE → F(Ω,E) +from the introduction, which we derive in the forthcoming, depend on assumptions +on different types of completeness of E. For this purpose we recapitulate some +definitions which are connected to completeness. We start with local completeness. +For a disk D ⊂ E, i.e. a bounded, absolutely convex set, the linear space ED ∶= +⋃n∈N nD becomes a normed space if it is equipped with the gauge functional of +D as a norm (see [89, p. 151]). The space E is called locally complete if ED is a +Banach space for every closed disk D ⊂ E (see [89, 10.2.1 Proposition, p. 197]). We +call a non-empty subset A of an lcHs E locally closed if every local limit point of +A belongs to A. Here, a point x ∈ E is called a local limit point of A if there is a +sequence (xn)n∈N in A that converges locally to x (see [138, Definition 5.1.14, p. +154–155]), i.e. there is a disk D ⊂ E such that (xn) converges to x in ED (see [138, +Definition 5.1.1, p. 151]). The local closure of a subset A of E is defined as the +smallest locally closed subset of E which contains A (see [138, Definition 5.1.18, p. +155]). Moreover, we note that every locally complete linear subspace of E is locally +closed and a locally closed linear subspace of a locally complete space is locally +complete by [138, Proposition 5.1.20 (i), p. 155]. +Moreover, a locally convex Hausdorff space is locally complete if and only if it +is convenient by [104, 2.14 Theorem, p. 20]. In particular, every complete locally +convex Hausdorff space is quasi-complete, every quasi-complete space is sequentially +complete and every sequentially complete space is locally complete and all these +implications are strict. The first two by [89, p. 58] and the third by [138, 5.1.8 +Corollary, p. 153] and [138, 5.1.12 Example, p. 154]. +Now, let us recall the following definition from [181, 9-2-8 Definition, p. 134] and +[175, p. 259]. A locally convex Hausdorff space is said to have the [metric] convex +compactness property ([metric] ccp) if the closure of the absolutely convex hull of + +VECTOR-VALUED CONTINUOUSLY PARTIALLY DIFFERENTIABLE FUNCTIONS +19 +every [metrisable] compact set is compact. Sometimes this condition is phrased +with the term convex hull instead of absolutely convex hull but these definitions +coincide. Indeed, the first definition implies the second since every convex hull of +a set A ⊂ E is contained in its absolutely convex hull. On the other hand, we have +acx(A) = cx(ch(A)) by [89, 6.1.4 Proposition, p. 103] and the circled hull ch(A) of a +[metrisable] compact set A is compact by [153, Chap. I, 5.2, p. 26] [and metrisable +by [34, Chap. IX, §2.10, Proposition 17, p. 159] since D × A is metrisable and +ch(A) = ME(D × A) where ME∶K × E → E is the continuous scalar multiplication +and D ∶= D1(0) the open unit disc], which yields the other implication. +In particular, every locally convex Hausdorff space with ccp has obviously met- +ric ccp, every quasi-complete locally convex Hausdorff space has ccp by [181, 9-2-10 +Example, p. 134], every sequentially complete locally convex Hausdorff space has +metric ccp by [23, A.1.7 Proposition (ii), p. 364] and every locally convex Hausdorff +space with metric ccp is locally complete by [175, Remark 4.1, p. 267]. All these +implications are strict. The second by [181, 9-2-10 Example, p. 134] and the others +by [175, Remark 4.1, p. 267]. For more details on the [metric] convex compactness +property and local completeness see [29, 175]. In addition, we remark that every +semi-Montel space is semi-reflexive by [89, 11.5.1 Proposition, p. 230] and every +semi-reflexive locally convex Hausdorff space is quasi-complete by [153, Chap. IV, +5.5, Corollary 1, p. 144] and these implications are strict as well. Summarizing, we +have the following diagram of strict implications: +semi-Montel ⇒ semi-reflexive +⇓ +complete ⇒ quasi-complete ⇒ sequentially complete ⇒ locally complete +⇓ +⇓ +�⇒ +ccp +⇒ +metric ccp +Vector-valued continuously partially differentiable functions +Since weighted spaces of continuously partially differentiable resp. holomorphic +vector-valued functions will serve as our standard examples, we recall the definition +of the spaces Ck(Ω,E) resp. O(Ω,E). A function f∶Ω → E on an open set Ω ⊂ Rd +to an lcHs E is called continuously partially differentiable (f is C1) if for the n-th +unit vector en ∈ Rd the limit +(∂en)Ef(x) ∶= +lim +h→0 +h∈R,h≠0 +f(x + hen) − f(x) +h +exists in E for every x ∈ Ω and (∂en)Ef is continuous on Ω ((∂en)Ef is C0) for +every 1 ≤ n ≤ d. For k ∈ N a function f is said to be k-times continuously partially +differentiable (f is Ck) if f is C1 and all its first partial derivatives are Ck−1. A +function f is called infinitely continuously partially differentiable (f is C∞) if f is +Ck for every k ∈ N. For k ∈ N∞ ∶= N∪{∞} the functions f∶Ω → E which are Ck form +a linear space which is denoted by Ck(Ω,E). For β ∈ Nd +0 with ∣β∣ ∶= ∑d +n=1 βn ≤ k +and a function f∶Ω → E on an open set Ω ⊂ Rd to an lcHs E we set (∂βn)Ef ∶= f +if βn = 0, and +(∂βn)Ef(x) ∶= (∂en)E⋯(∂en)E +������������������������������������������������������������������������������������������ +βn-times +f(x) +if βn ≠ 0 and the right-hand side exists in E for every x ∈ Ω. Further, we define +(∂β)Ef(x) ∶= ((∂β1)E⋯(∂βd)E)f(x) + +20 +2. NOTATION AND PRELIMINARIES +if the right-hand side exists in E for every x ∈ Ω. If E = K, we often just write +∂βf ∶= (∂β)Kf for β ∈ Nd +0, ∣β∣ ≤ k, and f ∈ Ck(Ω). Furthermore, we define the space +of bounded continuously partially differentiable functions by +C1 +b (Ω,E) ∶= {f ∈ C1(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣C1 +b (Ω),α ∶= +sup +x∈Ω +β∈Nd +0,∣β∣≤1 +pα((∂β)Ef(x)) < ∞}. +Vector-valued holomorphic functions +A function f∶Ω → E on an open set Ω ⊂ C to an lcHs E over C is called +holomorphic if the limit +(∂1 +C)Ef(z) ∶= +lim +h→0 +h∈C,h≠0 +f(z + h) − f(z) +h +, +z ∈ Ω, +exists in E. +We denote by O(Ω,E) the linear space of holomorphic functions +f∶Ω → E. Defining the vector-valued version of the Cauchy–Riemann operator by +∂ +Ef ∶= 1 +2((∂e1)E + i(∂e2)E)f +for f ∈ C(Ω,E) such that the partial derivatives (∂en)Ef, n = 1,2, exist in E, we +remark that +O(Ω,E) = {f ∈ C(Ω,E) ∣ f ∈ ker∂ +E} = {f ∈ C∞(Ω,E) ∣ f ∈ ker∂ +E} +(4) +by [113, Theorem 6.1, p. 267] if E is locally complete. Further, we set (∂0 +C)Ef ∶= f +and note that the (n + 1)-th complex derivative (∂n+1 +C +)Ef ∶= (∂1 +C)E((∂n +C)Ef) exists +for all n ∈ N0 and f ∈ O(Ω,E) by [79, 2.1 Theorem and Definition, p. 17–18] and +[79, 5.2 Theorem, p. 35] if E is locally complete. If E = C, we often just write +f (n) ∶= (∂n +C)Cf for n ∈ N0 and f ∈ O(Ω) ∶= O(Ω,C). We note that the real and +complex derivatives are related by +(∂β)Ef(z) = iβ2(∂∣β∣ +C )Ef(z), +z ∈ Ω, +(5) +for every f ∈ O(Ω,E) and β = (β1,β2) ∈ N2 +0 by [113, Proposition 7.1, p. 270] if E is +locally complete. + +CHAPTER 3 +The ε-product for weighted function spaces +3.1. ε-into-compatibility +In the introduction we already mentioned that linearisations of spaces of vector- +valued functions by means of ε-products are essential for our approach. +Here, +one of the important questions is which spaces of vector-valued functions can be +represented by ε-products. Let Ω be a non-empty set and E an lcHs. If F(Ω) ⊂ KΩ +is an lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω, then the map +S∶F(Ω)εE → EΩ, u �→ [x ↦ u(δx)], +is well-defined and linear. This leads to the following definition. +3.1.1. Definition (ε-into-compatible). Let Ω be a non-empty set and E an +lcHs. Let F(Ω) ⊂ KΩ and F(Ω,E) ⊂ EΩ be lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω. +We call the spaces F(Ω) and F(Ω,E) ε-into-compatible if the map +S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], +is a well-defined isomorphism into, i.e. to its range. We call F(Ω) and F(Ω,E) +ε-compatible if S is an isomorphism. We write SF(Ω) if we want to emphasise the +dependency on F(Ω). +In this section we introduce the weighted space FV(Ω,E) of E-valued functions +on Ω as a subspace of sections of domains in EΩ of linear operators T E +m equipped +with a generalised version of a weighted graph topology. This space is the role +model for many function spaces and an example for these operators are the partial +derivative operators. Then we treat the question whether FV(Ω,E) and FV(Ω)εE +are ε-into-compatible. This is deeply connected with the interplay of the pair of +operators (T E +m,T K +m) with the map S (see Definition 3.1.7). In our main theorem of +this section we give sufficient conditions such that S∶FV(Ω)εE → FV(Ω,E) is an +isomorphism into (see Theorem 3.1.12). In the next section we provide conditions +such that S becomes surjective (see Theorem 3.2.4). We start with the well-known +example Ck(Ω,E) of k-times continuously partially differentiable E-valued func- +tions to motivate our definition of FV(Ω,E). +3.1.2. Example. Let k ∈ N∞ and Ω ⊂ Rd be open. Consider the space C(Ω,E) +of continuous functions f∶Ω → E with the topology τc of compact convergence, i.e. +the topology given by the seminorms +∥f∥K,α ∶= sup +x∈K +pα(f(x)), +f ∈ C(Ω,E), +for compact K ⊂ Ω and α ∈ A. The usual topology on the space Ck(Ω,E) of k-times +continuously partially differentiable functions is the graph topology generated by +the partial derivative operators (∂β)E∶Ck(Ω,E) → C(Ω,E) for β ∈ Nd +0, ∣β∣ ≤ k, i.e. +the topology given by the seminorms +∥f∥K,β,α ∶= max(∥f∥K,α,∥(∂β)Ef∥K,α), +f ∈ Ck(Ω,E), +21 + +22 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +for compact K ⊂ Ω, β ∈ Nd +0, ∣β∣ ≤ k, and α ∈ A. The same topology is induced by +the directed system of seminorms given by +∣f∣K,m,α ∶= +sup +β∈Nd +0,∣β∣≤m +∥f∥K,β,α = +sup +x∈K +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x)), +f ∈ Ck(Ω,E), +for compact K ⊂ Ω, m ∈ N0, m ≤ k, and α ∈ A and may also be seen as a weighted +topology induced by the family (χK) of characteristic functions of the compact sets +K ⊂ Ω by writing +∣f∣K,m,α = +sup +x∈Ω +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x))χK(x), +f ∈ Ck(Ω,E). +This topology is inherited by linear subspaces of functions having additional prop- +erties like being holomorphic or harmonic. +We turn to the weight functions which we use to define a kind of weighted +graph topology. +3.1.3. Definition (weight function). Let J be a non-empty set and (ωm)m∈M +a family of non-empty sets. We call V ∶= (νj,m)j∈J,m∈M a family of weight functions +on (ωm)m∈M if it fulfils νj,m∶ωm → [0,∞) for all j ∈ J, m ∈ M and +∀ m ∈ M, x ∈ ωm ∃ j ∈ J ∶ 0 < νj,m(x). +(6) +From the structure of Example 3.1.2 we arrive at the following definition of the +weighted spaces of vector-valued functions we want to consider. +3.1.4. Definition. Let Ω be a non-empty set, V ∶= (νj,m)j∈J,m∈M a family +of weight functions on (ωm)m∈M and T E +m∶EΩ ⊃ domT E +m → Eωm a linear map for +every m ∈ M. Let AP(Ω,E) be a linear subspace of EΩ and define the space of +intersections +F(Ω,E) ∶= AP(Ω,E) ∩ ( ⋂ +m∈M +domT E +m) +as well as +FV(Ω,E) ∶= {f ∈ F(Ω,E) ∣ ∀ j ∈ J, m ∈ M, α ∈ A ∶ ∣f∣j,m,α < ∞} +where +∣f∣j,m,α ∶= sup +x∈ωm +pα(T E +m(f)(x))νj,m(x) = +sup +e∈Nj,m(f) +pα(e) +with +Nj,m(f) ∶= {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm}. +Further, we write F(Ω) ∶= F(Ω,K) and FV(Ω) ∶= FV(Ω,K). If we want to empha- +sise dependencies, we write M(E) instead of M, APFV(Ω,E) instead of AP(Ω,E) +and ∣f∣FV(Ω),j,m,α instead of ∣f∣j,m,α. If J, M or A are singletons, we omit the index +j, m resp. α in ∣f∣j,m,α. +Note that ωm need not be a subset of Ω. The space AP(Ω,E) is a placeholder +where we collect additional properties (AP) of our functions not being reflected by +the operators T E +m which we integrated in the topology. However, these additional +properties might come from being in the domain or kernel of additional operators, +e.g. harmonicity means being in the kernel of the Laplacian. But often AP(Ω,E) +can be chosen as EΩ or C(Ω,E). The space FV(Ω,E) is locally convex but need not +be Hausdorff. Since it is easier to work with Hausdorff spaces and a directed family +of seminorms plus the point evaluation functionals δx∶FV(Ω) → K, f ↦ f(x), for +x ∈ Ω and their continuity play a big role, we introduce the following definition. + +3.1. ε-INTO-COMPATIBILITY +23 +3.1.5. Definition (dom-space and T E +m,x). We call FV(Ω,E) a dom-space if it +is a locally convex Hausdorff space, the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A +is directed and, in addition, δx ∈ FV(Ω)′ for every x ∈ Ω if E = K. We define the +point evaluation of T E +m by T E +m,x∶domT E +m → E, T E +m,x(f) ∶= T E +m(f)(x), for m ∈ M and +x ∈ ωm. +3.1.6. Remark. +a) It is easy to see that FV(Ω,E) is Hausdorff if there +is m ∈ M such that ωm = Ω and T E +m = idEΩ since E is Hausdorff. +b) If E = K, then T K +m,x ∈ FV(Ω)′ for every m ∈ M and x ∈ ωm. Indeed, for +m ∈ M and x ∈ ωm there exists j ∈ J such that νj,m(x) > 0 by (6), implying +for every f ∈ FV(Ω) that +∣T K +m,x(f)∣ = +1 +νj,m(x)∣T K +m(f)(x)∣νj,m(x) ≤ +1 +νj,m(x)∣f∣j,m. +In particular, this implies δx ∈ FV(Ω)′ for all x ∈ Ω if there is m ∈ M such +that ωm = Ω and T K +m = idKΩ. +c) Let the family of weight functions V be directed, i.e. +∀ j1,j2 ∈ J,m1,m2 ∈ M ∃ j3 ∈ J, m3 ∈ M, C > 0 ∀ i ∈ {1,2} ∶ +(ωm1 ∪ ωm2) ⊂ ωm3 +and +νji,mi ≤ Cνj3,m3. +Then the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A is directed if V is +directed and additionally it holds with mi, i ∈ {1,2,3}, from above that +∀ f ∈ FV(Ω,E), i ∈ {1,2}, x ∈ ωmi ∶ T E +mi(f)(x) = T E +m3(f)(x), +since the system (pα)α∈A of E is already directed. +We point out that the additional condition in Remark 3.1.6 c) is missing in +[110, Remark 5 c), p. 1516] (resp. [106, 3.5 Remark, p. 6]), which we correct here. +For the lcHs E over K we want to define a natural E-valued version of a dom- +space FV(Ω) = FV(Ω,K). The natural E-valued version of FV(Ω) should be a +dom-space FV(Ω,E) such that there is a canonical relation between the families +(T K +m) and (T E +m). This canonical relation will be explained in terms of their interplay +with the map +S∶FV(Ω)εE → EΩ, u �→ [x ↦ u(δx)]. +Further, the elements of our E-valued version FV(Ω,E) of FV(Ω) should be com- +patible with a weak definition in the sense that e′ ○f ∈ FV(Ω) should hold for every +e′ ∈ E′ and f ∈ FV(Ω,E). +3.1.7. Definition (generator, consistent, strong). Let FV(Ω) and FV(Ω,E) +be dom-spaces such that M ∶= M(K) = M(E). +a) We call (T E +m,T K +m)m∈M a generator for (FV(Ω),E), in short, (FV,E). +b) We call (T E +m,T K +m)m∈M consistent if we have for all u ∈ FV(Ω)εE that +S(u) ∈ F(Ω,E) and +∀ m ∈ M, x ∈ ωm ∶ (T E +mS(u))(x) = u(T K +m,x). +c) We call (T E +m,T K +m)m∈M strong if we have for all e′ ∈ E′, f ∈ FV(Ω,E) that +e′ ○ f ∈ F(Ω) and +∀ m ∈ M, x ∈ ωm ∶ T K +m(e′ ○ f)(x) = (e′ ○ T E +m(f))(x). +More precisely, T K +m,x in b) means the restriction of T K +m,x to FV(Ω) and the +term u(T K +m,x) is well-defined by Remark 3.1.6 b). Consistency will guarantee that +the map S∶FV(Ω)εE → FV(Ω,E) is a well-defined isomorphism into, i.e. ε-into- +compatibility, and strength will help us to prove its surjectivity under some ad- +ditional assumptions on FV(Ω) and E. Let us come to a lemma which describes + +24 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +the topology of FV(Ω)εE in terms of the operators T K +m with m ∈ M. It was the +motivation for the definition of consistency and allows us to consider FV(Ω)εE as +a topological subspace of FV(Ω,E) via S, assuming consistency. +3.1.8. Lemma. Let FV(Ω) be a dom-space. Then the topology of FV(Ω)εE is +given by the system of seminorms defined by +∥u∥j,m,α ∶= sup +x∈ωm +pα(u(T K +m,x))νj,m(x), +u ∈ FV(Ω)εE, +for j ∈ J, m ∈ M and α ∈ A. +Proof. We define the sets Dj,m ∶= {T K +m,x(⋅)νj,m(x) ∣ x ∈ ωm} and Bj,m ∶= {f ∈ +FV(Ω) ∣ ∣f∣j,m ≤ 1} for every j ∈ J and m ∈ M. We claim that acx(Dj,m) is dense +in the polar B○ +j,m with respect to κ(FV(Ω)′,FV(Ω)). The observation +D○ +j,m = {T K +m,x(⋅)νj,m(x) ∣ x ∈ ωm}○ += {f ∈ FV(Ω) ∣ ∀x ∈ ωm ∶ ∣T K +m(f)(x)∣νj,m(x) ≤ 1} += {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} = Bj,m +yields +acx(Dj,m)κ(FV(Ω)′,FV(Ω)) = (Dj,m)○○ = B○ +j,m +by the bipolar theorem. By [89, 8.4, p. 152, 8.5, p. 156–157] the system of seminorms +defined by +qj,m,α(u) ∶= sup +y∈B○ +j,m +pα(u(y)), +u ∈ FV(Ω)εE, +for j ∈ J, m ∈ M and α ∈ A gives the topology on FV(Ω)εE (here it is used that +the system of seminorms (∣ ⋅ ∣j,m) of FV(Ω) is directed). As every u ∈ FV(Ω)εE +is continuous on B○ +j,m, we may replace B○ +j,m by a κ(FV(Ω)′,FV(Ω))-dense subset. +Therefore we obtain +qj,m,α(u) = sup{pα(u(y)) ∣ y ∈ acx(Dj,m)}. +For y ∈ acx(Dj,m) there are n ∈ N, λk ∈ K, xk ∈ ωm, 1 ≤ k ≤ n, with ∑n +k=1 ∣λk∣ ≤ 1 +such that y = ∑n +k=1 λkT K +m,xk(⋅)νj,m(xk). Then we have for every u ∈ FV(Ω)εE +pα(u(y)) ≤ +n +∑ +k=1 +∣λk∣pα(u(T K +m,xk))νj,m(xk) ≤ ∥u∥j,m,α, +thus qj,m,α(u) ≤ ∥u∥j,m,α. On the other hand, we derive +qj,m,α(u) ≥ sup +y∈Dj,m +pα(u(y)) = sup +x∈ωm +pα(u(T K +m,x))νj,m(x) = ∥u∥j,m,α. +□ +Let us turn to a more general version of Example 3.1.2, namely, to weighted +spaces of k-times continuously partially differentiable functions and kernels of linear +partial differential operators in these spaces. +3.1.9. Example. Let k ∈ N∞ and Ω ⊂ Rd be open. We consider the cases +(i) ωm ∶= Mm × Ω with Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0, or +(ii) ωm ∶= Nd +0 × Ω for all m ∈ N0 and k = ∞, +and let Vk ∶= (νj,m)j∈J,m∈N0 be a directed family of weights on (ωm)m∈N0. +a) We define the weighted space of k-times continuously partially differentiable +functions with values in an lcHs E as +CVk(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ j ∈ J, m ∈ N0, α ∈ A ∶ ∣f∣j,m,α < ∞} +where +∣f∣j,m,α ∶= +sup +(β,x)∈ωm +pα((∂β)Ef(x))νj,m(β,x). + +3.1. ε-INTO-COMPATIBILITY +25 +Setting domT E +m ∶= Ck(Ω,E) and +T E +m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)], +as well as AP(Ω,E) ∶= EΩ, we observe that CVk(Ω,E) is a dom-space by Remark +3.1.6 and +∣f∣j,m,α = sup +x∈ωm +pα(T E +mf(x))νj,m(x). +b) The space Ck(Ω,E) with its usual topology given in Example 3.1.2 is a special +case of a)(i) with J ∶= {K ⊂ Ω ∣ K compact}, νK,m(β,x) ∶= χK(x), (β,x) ∈ ωm, for +all m ∈ N0 and K ∈ J where χK is the characteristic function of K. In this case we +write Wk ∶= Vk for the family of weight functions. +c) The Schwartz space is defined by +S(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ m ∈ N0, α ∈ A ∶ ∣f∣m,α < ∞} +where +∣f∣m,α ∶= +sup +x∈Rd +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x))(1 + ∣x∣2)m/2. +This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶= {1} and ν1,m(β,x) ∶= +(1 + ∣x∣2)m/2, (β,x) ∈ ωm, for all m ∈ N0. +d) The multiplier space for the Schwartz space is defined by +OM(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ g ∈ S(Rd), m ∈ N0, α ∈ A ∶ ∥f∥g,m,α < ∞} +where +∥f∥g,m,α ∶= +sup +x∈Rd +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x))∣g(x)∣ +(see [158, 40), p. 97]). This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶= +{j ⊂ S(Rd) ∣ j finite} and νj,1,m(β,x) ∶= maxg∈j ∣g(x)∣, (β,x) ∈ ωm, for all m ∈ N0. +This choice of J guarantees that the family V∞ is directed and does not change the +topology. +e) Let K ∶= {K ⊂ Ω ∣ K compact} and (Mp)p∈N0 be a sequence of positive real +numbers. The space E(Mp)(Ω,E) of ultradifferentiable functions of class (Mp) of +Beurling-type is defined as +E(Mp)(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ K ∈ K, h > 0, α ∈ A ∶ ∣f∣(K,h),α < ∞} +where +∣f∣(K,h),α ∶= sup +x∈K +β∈Nd +0 +pα((∂β)Ef(x)) +1 +h∣β∣M∣β∣ +. +This is a special case of a)(ii) with J ∶= K × R>0 and ν(K,h),m(β,x) ∶= χK(x) +1 +h∣β∣M∣β∣ , +(β,x) ∈ ωm, for all (K,h) ∈ J and m ∈ N0 where R>0 ∶= (0,∞). +f) Let K and (Mp)p∈N0 be as in e). The space E{Mp}(Ω,E) of ultradifferentiable +functions of class {Mp} of Roumieu-type is defined as +E{Mp}(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ (K,H) ∈ J, α ∈ A ∶ ∣f∣(K,H),α < ∞} +where +J ∶= K × {H = (Hn)n∈N ∣ ∃ (hk)k∈N, hk > 0, hk ↗ ∞ ∀ n ∈ N ∶ Hn = h1 ⋅ ... ⋅ hn} +and +∣f∣(K,H),α ∶= sup +x∈K +β∈Nd +0 +pα((∂β)Ef(x)) +1 +H∣β∣M∣β∣ + +26 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +(see [101, Proposition 3.5, p. 675]). +Again, this is a special case of a)(ii) with +ν(K,H),m(β,x) ∶= χK(x) +1 +H∣β∣M∣β∣ , (β,x) ∈ ωm, for all (K,H) ∈ J and m ∈ N0. +g) Let n ∈ N, βi ∈ Nd +0 with ∣βi∣ ≤ k and ai∶Ω → K for 1 ≤ i ≤ n. We set +P(∂)E∶Ck(Ω,E) → EΩ, P(∂)E(f)(x) ∶= +n +∑ +i=1 +ai(x)(∂βi)E(f)(x) +and obtain the (topological) subspace of CVk(Ω,E) given by +CVk +P (∂)(Ω,E) ∶= {f ∈ CVk(Ω,E) ∣ f ∈ kerP(∂)E}. +Choosing AP(Ω,E) ∶= kerP(∂)E, we see that this is also a dom-space by a). If +P(∂)E is the Cauchy–Riemann operator (and E locally complete) or the Laplacian, +we obtain the weighted space of holomorphic resp. harmonic functions. +Let us show that the generators of these spaces are strong and consistent. In +order to obtain consistency for their generators we have to restrict to directed +families of weights which are locally bounded away from zero on Ω, i.e. +∀ K ⊂ Ω compact, m ∈ N0 ∃ j ∈ J ∀ β ∈ Nd +0, ∣β∣ ≤ min(m,k) ∶ inf +x∈K νj,m(β,x) > 0. +This condition on Vk guarantees that the map I∶CVk(Ω) → CWk(Ω), f ↦ f, is +continuous which is needed for consistency. +3.1.10. Proposition. Let E be an lcHs, k ∈ N∞, Vk be a directed family of +weights which is locally bounded away from zero on an open set Ω ⊂ Rd. +The +generator of (CVk,E) resp. (CVk +P (∂),E) from Example 3.1.9 is strong and consistent +if CVk(Ω) resp. CVk +P (∂)(Ω) is barrelled. +Proof. We recall the definitions from Example 3.1.9. We have ωm ∶= Mm × Ω +with Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0 or ωm ∶= Nd +0 × Ω for all m ∈ N0. +Further, APCVk(Ω,E) = EΩ, APCVk +P (∂)(Ω,E) = kerP(∂)E, domT E +m ∶= Ck(Ω,E) and +T E +m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)], +for all m ∈ N0 and the same with K instead of E. The family (T E +m,T K +m)m∈N0 is a +strong generator for (CVk,E) because +(∂β)K(e′ ○ f)(x) = e′((∂β)Ef(x)), +(β,x) ∈ ωm, +for all e′ ∈ E′, f ∈ CVk(Ω,E) and m ∈ N0 due to the linearity and continuity of +e′ ∈ E′. In addition, e′ ○ f ∈ kerP(∂)K for all e′ ∈ E′ and f ∈ CVk +P (∂)(Ω,E), which +implies that (T E +m,T K +m)m∈N0 is also a strong generator for (CVk +P (∂),E). +For consistency we need to prove that +(∂β)ES(u)(x) = u(δx ○ (∂β)K), +(β,x) ∈ ωm, +for all u ∈ CVk(Ω)εE resp. u ∈ CVk +P (∂)(Ω)εE. This follows from the subsequent +Proposition 3.1.11 b) since FV(Ω) = CVk(Ω) resp. FV(Ω) = CVk +P (∂)(Ω) is barrelled +and Vk locally bounded away from zero on Ω. Thus (T E +m,T K +m)m∈N0 is a consistent +generator for (CVk,E). In addition, we have with P(∂)E from Example 3.1.9 g) +that +P(∂)E(S(u))(x) = +n +∑ +i=1 +ai(x)(∂βi)E(S(u))(x) = u( +n +∑ +i=1 +ai(x)(δx ○ (∂βi)K)) += u(δx ○ P(∂)K) = 0, +x ∈ Ω, +(7) +for every u ∈ CVk +P (∂)(Ω)εE. This yields S(u) ∈ kerP(∂)E for all u ∈ CVk +P (∂)(Ω)εE. +Therefore (T E +m,T K +m)m∈N0 is a consistent generator for (CVk +P (∂),E) as well. +□ + +3.1. ε-INTO-COMPATIBILITY +27 +Let us turn to the postponed part in the proof of consistency. We denote by +CW(Ω) the space of scalar-valued continuous functions on an open set Ω ⊂ Rd +with the topology of uniform convergence on compact subsets, i.e. the weighted +topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and we +set δ(x) ∶= δx for x ∈ Ω. +3.1.11. Proposition. Let Ω ⊂ Rd be open, k ∈ N∞ and FV(Ω) a dom-space. +a) If T ∈ L(FV(Ω),CW(Ω)), then δ ○ T ∈ C(Ω,FV(Ω)′ +γ). +b) If T ∈ L(FV(Ω),CW1(Ω)) and FV(Ω) is barrelled, then +(∂en)FV(Ω)′ +κ(δ ○ T)(x) = lim +h→0 +δx+hen ○ T − δx ○ T +h += δx ○ (∂en)K ○ T, +x ∈ Ω, 1 ≤ n ≤ d, +and δ ○ T ∈ C1(Ω,FV(Ω)′ +κ). +c) If the inclusion I∶FV(Ω) → CWk(Ω), f ↦ f, is continuous and FV(Ω) +barrelled, then S(u) ∈ Ck(Ω,E) and +(∂β)ES(u)(x) = u(δx ○ (∂β)K), +β ∈ Nd +0, ∣β∣ ≤ k, x ∈ Ω, +for all u ∈ FV(Ω)εE. +Proof. a) First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we +observe that +(δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f) +for every f ∈ FV(Ω) as T(f) is continuous on Ω. Second, let K ⊂ Ω be compact. +Then there are j ∈ J, m ∈ M and C > 0 such that +sup +x∈K +∣(δx ○ T)(f)∣ = sup +x∈K +∣T(f)(x)∣ ≤ C∣f∣j,m +for every f ∈ FV(Ω). +This means that {δx ○ T ∣ x ∈ K} is equicontinuous in +FV(Ω)′. The topologies σ(FV(Ω)′,FV(Ω)) and γ(FV(Ω)′,FV(Ω)) coincide on +equicontinuous subsets of FV(Ω)′, implying that the restriction (δ ○ T)∣K∶K → +FV(Ω)′ +γ is continuous by our first observation. As δ ○ T is continuous on every +compact subset of the open set Ω ⊂ Rd, it follows that δ ○ T∶Ω → FV(Ω)′ +γ is well- +defined and continuous. +b) Let x ∈ Ω and 1 ≤ n ≤ d. Then there is ε > 0 such that x + hen ∈ Ω for all +h ∈ R with 0 < ∣h∣ < ε. We note that δ ○ T ∈ C(Ω,FV(Ω)′ +κ) by part a), which implies +δx+hen○T −δx○T +h +∈ FV(Ω)′. For every f ∈ FV(Ω) we have +lim +h→0 +δx+hen ○ T − δx ○ T +h +(f) = lim +h→0 +T(f)(x + hen) − T(f)(x) +h += (∂en)KT(f)(x) +in K as T(f) ∈ C1(Ω). Therefore 1 +h(δx+hen ○ T − δx ○ T) converges to δx ○ (∂en)K ○ T +in FV(Ω)′ +σ and thus in FV(Ω)′ +κ by the Banach–Steinhaus theorem as well. +In +particular, we obtain +δx ○ (∂en)K ○ T = lim +h→0 +δx+hen ○ T − δx ○ T +h += (∂en)FV(Ω)′ +κ(δ ○ T)(x) +in FV(Ω)′ +κ. Moreover, δ ○ (∂en)K ○ T ∈ C(Ω,FV(Ω)′ +κ) by part a) as (∂en)K ○ T ∈ +L(FV(Ω),CW(Ω)). Hence we deduce that δ ○ T ∈ C1(Ω,FV(Ω)′ +κ). +c) We prove our claim by induction on the order of differentiation. Let u ∈ +FV(Ω)εE. For β ∈ Nd +0 with ∣β∣ = 0 we get S(u) = u ○ δ ∈ C(Ω,E) from part a) with +T = I. Further, +(∂β)ES(u)(x) = S(u)(x) = u(δx) = u(δx ○ (∂β)K), +x ∈ Ω. + +28 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +Let m ∈ N0, m < k, such that S(u) ∈ Cm(Ω,E) and +(∂β)ES(u)(x) = u(δx ○ (∂β)K), +x ∈ Ω, +(8) +for all β ∈ Nd +0 with ∣β∣ ≤ m. +Let β ∈ Nd +0 with ∣β∣ = m + 1 ≤ k. +Then there is +1 ≤ n ≤ d and ̃β ∈ Nd +0 with ∣̃β∣ = m such that β = en + ̃β. The barrelledness of +FV(Ω) yields that 1 +h(δx+hen ○ (∂ +̃β)K − δx ○ (∂ +̃β)K) converges to δx ○ (∂en)K ○ (∂ +̃β)K +in FV(Ω)′ +κ for every x ∈ Ω by part b) with T ∶= (∂ +̃β)K. Therefore we derive from +δx ○ (∂en)K ○ (∂ +̃β)K = δx ○ (∂β)K by Schwarz’ theorem that +u(δx ○ (∂β)K) = lim +h→0 +1 +h(u(δx+hen ○ (∂ +̃β)K) − u(δx ○ (∂ +̃β)K)) += +(8) lim +h→0 +1 +h((∂ +̃β)ES(u)(x + hen) − (∂ +̃β)ES(u)(x)) += (∂en)E(∂ +̃β)ES(u)(x) +for every x ∈ Ω. Moreover, δ ○ (∂β)K = (∂en)FV(Ω)′ +κ(δ ○ T) ∈ C(Ω,FV(Ω)′ +κ) for +T = (∂ +̃β)K by part b). +Hence we have S(u) ∈ Cm+1(Ω,E) and it follows from +Schwarz’ theorem again that +u(δx ○ (∂β)K) = (∂en)E(∂ +̃β)ES(u)(x) = (∂β)ES(u)(x), +x ∈ Ω. +□ +Part a) of the preceding proposition is just a modification of [16, 4.1 Lemma, p. +198], where FV(Ω) = CV(Ω) is the Nachbin-weighted space of continuous functions +and T = id, and holds more general for kR-spaces Ω (see Lemma 4.1.2). +3.1.12. Theorem. Let (T E +m,T K +m)m∈M be a consistent generator for (FV,E). +Then the map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism into, i.e. the spaces +FV(Ω) and FV(Ω,E) are ε-into-compatible. +Proof. First, we show that S(FV(Ω)εE) ⊂ FV(Ω,E). Let u ∈ FV(Ω)εE. +Due to the consistency of (T E +m,T K +m)m∈M we have S(u) ∈ AP(Ω,E) ∩ domT E +m and +(T E +mS(u))(x) = u(T K +m,x), +m ∈ M x ∈ ωm. +Furthermore, we get by Lemma 3.1.8 for every j ∈ J, m ∈ M and α ∈ A +∣S(u)∣j,m,α = sup +x∈ωm +pα(T E +m(S(u))(x))νj,m(x) = ∥u∥j,m,α < ∞, +(9) +implying S(u) ∈ FV(Ω,E) and the continuity of S. Moreover, we deduce from (9) +that S is injective and that the inverse of S on the range of S is also continuous. +□ +3.1.13. Remark. If J, M and A are countable, then S is an isometry with +respect to the induced metrics on FV(Ω,E) and FV(Ω)εE by (9). +The basic idea for Theorem 3.1.12 was derived from analysing the proof of an +analogous statement for Bierstedt’s weighted spaces CV(Ω,E) and CV0(Ω,E) of +continuous functions already mentioned in the introduction (see [16, 4.2 Lemma, +4.3 Folgerung, p. 199–200] and [17, 2.1 Satz, p. 137]). +3.2. ε-compatibility +Now, we try to answer the natural question. When is S surjective? The strength +of a generator and a weaker concept to define a natural E-valued version of FV(Ω) +come into play to answer the question on the surjectivity of our key map S. Let +FV(Ω) be a dom-space. We define the linear space of E-valued weak FV-functions +by +FV(Ω,E)σ ∶= {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ FV(Ω)}. + +3.2. ε-COMPATIBILITY +29 +Moreover, for f ∈ FV(Ω,E)σ we define the linear map +Rf∶E′ → FV(Ω), Rf(e′) ∶= e′ ○ f, +and the dual map +Rt +f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], +where E′⋆ is the algebraic dual of E′. Furthermore, we set +FV(Ω,E)κ ∶= {f ∈ FV(Ω,E)σ ∣ ∀ α ∈ A ∶ Rf(B○ +α) relatively compact in FV(Ω)} +where Bα ∶= {x ∈ E ∣ pα(x) < 1} for α ∈ A. Next, we give a sufficient condition for +the inclusion FV(Ω,E) ⊂ FV(Ω,E)σ by means of the family (T E +m,T K +m)m∈M. +3.2.1. Lemma. If (T E +m,T K +m)m∈M is a strong generator for (FV,E), then we have +FV(Ω,E) ⊂ FV(Ω,E)σ and +sup +e′∈B○α +∣Rf(e′)∣j,m = ∣f∣j,m,α +(10) +for every f ∈ FV(Ω,E), j ∈ J, m ∈ M and α ∈ A. +Proof. Let f ∈ FV(Ω,E). +We have e′ ○ f ∈ F(Ω) for every e′ ∈ E′ since +(T E +m,T K +m)m∈M is a strong generator. Moreover, we have +∣Rf(e′)∣j,m = ∣e′ ○ f∣j,m = sup +x∈ωm +∣T K +m(e′ ○ f)(x)∣νj,m(x) += sup +x∈ωm +∣e′(T E +m(f)(x))∣νj,m(x) = +sup +x∈Nj,m(f) +∣e′(x)∣ +(11) +for every j ∈ J and m ∈ M with the set Nj,m(f) from Definition 3.1.4. We note +that Nj,m(f) is bounded in E by Definition 3.1.4 and thus weakly bounded, im- +plying that the right-hand side of (11) is finite. Hence we conclude f ∈ FV(Ω,E)σ. +Further, we observe that +sup +e′∈B○α +∣Rf(e′)∣j,m = ∣f∣j,m,α +for every j ∈ J, m ∈ M and α ∈ A due to [131, Proposition 22.14, p. 256]. +□ +Now, we phrase some sufficient conditions for FV(Ω,E) ⊂ FV(Ω,E)κ to hold +which is one of the key points regarding the surjectivity of S. +3.2.2. Lemma. If (T E +m,T K +m)m∈M is a strong generator for (FV,E) and one of +the following conditions is fulfilled, then FV(Ω,E) ⊂ FV(Ω,E)κ. +a) FV(Ω) is a semi-Montel space. +b) E is a semi-Montel or Schwartz space. +c) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ γ(E) ∶ Nj,m(f) ⊂ K. +Proof. Let f ∈ FV(Ω,E). By virtue of Lemma 3.2.1 we already have f ∈ +FV(Ω,E)σ. +a) For every j ∈ J, m ∈ M and α ∈ A we derive from +sup +e′∈B○α +∣Rf(e′)∣j,m = +(10) ∣f∣j,m,α < ∞ +that Rf(B○ +α) is bounded and thus relatively compact in the semi-Montel space +FV(Ω). +c) It follows from (11) that Rf ∈ L(E′ +γ,FV(Ω)). +Further, the polar B○ +α is +relatively compact in E′ +γ for every α ∈ A by the Alaoğlu–Bourbaki theorem. The +continuity of Rf implies that Rf(B○ +α) is relatively compact as well. +b) Let j ∈ J and m ∈ M. The set K ∶= Nj,m(f) is bounded in E by Definition +3.1.4. We deduce that K is already precompact in E by [89, 10.4.3 Corollary, p. +202] if E is a Schwartz space resp. since it is relatively compact if E is a semi-Montel +space. Hence the statement follows from c). +□ + +30 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +Let us turn to sufficient conditions for FV(Ω,E) ≅ FV(Ω)εE. For the lcHs E +we denote by J ∶E → E′⋆, x �→ [e′ ↦ e′(x)], the canonical injection. +3.2.3. Condition. Let (T E +m,T K +m)m∈M be a strong generator for (FV,E). Define +the following conditions: +a) E is complete. +b) E is quasi-complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there is +a bounded net (f ′ +τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′ +κ such that +Rt +f(f ′ +τ) ∈ J (E) for every τ ∈ T . +c) E is sequentially complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ +there is a sequence (f ′ +n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′ +κ such +that Rt +f(f ′ +n) ∈ J (E) for every n ∈ N. +d) E is locally complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there +is a sequence (f ′ +n)n∈N in FV(Ω)′ locally converging to f ′ in FV(Ω)′ +κ such +that Rt +f(f ′ +n) ∈ J (E) for every n ∈ N. +e) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ τ(E) ∶ Nj,m(f) ⊂ K. +3.2.4. Theorem. Let (T E +m,T K +m)m∈M be a consistent generator for (FV,E) and +let FV(Ω,E) ⊂ FV(Ω,E)κ. If one of the Conditions 3.2.3 is fulfilled, then the +map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism, i.e. FV(Ω) and FV(Ω,E) are +ε-compatible. The inverse of S is given by the map +Rt∶FV(Ω,E) → FV(Ω)εE, f ↦ J −1 ○ Rt +f, +where J ∶E → E′⋆ is the canonical injection and +Rt +f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], +with Rf(e′) = e′ ○ f. +Proof. Due to Theorem 3.1.12 we only have to show that S is surjective. We +equip J (E) with the system of seminorms given by +pB○α(J (x)) ∶= sup +e′∈B○α +∣J (x)(e′)∣ = pα(x), +x ∈ E, +(12) +for every α ∈ A. Let f ∈ FV(Ω,E). We consider the dual map Rt +f and claim that +Rt +f ∈ L(FV(Ω)′ +κ,J (E)). Indeed, we have +pB○α(Rt +f(y)) = sup +e′∈B○α +∣y(Rf(e′))∣ = +sup +x∈Rf (B○α) +∣y(x)∣ ≤ sup +x∈Kα +∣y(x)∣ +(13) +for all y ∈ FV(Ω)′ where Kα ∶= Rf(B○α). Since FV(Ω,E) ⊂ FV(Ω,E)κ, the set +Rf(B○ +α) is absolutely convex and relatively compact, implying that Kα is absolutely +convex and compact in FV(Ω) by [89, 6.2.1 Proposition, p. 103]. Further, we have +for all e′ ∈ E′ and x ∈ Ω +Rt +f(δx)(e′) = δx(e′ ○ f) = e′(f(x)) = J (f(x))(e′) +(14) +and thus Rt +f(δx) ∈ J (E). +a) Let E be complete and f ′ ∈ FV(Ω)′. Since the span of {δx ∣ x ∈ Ω} is dense +in F(Ω)′ +κ by the bipolar theorem, there is a net (f ′ +τ) converging to f ′ in FV(Ω)′ +κ +with Rt +f(f ′ +τ) ∈ J (E) by (14). As +pB○α(Rt +f(f ′ +τ) − Rt +f(f ′)) ≤ +(13) sup +x∈Kα +∣(f ′ +τ − f ′)(x)∣ → 0, +(15) +for all α ∈ A, we gain that (Rt +f(f ′ +τ)) is a Cauchy net in the complete space J (E). +Hence it has a limit g ∈ J (E) which coincides with Rt +f(f ′) since +pB○α(g − Rt +f(f ′)) ≤ pB○α(g − Rt +f(f ′ +τ)) + pB○α(Rt +f(f ′ +τ) − Rt +f(f ′)) + +3.2. ε-COMPATIBILITY +31 +≤ +(15)pB○α(g − Rt +f(f ′ +τ)) + sup +x∈Kα +∣(f ′ +τ − f ′)(x)∣ → 0 +for all α ∈ A. We conclude that Rt +f(f ′) ∈ J (E) for every f ′ ∈ FV(Ω)′. +b) Let Condition 3.2.3 b) hold and f ′ ∈ FV(Ω)′. Then there is a bounded +net (f ′ +τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′ +κ such that Rt +f(f ′ +τ) ∈ J (E) for +every τ ∈ T . Due to (13) we obtain that (Rt +f(f ′ +τ)) is a bounded Cauchy net in the +quasi-complete space J (E) converging to Rt +f(f ′) ∈ J (E). +c) Let Condition 3.2.3 c) hold and f ′ ∈ FV(Ω)′. +Then there is a sequence +(f ′ +n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′ +κ such that Rt +f(f ′ +n) ∈ J (E) for every +n ∈ N. Again (13) implies that (Rt +f(f ′ +n)) is a Cauchy sequence in the sequentially +complete space J (E) which converges to Rt +f(f ′) ∈ J (E). +d) Let Condition 3.2.3 d) hold and f ′ ∈ FV(Ω)′. Then there is an absolutely +convex, bounded subset D ⊂ FV(Ω)′ +κ and a sequence (f ′ +n)n∈N in FV(Ω)′ converging +to f ′ in (FV(Ω)′ +κ)D such that Rt +f(fn) ∈ J (E) for every n ∈ N. Let r > 0 and +f ′ +n − f ′ +k ∈ rD. Then Rt +f(f ′ +n − f ′ +k) ∈ r(Rt +f(D) ∩ J (E)), implying +{r > 0 ∣ f ′ +n − f ′ +k ∈ rD} ⊂ {r > 0 ∣ Rt +f(f ′ +n − f ′ +k) ∈ r(Rt +f(D) ∩ J (E)) +J (E)}. +Setting B ∶= Rt +f(D) ∩ J (E) +J (E), we derive +qB(Rt +f(f ′ +n − f ′ +k)) ≤ qD(f ′ +n − f ′ +k) +where qB and qD are the gauge functionals of B resp. D. The set Rt +f(D)∩J (E) is +absolutely convex as the intersection of two absolutely convex sets and it is bounded +by (13) and the boundedness of D. So B, being the closure of a disk, is a disk as +well. Since (f ′ +n) is a Cauchy sequence in (FV(Ω)′ +κ)D, we conclude that (R′ +f(f ′ +n)) +is a Cauchy sequence in J (E)B. The set B is a closed disk in the locally complete +space J (E) and hence a Banach disk by [89, 10.2.1 Proposition, p. 197]. Thus +J (E)B is a Banach space and (Rt +f(f ′ +n)) has a limit g ∈ J (E)B. The continuity +of the canonical injection J (E)B ↪ J (E) implies that (Rt +f(f ′ +n)) converges to g in +J (E) as well. As in a) we obtain that Rt +f(f ′) = g ∈ J (E). +e) Let Condition 3.2.3 e) be fulfilled. Let f ∈ FV(Ω,E) and e′ ∈ E′. For every +f ′ ∈ FV(Ω)′ there are j ∈ J, m ∈ M and C > 0 such that +∣Rt +f(f ′)(e′)∣ ≤ C∣Rf(e′)∣j,m = +(11) C +sup +x∈Nj,m(f) +∣e′(x)∣ +because (T E +m,T K +m)m∈M is a strong generator. Since there is K ∈ τ(E) such that +Nj,m(f) ⊂ K, we have +∣Rt +f(f ′)(e′)∣ ≤ C sup +x∈K +∣e′(x)∣, +implying Rt +f(f ′) ∈ (E′ +τ)′ = J (E) by the Mackey–Arens theorem. +Therefore we obtain that Rt +f ∈ L(FV(Ω)′ +κ,J (E)). So we get for all α ∈ A and +y ∈ F(Ω)′ +pα((J −1 ○ Rt +f)(y)) = +(12) pB○α(J ((J −1 ○ Rt +f)(y))) = pB○α(Rt +f(y)) ≤ +(13) sup +x∈Kα +∣y(x)∣. +This implies J −1 ○ Rt +f ∈ L(FV(Ω)′ +κ,E) = FV(Ω)εE (as linear spaces) and we gain +S(J −1 ○ Rt +f)(x) = J −1(Rt +f(δx)) = +(14) J −1(J (f(x))) = f(x) +for every x ∈ Ω. Thus S(J −1 ○ Rt +f) = f, proving the surjectivity of S. +□ +Further sufficient conditions for S being a topological isomorphism can be found +in Proposition 5.2.10, Proposition 5.6.6 and Theorem 5.7.1. In particular, we get +the following corollary as a special case of Theorem 3.2.4. + +32 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +3.2.5. Corollary. Let (T E +m,T K +m)m∈M be a strong, consistent generator for +(FV,E). If +(i) FV(Ω) is a semi-Montel space and E complete, or +(ii) FV(Ω) is a Fréchet–Schwartz space and E locally complete, or +(iii) E is a semi-Montel space, or +(iv) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ κ(E) ∶ Nj,m(f) ⊂ K, +then FV(Ω) and FV(Ω,E) are ε-compatible, in particluar, FV(Ω,E) ≅ FV(Ω)εE. +Proof. (i) Follows from Lemma 3.2.2 a) and Theorem 3.2.4 with Condition +3.2.3 a). +(ii) If FV(Ω) is a Fréchet–Schwartz space, then we have +span{δx ∣ x ∈ Ω} +lc = span{δx ∣ x ∈ Ω} +FV(Ω)′ +b = span{δx ∣ x ∈ Ω} +FV(Ω)′ +κ = FV(Ω)′ +by [30, Lemma 6 (b), p. 231] and the bipolar theorem where span{δx ∣ x ∈ Ω} +lc is the +local closure of span{δx ∣ x ∈ Ω} in FV(Ω)′ +b. Hence for every f ′ ∈ FV(Ω)′ there is a +sequence (f ′ +n) in the span of {δx ∣ x ∈ Ω} which converges locally to f ′ in FV(Ω)′ +κ. +Due to (14) we know that Rt +f(f ′ +n) ∈ J (E) for every f ∈ FV(Ω,E) and n ∈ N. Since +Fréchet–Schwartz spaces are also semi-Montel spaces, the statement follows from +Lemma 3.2.2 a) and Theorem 3.2.4 with Condition 3.2.3 d). +(iv) Follows from Lemma 3.2.2 c) and Theorem 3.2.4 with Condition 3.2.3 e). +(iii) Is a special case of (iv) since the set K ∶= acx(Nj,m(f)) is absolutely convex +and compact in the semi-Montel space E by [89, 6.2.1 Proposition, p. 103] and [89, +6.7.1 Proposition, p. 112] for every f ∈ FV(Ω,E), j ∈ J and m ∈ M. +□ +3.2.6. Remark. Linearisations of spaces FV(Ω,E)σ of weak E-valued func- +tions, where FV(Ω) need not be a dom-space, are treated in [118]. +Let us apply our preceding results to our weighted spaces of k-times continu- +ously partially differentiable functions on an open set Ω ⊂ Rd with k ∈ N∞. +3.2.7. Example. Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights +which is locally bounded away from zero on an open set Ω ⊂ Rd. +a) CVk(Ω,E) ≅ CVk(Ω)εE if E is a semi-Montel space and CVk(Ω) barrelled. +b) CVk +P (∂)(Ω,E) ≅ CVk +P (∂)(Ω)εE if E is a semi-Montel space and CVk +P (∂)(Ω) +barrelled. +c) CVk(Ω,E) ≅ CVk(Ω)εE if E is complete and CVk(Ω) a Montel space. +d) CVk +P (∂)(Ω,E) ≅ CVk +P (∂)(Ω)εE if E is complete and CVk +P (∂)(Ω) a Montel +space. +e) CVk(Ω,E) ≅ CVk(Ω)εE if E is locally complete and CVk(Ω) a Fréchet– +Schwartz space. +f) CVk +P (∂)(Ω,E) ≅ CVk +P (∂)(Ω)εE if E is locally complete and CVk +P (∂)(Ω) a +Fréchet–Schwartz space. +Proof. The generator of (CVk,E) and (CVk +P (∂),E) is strong and consistent +by Proposition 3.1.10. From Corollary 3.2.5 (iii) we deduce part a) and b), from +(i) part c) and d) and from (ii) part e) and f). +□ +Closed subspaces of Fréchet–Schwartz spaces are also Fréchet–Schwartz spaces +by [131, Proposition 24.18, p. 284]. The space CV∞ +P (∂)(Ω) is closed in CV∞(Ω) if +there is an lcHs Y such that P(∂)∣CV∞(Ω)∶CV∞(Ω) → Y is continuous. For example, +this is fulfilled if the coefficients of P(∂) belong to C(Ω), in particular, if P(∂) ∶= ∆ +or ∂, with Y ∶= (C(Ω),τc) due to V∞ being locally bounded away from zero. The +spaces CVk(Ω) from Example 3.1.9 a)(i) with ωm ∶= Mm × Ω for all m ∈ N0, where +Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ min(m,k)}, are Fréchet spaces and thus barrelled if the J + +3.2. ε-COMPATIBILITY +33 +in Vk ∶= (νj,m)j∈J,m∈N0 is countable by [107, Proposition 3.7, p. 240]. Sufficient +conditions on the weights that guarantee that CV∞(Ω) is a nuclear Fréchet space +and hence a Schwartz space as well can be found in [111, Theorem 3.1, p. 188]. For +the case ωm = Nd +0 × Ω see the references given in [111, p. 1]. +If Vk = Wk, i.e. Ck(Ω,E) is equipped with its usual topology of uniform conver- +gence of all partial derivatives up to order k on compact subsets of Ω, Example 3.2.7 +c)+d) can be improved to quasi-complete E. For Ω = Rd this can be found in [158, +Proposition 9, p. 108, Théorème 1, p. 111] and for general open Ω ⊂ Rd it is already +mentioned in [94, (9), p. 236] (without a proof) that CWk(Ω,E) ≅ CWk(Ω)εE for +k ∈ N∞ and quasi-complete E. For k = ∞ we even have CW∞(Ω,E) ≅ CW∞(Ω)εE +for locally complete E by [30, p. 228]. Our technique allows us to generalise the +first result and to get back the second result. +3.2.8. Example. Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open. If k < ∞ and E +has metric ccp, or if k = ∞ and E is locally complete, then +a) CWk(Ω,E) ≅ CWk(Ω)εE, and +b) CWk +P (∂)(Ω,E) ≅ CWk +P (∂)(Ω)εE if CWk +P (∂)(Ω) is closed in CWk(Ω). +Proof. We recall from Example 3.1.9 b) that Wk is the family of weights +given by νK,m(β,x) ∶= χK(x), (β,x) ∈ Mm × Ω, for all m ∈ N0 and compact K ⊂ Ω +where Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ min(m,k)} and χK is the characteristic function of K. +We already know that the generator for (CWk,E) and (CWk +P (∂),E) is strong and +consistent by Proposition 3.1.10 because Wk is locally bounded away from zero +on Ω, and CWk(Ω) and its closed subspace CWk +P (∂)(Ω) are Fréchet spaces. Let +f ∈ CWk(Ω,E), K ⊂ Ω be compact, m ∈ N0 and consider +NK,m(f) = {(∂β)Ef(x)νK,m(β,x) ∣ x ∈ Ω, β ∈ Mm} = {0} ∪ +⋃ +β∈Mm +(∂β)Ef(K). +NK,m(f) is compact since it is a finite union of compact sets. Furthermore, the +compact sets {0} and (∂β)Ef(K) are metrisable by [34, Chap. IX, §2.10, Proposi- +tion 17, p. 159] and thus their finite union NK,m(f) is metrisable as well by [169, +Theorem 1, p. 361] since the compact set NK,m(f) is collectionwise normal and +locally countably compact by [63, 5.1.18 Theorem, p. 305]. If E has metric ccp, +then the set acx(NK,m(f)) is absolutely convex and compact. Thus Corollary 3.2.5 +(iv) settles the case for k < ∞. If k = ∞ and E is locally complete, we observe that +Kβ ∶= acx((∂β)Ef(K)) for f ∈ CW∞(Ω,E) is absolutely convex and compact by +[29, Proposition 2, p. 354]. Then we have +NK,m(f) ⊂ acx( ⋃ +β∈Mm +Kβ) +and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3 +Proposition, p. 113]. Again, the statement follows from Corollary 3.2.5 (iv). +□ +The statement above for k = ∞ follows from Example 3.2.7 e)+f) as well because +CW∞(Ω) and its closed subspaces are Fréchet–Schwartz spaces. In the context of +differentiability on infinite dimensional spaces the preceding example a) remains +true for an open subset Ω of a Fréchet space or DFM-space and quasi-complete +E by [129, 3.2 Corollary, p. 286]. +Like here this can be generalised to E with +[metric] ccp. A special case of example b) is already known to be a consequence of +[30, Theorem 9, p. 232], namely, if k = ∞ and P(∂)K is hypoelliptic with constant +coefficients. In particular, this covers the space of holomorphic functions and the +space of harmonic functions. Holomorphy on infinite dimensional spaces is treated +in [52, Corollary 6.35, p. 332–333] where V = W0, Ω is an open subset of a locally + +34 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +convex Hausdorff k-space and E a quasi-complete locally convex Hausdorff space, +both over C, which can be generalised to E with [metric] ccp in a similar way. +For a second improvement of Example 3.2.7 for k = ∞ to locally complete E +without the condition that CV∞(Ω) resp. CV∞ +P (∂)(Ω) is a Fréchet–Schwartz space +we introduce the following conditions on the family V∞ on (Mm × Ω)m∈N0. We say +that a family V∞ of weights on (Mm × Ω)m∈N0 is C1-controlled if +(i) ∀ j ∈ J, m ∈ N0, β ∈ Mm ∶ νj,m(β,⋅) ∈ C1(Ω), +(ii) ∀ j ∈ J, m ∈ N0, β,γ ∈ Mm,x ∈ Ω ∶ νj,m(β,x) = νj,m(γ,x), +(iii) ∀ j ∈ J, m ∈ N0 ∃ i ∈ J, k ∈ N0, k ≥ m, C > 0 ∀ β ∈ Mm, x ∈ Ω, 1 ≤ n ≤ d ∶ +∣∂enνj,m(β,⋅)∣(x) ≤ Cνi,k(β,x). +We say that family Vk, k ∈ N∞, fulfils condition (V∞) if +∀ m ∈ N0, j ∈ J ∃ n ∈ N≥m, i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ β ∈ Mm, x ∈ Ω ∖ K ∶ +νj,m(β,x) ≤ ενi,n(β,x) +where N≥m ∶= {n ∈ N0 ∣ n ≥ m}. Here (V∞) stands for vanishing at infinity and +the condition was introduced in [107, Remark 3.4, p. 239] and for k = 0 in [16, 1.3 +Bemerkung, p. 189]. +3.2.9. Example. Let E be an lcHs and V∞ a directed C1-controlled family of +weights on an open convex set Ω ⊂ Rd which fulfils (V∞). If E is locally complete, +then +a) CV∞(Ω,E) ≅ CV∞(Ω)εE if CV∞(Ω) is barrelled, and +b) CV∞ +P (∂)(Ω,E) ≅ CV∞ +P (∂)(Ω)εE if CV∞ +P (∂)(Ω) is barrelled. +Proof. We already know that the generator for (CV∞,E) and (CV∞ +P (∂),E) is +strong and consistent by Proposition 3.1.10 because V∞ is locally bounded away +from zero on Ω as νj,m(β,⋅) is continuous for all j ∈ J, m ∈ N0 and β ∈ Mm. +Let f ∈ CV∞(Ω,E), j ∈ J, m ∈ N0 and β ∈ Mm. We set g∶Ω → E, g(x) ∶= +(∂β)Ef(x)νj,m(β,x), and note that +(∂en)Eg(x) = (∂β+en)Ef(x)νj,m(β,x) + (∂β)Ef(x)((∂en)Rνj,m(β,⋅))(x), +x ∈ Ω, +for all 1 ≤ n ≤ d. +Since V∞ is directed and C1-controlled there are i1,i2 ∈ J, +k1,k2 ∈ N0, k1 > m, k2 ≥ m, and C1,C2 > 0 such that +pα((∂en)Eg(x)) +≤ pα((∂β+en)Ef(x))νj,m(β,x) + pα((∂β)Ef(x))∣(∂en)Rνj,m(β,⋅)∣(x) +≤ C1pα((∂β+en)Ef(x))νi1,k1(β,x) + C2pα((∂β)Ef(x))νi2,k2(β,x) += C1pα((∂β+en)Ef(x))νi1,k1(β + en,x) + C2pα((∂β)Ef(x))νi2,k2(β,x) +for all 1 ≤ n ≤ d and α ∈ A, which implies +sup +x∈Ω +γ∈Nd +0,∣γ∣≤1 +pα((∂γ)Eg(x)) ≤ ∣f∣j,m,α + C1∣f∣i1,k1,α + C2∣f∣i2,k2,α. +Thus g is (weakly) C1 +b . +Due to (V∞) there are n ∈ N≥m and i ∈ J such that for all ε > 0 there is a +compact set K ⊂ Ω such that for all β ∈ Mm and x ∈ Ω ∖ K we have +νj,m(β,x) ≤ ενi,n(β,x). +Since V∞ is directed, we may assume w.l.o.g. that νj,m(β,x) ≤ νi,n(β,x) for all +x ∈ Ω. This implies that the zeros of νi,n(β,⋅) are zeros of νj,m(β,⋅). We define +h∶Ω → [0,∞) by h(x) ∶= νi,n(β,x)/νj,m(β,x) for x ∈ Ω with νj,m(β,x) ≠ 0 and + +3.2. ε-COMPATIBILITY +35 +h(x) ∶= 1 if νj,m(β,x) = 0. We note that h(x) > 0 for all x ∈ Ω as the zeros of +νi,n(β,⋅) are contained in the zeros of νj,m(β,⋅). It follows that +(∂β)Ef(x)νj,m(β,x)h(x) = (∂β)Ef(x)νi,n(β,x) +for x ∈ Ω with νj,m(β,x) ≠ 0 and (∂β)Ef(x)νj,m(β,x)h(x) = 0 for x ∈ Ω with +νj,m(β,x) = 0. Therefore (∂β)Efνj,m(β,⋅)h is bounded on Ω. Further, +εh(x) = ενi,n(β,x)/νj,m(β,x) ≥ 1 +for x ∈ Ω ∖ K with νj,m(β,x) ≠ 0 because (V∞) is fulfilled. Further, the zeros of +νj,m(β,⋅) are contained in N ∶= {x ∈ Ω ∣ (∂β)Ef(x)νj,m(β,x) = 0}. This yields that +Kβ ∶= acx((∂β)Efνj,m(β,⋅)(Ω)) is absolutely convex and compact by Proposition +A.1.4 and A.1.5. Furthermore, +Nj,m(f) = {(∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} ⊂ acx( ⋃ +β∈Mm +Kβ) +and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3 +Proposition, p. 113]. Finally, our statement follows from Corollary 3.2.5 (iv). +□ +For the Schwartz space S(Rd,E) and the multiplier space OM(Rd,E) from +Example 3.1.9 c) and d) an improvement of Example 3.2.7 c) to quasi-complete E +is already known, see e.g. [158, Proposition 9, p. 108, Théorème 1, p. 111]. However, +due to Example 3.2.9 it is even allowed that E is only locally complete. +3.2.10. Corollary. If E is a locally complete lcHs, then S(Rd,E) ≅ S(Rd)εE +and OM(Rd,E) ≅ OM(Rd)εE. +Proof. We start with the Schwartz space. Due to Example 3.2.9 a) and the +barrelledness of the Fréchet space S(Rd) we only need to check that its directed +family V∞ ∶= (ν1,m)m∈N0 of weights given by ν1,m(β,x) ∶= (1 + ∣x∣2)m/2, x ∈ Rd, for +m ∈ N0 and β ∈ Mm is C1-controlled and fulfils (V∞). Obviously, condition (i) and +(ii) are fulfilled. Since +∣∂enν1,m(β,⋅)∣(x) = (m/2)(1 + ∣x∣2)(m/2)−12∣xn∣ ≤ m(1 + ∣x∣2)m/2 = mν1,m(β,x) +for all x ∈ Rd and 1 ≤ n ≤ d, condition (iii) is also fulfilled. Thus V∞ is C1-controlled. +Noting that for every m ∈ N and ε > 0 there is r > 0 such that +(1 + ∣x∣2)m/2 +(1 + ∣x∣2)m = (1 + ∣x∣2)−m/2 ≤ ε +for all x ∉ Br(0), we obtain that +ν1,m(β,x) ≤ εν1,2m(β,x) +for all x ∉ Br(0) and β ∈ Mm. Hence V∞ fulfils condition (V∞). +Now, let us consider the multiplier space. We already know that the generator +for (OM,E) is strong and consistent by Proposition 3.1.10 because OM(R) is a +Montel space, thus barrelled, by [83, Chap. II, §4, n○4, Théorème 16, p. 131] and +its family of weights is continuous on Rd, thus locally bounded away from zero. +Let f ∈ OM(R,E), g ∈ S(Rd), m ∈ N0 and β ∈ Mm. Then (∂β)Ef ∈ OM(R,E) +and hence ((∂β)Ef)g ∈ S(Rd,E), which implies that ((∂β)Ef)g ∈ C1 +b (Rd,E). More- +over, we choose h∶Rd → (0,∞), h(x) ∶= 1 + ∣x∣2. Then ((∂β)Ef)gh is bounded on +Rd and for ε > 0 there is r > 0 such that (1 + ∣x∣2)−1 ≤ ε for all x ∉ Br(0), yielding +that Kβ,g ∶= acx(((∂β)Ef)g(Rd)) is absolutely convex and compact by Proposi- +tion A.1.4 and A.1.5. +Let j ⊂ S(Rd) be finite. +Since for each x ∈ Rd we have +(∂β)Ef(x)maxg∈j ∣g(x)∣ = eiθ(∂β)Ef(x)̃g(x) for some ̃g ∈ j and θ ∈ [0,2π), we get +Nj,m(f) = {(∂β)Ef(x)max +g∈j ∣g(x)∣ ∣ x ∈ Rd, β ∈ Mm} ⊂ acx( +⋃ +β∈Mm,g∈j +Kβ,g). + +36 +3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES +The set on the right-hand side is absolutely convex and compact by [89, 6.7.3 +Proposition, p. 113]. Finally, our statement follows from Corollary 3.2.5 (iv). +□ +For an alternative proof in the case of the Schwartz space we may also use +Example 3.2.7 e) since S(Rd) is a Fréchet–Schwartz space. +Example 3.2.9 can +also be used for an alternative proof of Example 3.2.8 if k = ∞ by observing that +CW∞(Ω,E) = CV∞(Ω,E) for any lcHs E where V∞ ∶= {ν ∈ C∞ +c (Ω) ∣ ν ≥ 0} and +C∞ +c (Ω) is the space of functions in C∞(Ω) with compact support. +Now, we improve Example 3.2.7 for the special case of spaces of ultradifferen- +tiable functions E(Mp)(Ω,E) and E{Mp}(Ω,E) from Example 3.1.9 e) and f) where +ωm ∶= Nd +0 × Ω for all m ∈ N0. For this purpose we recall the following conditions of +Komatsu for the sequence (Mp)p∈N0 (see [99, p. 26] and [101, p. 653]): +(M.0) M0 = M1 = 1, +(M.1) ∀ p ∈ N ∶ M 2 +p ≤ Mp−1Mp+1, +(M.2)’ ∃ A,C > 0 ∀ p ∈ N0 ∶ Mp+1 ≤ ACp+1Mp, +(M.3)’ ∑∞ +p=1 +Mp−1 +Mp < ∞. +3.2.11. Example. Let E be an lcHs, Ω ⊂ Rd open and (Mp)p∈N0 a sequence of +positive real numbers. +a) E(Mp)(Ω,E) ≅ E(Mp)(Ω)εE if E is locally complete. +b) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is complete or semi-Montel and in both +cases (Mp)p∈N0 fulfils (M.1) and (M.3)’. +c) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is sequentially complete and (Mp)p∈N0 +fulfils (M.0), (M.1), (M.2)’ and (M.3)’. +Proof. The generator is strong and consistent by Proposition 3.1.10 since the +family of weights given in Example 3.1.9 e) resp. f) is locally bounded away from +zero on Ω and E(Mp)(Ω) is a Fréchet–Schwartz space in a) by [99, Theorem 2.6, +p. 44] whereas E{Mp}(Ω) is a Montel space in b) and c) by [99, Theorem 5.12, p. +65–66]. Hence the statements a) and b) follow from Example 3.2.7. +Let us turn to c). We note that E{Mp}(Ω,E) ⊂ E{Mp}(Ω,E)κ by Lemma 3.2.2 +a) for any lcHs E. +Further, we claim that Condition 3.2.3 c) is fulfilled. +Let +f ′ ∈ E{Mp}(Ω)′. Due to [101, Proposition 3.7, p. 677] there is a sequence (fn)n∈N in +the space D{Mp}(Ω) of ultradifferentiable functions of class {Mp} of Roumieu-type +with compact support which converges to f ′ in E{Mp}(Ω)′ +b. Let f ∈ E{Mp}(Ω,E). +We observe that for every e′ ∈ E′ +∣Rt +f(fn)(e′)∣ = ∣∫ +Ω +fn(x)e′(f(x))dx∣ ≤ λ(supp(fn)) +sup +y∈Kn(f) +∣e′(y)∣ +where λ is the Lebesgue measure, supp(fn) is the support of fn and Kn(f) ∶= +{fn(x)f(x) ∣ x ∈ supp(fn)}. The set Kn(f) is compact and metrisable by [34, +Chap. IX, §2.10, Proposition 17, p. 159] and thus the closure of its absolutely +convex hull is compact in E as the sequentially complete space E has metric ccp. +We conclude that Rt +f(fn) ∈ (E′ +κ)′ = J (E) for every n ∈ N. Therefore Condition +3.2.3 c) is fulfilled, implying statement c) for sequentially complete E by Theorem +3.2.4. +□ +The results a) and b) in this example are new whereas c) is already proved in +[101, Theorem 3.10, p. 678] in a different way. In particular, part a) improves [101, +Theorem 3.10, p. 678] since Komatsu’s conditions (M.0), (M.1), (M.2)’ and (M.3)’ +are not needed and the condition that E is sequentially complete is weakened to +local completeness. We included c) to demonstrate an application of Condition +3.2.3 c). + +CHAPTER 4 +Consistency +4.1. The spaces AP(Ω,E) and consistency +This section is dedicated to the properties of functions which are compatible +with the ε-product in the sense that the space of functions having these properties +can be chosen as the space AP(Ω,E) or ⋂m∈M domT E +m in the Definition 3.1.7 b) of +consistency. This is done in a quite general way so that we are not tied to certain +spaces and have to redo our argumentation, for example, if we consider the same +generator (T E +m,T K +m)m∈M for two different spaces of functions. +Due to the linearity and continuity of u ∈ FV(Ω)εE for a dom-space FV(Ω) +and S(u) = u ○ δ with δ∶Ω → FV(Ω)′, x ↦ δx, these are properties which are purely +pointwise or given by pointwise approximation. Among such properties of func- +tions are continuity by Proposition 4.1.1, Cauchy continuity by Proposition 4.1.3, +uniform continuity by Proposition 4.1.5, continuous extendability by Proposition +4.1.7, continuous differentiability by Proposition 3.1.10, vanishing at infinity by +Proposition 4.1.9 and purely pointwise properties of a function like vanishing on a +set by Proposition 4.1.10. +We collect these properties in propositions and in follow-up lemmas we handle +properties which can be described by compositions of defining operators T E +m1 ○ T E +m2 +like continuous differentiability (of higher order) of Fourier transformations (see +Example 4.2.26). We fix the following notation for this section. For a dom-space +FV(Ω) and linear T∶FV(Ω) → KΩ we set (δ ○ T)(x)(f) ∶= (δx ○ T)(f) ∶= T(f)(x) +for all x ∈ Ω and f ∈ FV(Ω). +4.1.1. Proposition (continuity). Let Ω be a topological Hausdorff space and +FV(Ω) a dom-space such that FV(Ω) ⊂ C(Ω) as a linear subspace. Then S(u) ∈ +C(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ C(Ω,FV(Ω)′ +κ). +Proof. Let u ∈ FV(Ω)εE. Since S(u) = u○δ and δ ∈ C(Ω,FV(Ω)′ +κ), we obtain +that S(u) is in C(Ω,E). +□ +Now, we tackle the problem of the continuity of δ∶Ω → FV(Ω)′ +κ in the proposi- +tion above and phrase our solution in a way such that it can be applied to show the +continuity of the partial derivative (∂β)E(S(u)) as well (see Proposition 3.1.11). +We recall that a topological space Ω is called completely regular if for any non-empty +closed subset A ⊂ Ω and x ∈ Ω ∖ A there is f ∈ C(Ω,[0,1]) such that f(x) = 0 and +f(z) = 1 for all z ∈ A (see [88, Definition 11.1, p. 180]). Examples of completely +regular spaces are uniformisable, particularly metrisable, spaces by [88, Proposition +11.5, p. 181] and locally convex Hausdorff spaces by [65, Proposition 3.27, p. 95]. +A completely regular space Ω is a kR-space if for any completely regular space Y +and any map f∶Ω → Y , whose restriction to each compact K ⊂ Ω is continuous, +the map is already continuous on Ω (see [37, (2.3.7) Proposition, p. 22]). Examples +of kR-spaces are completely regular k-spaces by [63, 3.3.21 Theorem, p. 152]. A +topological space Ω is called k-space (compactly generated space) if it satisfies the +following condition: A ⊂ Ω is closed if and only if A ∩ K is closed in K for every +compact K ⊂ Ω. Every locally compact Hausdorff space is a completely regular +37 + +38 +4. CONSISTENCY +k-space. Further, every sequential Hausdorff space is a k-space by [63, 3.3.20 The- +orem, p. 152], in particular, every first-countable Hausdorff space. Thus metrisable +spaces are completely regular Hausdorff k-spaces. Moreover, the dual space (X′,τc) +with the topology of compact convergence τc is an example of a completely regular +Hausdorff k-space that is neither locally compact nor metrisable by [178, p. 267] if +X is an infinite-dimensional Fréchet space. +We denote by CW(Ω) the space of scalar-valued continuous functions on a +topological Hausdorff space Ω with the topology τc of compact convergence, i.e. the +topology of uniform convergence on compact subsets, which itself is the weighted +topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and +by Cb(Ω) the space of scalar-valued bounded, continuous functions on Ω with the +topology of uniform convergence on Ω. +4.1.2. Lemma. Let Ω be a topological Hausdorff space, FV(Ω) a dom-space +and T∶FV(Ω) → C(Ω) linear. Then δ ○T ∈ C(Ω,FV(Ω)′ +γ) in each of the subsequent +cases: +(i) Ω is a kR-space and T∶FV(Ω) → CW(Ω) is continuous. +(ii) T∶FV(Ω) → Cb(Ω) is continuous. +Proof. First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we +observe that +(δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f) +for every f ∈ FV(Ω) as T(f) is continuous on Ω. +(i) Verbatim as in Proposition 3.1.11 a). +(ii) There are j ∈ J, m ∈ M and C > 0 such that +sup +x∈Ω +∣(δx ○ T)(f)∣ = sup +x∈Ω +∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m +for every f ∈ FV(Ω). This means that {δx ○T ∣ x ∈ Ω} is equicontinuous in FV(Ω)′, +yielding the statement like before. +□ +The preceding lemma is just a modification of [16, 4.1 Lemma, p. 198] where +FV(Ω) = CV(Ω), the Nachbin-weighted space of continuous functions, and T = id. +Next, we turn to Cauchy continuity. A function f∶Ω → E from a metric space +Ω to an lcHs E is called Cauchy continuous if it maps Cauchy sequences to Cauchy +sequences. We write CC(Ω,E) for the space of Cauchy continuous functions from +Ω to E and set CC(Ω) ∶= CC(Ω,K). +4.1.3. Proposition (Cauchy continuity). Let Ω be a metric space and FV(Ω) +a dom-space such that FV(Ω) ⊂ CC(Ω) as a linear subspace. Then S(u) ∈ CC(Ω,E) +for all u ∈ FV(Ω)εE if δ ∈ CC(Ω,FV(Ω)′ +κ). +Proof. Let u ∈ FV(Ω)εE and (xn) a Cauchy sequence in Ω. Then (δxn) is a +Cauchy sequence in FV(Ω)′ +κ since δ ∈ CC(Ω,FV(Ω)′ +κ). It follows that (S(u)(xn)) is +a Cauchy sequence in E because u is uniformly continuous and u(δxn) = S(u)(xn). +Hence we conclude that S(u) ∈ CC(Ω,E). +□ +For the next lemma we equip the space CC(Ω) with the topology of uniform +convergence on precompact subsets of Ω. +4.1.4. Lemma. Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),CC(Ω)) for a +metric space Ω. Then δ ○ T ∈ CC(Ω,FV(Ω)′ +γ). +Proof. Let (xn) be a Cauchy sequence in Ω. We have (δxn○T)(f) = T(f)(xn) +for every f ∈ FV(Ω), which implies that ((δxn ○ T)(f)) is a Cauchy sequence in +K because T(f) ∈ CC(Ω) by assumption. +Since K is complete, it has a unique +limit T∞(f) ∶= limn→∞(δxn ○ T)(f) defining a linear functional in f. The set N ∶= + +4.1. THE SPACES AP(Ω, E) AND CONSISTENCY +39 +{xn ∣ n ∈ N} is precompact in Ω since Cauchy sequences are precompact. Hence +there are j ∈ J, m ∈ M and C > 0 such that +sup +n∈N +∣(δxn ○ T)(f)∣ = sup +x∈N +∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m +for every f ∈ FV(Ω). Therefore the set {δxn○T ∣ n ∈ N} is equicontinuous in FV(Ω)′, +which implies that T∞ ∈ FV(Ω)′ and the convergence of (δxn ○T) to T∞ in FV(Ω)′ +γ +due to the observation in the beginning and the fact that γ(FV(Ω)′,FV(Ω)) and +σ(FV(Ω)′,FV(Ω)) coincide on equicontinuous sets. In particular, (δxn ○ T) is a +Cauchy sequence in FV(Ω)′ +γ. Furthermore, for every x ∈ Ω we obtain from the +choice xn = x for all n ∈ N that δx ○ T ∈ FV(Ω)′. Thus the map δ ○ T∶Ω → FV(Ω)′ +γ +is well-defined and Cauchy continuous. +□ +The subsequent proposition and lemma handle the analogous statements for +uniform continuity. +For a metric space Ω we denote by Cu(Ω,E) the space of +uniformly continuous functions from Ω to E and set Cu(Ω) ∶= Cu(Ω,K). +4.1.5. Proposition (uniform continuity). Let (Ω,d) be a metric space and +FV(Ω) a dom-space such that FV(Ω) ⊂ Cu(Ω) as a linear subspace. Then S(u) ∈ +Cu(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cu(Ω,FV(Ω)′ +κ).1 +Proof. Let (zn), (xn) be sequences in Ω with limn→∞ d(zn,xn) = 0 and u ∈ +FV(Ω)εE. Then (δzn −δxn) converges to 0 in FV(Ω)′ +κ because δ ∈ Cu(Ω,FV(Ω)′ +κ). +As a consequence (S(u)(zn) − S(u)(xn)) converges to 0 in E since u is uniformly +continuous and u(δzn −δxn) = S(u)(zn)−S(u)(xn). Hence we conclude that S(u) ∈ +Cu(Ω,E). +□ +For the next lemma we mean by Cbu(Ω) the space of scalar-valued bounded, +uniformly continuous functions equipped with the topology of uniform convergence +on a metric space Ω. +4.1.6. Lemma. Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),Cbu(Ω)) for a +metric space (Ω,d). Then δ ○ T ∈ Cu(Ω,FV(Ω)′ +γ). +Proof. Let (zn) and (xn) be sequences in Ω such that limn→∞ d(zn,xn) = 0. +We have +(δzn ○ T − δxn ○ T)(f) = T(f)(zn) − T(f)(xn) +for every f ∈ FV(Ω), which implies that (δzn ○ T − δxn ○ T)(f) converges to 0 in K +for every f ∈ FV(Ω) because T(f) ∈ Cu(Ω). There exist j ∈ J, m ∈ M and C > 0 +such that +sup +n∈N +∣(δzn ○ T − δxn ○ T)(f)∣ ≤ 2sup +x∈Ω +∣T(f)(x)∣ ≤ 2C∣f∣FV(Ω),j,m +for every f ∈ FV(Ω). Therefore the set {δzn ○ T − δxn ○ T ∣ n ∈ N} is equicontinuous +in FV(Ω)′ and we conclude the statement like before. +□ +Let us turn to continuous extensions. Let X be a metric space and Ω ⊂ X. We +write Cext(Ω,E) for the space of functions f ∈ C(Ω,E) which have a continuous +extension to Ω and set Cext(Ω) ∶= Cext(Ω,K). +4.1.7. Proposition (continuous extendability). Let X be a metric space, Ω ⊂ X +and FV(Ω) a dom-space such that FV(Ω) ⊂ Cext(Ω) as a linear subspace. Then +S(u) ∈ Cext(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cext(Ω,FV(Ω)′ +κ). +1Here, we use the symbol u for elements in FV(Ω)εE instead of the usual u to avoid confusion +with the index u of Cu(Ω) resp. Cu(Ω, E). + +40 +4. CONSISTENCY +Proof. Let u ∈ FV(Ω)εE. There is δext ∈ C(Ω,FV(Ω)′ +κ) such that δext = δ on +Ω since δ ∈ Cext(Ω,FV(Ω)′ +κ). Moreover, u○δext ∈ C(Ω,E) and equal to S(u) = u○δ +on Ω, yielding S(u) ∈ Cext(Ω,E). +□ +For the next lemma we equip Cext(Ω) with the topology of uniform convergence +on compact subsets of Ω. +4.1.8. Lemma. Let X be a metric space, Ω ⊂ X, FV(Ω) a dom-space and +T ∈ L(FV(Ω),Cext(Ω)). Then δ ○ T ∈ Cext(Ω,FV(Ω)′ +γ) if FV(Ω) is barrelled. +Proof. From Lemma 4.1.2 (i) we derive that δ○T ∈ C(Ω,FV(Ω)′ +γ). Let x ∈ ∂Ω +and (xn) be a sequence in Ω with xn → x. Then (δxn ○ T) is a sequence in FV(Ω)′ +and +lim +n→∞(δxn ○ T)(f) = lim +n→∞T(f)(xn) =∶ (δext +x +○ T)(f) +in K for every f ∈ FV(Ω), which implies that (δxn○T) converges to δext +x +○T pointwise +on FV(Ω) because T(f) ∈ Cext(Ω). As a consequence of the Banach–Steinhaus +theorem we get (δext +x +○ T) ∈ FV(Ω)′ and the convergence in FV(Ω)′ +γ. +□ +Let FV(Ω,E) be a dom-space, X a set, K a family of sets and π∶⋃m∈M ωm → X +such that ⋃K∈K K ⊂ X. +We say that a function f ∈ ⋂m∈M domT E +m vanishes at +infinity in the weighted topology w.r.t. (π,K) if +∀ ε > 0, j ∈ J, m ∈ M, α ∈ A ∃ K ∈ K ∶ +sup +x∈ωm, +π(x)∉K +pα(T E +m(f)(x))νj,m(x) < ε. +(16) +Further, we set +APπ,K(Ω,E) ∶= {f ∈ ⋂ +m∈M +domT E +m ∣ f fulfils (16)}. +4.1.9. Proposition (vanishing at ∞ w.r.t. to (π,K)). Let (T E +m,T K +m)m∈M be +the generator for (FV,E), let FV(Ω,Y ) ⊂ APπ,K(Ω,Y ) as a linear subspace for +Y ∈ {K,E} and K be closed under taking finite unions. +(i) If for all u ∈ FV(Ω)εE it holds that S(u) ∈ ⋂m∈M dom(T E +m) and +∀ m ∈ M, x ∈ ωm ∶ (T E +mS(u))(x) = u(T K +m,x), +(17) +then S(u) ∈ APπ,K(Ω,E) for all u ∈ FV(Ω)εE. +(ii) If for all e′ ∈ E′ and f ∈ FV(Ω,E) it holds that e′ ○ f ∈ ⋂m∈M dom(T K +m) +and +∀ m ∈ M, x ∈ ωm ∶ T K +m(e′ ○ f)(x) = (e′ ○ T E +m(f))(x), +(18) +then e′ ○ f ∈ APπ,K(Ω) for all e′ ∈ E′ and f ∈ FV(Ω,E). +Proof. (i) We set Bj,m ∶= {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} for j ∈ J and m ∈ M. Let +u ∈ FV(Ω)εE. The topologies σ(FV(Ω)′,FV(Ω)) and κ(FV(Ω)′,FV(Ω)) coincide +on the equicontinuous set B○ +j,m and we deduce that the restriction of u to B○ +j,m is +σ(FV(Ω)′,FV(Ω))-continuous. +Let ε > 0, j ∈ J, m ∈ M, α ∈ A and set Uα,ε ∶= {x ∈ E ∣ pα(x) < ε}. Then there +are a finite set N ⊂ FV(Ω) and η > 0 such that u(f ′) ∈ Uα,ε for all f ′ ∈ VN,η where +VN,η ∶= {f ′ ∈ FV(Ω)′ ∣ sup +f∈N +∣f ′(f)∣ < η} ∩ B○ +j,m +because the restriction of u to B○ +j,m is σ(FV(Ω)′,FV(Ω))-continuous. Since N ⊂ +FV(Ω) is finite, FV(Ω) ⊂ APπ,K(Ω) and K is closed under taking finite unions, +there is K ∈ K such that +sup +x∈ωm +π(x)∉K +∣T K +m(f)(x)∣νj,m(x) < η +(19) + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +41 +for every f ∈ N. It follows from (19) and (the proof of) Lemma 3.1.8 that +Dπ⊄K,j,m ∶= {T K +m,x(⋅)νj,m(x) ∣ x ∈ ωm, π(x) ∉ K} ⊂ VN,η +and thus u(Dπ⊄K,j,m) ⊂ Uα,ε. Therefore we have +sup +x∈ωm +π(x)∉K +pα(T E +m(S(u))(x))νj,m(x) = +(17) +sup +x∈ωm +π(x)∉K +pα(u(T K +m,x))νj,m(x) < ε. +Hence we conclude that S(u) ∈ APπ,K(Ω,E). +(ii) Let ε > 0, f ∈ FV(Ω,E) and e′ ∈ E′. Then there exist α ∈ A and C > 0 such +that ∣e′(x)∣ ≤ Cpα(x) for every x ∈ E. For j ∈ J and m ∈ M there is K ∈ K such that +sup +x∈ωm +π(x)∉K +pα(T E +m(f)(x))νj,m(x) < ε +C +since FV(Ω,E) ⊂ APπ,K(Ω,E). It follows that +sup +x∈ωm +π(x)∉K +∣T K +m(e′ ○ f)(x)∣νj,m(x) = +(18) +sup +x∈ωm +π(x)∉K +∣e′(T E +m(f)(x))∣νj,m(x) < C ε +C = ε, +yielding e′ ○ f ∈ APπ,K(Ω). +□ +The first part of the proof above adapts an idea in the proof of [16, 4.4 Theo- +rem, p. 199–200] where (T E +m,T K +m)m∈M = (idEΩ,idKΩ) which is a special case of our +proposition. +Our last proposition of this section is immediate. For ω ⊂ Ω we set APω(Ω,E) ∶= +{f ∈ EΩ ∣ ∀ x ∈ ω ∶ f(x) = 0} and APω(Ω) ∶= APω(Ω,K). +4.1.10. Proposition (vanishing on a subset). Let ω ⊂ Ω and FV(Ω) a dom- +space such that FV(Ω) ⊂ APω(Ω) as a linear subspace. Then S(u) ∈ APω(Ω,E) +for all u ∈ FV(Ω)εE. +4.2. Further examples of ε-products +In Chapter 3 we dealt with weighted spaces of continuously partially differen- +tiable functions. Now, we treat many examples of weighted spaces FV(Ω,E) of +functions with less regularity on a set Ω with values in a locally convex Hausdorff +space E over the field K. Applying the results of the preceding sections, we give +conditions on E such that FV(Ω) and FV(Ω,E) are ε-compatible, in particular, +that +FV(Ω,E) ≅ FV(Ω)εE +holds. We start with the simplest example of all. Let Ω be a non-empty set and +equip the space EΩ with the topology of pointwise convergence, i.e. the locally +convex topology given by the seminorms +∣f∣K,α ∶= sup +x∈K +pα(f(x))χK(x), +f ∈ EΩ, +for finite K ⊂ Ω and α ∈ A. To prove EN0 ≅ KN0εE for complete E is given as an +exercise in [94, Aufgabe 10.5, p. 259], which we generalise now. +4.2.1. Example. Let Ω be a non-empty set and E an lcHs. Then EΩ ≅ KΩεE. +Proof. The strength and consistency of the generator (idEΩ,idKΩ) is obvious. +Let f ∈ EΩ, K ⊂ Ω be finite and set NK(f) ∶= f(Ω)χK(Ω). Then we have NK(f) = +f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω. Thus NK(f) is finite, hence +compact, NK(f) ⊂ acx(f(K)) and acx(f(K)) is a subset of the finite dimensional +subspace span(f(K)) of E. It follows that acx(f(K)) is compact by [89, 6.7.4 +Proposition, p. 113], implying our statement by virtue of Corollary 3.2.5 (iv). +□ + +42 +4. CONSISTENCY +The next example will give us the counterpart of Example 3.2.9 a) on the level +of sequence spaces. Let Ω be a set, E an lcHs and V ∶= (νj)j∈J a directed family of +weights νj∶Ω → [0,∞) on Ω. We set +ℓV(Ω,E) ∶= {f ∈ EΩ ∣ ∀ j ∈ J, α ∈ A ∶ ∣f∣j,α ∶= sup +x∈Ω +pα(f(x))νj(x) < ∞} +and ℓV(Ω) ∶= ℓV(Ω,K). +4.2.2. Example. Let E be an lcHs, (Ω,d) a uniformly discrete metric space, +i.e. there is r > 0 such that d(x,y) ≥ r for all x,y ∈ Ω, x ≠ y, and V ∶= (νj)j∈J a +directed family of weights on Ω such that +∀ j ∈ J ∃ i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ K ∶ νj(x) ≤ ενi(x). +(20) +If E is locally complete, then ℓV(Ω,E) ≅ ℓV(Ω)εE. +Proof. Let f ∈ ℓV(Ω,E) and j ∈ J. Then fνj is bounded on Ω by definition +of ℓV(Ω,E). Since (Ω,d) is uniformly discrete, there is r > 0 such that +pα(f(x)νj(x) − f(y)νj(y)) +d(x,y) +≤ 2 +r ∣f∣j,α < ∞, +x,y ∈ Ω, x ≠ y, +for every α ∈ A. Therefore fνj ∈ C[1] +b (Ω,E) where +C[1] +b (Ω,E) ∶= {g ∈ EΩ ∣ ∀α ∈ A ∶ sup +x∈Ω +pα(g(x)) < ∞ and sup +x,y∈Ω +x≠y +pα(g(x) − g(y)) +d(x,y) +< ∞}. +Due to (20) there is i ∈ J such that for all ε > 0 there exists a compact set K ⊂ Ω +such that νj(x) ≤ ενi(x) for all x ∈ Ω∖K. As V is directed, we may assume w.l.o.g. +that νj(x) ≤ νi(x) for all x ∈ Ω. This implies that the zeros of νi are zeros of νj. We +define h∶Ω → [0,∞) by h(x) ∶= νi(x)/νj(x) for x ∈ Ω with νj(x) ≠ 0 and h(x) ∶= 1 +if νj(x) = 0. We observe that h(x) > 0 for all x ∈ Ω as the zeros of νi are contained +in the zeros of νj. It follows that +f(x)νj(x)h(x) = f(x)νi(x) +for x ∈ Ω with νj(x) ≠ 0 and f(x)νj(x)h(x) = 0 for x ∈ Ω with νj(x) = 0. Hence +fνjh is bounded on Ω. Further, +εh(x) = ενi(x)/νj(x) ≥ 1, +for x ∈ Ω ∖ K with νj(x) ≠ 0 because (20) is fulfilled. Moreover, the zeros of νj +are contained in N ∶= {x ∈ Ω ∣ f(x)νj(x) = 0}. This yields that acx(fνj(Ω)) is +absolutely convex and compact by Proposition A.1.4. +So our statement follows +from Corollary 3.2.5 (iv). +□ +Let us apply the preceding result to some known sequence spaces. We recall +that a matrix A ∶= (ak,j)k,j∈N of non-negative numbers is called Köthe matrix if it +fulfils: +(1) ∀ k ∈ N ∃ j ∈ N ∶ ak,j > 0, +(2) ∀ k,j ∈ N ∶ ak,j ≤ ak,j+1. +We note that what we call k is usually called j and vice-versa (see e.g. [131, Def- +inition, p. 326]). But the notation we chose is more in line with the meaning of +j in our Definition 3.1.3 of a weight function and therefore we prefer to keep our +notation consistent. For an lcHs E we define the Köthe space +λ∞(A,E) ∶= {x = (xk) ∈ EN ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup +k∈N +pα(xk)ak,j < ∞} + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +43 +and the spaces of E-valued rapidly decreasing sequences which we need for some +theorems on Fourier expansions (see Theorem 5.6.13, Theorem 5.6.14) by +s(Ω,E) ∶= {x = (xk) ∈ EΩ ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup +k∈Ω +pα(xk)(1 + ∣k∣2)j/2 < ∞} +with Ω = Nd, Nd +0, Zd. Further, we set λ∞(A) ∶= λ∞(A,K) and s(Ω) ∶= s(Ω,K). +4.2.3. Corollary. Let E be a locally complete lcHs. +a) If A ∶= (ak,j)k,j∈N is a Köthe matrix such that +∀ j ∈ N ∃ i ∈ N ∀ ε > 0 ∃ K ∈ N ∀ k ∈ N, k > K ∶ ak,j ≤ εak,i, +(21) +then λ∞(A,E) ≅ λ∞(A)εE. +b) s(Ω,E) ≅ s(Ω)εE for Ω = Nd, Nd +0, Zd. +Proof. We observe that N and Ω are uniformly discrete metric spaces if they +are equipped with the metric induced by the absolute value. Further, a set in a +discrete space is compact if and only if it is finite. In case b) we set νj∶Ω → (0,∞), +νj(k) ∶= (1 + ∣k∣2)j/2 for j ∈ N. Then for ε > 0 there is K ∈ N such that +(1 + ∣k∣2)j/2 +(1 + ∣k∣2)j += (1 + ∣k∣2)−j/2 ≤ ε +for all k ∈ Ω with ∣k∣ > K. In both cases the family of weights are directed, in case +a) due to condition (2) of the definition of a Köthe matrix. Hence we can apply +Example 4.2.2 in both cases. +□ +Due to [131, Proposition 27.10, p. 330–331] condition (21) is equivalent to +λ∞(A) being a Schwartz space. +Since λ∞(A) is also a Fréchet space by [131, +Lemma 27.1, p. 326], another way to prove Corollary 4.2.3 a) (and b) as well) is +given by Corollary 3.2.5 (ii). +Our next examples are Favard-spaces. Let E be an lcHs, 0 < γ ≤ 1, Ω a compact +Hausdorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow, i.e. +ϕ(t + r,s) = ϕ(t,ϕ(r,s)) +and +ϕ(0,s) = s, +t,r ∈ [0,∞), s ∈ Ω, +and (̃T E +t )t≥0 the induced semigroup given by ̃T E +t ∶C(Ω,E) → C(Ω,E), ̃T E +t (f) ∶= +f(ϕ(t,⋅)). The semigroup (̃T K +t )t≥0 is (equi-)bounded and strongly continuous by +[62, Chap. II, 3.31 Exercises (1), p. 95]. The vector-valued Favard space of order γ +of the semigroup (̃T E +t )t≥0 is defined by +Fγ(Ω,E) ∶= {f ∈ C(Ω,E) ∣ ∀ α ∈ A ∶ +sup +x∈Ω,t>0 +pα(̃T E +t (f)(x) − f(x))t−γ < ∞} +equipped with the system of seminorms given by +∣f∣α ∶= max(sup +x∈Ω +pα(f(x)), sup +x∈Ω,t>0 +pα(̃T E +t (f)(x) − f(x))t−γ), +f ∈ Fγ(Ω,E), +for α ∈ A (see [39, Definition 3.1.2, p. 160] and [39, Proposition 3.1.3, p. 160]). +Further, we set Fγ(Ω) ∶= Fγ(Ω,K). Fγ(Ω,E) is a dom-space, which follows from +the setting ω ∶= [0,∞) × Ω, domT E ∶= C(Ω,E) and T E∶C(Ω,E) → Eω given by +T E(f)(0,x) ∶= f(x) and T E(f)(t,x) ∶= ̃T E +t (f)(x) − f(x), +t > 0, x ∈ Ω, +as well as AP(Ω,E) ∶= EΩ and the weight given by ν(0,x) ∶= 1 and ν(t,x) ∶= t−γ +for t > 0 and x ∈ Ω. +4.2.4. Example. Let E be a semi-Montel space, 0 < γ ≤ 1, Ω a compact Haus- +dorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow. Then Fγ(Ω,E) ≅ Fγ(Ω)εE +holds for the Favard space of order γ of the induced semigroup (̃T E +t )t≥0. + +44 +4. CONSISTENCY +Proof. The generator (T E,T K) for (Fγ,E) is consistent by Proposition 4.1.1 +and Lemma 4.1.2 b)(ii). Its strength is clear. Thus our statement follows from +Corollary 3.2.5 (iii). +□ +The space of càdlàg functions on a set Ω ⊂ R with values in an lcHs E is defined +by +D(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ Ω ∶ +lim +w→x+f(w) = f(x) and lim +w→x−f(w) exists}.2 +Further, we set D(Ω) ∶= D(Ω,K). Due to Proposition A.1.1 the maps given by +∣f∣K,α ∶= sup +x∈Ω +pα(f(x))χK(x), +f ∈ D(Ω,E), +for compact K ⊂ Ω and α ∈ A form a system of seminorms inducing a locally convex +Hausdorff topology on D(Ω,E). +4.2.5. Example. Let E be an lcHs and Ω ⊂ R locally compact. If E is quasi- +complete, then D(Ω)εE ≅ D(Ω,E). +Proof. First, we show that the generator (idEΩ,idKΩ) for (D,E) is strong +and consistent. The strength is a consequence of a simple calculation, so we only +prove the consistency explicitly. +We have to show that S(u) ∈ D(Ω,E) for all +u ∈ D(Ω)εE. Let x ∈ Ω be an accumulation point of [x,∞) ∩ Ω resp. (−∞,x] ∩ Ω, +(xn) be a sequence in Ω such that xn → x+ resp. xn → x−. We have +δxn(f) = f(xn) → f(x) = δx(f), +xn → x+, +and +δxn(f) = f(xn) → lim +n→∞f(xn) =∶ T(f)(x), +xn → x−, +for every f ∈ D(Ω), which implies that (δxn) converges to δx if xn → x+, and to +δx ○ T if xn → x− in D(Ω)′ +σ. +Since Ω is locally compact, there are a compact +neighbourhood U(x) ⊂ Ω of x and n0 ∈ N such that xn ∈ U(x) for all n ≥ n0. Hence +we deduce +sup +n≥n0 +∣δxn(f)∣ ≤ ∣f∣U(x) +for every f ∈ D(Ω). Therefore the set {δxn ∣ n ≥ n0} is equicontinuous in D(Ω)′, +which implies that (δxn) converges to δx if xn → x+ and to δx ○ T if xn → x− in +D(Ω)′ +γ and thus in D(Ω)′ +κ. From +S(u)(x) = u(δx) = lim +n→∞u(δxn) = lim +n→∞S(u)(xn), +xn → x+, +and +u(δx ○ T) = lim +n→∞u(δxn) = lim +n→∞S(u)(xn), +xn → x−, +for every u ∈ D(Ω)εE follows the consistency. Second, let f ∈ D(Ω,E), K ⊂ Ω be +compact and consider NK(f) = f(Ω)χK(Ω). We observe that NK(f) = f(K)∪{0} +if K ≠ Ω, and NK(f) = f(K) if K = Ω. We note that NK(f) ⊂ acx(f(K)) and +acx(f(K)) is absolutely convex and compact by Proposition A.1.1 because E is +quasi-complete. Thus we derive our statement from Corollary 3.2.5 (iv). +□ +We turn to Cauchy continuous functions. Let Ω be a metric space, E an lcHs +and the space CC(Ω,E) of Cauchy continuous functions from Ω to E be equipped +with the system of seminorms given by +∣f∣K,α ∶= sup +x∈K +pα(f(x))χK(x), +f ∈ CC(Ω,E), +for K ⊂ Ω precompact and α ∈ A. +2We note that for x ∈ Ω we only demand limw→x+ f(w) = f(x) if x is an accumulation point of +[x, ∞)∩Ω, and the existence of the limit limw→x− f(w) if x is an accumulation point of (−∞, x]∩Ω. + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +45 +4.2.6. Example. Let E be an lcHs and Ω a metric space. If E is a Fréchet +space or a semi-Montel space, then CC(Ω,E) ≅ CC(Ω)εE. +Proof. The generator (idEΩ,idKΩ) for (CC,E) is consistent by Proposition +4.1.3 with Lemma 4.1.4. Its strength follows from the uniform continuity of every +e′ ∈ E′. First, we consider the case that E is a Fréchet space. Let f ∈ CC(Ω,E), K ⊂ +Ω be precompact and consider NK(f) = f(Ω)χK(Ω). Then NK(f) = f(K) ∪ {0} if +K ≠ Ω, and NK(f) = f(K) if K = Ω. The set f(K) is precompact in the metrisable +space E by [13, Proposition 4.11, p. 576]. Thus we obtain CC(Ω,E) ⊂ CC(Ω,E)κ +by virtue of Lemma 3.2.2 c). Since E is complete, the first part of the statement +follows from Theorem 3.2.4 with Condition 3.2.3 a). If E is a semi-Montel space, +then it is a consequence of Corollary 3.2.5 (iii). +□ +Let (Ω,d) be a metric space, E an lcHs and the space Cbu(Ω,E) of bounded +uniformly continuous functions from Ω to E be equipped with the system of semi- +norms given by +∣f∣α ∶= sup +x∈Ω +pα(f(x)), +f ∈ Cbu(Ω,E), +for α ∈ A. +4.2.7. Example. Let E be an lcHs and (Ω,d) a metric space. If E is a semi- +Montel space, then Cbu(Ω,E) ≅ Cbu(Ω)εE. +Proof. The generator (idEΩ,idKΩ) for (Cbu,E) is consistent by Proposition +4.1.5 with Lemma 4.1.6. It is also strong due to the uniform continuity of every +e′ ∈ E′, yielding our statement by Corollary 3.2.5 (iii). +□ +4.2.8. Remark. If N is equipped with the metric induced by the absolut value, +then Cbu(N,E) = ℓ∞(N,E) where ℓ∞(N,E) is the space of bounded E-valued +sequences. If E is a separable infinite-dimensional Hilbert space, then the map +S∶Cbu(N)εE → Cbu(N,E) is not surjective by [17, 2.8 Beispiel, p. 140] and [94, Satz +10.5, p. 235–236]. Hence one cannot drop the condition that E is a semi-Montel +space in Example 4.2.7. +Let (Ω,d) be a metric space, z ∈ Ω, E an lcHs, 0 < γ ≤ 1 and define the space +of E-valued γ-Hölder continuous functions on Ω that vanish at z by +C[γ] +z (Ω,E) ∶= {f ∈ EΩ ∣ f(z) = 0 and ∀ α ∈ A ∶ ∣f∣α < ∞} +where +∣f∣α ∶= sup +x,w∈Ω +x≠w +pα(f(x) − f(w)) +d(x,w)γ +. +The topological subspace C[γ] +z,0(Ω,E) of γ-Hölder continuous functions that vanish +at infinity consists of all f ∈ C[γ] +z (Ω,E) such that for all ε > 0 there is δ > 0 with +sup +x,w∈Ω +0 0}, and let π∶ω1 → ω1 be the identity. Then +C[γ] +z,0(Ω,E) = C[γ] +z (Ω,E) ∩ APπ,K(Ω,E) with APπ,K(Ω,E) from Proposition 4.1.9 +and the generator (T E +1 ,T K +1 ) for (C[γ] +z,0,E) is strong and consistent by Proposition +4.1.10 for vanishing at z and Proposition 4.1.9 for vanishing at infinity w.r.t. (π,K). +Let f ∈ C[γ] +z,0(Ω,E) and Kδ ∶= {(x,w) ∈ Ω2 ∣ d(x,w) ≥ δ} for δ > 0. For +Nπ⊂Kδ,1,1(f) = {T E +1 (f)(x,w)ν1,1(x,w) ∣ (x,w) ∈ Kδ} = { f(x)−f(w) +d(x,w)γ +∣ (x,w) ∈ Kδ} +we have +Nπ⊂Kδ,1,1(f) ⊂ δ−γ{c(f(x) − f(w)) ∣ x,w ∈ Ω, ∣c∣ ≤ 1} += δ−γ ch(f(Ω) − f(Ω)). +The set f(Ω) is precompact because Ω is precompact and the γ-Hölder continuous +function f is uniformly continuous. It follows that the linear combination f(Ω) − +f(Ω) is precompact and the circled hull of a precompact set is still precompact by +[153, Chap. I, 5.1, p. 25]. Therefore Nπ⊂Kδ,1,1(f) is precompact for every δ > 0, +giving the precompactness of +N1,1(f) = {T E +1 (f)(x,w)ν1,1(x,w) ∣ (x,w) ∈ ω1} +by Proposition A.1.6. Hence statement b) is a consequence of Corollary 3.2.5 (iv), +Proposition A.1.6 and the quasi-completeness of E. +□ +Let Ω be a topological Hausdorff space and V ∶= (νj)j∈J a directed family of +weights νj∶Ω → [0,∞). +The weighted space of continuous functions on Ω with +values in an lcHs E is given by +CV(Ω,E) ∶= {f ∈ C(Ω,E) ∣ ∀ j ∈ J, α ∈ A ∶ ∣f∣j,α < ∞} +where +∣f∣j,α ∶= sup +x∈Ω +pα(f(x))νj(x). +Its topological subspace of functions that vanish at infinity in the weighted topology +is defined by +CV0(Ω,E) ∶= {f ∈ CV(Ω,E) ∣ ∀ j ∈ J, α ∈ A, ε > 0 +∃ K ⊂ Ω compact ∶ ∣f∣Ω∖K,j,α < ε} +where +∣f∣Ω∖K,j,α ∶= sup +x∈Ω∖K +pα(f(x))νj(x). +Further, we define CV(Ω) ∶= CV(Ω,K) and CV0(Ω) ∶= CV(Ω,K). In particular, we +set Cb(Ω,E) ∶= CV(Ω,E), i.e. the space of bounded continuous functions, and have +CV0(Ω,E) = C0(Ω,E) if V ∶= {1}. In [15, 16, 17] Bierstedt studies these spaces in the +case that V is a Nachbin-family which means that the functions νj are upper semi- +continuous for all j ∈ J and directed in the sense that for j1,j2 ∈ J and λ ≥ 0 there +is j3 ∈ J such that λνj1,λνj2 ≤ νj3. Formally this is stronger than our definition +of being directed in Remark 3.1.6 c). The notion U ≤ V for two Nachbin-families + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +47 +means that for every µ ∈ U there is ν ∈ V such that µ ≤ ν. One of his main results +from [17] is the following theorem. +4.2.10. Theorem ([17, 2.4 Theorem (2), p. 138–139]). Let E be a quasi- +complete lcHs, Ω a completely regular Hausdorff space and V a Nachbin-family +on Ω. If +(i) Z ∶= {v∶Ω → R ∣ v constant, v ≥ 0} ≤ V, or +(ii) ̃ +W ∶= {µχK ∣ µ > 0, K ⊂ Ω compact} ≤ V and Ω is a kR-space, +then CV0(Ω,E) ≅ CV0(Ω)εE. +We note that C̃ +W(Ω,E) = CW(Ω,E) with our definition of W = {χK ∣ K ⊂ +Ω compact} from above Lemma 4.1.2. +The only difference is that W is not a +Nachbin-family because it is not directed in the sense of Nachbin-families but in +the sense of Remark 3.1.6 c). We improve this result by strengthening the conditions +on Ω and V which allows us to weaken the assumptions on E. +4.2.11. Example. Let E be an lcHs, Ω a locally compact topological Hausdorff +space and V a directed family of continuous weights on Ω. +(i) If E has ccp, or +(ii) if E has metric ccp and Ω is second-countable, +then CV0(Ω,E) ≅ CV0(Ω)εE. +Proof. We set K ∶= {K ⊂ Ω ∣ K compact} and π∶Ω → Ω, π(x) ∶= x. It follows +from Proposition 4.1.1 combined with Lemma 4.1.2 (i) (continuity) and Proposition +4.1.9 (vanish at infinity w.r.t. (π,K)) that the generator (idEΩ,idKΩ) is strong and +consistent since V is a family of continuous weights and Ω a kR-space due to local +compactness. +Let f ∈ CV0(Ω,E), j ∈ J and consider Nj(f) = (fνj)(Ω). +By Proposition +A.1.3 the set K ∶= acx(Nj(f)) is absolutely convex and compact as fνj ∈ C0(Ω,E), +implying our statement by Corollary 3.2.5 (iv). +□ +4.2.12. Example. Let E be an lcHs and Ω a [metrisable] kR-space. If E has +[metric] ccp, then CW(Ω,E) ≅ CW(Ω)εE. +Proof. First, we observe that the generator (idEΩ,idKΩ) for (CW,E) is con- +sistent by Proposition 4.1.1 and Lemma 4.1.2 b)(i). Its strength is obvious. Let +f ∈ CW(Ω,E), K ⊂ Ω be compact and consider NK(f) = f(Ω)νK(Ω). +Then +NK(f) = f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω, which yields that +NK(f) is compact in E. If Ω is even metrisable, then f(K) is also metrisable by +[34, Chap. IX, §2.10, Proposition 17, p. 159] and thus the finite union NK(f) as +well by [169, Theorem 1, p. 361] since the compact set NK(f) is collectionwise +normal and locally countably compact by [63, 5.1.18 Theorem, p. 305]. Further, +acx(NK(f)) is absolutely convex and compact in E if E has ccp resp. if Ω is metris- +able and E has metric ccp. We conclude that CW(Ω,E) ≅ CW(Ω)εE if E has ccp +resp. if Ω is metrisable and E has metric ccp by Corollary 3.2.5 (iv). +□ +Bierstedt also considers closed subspaces of CV(Ω) and CV0(Ω), for instance +subspaces of holomorpic functions on open Ω, and of holomorpic functions on the +inner points of Ω which are continuous on the boundary in [17, 3.1 Bemerkung, p. +141] and [17, 3.7 Satz, p. 144]. +Let Ω ⊂ C be open and bounded and E an lcHs over C. We denote by A(Ω,E) +the space of continuous functions from Ω to an lcHs E which are holomorphic on +Ω and equip A(Ω,E) with the system of seminorms given by +∣f∣α ∶= sup +x∈Ω +pα(f(x)), +f ∈ A(Ω,E), + +48 +4. CONSISTENCY +for α ∈ A. We set A(Ω) ∶= A(Ω,C), J ∶= M ∶= {1} and ν1,1 ∶= 1 on Ω. +4.2.13. Example. Let E be an lcHs and Ω ⊂ C open and bounded. +Then +A(Ω,E) ≅ A(Ω)εE if E has metric ccp. +Proof. The space A(Ω) is a Banach space and hence barrelled. The inclusion +I∶A(Ω) → CW∞ +∂ (Ω) is continuous due to the Cauchy inequality (I is an inclusion +due to the identity theorem). It follows from Proposition 4.1.1, Lemma 4.1.2 b)(i), +Proposition 3.1.11 c) and (4) that the generator (idEΩ,idCΩ) is consistent and as +in Proposition 3.1.10 that it is strong, too. +Let f ∈ A(Ω,E) and N1,1(f) = f(Ω). The set K ∶= acx(N1,1(f)) is absolutely +convex and compact by Proposition A.1.3 since f ∈ C(Ω,E) = C0(Ω,E), implying +our statement by Corollary 3.2.5 (iv). +□ +For quasi-complete E this is already covered by [17, 3.1 Bemerkung, p. 141]. +More general than holomorphic functions, we may also consider kernels of hypoel- +liptic linear partial differential operators in CV(Ω) and CV0(Ω). For an open set +Ω ⊂ Rd, a directed family V ∶= (νj)j∈N of weights νj∶Ω → [0,∞), an lcHs E and a +linear partial differential operator P(∂)E which is hypoelliptic if E = K we define +the space of zero solutions +CVP (∂)(Ω,E) ∶= {f ∈ C∞ +P (∂)(Ω,E) ∣ ∀ j ∈ N, α ∈ A ∶ ∣f∣j,α < ∞}, +where C∞ +P (∂)(Ω,E) is the kernel of P(∂)E in C∞(Ω,E), +∣f∣j,α ∶= sup +x∈Ω +pα(f(x))νj(x), +and its topological subspace +CV0,P (∂)(Ω,E) ∶= CVP (∂)(Ω,E) ∩ CV0(Ω,E). +Further, we set CVP (∂)(Ω) ∶= CVP (∂)(Ω,K) and CV0,P (∂)(Ω) ∶= CV0,P (∂)(Ω,K). We +say that V is locally bounded away from zero on Ω if +∀ K ⊂ Ω compact ∃ j ∈ N ∶ inf +x∈K νj(x) > 0. +This is an extension of the definition of being locally bounded away from zero from +Vk with k ∈ N∞ to the case k = 0 (see Proposition 3.1.10). If V is a Nachbin-family, +this means that ̃ +W ≤ V (see Theorem 4.2.10 (ii)). +4.2.14. Proposition. Let Ω ⊂ Rd be open, V ∶= (νj)j∈N an increasing family +of weights which is locally bounded away from zero on Ω and P(∂)K a hypoelliptic +linear partial differential operator. Then CVP (∂)(Ω) and CV0,P (∂)(Ω) are Fréchet +spaces. +Proof. We note that CVP (∂)(Ω) is metrisable as V is countable. Let (fn) be +a Cauchy sequence in CVP (∂)(Ω). From V being locally bounded away from zero +it follows that for every compact K ⊂ Ω there is j ∈ N such that +sup +x∈K +∣f(x)∣ ≤ sup +z∈K +νj(z)−1 sup +x∈K +∣f(x)∣νj(x) ≤ sup +z∈K +νj(z)−1∣f∣j, +f ∈ CVP (∂)(Ω), +(22) +which means that the inclusion I∶CVP (∂)(Ω) → CWP (∂)(Ω) is continuous. Thus +(fn) is also a Cauchy sequence in CWP (∂)(Ω) and has a limit f there as CWP (∂)(Ω) +is complete due to the hypoellipticity of P(∂)K. Let j ∈ N, ε > 0 and x ∈ Ω. Then +there is mj,ε,x ∈ N such that for all m ≥ mj,ε,x it holds that +∣fm(x) − f(x)∣ < +ε +2νj(x) + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +49 +if νj(x) ≠ 0. Further, there is mj,ε ∈ N such that for all n,m ≥ mj,ε it holds that +∣fn − fm∣j < ε +2. +Hence for n ≥ mj,ε we choose m ≥ max(mj,ε,mj,ε,x) and derive +∣fn(x)−f(x)∣νj(x) ≤ ∣fn(x)−fm(x)∣νj(x)+∣fm(x)−f(x)∣νj(x) < ε +2+ +ε +2νj(x)νj(x) = ε. +It follows that ∣fn − f∣j ≤ ε and ∣f∣j ≤ ε + ∣fn∣j for all n ≥ mj,ε, implying the +convergence of (fn) to f in CVP (∂)(Ω). Therefore CVP (∂)(Ω) is a Fréchet space. +CV0,P (∂)(Ω) is a closed subspace of CVP (∂)(Ω) and so a Fréchet space as well. +□ +Due to the proposition above the spaces CVP (∂)(Ω) and CV0,P (∂)(Ω) are closed +subspaces of CV(Ω) resp. CV0(Ω). Hence we have the following consequence of +Theorem 4.2.10 (ii), [17, 2.12 Satz (1), p. 141] and [17, 3.1 Bemerkung, p. 141]. +4.2.15. Corollary. Let E be an lcHs, Ω ⊂ Rd open, V a Nachbin-family on +Ω which is locally bounded away from zero and P(∂)K a hypoelliptic linear partial +differential operator. +a) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is a semi-Montel space. +b) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E is quasi-complete. +Like before we may improve this result by strengthening the conditions on V +and CVP (∂)(Ω) resp. CV0,P (∂)(Ω) which allows us to weaken the assumptions on +E. +4.2.16. Example. Let E be an lcHs, Ω ⊂ Rd open, V ∶= (νj)j∈N an increasing +family of weights which is locally bounded away from zero on Ω and P(∂)K a +hypoelliptic linear partial differential operator. +a) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is complete and CVP (∂)(Ω) a semi- +Montel space. +b) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is locally complete and CVP (∂)(Ω) a +Schwartz space. +c) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E has metric ccp and νj ∈ C(Ω) for all +j ∈ N. +d) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E is locally complete and CV0,P (∂)(Ω) +a Schwartz space. +Proof. Let F stand for CVP (∂) or CV0,P (∂). The space F(Ω) is a Fréchet space +and hence barrelled by Proposition 4.2.14. The inclusion I∶F(Ω) → CWP (∂)(Ω) is +continuous since V is locally bounded away from zero on Ω. The hypoellipticity +of P(∂)K (see e.g. [70, p. 690]) yields that CWP (∂)(Ω) = CW∞ +P (∂)(Ω) as locally +convex spaces. Thus the inclusion I∶F(Ω) → CW∞ +P (∂)(Ω) is continuous. It follows +from Proposition 3.1.11 c) that the generator (idEΩ,idKΩ) is consistent if F = +CVP (∂), and combined with Proposition 4.1.9 (vanish at infinity w.r.t. (π,K)) if +F = CV0,P (∂) where K and π are chosen as in Example 4.2.11. The strength of the +generator follows as in Proposition 3.1.10 and, if F = CV0,P (∂), in combination with +Proposition 4.1.9 b). This proves part a), b) and d) due to Corollary 3.2.5 (i) and +(ii). +Let us turn to part c). Let f ∈ CV0,P (∂)(Ω,E), j ∈ N and Nj(f) ∶= (fνj)(Ω). +The set K ∶= acx(Nj(f)) is absolutely convex compact by Proposition A.1.3 as +fνj ∈ C0(Ω,E), implying our statement by Corollary 3.2.5 (iv). +□ +At least for some weights and operators P(∂) we can show that CVP (∂)(Ω,E) +coincides with a corresponding space CV∞ +P (∂)(Ω,E) from Example 3.1.9 if E is +locally complete. + +50 +4. CONSISTENCY +4.2.17. Proposition. Let E be a locally complete lcHs, Ω ⊂ Rd and P(∂)K +a hypoelliptic linear partial differential operator. Then we have CWP (∂)(Ω)εE ≅ +CWP (∂)(Ω,E) and CWP (∂)(Ω,E) = CW∞ +P (∂)(Ω,E) as locally convex spaces. +Proof. We already know that +SCW∞ +P (∂)(Ω)∶CW∞ +P (∂)(Ω)εE → CW∞ +P (∂)(Ω,E) +is an isomorphism by Example 3.2.8 b). +The hypoellipticity of P(∂)K (see e.g. +[70, p. 690]) yields that CWP (∂)(Ω)εE = CW∞ +P (∂)(Ω)εE. Thus SCWP (∂)(Ω)(u) = +SCW∞ +P (∂)(Ω)(u) ∈ C∞ +P (∂)(Ω,E) for all u ∈ CWP (∂)(Ω)εE. In particular, we obtain +that +SCWP (∂)(Ω)∶CWP (∂)(Ω)εE → CW∞ +P (∂)(Ω,E) +is an isomorphism. From Proposition 3.1.11 c) and Theorem 3.1.12 with (T E,T K) ∶= +(idEΩ,idKΩ) we deduce that +SCWP (∂)(Ω)∶CWP (∂)(Ω)εE → CWP (∂)(Ω,E) +is an isomorphism into, and from +SCWP (∂)(Ω)(CWP (∂)(Ω)εE) = C∞ +P (∂)(Ω,E) +that CWP (∂)(Ω,E) = CW∞ +P (∂)(Ω,E) as locally convex spaces, which proves our +statement. +□ +Hence the topology τc of compact convergence induced by C(Ω,E) and the +usual topology from Example 3.1.2 induced by C∞(Ω,E) coincide on CP (∂)(Ω,E) if +P(∂)K is hypoelliptic and E locally complete by Proposition 4.2.17. In particular, +we have +(O(Ω,E),τc) = +(4) CW∂(Ω,E) = CW∞ +∂ (Ω,E) +(23) +if E is locally complete. For more interesting weights than W we introduce the +following condition. +4.2.18. Condition. Let V ∶= (νj)j∈N be an increasing family of continuous +weights on Rd. Let there be r∶Rd → (0,1] and for any j ∈ N let there be ψj ∈ L1(Rd), +ψj > 0, and N ∋ Im(j) ≥ j and Am(j) > 0, m ∈ {1,2,3}, such that for any x ∈ Rd: +(α.1) supζ∈Rd, ∥ζ∥∞≤r(x) νj(x + ζ) ≤ A1(j)infζ∈Rd, ∥ζ∥∞≤r(x) νI1(j)(x + ζ), +(α.2) νj(x) ≤ A2(j)ψj(x)νI2(j)(x), +(α.3) νj(x) ≤ A3(j)r(x)νI3(j)(x). +Here, ∥ζ∥∞ ∶= sup1≤n≤d ∣ζn∣ for ζ = (ζn) ∈ Rd. The preceding condition is a +special case of [111, Condition 2.1, p. 176] with Ω ∶= Ωn ∶= Rd for all n ∈ N. If V +fulfils Condition 4.2.18 and we set V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd +0 ∣ ∣β∣ ≤ +m} × Rd → [0,∞), νj,m(β,x) ∶= νj(x), then CV∞(Rd) and its closed subspace +CV∞ +P (∂)(Rd) for P(∂) with continuous coefficients are nuclear by [111, Theorem +3.1, p. 188] in combination with [111, Remark 2.7, p. 178–179] and Fréchet spaces +by [107, Proposition 3.7, p. 240]. +4.2.19. Proposition. Let E be a locally complete lcHs, V ∶= (νj)j∈N an in- +creasing family of continuous weights on Rd and V∞ defined as above. If V ful- +fils Condition 4.2.18, then CV∂(C) and CV∆(Rd) are nuclear Fréchet spaces and +CV∂(C,E) = CV∞ +∂ (C,E) and CV∆(Rd,E) = CV∞ +∆(Rd,E) as locally convex spaces. +Proof. Let P(∂) ∶= ∂ (d ∶= 2 and K ∶= C) or P(∂) ∶= ∆. First, we show that +CVP (∂)(Rd) = CV∞ +P (∂)(Rd) as locally convex spaces, which implies that CVP (∂)(Rd) +is a nuclear Fréchet space as CV∞ +P (∂)(Rd) is such a space. Let f ∈ CV∂(C), j ∈ N, + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +51 +m ∈ N0, z ∈ C and β ∶= (β1,β2) ∈ N2 +0. Then it follows from ∥ ⋅ ∥∞ ≤ ∣ ⋅ ∣ and Cauchy’s +inequality that +∣∂βf(z)∣νj(z) = +(5) ∣iβ2∂∣β∣ +C f(z)∣νj(z) ≤ +∣β∣! +r(z)∣β∣ +sup +∣w−z∣=r(z) +∣f(w)∣νj(z) +≤ +(α.3)∣β∣!C(j,∣β∣) +sup +∣w−z∣=r(z) +∣f(w)∣νB3(j)(z) +≤ +(α.1)∣β∣!C(j,∣β∣)A1(B3(j)) +sup +∣w−z∣=r(z) +∣f(w)∣νI1B3(j)(w) +≤ ∣β∣!C(j,∣β∣)A1(B3(j))∣f∣CV∂(C),I1B3(j) +where C(j,∣β∣) ∶= A3(j)A3(I3(j))⋯A3((B3 − 1)(j)) and B3 − 1 is the (∣β∣ − 1)-fold +composition of I3. Choosing k ∶= max∣β∣≤m I1B3(j), it follows that +∣f∣CV∞ +∂ (C),j,m ≤ sup +∣β∣≤m +∣β∣!C(j,∣β∣)A1(B3(j))∣f∣CV∂(C),k < ∞ +and thus f ∈ CV∞ +∂ (C) and CV∂(C) = CV∞ +∂ (C) as locally convex spaces. +In the +case P(∂) = ∆ an analogous proof works due to Cauchy’s inequality for harmonic +functions, i.e. for all f ∈ CV∆(Rd), j ∈ N, x ∈ Rd and β ∈ Nd +0 it holds that +∣∂βf(x)∣νj(x) ≤ ( d∣β∣ +r(x)) +∣β∣ +sup +∣w−x∣ 0 +∃ K ⊂ Ω compact ∶ ∣f∣Ω∖K,j,m,α < ε} + +52 +4. CONSISTENCY +where +∣f∣Ω∖K,j,m,α ∶= sup +x∈Ω∖K +β∈Mm +pα((∂β)Ef(x))νj,m(β,x). +Further, we define its subspace CVk +0,P (∂)(Ω,E) ∶= {f ∈ CVk +0(Ω,E) ∣ f ∈ kerP(∂)E} +where +P(∂)E∶Ck(Ω,E) → EΩ, P(∂)E(f)(x) ∶= +n +∑ +i=1 +ai(x)(∂βi)E(f)(x), +with n ∈ N, βi ∈ Nd +0 such that ∣βi∣ ≤ k and ai∶Ω → K for 1 ≤ i ≤ n. +4.2.21. Remark. If Vk fulfils condition (V∞) from Example 3.2.9, then we have +CVk +0(Ω,E) = CVk(Ω,E) (see [107, Remark 3.4, p. 239]). +So CWk(Ω,E), S(Rd,E) and OM(Rd,E) are concrete examples of spaces +CVk +0(Ω,E) (see Corollary 3.2.10). +We present the counterpart for differentiable +functions to Bierstedt’s Theorem 4.2.10 for the space CV0(Ω,E) of continuous func- +tions from a completely regular Hausdorff space Ω to an lcHs E weighted with a +Nachbin-family V that vanish at infinity in the weighted topology. For this purpose +we need the following definition. We call Vk locally bounded on Ω if +∀ K ⊂ Ω compact, j ∈ J, m ∈ N0, β ∈ Mm ∶ sup +x∈K +νj,m(β,x) < ∞. +4.2.22. Example. Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights +which is locally bounded away from zero on an open set Ω ⊂ Rd. +a) CVk +0(Ω,E) ≅ CVk +0(Ω)εE if E is quasi-complete, Vk locally bounded and +CVk +0(Ω) barrelled. +b) CVk +0(Ω,E) ≅ CVk +0(Ω)εE if E has metric ccp, CVk +0(Ω) is barrelled and +νj,m(β,⋅) ∈ C(Ω) for all j ∈ J, m ∈ N0, β ∈ Nd +0, ∣β∣ ≤ min(m,k). +c) CVk +0(Ω,E) ≅ CVk +0(Ω)εE if E is locally complete and CVk +0(Ω) a Fréchet– +Schwartz space. +d) CVk +0,P (∂)(Ω,E) ≅ CVk +0,P (∂)(Ω)εE if E is quasi-complete, Vk loc. bounded +and CVk +0,P (∂)(Ω) barrelled. +e) CVk +0,P (∂)(Ω,E) ≅ CVk +0,P (∂)(Ω)εE if E has metric ccp, CVk +0,P (∂)(Ω) is bar- +relled and νj,m(β,⋅) ∈ C(Ω) for all j ∈ J, m ∈ N0, β ∈ Nd +0, ∣β∣ ≤ min(m,k). +f) CVk +0,P (∂)(Ω,E) ≅ CVk +0,P (∂)(Ω)εE if E is locally complete and CVk +0,P (∂)(Ω) +a Fréchet–Schwartz space. +Proof. The generator (T E +m,T K +m)m∈N0 for (CVk +0,E) and (CVk +0,P (∂),E) is given +by domT E +m ∶= Ck(Ω,E) and +T E +m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)], +for all m ∈ N0 and the same with K instead of E. +Set X ∶= Ω, K ∶= {K ⊂ Ω ∣ K compact} and π∶⋃m∈N0 ωm → X, π(β,x) ∶= x. We +have +∣f∣Ω∖K,j,m,α = +sup +x∈ωm +π(x)∉K +pα(T E +m(f)(x))νj,m(x), +for f ∈ CVk +0(Ω,E), K ∈ K, j ∈ J and m ∈ N0, implying that (16) is satisfied. With +APπ,K(Ω,E) from Proposition 4.1.9 we note that +CVk +0(Ω,E) = CVk(Ω,E) ∩ APπ,K(Ω,E). +As in Proposition 3.1.10 it follows that the generator (T E +m,T K +m)m∈N0 fulfils (17) and +(18) where we use Proposition 3.1.11, the barrelledness of CVk +0(Ω) resp. CVk +0,P (∂)(Ω) + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +53 +and the assumption that Vk is locally bounded away from zero on Ω. Therefore the +generator is strong and consistent by virtue of Proposition 4.1.9. +a)+d) Let f ∈ CVk +0(Ω,E), K ∈ K, j ∈ J and m ∈ N0. We claim that the set +Nj,m(f) = {∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} +is precompact in E by Proposition A.1.6. Since f vanishes at infinity in the weighted +topology, condition (i) of Proposition A.1.6 is fulfilled. Hence we only need to show +that condition (ii) is satisfied as well, i.e. we have to show that +Nπ⊂K,j,m(f) = +⋃ +β∈Mm +(∂β)Efνj,m(β,⋅)(K) +is precompact in E. Thus we only have to prove that the sets (∂β)Efνj,m(β,⋅)(K) +are precompact since Nπ⊂K,j,m(f) is a finite union of these sets. +But this is a +consequence of the proof of [15, §1, 16. Lemma, p. 15] using the continuity of +(∂β)Ef and the boundedness of νj,m(β,K), which follows from Vk being locally +bounded. So we deduce statements a) and d) from Corollary 3.2.5 (iv), Proposition +A.1.6 and the quasi-completeness of E. +b)+e) The set Kβ ∶= acx((∂β)Efνj,m(β,⋅)(Ω)) is absolutely convex and com- +pact by Proposition A.1.3 (ii) for every f ∈ CVk +0(Ω,E), j ∈ J, m ∈ N0 and β ∈ Mm +as E has metric ccp and νj,m(β,⋅) ∈ C(Ω). We have +Nj,m(f) = {(∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} ⊂ acx( ⋃ +β∈Mm +Kβ) +and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3 +Proposition, p. 113]. Now, statements b)+e) follow from Corollary 3.2.5 (iv). +c)+f) They follow from Corollary 3.2.5 (ii). +□ +The spaces CVk +0(Ω) are Fréchet spaces and thus barrelled if J is countable by +[107, Proposition 3.7, p. 240]. In [107, Theorem 5.2, p. 255] the question is answered +when they have the approximation property. The spaces CV∞ +0 (Ω) and CV∞ +P (∂),0(Ω) +are closed subspaces of CV∞(Ω) and CV∞ +P (∂)(Ω), respectively. For conditions that +they are Fréchet–Schwartz spaces see the remarks below Example 3.2.7. +We already saw different choices for K in Example 4.2.9 b) and Example 4.2.22. +For holomorphic functions on an open subset Ω of an infinite dimensional Banach +space X the family K of Ω-bounded sets, i.e. bounded sets K ⊂ Ω with positive +distance to X ∖ Ω, is used in [71, p. 2] and [93, p. 2]. This family is clearly closed +under taking finite unions, so Proposition 4.1.9 is applicable as well. +Now, we consider an example of weighted smooth functions where the corre- +sponding space of scalar-valued functions may not be barrelled. For an open set +Ω ⊂ Rd, an lcHs E and a linear partial differential operator P(∂)E which is hypoel- +liptic if E = K we define the space of bounded zero solutions +C∞ +P (∂),b(Ω,E) ∶= {f ∈ C∞ +P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∥f∥∞,α ∶= sup +x∈Ω +pα(f(x)) < ∞} +where C∞ +P (∂)(Ω,E) is the kernel of P(∂)E in C∞(Ω,E). Further, we set C∞ +P (∂),b(Ω) ∶= +C∞ +P (∂),b(Ω,K). +Apart from the topology given by (∥ ⋅ ∥∞,α)α∈A there is another +weighted locally convex topology on C∞ +P (∂),b(Ω,E) which is of interest, namely, the +one induced by the seminorms +∣f∣ν,α ∶= sup +x∈Ω +pα(f(x))∣ν(x)∣, +f ∈ C∞ +P (∂),b(Ω,E), +for ν ∈ C0(Ω) and α ∈ A. We denote by (C∞ +P (∂),b(Ω,E),β) the space C∞ +P (∂),b(Ω,E) +equipped with the topology β induced by the seminorms (∣ ⋅ ∣ν,α)ν∈C0(Ω),α∈A. The +topology β is called the strict topology. It is a bit tricky to prove the ε-compatibility + +54 +4. CONSISTENCY +of (C∞ +P (∂),b(Ω),β) and (C∞ +P (∂),b(Ω,E),β) because (C∞ +P (∂),b(Ω),β) may not be bar- +relled. +4.2.23. Remark. Let Ω ⊂ Rd be open and P(∂)K a hypoelliptic linear partial +differential operator. Then (C∞ +P (∂),b(Ω),β) is non-barrelled if τc does not coincide +with the ∥⋅∥∞-topology by [46, Section I.1, 1.15 Proposition, p. 12], e.g. (C∞ +∂,b(D),β) +is non-barrelled. +Hence we cannot use Proposition 3.1.11 c) directly. +4.2.24. Proposition. Let Ω ⊂ Rd be open, P(∂)K a hypoelliptic linear partial +differential operator and E an lcHs. Then (C∞ +P (∂),b(Ω),β)εE ≅ (C∞ +P (∂),b(Ω,E),β) if +E has metric ccp. +Proof. We set AP(Ω,E) ∶= C∞ +P (∂),b(Ω,E) and observe that (idEΩ,idΩK) is the +generator of ((C∞ +P (∂),b(Ω),β),E). First, we prove that the generator is consistent. +Clearly, we only need to show that S(u) ∈ AP(Ω,E) for every u ∈ (C∞ +P (∂),b(Ω),β)εE. +Let u ∈ (C∞ +P (∂),b(Ω),β)εE. Next, we show that u ∈ CW∞ +P (∂)(Ω)εE with CW∞ +P (∂)(Ω) +from Example 3.1.9 b). For α ∈ A there are an absolutely convex, compact K ⊂ +(C∞ +P (∂),b(Ω),β) and C > 0 such that for all f ′ ∈ (C∞ +P (∂),b(Ω),β)′ it holds that +pα(u(f ′)) ≤ C sup +f∈K +∣f ′(f)∣. +(24) +We denote by τc the topology of compact convergence on C∞ +P (∂),b(Ω), i.e. the topol- +ogy of uniform convergence on compact subsets of Ω. From the compactness of K in +(C∞ +P (∂),b(Ω),β) it follows that K is ∥ ⋅ ∥∞-bounded and τc-compact by [45, Proposi- +tion 1 (viii), p. 586] since (C∞ +P (∂),b(Ω),β) carries the induced topology of (Cb(Ω),β) +and the strict topology β is the mixed topology γ(τc,∥ ⋅ ∥∞) by [45, Proposition 3, +p. 590]. Let f ′ ∈ (C∞ +P (∂)(Ω),τc)′. Then there are M ⊂ Ω compact and C0 > 0 such +that +∣f ′(f)∣ ≤ C0 sup +x∈M +∣f(x)∣ +for all f ∈ C∞ +P (∂)(Ω). Choosing a compactly supported cut-off function ν ∈ C∞ +c (Ω) +with ν = 1 near M, we obtain +∣f ′(f)∣ ≤ C0 sup +x∈Ω +∣f(x)∣∣ν(x)∣ = C0∣f∣ν +for all f ∈ C∞ +P (∂)(Ω). +Therefore f ′ ∈ (C∞ +P (∂)(Ω),β)′. +In combination with the +τc-compactness of K it follows from (24) that u ∈ (C∞ +P (∂)(Ω),τc)εE. Using that +(C∞ +P (∂)(Ω),τc) = CW∞ +P (∂)(Ω) as locally convex spaces by the hypoellipticity of +P(∂)K (see e.g. [70, p. 690]), we obtain that u ∈ CW∞ +P (∂)(Ω)εE. Due to Propo- +sition 3.1.11 c) this yields that S(u) ∈ C∞ +P (∂)(Ω,E). Furthermore, we note that +∥S(u)∥∞,α = sup +x∈Ω +pα(S(u)(x)) = sup +x∈Ω +pα(u(δx)) ≤ +(24) C sup +x∈Ω +sup +f∈K +∣δx(f)∣ += C sup +f∈K +∥f∥∞ < ∞ +as K is ∥ ⋅ ∥∞-bounded, implying that S(u) ∈ C∞ +P (∂),b(Ω,E) = AP(Ω,E). Hence the +generator (idEΩ,idΩK) is consistent. +It is easily seen that e′ ○ f ∈ C∞ +P (∂),b(Ω) = AP(Ω) for all e′ ∈ E′ and f ∈ +C∞ +P (∂),b(Ω,E) (see the proof of Proposition 3.1.10), which proves that the gener- +ator is strong as well. +Moreover, we define Nν(f) ∶= {f(x)∣ν(x)∣ ∣ x ∈ Ω} for +f ∈ (C∞ +P (∂),b(Ω,E),β) and ν ∈ C0(Ω). The set K ∶= acx(Nν(f)) is absolutely con- +vex and compact in E by Proposition A.1.3 (ii) because f∣ν∣ ∈ C0(Ω,E) and Ω +second-countable, yielding our statement by Corollary 3.2.5 (iv). +□ + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +55 +If Ω ⊂ C is an open, simply connected set, P(∂) = ∂ and E is complete, then +the preceding result is also a consequence of [17, 3.10 Satz, p. 146]. +Next, we consider the vector-valued Beurling–Björck space Sµ(Rd,E) which +generalises the Schwartz space and whose scalar-valued counterpart was studied by +Björck in [20], by Schmeisser and Triebel in [155] (see [20, Definition 1.8.1, p. 375], +[155, 1.2.1.2 Definition, p. 15]) whereas semigroups on its toplogical dual space +were treated by Alvarez et al. in [5]. Since Fourier transformation is involved in the +definition of Sµ(Rd,E), we start with the following statement. +4.2.25. Proposition. Let E be a locally complete lcHs over C, f ∈ S(Rd,E) +and x ∈ Rd. Then fe−i⟨x,⋅⟩ is Pettis-integrable on Rd where ⟨⋅,⋅⟩ is the usual scalar +product on Rd. +Proof. We choose m ∶= d + 1 and set ψ∶Rd → [0,∞), ψ(ζ) ∶= (1 + ∣ζ∣2)−m/2, as +well as g∶Rd → [0,∞), g(ζ) ∶= ψ(ζ)−1. Then ψ ∈ L1(Rd,λ) and ψg = 1. Moreover, +let x = (xi) ∈ Rd and set u∶Rd → E, u(ζ) ∶= f(ζ)e−i⟨x,ζ⟩g(ζ). We note that +(∂en)Eu(ζ) += (∂en)Ef(ζ)e−i⟨x,ζ⟩g(ζ) − ixnf(ζ)e−i⟨x,ζ⟩g(ζ) + mf(ζ)e−i⟨x,ζ⟩(1 + ∣ζ∣2)(m/2)−1ζn +for all ζ = (ζi) ∈ Rd and 1 ≤ n ≤ d, which implies +pα((∂en)Eu(ζ)) ≤ pα((∂en)Ef(ζ))g(ζ) + ∣xn∣pα(f(ζ))g(ζ) + mpα(f(ζ))g(ζ) +for all α ∈ A and hence +sup +ζ∈Rd +β∈Nd +0,∣β∣≤1 +pα((∂β)Eu(ζ)) ≤ (1 + ∣xn∣ + m)∣f∣S(Rd),m,α. +Therefore u = fe−i⟨x,⋅⟩g is (weakly) C1 +b , which yields u ∈ C[1] +b (Rd,E) by Proposition +A.1.5. Now, we choose h∶Rd → (0,∞), h(ζ) ∶= 1 + ∣ζ∣2. Then +sup +ζ∈Rd pα(u(ζ)h(ζ)) ≤ sup +ζ∈Rd pα(f(ζ))(1 + ∣ζ∣2)(m+2)/2 ≤ ∣f∣S(Rd),m+2,α < ∞ +for all α ∈ A, and for every ε > 0 there is r > 0 such that 1 ≤ εh(ζ) for all ζ ∉ Br(0) =∶ +K. +We deduce from Proposition A.2.7 (iii) that fe−i⟨x,⋅⟩ is Pettis-integrable on +Rd. +□ +Thus, for f ∈ S(Rd,E) with locally complete E the Fourier transformation +FE(f)∶Rd → E, FE(f)(x) ∶= (2π)−d/2 ∫ +Rd +f(ζ)e−i⟨x,ζ⟩dζ, +is defined. From the Pettis-integrability we get (e′ ○ FE)(f) = FC(e′ ○ f) for every +e′ ∈ E′. As FC(e′ ○ f) ∈ S(Rd) for every e′ ∈ E′ by [20, Proposition 1.8.2, p. 375], +we obtain from the weak-strong principle Corollary 5.2.21 (or [30, Theorem 9, p. +232] and [131, Mackey’s theorem 23.15, p. 268]) that FE(f) ∈ S(Rd,E). +For a locally complete lcHs E over C and a continuous function µ∶Rd → [0,∞) +such that +(γ) there are a ∈ R, b > 0 with µ(x) ≥ a + bln(1 + ∣x∣) for all x ∈ Rd, +we set +Sµ(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ m,j ∈ N0, α ∈ A ∶ ∣f∣j,m,α < ∞} +where ∣f∣m,j,α ∶= max(qm,j,α(f),qm,j,α(FE(f))) with +qm,j,α(f) ∶= +sup +x∈Rd +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x))ejµ(x). + +56 +4. CONSISTENCY +We note that from qm,j,α(f) < ∞ for all m,j ∈ N0, α ∈ A and condition (γ) it follows +that f ∈ S(Rd,E) and hence qm,j,α(FE(f)) is defined. Further, we set Sµ(Rd) ∶= +Sµ(Rd,C). We observe that Sµ(Rd,E) is a dom-space. Indeed, let ωm ∶= ̃ωm ∪ ̃ωm,1 +where ̃ωm ∶= Mm × Rd with Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ m} and ̃ωm,1 ∶= ̃ωm × {1} for all +m ∈ N0. Setting domT E +m ∶= S(Rd,E) and T E +m∶S(Rd,E) → Eωm by +T E +m(f)(β,x) ∶= (∂β)Ef(x) and T E +m(f)(β,x,1) ∶= ((∂β)E ○ FE)f(x), +(β,x) ∈ ̃ωm, +for every m ∈ N0 as well as AP(Rd,E) ∶= ERd, we have that Sµ(Rd,E) is a dom- +space with weights given by νj,m(β,x) ∶= νj,m(β,x,1) ∶= ejµ(x) for all (β,x) ∈ ̃ωm +and m,j ∈ N0. +The condition (γ) is introduced in [20, p. 363]. Choosing µ(x) ∶= ln(1 + ∣x∣), +x ∈ Rd, we get the Schwartz space Sµ(Rd,E) = S(Rd,E) back. +4.2.26. Example. Let E be a locally complete lcHs over C and µ∶Rd → [0,∞) +continuous such that condition (γ) is fulfilled. +(i) If E has metric ccp, or +(ii) if µ ∈ C1(Rd) and there are k ∈ N0, C > 0 such that ∣∂enµ(x)∣ ≤ Cekµ(x) +for all x ∈ Rd and 1 ≤ n ≤ d, +then Sµ(Rd,E) ≅ Sµ(Rd)εE. +Proof. First, we show that the generator (T E +m,T C +m)m∈N0 for (Sµ,E) is strong +and consistent. From +(∂β)C(e′ ○ f)(x) = e′((∂β)Ef(x)), +(β,x) ∈ ̃ωm, +where ̃ωm = {β ∈ Nd +0 ∣ ∣β∣ ≤ m} × Rd, we get in combination with the Pettis- +integrability by Proposition 4.2.25 that +((∂β)C ○ FC)(e′ ○ f)(x) = e′(((∂β)E ○ FE)f(x)), +(β,x) ∈ ̃ωm +(25) +for all e′ ∈ E′, f ∈ Sµ(Rd,E) and m ∈ N0, which means that the generator is +strong. For consistency we consider the case µ(x) = ln(1 + ∣x∣), x ∈ Rd, i.e. the +Schwartz space, first. Due to Corollary 3.2.10 the map S∶S(Rd)εE → S(Rd,E) is +an isomorphism and according to Theorem 3.2.4 its inverse is given by +Rt∶S(Rd,E) → S(Rd)εE, f ↦ J −1 ○ Rt +f. +Let u ∈ S(Rd)εE. Thanks to the proof of Corollary 3.2.10 we only need to show +that +u(δx ○ (∂β ○ FC)) = (∂β)EFE(S(u))(x), +x ∈ Rd. +We set f ∶= S(u) ∈ S(Rd,E) and from (25) we obtain +Rt +f(δx ○ ((∂β)C ○ FC))(e′) = (∂β)CFC(e′ ○ f)(x) = e′((∂β)EFE(f)(x)), +e′ ∈ E′, +for all x ∈ Rd and β ∈ Nd +0, which results in +u(δx ○ ((∂β)C ○ FC)) = S−1(f)(δx ○ ((∂β)C ○ FC)) = J −1(Rt +f(δx ○ ((∂β)C ○ FC))) += (∂β)EFE(f)(x) = (∂β)EFE(S(u))(x). +(26) +Thus (T E +m,T C +m)m∈N0 is a consistent generator for (S,E). +Let us turn to general µ. Let u ∈ Sµ(Rd)εE. We show that u ∈ S(Rd)εE. Then +it follows from the first part of the proof that (T E +m,T C +m)m∈N0 is a consistent generator +for (Sµ,E). For α ∈ A there are an absolutely convex compact set K ⊂ Sµ(Rd) and +C > 0 such that for all f ′ ∈ Sµ(Rd)′ it holds +pα(u(f ′)) ≤ C sup +f∈K +∣f ′(f)∣. +(27) + +4.2. FURTHER EXAMPLES OF ε-PRODUCTS +57 +The compactness of K in Sµ(Rd) and the estimate +sup +x∈Rd +β∈Nd +0,∣β∣≤m +∣(∂β)Cf(x)∣(1 + ∣x∣2)j/2 ≤ +sup +x∈Rd +β∈Nd +0,∣β∣≤m +∣(∂β)Cf(x)∣e(j/2)(2/b)(µ(x)−a) +≤ e−(aj)/b∣f∣j,m, +f ∈ Sµ(Rd), +for all j,m ∈ N0 by condition (γ) imply that the inclusion Sµ(Rd) ↪ S(Rd) is +continuous and thus that K is compact in S(Rd). Let f ′ ∈ S(Rd)′. Then there are +j,m ∈ N0 and C0 > 0 such that +∣f ′(f)∣ ≤ C0 +sup +x∈Rd +β∈Nd +0,∣β∣≤m +∣(∂β)Cf(x)∣(1 + ∣x∣2)j/2 ≤ C0e−(aj)/b∣f∣j,m +for all f ∈ Sµ(Rd). Hence f ′ ∈ Sµ(Rd)′ and from (27) we obtain that u ∈ S(Rd)εE +because K is absolutely convex and compact in S(Rd). +Condition (γ) implies that µ(x) → ∞ for ∣x∣ → ∞. Noting that for every j ∈ N +and ε > 0 there is r > 0 such that +ejµ(x) +e2jµ(x) = e−jµ(x) < ε +(28) +for all x ∉ Br(0), we deduce ∣f∣Rd∖Br(0),m,j,α ≤ ε∣f∣m,2j,α for every f ∈ Sµ(Rd,E), +m ∈ N0 and α ∈ A. +(i) Thus, if E has metric ccp, then the sets Kβ ∶= acx((∂β)Efejµ(Rd)) and +Kβ,1 ∶= acx((∂β)EFE(f)ejµ(Rd)) are absolutely convex and compact by Propo- +sition A.1.3 (ii) for every f ∈ Sµ(Rd,E), j,m ∈ N0 and β ∈ Mm as (∂β)Efejµ ∈ +C0(Rd,E) and (∂β)EFE(f)ejµ ∈ C0(Rd,E). +(ii) We set g0∶Rd → E, g0(x) ∶= (∂β)Ef(x)ejµ(x), and g1∶Rd → E, g1(x) ∶= +(∂β)EFE(f)(x)ejµ(x), for j,m ∈ N0 and β ∈ Mm. We observe that +(∂en)Eg0(x) = (∂β+en)Ef(x)ejµ(x) + j(∂β)Ef(x)ejµ(x)∂enµ(x) +and +(∂en)Eg1(x) = (∂β+en)EFE(f)(x)ejµ(x) + j(∂β)EFE(f)(x)ejµ(x)∂enµ(x) +for all x ∈ Rd and 1 ≤ n ≤ d. As in Example 3.2.9 it follows from condition (ii) that +there are k ∈ N0, C > 0 such that +sup +x∈Rd +γ∈Nd +0,∣γ∣≤1 +pα((∂γ)Egi(x)) ≤ ∣f∣m+1,j,α + Cj∣f∣m,j+k,α +for all α ∈ A and i = 0,1. Thus g0 and g1 are (weakly) C1 +b . We set h ∶= ejµ and note +that +sup +x∈Rd pα(gi(x)h(x)) ≤ ∣f∣m,2j,α < ∞ +for all α ∈ A and i = 0,1. This yields that Kβ = acx(g0(Rd)) and Kβ,1 = acx(g1(Rd)) +are absolutely convex and compact by Proposition A.1.4 with (28) and Proposition +A.1.5. +Then we have in both cases +Nj,m(f) = ({(∂β)Ef(x)ejµ(x) ∣ x ∈ Rd, β ∈ Mm} +∪ {(∂β)EFE(f)(x)ejµ(x) ∣ x ∈ Rd, β ∈ Mm}) +⊂acx( ⋃ +β∈Mm +(Kβ ∪ Kβ,1)) + +58 +4. CONSISTENCY +and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3 +Proposition, p. 113], which implies that Sµ(Rd,E) ≅ Sµ(Rd)εE by Corollary 3.2.5 +(iv). +□ +We come back to these spaces in Theorem 5.1.5. +Another example that is +related to Fourier transformation is the space of vector-valued smooth functions +that are 2π-periodic in each variable. We equip the space C∞(Rd,E) for an lcHs E +with the system of seminorms generated by +∣f∣K,m,α ∶= +sup +x∈Rd +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x))χK(x) = +sup +x∈K +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x)), +f ∈ C∞(Rd,E), +for K ⊂ Rd compact, m ∈ N0 and α ∈ A, i.e. we consider CW∞(Rd,E). +By +C∞ +2π(Rd,E) we denote the topological subspace of C∞(Rd,E) consisting of the func- +tions which are 2π-periodic in each variable. Further, we set C∞ +2π(Rd) ∶= C∞ +2π(Rd,K). +4.2.27. Example. If E is a locally complete lcHs, then C∞ +2π(Rd,E) ≅ C∞ +2π(Rd)εE. +Proof. First, we note that for each x ∈ Rd and 1 ≤ n ≤ d we have δx = δx+2πen +in C∞ +2π(Rd)′ and thus +SC∞ +2π(Rd)(u)(x) − SC∞ +2π(Rd)(u)(x + 2πen) = u(δx − δx+2πen) = 0, +u ∈ C∞ +2π(Rd)εE, +implying that SC∞ +2π(Rd)(u) is 2π-periodic in each variable. In addition, we observe +that e′ ○ f is 2π-periodic in each variable for all e′ ∈ E′ and f ∈ C∞ +2π(Rd,E). Now, +we obtain as in Example 3.2.8 a) for k = ∞ that SC∞ +2π(Rd)∶C∞ +2π(Rd)εE → C∞ +2π(Rd,E) +is an isomorphism. +□ +We return to C∞ +2π(Rd,E) in Theorem 5.1.4 and Theorem 5.6.14. Now, we direct +our attention to spaces of continuously partially differentiable functions on an open +bounded set such that all derivatives can be continuously extended to the boundary. +Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open and bounded. The space Ck(Ω,E) is +given by +Ck(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ (∂β)Ef cont. extendable on Ω for all β ∈ Nd +0, ∣β∣ ≤ k} +and equipped with the system of seminorms given by +∣f∣α ∶= +sup +x∈Ω +β∈Nd +0,∣β∣≤k +pα((∂β)Ef(x)), +f ∈ Ck(Ω,E), +for α ∈ A if k < ∞, and by +∣f∣m,α ∶= +sup +x∈Ω +β∈Nd +0,∣β∣≤m +pα((∂β)Ef(x)), +f ∈ C∞(Ω,E), +for m ∈ N0 and α ∈ A if k = ∞. Further, we set Ck(Ω) ∶= Ck(Ω,K). +4.2.28. Example. Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open and bounded. +(i) If E has metric ccp, or +(ii) if E is locally complete, k = ∞ and there exists C > 0 such that for each +x,y ∈ Ω there is a continuous path from x to y in Ω whose length is +bounded by C∣x − y∣, +then Ck(Ω,E) ≅ Ck(Ω)εE. +Proof. The generator coincides with the one of Example 4.2.22. +Due to +Proposition 3.1.11 we have S(u) ∈ Ck(Ω,E) and +(∂β)ES(u)(x) = u(δx ○ (∂β)K), +β ∈ Nd +0, ∣β∣ ≤ k, x ∈ Ω, + +4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS +59 +for all u ∈ Ck(Ω)εE since Ck(Ω) is a Banach space if k < ∞, and a Fréchet space +if k = ∞, in particular, both are barrelled. As a consequence of Proposition 4.1.7 +and Lemma 4.1.8 with T = (∂β)K for β ∈ Nd +0, ∣β∣ ≤ k, we obtain that (∂β)ES(u) ∈ +Cext(Ω,E) for all u ∈ Ck(Ω)εE. Thus the generator is consistent. It is easy to check +that it is strong, too. This yields (ii) by Corollary 3.2.5 (ii) since C∞(Ω) is a nuclear +Fréchet space by [131, Examples 28.9 (5), p. 350] under the conditions on Ω. +Let us turn to part (i). Let f ∈ Ck(Ω,E), J ∶= {1}, m ∈ N0 and set Mm ∶= {β ∈ +Nd +0 ∣ ∣β∣ ≤ k} if k < ∞, and Mm ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ m} if k = ∞. We denote by fβ the +continuous extension of (∂β)Ef on the compact metrisable set Ω. The set +N1,m(f) = {(∂β)Ef(x) ∣ x ∈ Ω, β ∈ Mm} ⊂ +⋃ +β∈Mm +fβ(Ω) +is relatively compact and metrisable since it is a subset of a finite union of the +compact metrisable sets fβ(Ω) as in Example 3.2.8. Due to Corollary 3.2.5 (iv) we +obtain our statement (i) as E has metric ccp. +□ +We close this section by an examination of the topological subspace +E0(E) ∶= {f ∈ C∞([0,1],E) ∣ ∀ k ∈ N0 ∶ (∂k)Ef(1) = 0} +where (∂k)Ef(1) ∶= limx→1+(∂k)Ef(x). Further, we set E0 ∶= E0(K). +4.2.29. Example. Let E be a locally complete lcHs. Then E0εE ≅ E0(E). +Proof. We note that Ω ∶= (0,1) satisfies the condition on Ω in Example 4.2.28 +(ii) with C ∶= 1 and thus C∞([0,1]) and its closed subspace E0 are nuclear Fréchet +spaces. The generator coincides with the one of Example 4.2.28. From the proof of +Example 4.2.28 we know that +lim +x→1+(∂k)ES(u)(x) = u(δ1 ○ (∂k)K) = u(0) = 0, +k ∈ N0, +for all u ∈ E0εE. In combination with Example 4.2.28 this yields the consistency +of the generator. Again, its strength is easy to check. Therefore our statement is +valid by Corollary 3.2.5 (ii). +□ +4.3. Riesz–Markov–Kakutani representation theorems +In this subsection we generalise the concept of strength and consistency such +that it is not strictly bounded to dom-spaces and their generators anymore. This +allows us to answer the question: Given T K +m ∈ F(Ω)′ is there T E +m ∈ L(F(Ω,E),E) +such that (T E +m,T K +m) is strong and consistent? Furthermore, we will see that the +operators T E +m are usually the ones that can be obtained from integral representations +of T K +m, i.e. we transfer Riesz–Markov–Kakutani theorems from the scalar-valued to +the vector-valued case. +We recall that the Riesz–Markov–Kakutani theorem for +compact topological Hausdorff spaces Ω says that for every T R ∈ Cb(Ω)′ there is a +unique regular R-valued Borel measure µ on Ω such that +T R(f) = ∫ +Ω +f(x)dµ(x), +f ∈ Cb(Ω), +(29) +which was proved by Riesz [146, p. 976] in the case Ω ∶= [0,1] and by Kakutani [95, +Theorem 9, p. 1009] for general compact Hausdorff Ω (see Saks [152, Eq. (1.1), 6., p. +408, 411] for compact metric Ω). Markov treated the case where Ω is a normal (not +necessarily Hausdorff) topological space and the T R are positive linear functionals +on Cb(Ω) such that T R(1) = 1 [127, Definition 2, p. 167]. In this case, for every +such T R there is a unique exterior density µ on Ω in the sense of [127, Definition +3, p. 167] such that (29) holds by [127, Theorem 22, p. 184] and the right-hand +side is read in the sense of [127, Eq. (71), (72), (80), p. 180–181] (see also [60, + +60 +4. CONSISTENCY +IV.6.2 Theorem, p. 262] for a more familiar version with regular (finitely) additive +bounded Borel measures µ). +4.3.1. Definition (strong, consistent). Let E be an lcHs and Ω a non-empty +set. +Let F(Ω) ⊂ KΩ and F(Ω,E) ⊂ EΩ be lcHs such that δx ∈ F(Ω)′ for all +x ∈ Ω. +Let (ωm)m∈M be a family of non-empty sets, T K +m∶domT K +m → Kωm and +T E +m∶domT E +m → Eωm be linear with F(Ω) ⊂ domT K +m ⊂ KΩ and F(Ω,E) ⊂ domT E +m ⊂ +EΩ for all m ∈ M. +a) We call (T E +m,T K +m)m∈M a consistent family for (F(Ω),E), in short (F,E), +if we have for every u ∈ F(Ω)εE, m ∈ M and x ∈ ωm that +(i) S(u) ∈ F(Ω,E) and T K +m,x ∶= δx ○ T K +m ∈ F(Ω)′, +(ii) T E +mS(u)(x) = u(T K +m,x). +b) We call (T E +m,T K +m)m∈M a strong family for (F(Ω),E), in short (F,E), if +we have for every e′ ∈ E′, f ∈ F(Ω,E), m ∈ M and x ∈ ωm that +(i) e′ ○ f ∈ F(Ω), +(ii) T K +m(e′ ○ f)(x) = (e′ ○ T E +m(f))(x). +Note that ωm need not be a subset of Ω. As a convention we omit the index m +of the set ωm, the operators T E +m and T K +m if M is a singleton. The following remark +shows that the preceding definition of a consistent resp. strong family coincides with +the usual definition in the case of generators of dom-spaces (see Definition 3.1.7). +4.3.2. Remark. Let (T E +m,T K +m)m∈M be a generator for (FV,E). We note that +the condition T K +m,x ∈ FV(Ω)′ for all m ∈ M and x ∈ ωm in a)(i) of Definition +4.3.1 is always satisfied for generators by Remark 3.1.6 b). Moreover, if S(u) ∈ +AP(Ω,E) ∩ domT E +m for u ∈ FV(Ω)εE and all m ∈ M and a)(ii) of Definition 4.3.1 +is fulfilled, then S(u) ∈ FV(Ω,E) by Lemma 3.1.8, implying that a)(i) is satisfied. +Further, if f ∈ FV(Ω,E) and e′ ○f ∈ AP(Ω)∩domT K +m for all e′ ∈ E′ and m ∈ M and +b)(ii) of Definition 4.3.1 is fulfilled, then e′ ○ f ∈ FV(Ω) by Lemma 3.2.1, implying +that b)(i) is satisfied. +The next proposition is the key result in transferring Riesz–Markov–Kakutani +theorems from the scalar-valued to the vector-valued case. To state this proposition +we need that our map S∶F(Ω)εE → F(Ω,E) is an isomorphism and that its inverse +is given as in Theorem 3.2.4, i.e. that +Rt∶F(Ω,E) → F(Ω)εE, f ↦ J −1 ○ Rt +f, +is the inverse of S where Rt +f(f ′)(e′) = f ′(e′ ○ f), for f ′ ∈ F(Ω)′ and e′ ∈ E′, and +J ∶E → E′⋆ is the canonical injection in the algebraic dual E′⋆ of E′. +4.3.3. Proposition. Let E be an lcHs, (Ω,Σ,µ) a measure space and F(Ω) +and F(Ω,E) ε-compatible with inverse Rt of S and (T E +0 ,T K +0 ) a strong family for +(F,E) with ω0 ∶= Ω. +If T K +0 (f) is integrable for every f ∈ F(Ω) and T E +0 (f) is +Pettis-integrable on Ω for every f ∈ F(Ω,E) and +T K∶F(Ω) → K, T K(f) ∶= ∫ +Ω +T K +0 (f)(x)dµ(x), +is continuous, then +u(T K) = ∫ +Ω +T E +0 S(u)(x)dµ(x), +u ∈ F(Ω)εE. +Proof. Let u ∈ F(Ω)εE and set f ∶= S(u) ∈ F(Ω,E). We have +Rt +f(T K)(e′) = T K(e′○f) = ∫ +Ω +T K +0 (e′○f)(x)dµ(x) = ⟨e′,∫ +Ω +T E +0 f(x)dµ(x)⟩, +e′ ∈ E′, + +4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS +61 +by the strength of (T E +0 ,T K +0 ) and the Pettis-integrability of T E +0 (f), which yields +u(T K) = S−1(f)(T K) = J −1(Rt +f(T K)) = ∫ +Ω +T E +0 f(x)dµ(x) = ∫ +Ω +T E +0 S(u)(x)dµ(x) +due to Rt being the inverse of S. +□ +4.3.4. Proposition. Let E be an lcHs, (Ω,Σ,µ) a measure space, (T E +0 ,T K +0 ) a +strong family for (F,E) with ω0 ∶= Ω such that T E +0 (f) is Pettis-integrable on Ω for +every f ∈ F(Ω,E), and (T E,T K) a consistent family for (F,E) such that +T E(f) = ∫ +Ω +T E +0 f(x)dµ(x), +f ∈ F(Ω,E). +Then (T E,T K) is a strong family for (F,E), T K +0 (f) is integrable for every f ∈ F(Ω) +and +T K = ∫ +Ω +T K +0 (f)(x)dµ(x), +f ∈ F(Ω). +Proof. We set f ⋅ e∶Ω → E, (f ⋅ e)(x) ∶= f(x)e, for e ∈ E and f ∈ F(Ω). Since +(T E,T K) is a consistent family for (F,E), we get f ⋅e = S(Θ(e⊗f)) ∈ F(Ω,E) and +T E(f ⋅ e) = Θ(e ⊗ f)(T K) = T K(f)e +(30) +with the map Θ from (3). From the strength of (T E +0 ,T K +0 ) we deduce that +e′ ○ T E +0 (f ⋅ e) = T K +0 (e′ ○ (f ⋅ e)) = T K +0 (e′(e)f) = e′(e)T K +0 (f) +and from the Pettis-integrability of T E +0 (f ⋅ e) that +T K(f)e′(e) = +(30) ⟨e′,T E(f ⋅ e)⟩ = ∫ +Ω +e′(e)T K +0 (f)(x)dµ(x) +for all e′ ∈ E′. This implies that e′(e)T K +0 (f) is integrable for all e′ ∈ E′. Further, +since E is non-trivial by our assumptions in Chapter 2, there is some e0 ∈ E, e0 ≠ 0. +By the Hahn–Banach theorem there is some e′ +0 ∈ E′ with e′ +0(e0) ≠ 0, which yields +that T K +0 (f) is integrable and +T K(f) = ∫ +Ω +T K +0 (f)(x)dµ(x). +Furthermore, we conclude in combination with the strength of (T E +0 ,T K +0 ) and the +Pettis-integrability of T E +0 (f) for all f ∈ F(Ω,E) that +⟨e′,T E(f)⟩ = ∫ +Ω +T K +0 (e′ ○ f)(x)dµ(x) = T K(e′ ○ f) +for all f ∈ F(Ω,E) and e′ ∈ E′, which means that (T E,T K) is a strong family for +(F,E). +□ +Let us apply the preceding propositions to the space D([0,1],E) of E-valued +càdlàg functions on [0,1]. For f ∈ D([0,1],E) we set f(x−) ∶= limw→x− f(w) if +x ∈ (0,1], and f(0−) ∶= 0. +4.3.5. Proposition. Let E be a quasi-complete lcHs. Then for every T K ∈ +D([0,1])′ there is T E ∈ L(D([0,1],E),E) such that (T E,T K) is a consistent family +for (D,E) and there are a unique regular K-valued Borel measure µ on [0,1] and +a unique ϕ ∈ ℓ1([0,1],K) such that +T E(f) = ∫ +[0,1] +f(x)dµ(x) + +∑ +x∈[0,1] +(f(x) − f(x−))ϕ(x), +f ∈ D([0,1],E). +(31) + +62 +4. CONSISTENCY +On the other hand, if (T E,T K) is a consistent family, there is a unique regular K- +valued Borel measure µ on [0,1] such that (31) holds and T E ∈ L(D([0,1],E),E). +Proof. Due to the representation theorem [139, Theorem 1, p. 383] there are +a unique regular K-valued Borel measure µ on [0,1] and a unique ϕ ∈ ℓ1([0,1],K) +such that +T K(f) = ∫ +[0,1] +f(x)dµ(x) + +∑ +x∈[0,1] +(f(x) − f(x−))ϕ(x), +f ∈ D([0,1],K). +(32) +By Example 4.2.5 S∶D([0,1])εE → D([0,1],E) is an isomorphism with inverse +Rt∶f ↦ J ○ Rt +f. +The next part is the analogon of Proposition 4.3.3 for ∑x∈[0,1]. We set +T K +1 ∶D([0,1]) → K, T K +1 (f) ∶= +∑ +x∈[0,1] +(f(x) − f(x−))ϕ(x), +and note that T K +1 ∈ D([0,1])′. Let u ∈ D([0,1])εE and set f ∶= S(u) ∈ D([0,1],E). +We have +Rt +f(T K +1 )(e′) = T K +1 (e′ ○ f) = +∑ +x∈[0,1] +((e′ ○ f)(x) − (e′ ○ f)(x−))ϕ(x) += ⟨e′, +∑ +x∈[0,1] +(f(x) − f(x−))ϕ(x)⟩, +e′ ∈ E′, +due to the Pettis-summability of x ↦ (f(x) − f(x−))ϕ(x) on [0,1] by Proposition +A.2.6, which yields +u(T K +1 ) = S−1(f)(T K +1 ) = J −1(Rt +f(T K +1 )) = +∑ +x∈[0,1] +(f(x) − f(x−))ϕ(x) += +∑ +x∈[0,1] +(S(u)(x) − S(u)(x−))ϕ(x). +(33) +We note that every f ∈ D([0,1],E) is Pettis-integrable and that x ↦ (f(x) − +f(x−))ϕ(x) is Pettis-summable on [0,1] by Proposition A.2.6. Further, +pα( ∫ +[0,1] +f(x)dµ(x)+ ∑ +x∈[0,1] +(f(x)−f(x−))ϕ(x)) ≤ (∣µ∣([0,1])+2∥ϕ∥ℓ1) sup +x∈[0,1] +pα(f(x)) +for all f ∈ D([0,1],E) and α ∈ A. The rest follows from Proposition 4.3.3 with +(T E +0 ,T K +0 ) ∶= (idE[0,1],idK[0,1]) combined with (33). For the uniqueness of µ in (31) +use that the µ in (32) is unique and Proposition 4.3.4 (and for the uniqueness of ϕ +use an analogon of Proposition 4.3.4 for (T E +1 ,T K +1 )). +□ +Let us turn to continuous functions that vanish at infinity. +4.3.6. Proposition. Let Ω be a locally compact [second countable] topological +Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ C0(Ω)′ there +is T E ∈ L(C0(Ω,E),E) such that (T E,T K) is a consistent family for (C0,E) and +there is a unique regular K-valued Borel measure µ on Ω such that +T E(f) = ∫ +Ω +f(x)dµ(x), +f ∈ C0(Ω,E). +(34) +On the other hand, if (T E,T K) is a consistent family, then there is a unique regular +K-valued Borel measure µ on Ω such that (34) holds and T E ∈ L(C0(Ω,E),E). + +4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS +63 +Proof. Due to the Riesz–Markov–Kakutani representation theorem (see [149, +6.19 Theorem, p. 130]) there is a unique regular K-valued Borel measure µ on Ω +such that +T K(f) = ∫ +Ω +f(x)dµ(x), +f ∈ C0(Ω). +(35) +By Example 4.2.11 S∶C0(Ω)εE → C0(Ω,E) is an isomorphism with inverse Rt∶f ↦ +J ○ Rt +f. We note that every f ∈ C0(Ω,E) is Pettis-integrable by Proposition A.2.7 +(i) resp. (ii) with ψ ∶= g ∶= 1 since +∫ +Ω +∣ψ(x)∣d∣µ∣(x) = ∣µ∣(Ω) < ∞ +and +pα(∫ +Ω +f(x)dµ(x)) ≤ ∣µ∣(Ω)sup +x∈Ω +pα(f(x)), +f ∈ C0(Ω,E), +for all α ∈ A. The rest follows from Proposition 4.3.3 with (T E +0 ,T K +0 ) ∶= (idEΩ,idKΩ). +For the uniqueness of µ in (34) use that the µ in (35) is unique and Proposition +4.3.4. +□ +Next, we consider the space of bounded continuous E-valued functions on a +locally compact topological Hausdorff space Ω, i.e. +Cb(Ω,E) = {f ∈ C(Ω,E) ∣ ∀ α ∈ A ∶ sup +x∈Ω +pα(f(x)) < ∞}, +but equipped with the strict topology β (see Remark 4.2.23) which is induced by +the seminorms +∣f∣ν,α ∶= sup +x∈Ω +pα(f(x))∣ν(x)∣, +f ∈ Cb(Ω,E), +for ν ∈ C0(Ω) and α ∈ A. +4.3.7. Proposition. Let Ω be a locally compact [second countable] topological +Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ (Cb(Ω),β)′ +there is T E ∈ L((Cb(Ω,E),β),E) such that (T E,T K) is a consistent family for +((Cb(Ω),β),E) and there is a unique regular K-valued Borel measure µ on Ω such +that +T E(f) = ∫ +Ω +f(x)dµ(x), +f ∈ Cb(Ω,E). +(36) +On the other hand, if (T E,T K) is a consistent family, then there is a unique regular +K-valued Borel measure µ on Ω such that (36) holds and T E ∈ L((Cb(Ω,E),β),E). +Proof. Due to the Riesz–Markov–Kakutani representation theorem [89, 7.6.3 +Theorem, p. 141] for the strict topology there is a unique regular K-valued Borel +measure µ on Ω such that +T K(f) = ∫ +Ω +f(x)dµ(x), +f ∈ Cb(Ω). +Since T K is continuous, there are ν ∈ C0(Ω) and C > 0 such that +∣∫ +Ω +⟨e′,f(x)⟩dµ(x)∣ = ∣T K(e′ ○ f)∣ ≤ C sup +x∈Ω +∣(e′ ○ f)(x)ν(x)∣ ≤ C sup +x∈K +∣e′(x)∣, +e′ ∈ E′, +for f ∈ Cb(Ω,E) with K ∶= acx(fν(Ω)). As K is absolutely convex and compact by +Proposition A.1.3, f is Pettis-integrable on Ω w.r.t. µ by the Mackey–Arens theo- +rem. The remaining parts of the proof follow from Example 4.2.11 and Proposition +4.3.3 as in Proposition 4.3.6. +□ + +64 +4. CONSISTENCY +4.3.8. Proposition. Let Ω be a locally compact [second countable] topological +Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ CW(Ω)′ there +is T E ∈ L(CW(Ω,E),E) such that (T E,T K) is a consistent family for (CW,E) and +there is a unique regular K-valued Borel measure µ on Ω with compact support such +that +T E(f) = ∫ +Ω +f(x)dµ(x), +f ∈ C(Ω,E). +(37) +On the other hand, if (T E,T K) is a consistent family, then there is a unique regular +K-valued Borel measure µ on Ω with compact support such that (37) holds and +T E ∈ L(CW(Ω,E),E). +Proof. By the Riesz–Markov–Kakutani representation theorem given in the +remark after [35, Chap. 4, §4.8, Proposition 14, p. INT IV.48] for the topology of +compact convergence there is a unique regular K-valued Borel measure µ on Ω with +compact support such that +T K(f) = ∫ +Ω +f(x)dµ(x), +f ∈ C(Ω). +Since T K is continuous, there are a compact set M ⊂ Ω and C > 0 such that +∣∫ +Ω +⟨e′,f(x)⟩dµ(x)∣ = ∣T K(e′ ○ f)∣ ≤ C sup +x∈M +∣(e′ ○ f)(x)∣ ≤ C sup +x∈K +∣e′(x)∣, +e′ ∈ E′, +for f ∈ C(Ω,E) with the absolutely convex and compact set K ∶= acx(f(M)), +implying that f is Pettis-integrable on Ω w.r.t. µ by the Mackey–Arens theorem. +The rest of the proof is identical to the one of Proposition 4.3.7. +□ +4.3.9. Proposition. Let Ω ⊂ Rd be open and E a locally complete lcHs. Then +for every T K ∈ CW∞(Ω)′ there is T E ∈ L(CW∞(Ω,E),E) such that (T E,T K) is +a consistent family for (CW∞,E). Given any open neighbourhood U ⊂ Ω of the +compact distributional support suppT K of T K there are m ∈ N0 and a family of +K-valued Radon measures (µβ)β∈Nd +0,∣β∣≤m on Ω such that suppµβ ⊂ U for all β ∈ Nd +0, +∣β∣ ≤ m, and +T E(f) = ∑ +∣β∣≤m +∫ +Ω +(∂β)Ef(x)dµβ(x), +f ∈ C∞(Ω,E). +(38) +On the other hand, if (T E,T K) is a consistent family, then given any open neigh- +bourhood U ⊂ Ω of the compact distributional support suppT K of T K there are +m ∈ N0 and a family of K-valued Radon measures (µβ)β∈Nd +0,∣β∣≤m on Ω such that +(38) holds, suppµβ ⊂ U for all β ∈ Nd +0, ∣β∣ ≤ m and T E ∈ L(CW∞(Ω,E),E). +Proof. T K ∈ CW∞(Ω)′ is a distribution with compact support and thus has +finite order by [171, Corollary, p. 259]. Denote by m ∈ N0 the order of T K. Given +any open neighbourhood U ⊂ Ω of suppT K there is a family of K-valued Radon +measures (µβ)β∈Nd +0,∣β∣≤m on Ω such that +T K(f) = ∑ +∣β∣≤m +∫ +Ω +(∂β)Kf(x)dµβ(x), +f ∈ C∞(Ω), +and suppµβ ⊂ U for all β ∈ Nd +0, ∣β∣ ≤ m by [171, Theorem 24.4, p. 259]. Since the +support Kβ ∶= suppµβ of µβ is compact +∫ +Ω +(∂β)Kf(x)dµβ(x) = ∫ +Kβ +(∂β)Kf(x)dµβ(x), +f ∈ C∞(Ω), + +4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS +65 +and (∂β)Ef ∈ C1(Ω,E) for f ∈ C∞(Ω,E), it follows from Lemma A.2.2 and Remark +A.2.4 that (∂β)Ef is Pettis-integrable on Ω w.r.t. µβ for all β and that +pα(∫ +Ω +(∂β)Ef(x)dµβ(x)) ≤ ∣µβ∣(Kβ) sup +x∈Kβ +pα((∂β)Ef(x)), +α ∈ A. +By Example 3.2.8 a) the map S∶CW∞(Ω)εE → CW∞(Ω,E) is an isomorphism with +inverse Rt∶f ↦ J ○ Rt +f. The remaining parts of the proof follow from Proposition +4.3.3 with (T E +0 ,T K +0 ) ∶= ((∂β)E,(∂β)K). +□ +4.3.10. Proposition. Let E be a locally complete lcHs. Then for every T K ∈ +S(Rd)′ there is T E ∈ L(S(Rd,E),E) such that (T E,T K) is a consistent family for +(S,E) and there are m ∈ N0 and a family of continuous functions (gβ)β∈Nd +0,∣β∣≤m on +Rd growing at infinity slower than some polynomial such that +T E(f) = ∑ +∣β∣≤m +∫ +Rd +gβ(x)(∂β)Ef(x)dx, +f ∈ S(Rd,E). +(39) +On the other hand, if (T E,T K) is a consistent family, then there are m ∈ N0 and a +family of continuous functions (gβ)β∈Nd +0,∣β∣≤m on Rd growing at infinity slower than +some polynomial such that (39) holds and T E ∈ L(S(Rd,E),E). +Proof. Let T K ∈ S(Rd)′. Then there are m ∈ N0 and a family of continuous +functions (gβ)β∈Nd +0,∣β∣≤m on Rd growing at infinity slower than some polynomial such +that +T K(f) = ∑ +∣β∣≤m +∫ +Rd +gβ(x)(∂β)Kf(x)dx, +f ∈ S(Rd), +by [171, Theorem 25.4, p. 272]. +Here, gβ growing at infinity slower than some +polynomial means that there are k ∈ N0 and C > 0 such that ∣gβ(x)∣ ≤ C(1+∣x∣2)k/2 +for all x ∈ Rd. Since the family (gβ) is finite, we can take one k and one C for all +β. Due to the proof of Example 3.2.9 and Corollary 3.2.10 we know that Kβ ∶= +acx(((∂β)Ef)(1 + ∣ ⋅ ∣2)k/2(Rd)) is absolutely convex and compact for f ∈ S(Rd,E). +The estimate +∣∫ +Rd +⟨e′,gβ(x)(∂β)Ef(x)⟩dx∣ ≤ C sup +x∈Rd ∣e′((∂β)Ef(x))∣(1 + ∣x∣2)k/2 = C sup +x∈Kβ +∣e′(x)∣ +for all e′ ∈ E′ and f ∈ S(Rd,E) yields that gβ(∂β)Ef is Pettis-integrable on Rd +w.r.t. the Lebesgue measure by the Mackey–Arens theorem. Further, it implies +that +pα(∫ +Rd +gβ(x)(∂β)Ef(x)dx) ≤ C sup +x∈Rd pα((∂β)Ef(x))(1 + ∣x∣2)k/2, +α ∈ A, +as in Lemma A.2.2. By Corollary 3.2.10 the map S∶S(Rd)εE → S(Rd,E) is an +isomorphism with inverse Rt∶f ↦ J ○ Rt +f. The remaining parts of the proof follow +from Proposition 4.3.3 with (T E +0 ,T K +0 ) ∶= (gβ(∂β)E,gβ(∂β)K). +□ +4.3.11. Remark. +a) Let Ω ⊂ Rd be open and E a locally complete lcHs. +Then Proposition 4.3.9 is still valid with CW∞ replaced by CW∞ +P (∂) due +to the Hahn–Banach theorem and Example 3.2.8 b). If P(∂)K is a hy- +poelliptic linear partial differential operator, then one can represent T E +as in (37) due to Proposition 4.2.17 but the measure µ need not be unique +anymore. + +66 +4. CONSISTENCY +b) Let Ω ⊂ Rd be open and E an lcHs with metric ccp. Then Proposition +4.3.7 is still valid with Cb replaced by C∞ +P (∂),b for a hypoelliptic linear +partial differential operator P(∂)K due to the Hahn–Banach theorem and +Proposition 4.2.24 but the measure µ need not be unique anymore. +c) All families (T E,T K) considered in this section are strong which is a con- +sequence of Proposition 4.3.4 (and of Pettis-summability in Proposition +4.3.5). + +CHAPTER 5 +Applications +5.1. Lifting the properties of maps from the scalar-valued case +In this section we briefly show how to use the ε-compatibility of spaces F(Ω) +and F(Ω,E) to lift properties like injectivity, surjectivity, bijectivity and continu- +ity from a map T K to a map T E if (T E,T K) forms a consistent family. Especially, +we pay attention to surjectivity whose transfer to the vector-valued case is ac- +complished by Grothendieck’s classical theory of tensor products of Fréchet spaces +[83] and by the splitting theory of Vogt for Fréchet spaces [173] and of Bonet and +Domański for PLS-spaces [54]. In order to apply splitting theory, we recall the +definitions of the topological invariants (Ω), (DN) and (PA). +Let us recall that a Fréchet space F with an increasing fundamental system of +seminorms (∣∣∣⋅∣∣∣k)k∈N satisfies (Ω) if +∀ p ∈ N ∃ q ∈ N ∀ k ∈ N ∃ n ∈ N, C > 0 ∀ r > 0 ∶ Uq ⊂ CrnUk + 1 +r Up +where Uk ∶= {x ∈ F ∣ ∣∣∣x∣∣∣k ≤ 1} (see [131, Chap. 29, Definition, p. 367]). +We recall that a Fréchet space (F,(∣∣∣⋅∣∣∣k)k∈N) satisfies (DN) by [131, Chap. 29, +Definition, p. 359] if +∃ p ∈ N ∀ k ∈ N ∃ n ∈ N, C > 0 ∀ x ∈ F ∶ ∣∣∣x∣∣∣2 +k ≤ C∣∣∣x∣∣∣p∣∣∣x∣∣∣n. +A PLS-space is a projective limit X = lim +←� +N∈N +XN, where the XN given by inductive +limits XN = lim +�→ +n∈N +(XN,n,∣∣∣⋅∣∣∣N,n) are DFS-spaces (which are also called LS-spaces), +and it satisfies (PA) if +∀ N ∃ M ∀ K ∃ n ∀ m ∀ η > 0 ∃ k,C,r0 > 0 ∀ r > r0 ∀ x′ ∈ X′ +N ∶ +∣∣∣x′ ○ iM +N ∣∣∣ +∗ +M,m ≤ C(rη∣∣∣x′ ○ iK +N∣∣∣ +∗ +K,k + 1 +r ∣∣∣x′∣∣∣ +∗ +N,n) +where ∣∣∣⋅∣∣∣∗ denotes the dual norm of ∣∣∣⋅∣∣∣ and iM +N , iK +N the linking maps (see [26, +Section 4, Eq. (24), p. 577]). +Examples of Fréchet spaces with (DN) are the spaces of rapidly decreasing +sequences s(Nd), s(Nd +0) and s(Zd), the space C∞([a,b]) of all C∞-smooth functions +on (a,b) such that all derivatives can be continuously extended to the boundary +and the space of smooth functions C∞ +2π(Rd) that are 2π-periodic in each variable. +Examples of ultrabornological PLS-space with (PA) are Fréchet–Schwartz spaces, +the space of tempered distributions S(Rd)′ +b, the space of distributions D(Ω)′ +b and +ultradistributions of Beurling type D(ω)(Ω)′ +b on an open set Ω ⊂ Rd. These and +many more examples may be found in [26], [54, Corollary 4.8, p. 1116] and [112, +Example 3, p. 7]. +5.1.1. Proposition. +a) Let Y be a Fréchet space and X a semi-reflexive +lcHs. Then Lb(X′ +b,Y ′ +b ) ≅ Lb(Y,(X′ +b)′ +b) via taking adjoints. +b) Let E be an lcHs and X a Montel space. Then Lb(X′ +b,E) ≅ XεE where +the isomorphism is the identity map. +67 + +68 +5. APPLICATIONS +Proof. a) We consider the map +t(⋅)∶Lb(X′ +b,Y ′ +b ) → Lb(Y,(X′ +b)′ +b), u ↦ tu, +defined by tu(y)(x′) ∶= u(x′)(y) for y ∈ Y and x′ ∈ X′. First, we prove that t(⋅) is +well-defined. Let u ∈ L(X′ +b,Y ′ +b ) and y ∈ Y . Since u ∈ L(X′ +b,Y ′ +b ) and {y} is bounded +in Y , there are a bounded set B ⊂ X and C > 0 such that +∣tu(y)(x′)∣ = ∣u(x′)(y)∣ ≤ C sup +x∈B +∣x′(x)∣ +for all x′ ∈ X′, implying tu(y) ∈ (X′ +b)′. +Let us denote by (∥⋅∥Y,n)n∈N the (directed) system of seminorms generating the +metrisable locally convex topology of Y . The canonical embedding J∶Y → (Y ′ +b )′ +b is +an isomorphism between Y and J(Y ) by [131, Corollary 25.10, p. 298] because Y +is a Fréchet space. For a bounded set M ⊂ X′ +b we note that +sup +x′∈M +∣tu(y)(x′)∣ = sup +x′∈M +∣u(x′)(y)∣ = sup +x′∈M +∣⟨J(y),u(x′)⟩∣. +The next step is to prove that u(M) is bounded in Y ′ +b . Let N ⊂ Y be bounded. +Since u ∈ L(X′ +b,Y ′ +b ), there are again a bounded set B ⊂ X and a constant C > 0 +such that +sup +x′∈M +sup +y∈N +∣u(x′)(y)∣ ≤ C sup +x′∈M +sup +x∈B +∣x′(x)∣ < ∞ +where the last estimate follows from the boundedness of M ⊂ X′ +b. Hence u(M) is +bounded in Y ′ +b . By the remark about the canonical embedding there are n ∈ N and +C0 > 0 such that +sup +x′∈M +∣tu(y)(x′)∣ = +sup +y′∈u(M) +∣⟨J(y),y′⟩∣ ≤ C0∥y∥Y,n, +so tu ∈ L(Y,(X′ +b)′ +b) and the map t(⋅) is well-defined. +Let us turn to injectivity. Let u,v ∈ L(X′ +b,Y ′ +b ) with tu = tv. This is equivalent +to +u(x′)(y) = tu(y)(x′) = tv(y)(x′) = v(x′)(y) +for all y ∈ Y and x′ ∈ X′. This implies u(x′) = v(x′) for all x′ ∈ X′, hence u = v. +Next, we turn to surjectivity. We consider the map +t(⋅)∶Lb(Y,(X′ +b)′ +b) → Lb(X′ +b,Y ′ +b ), u ↦ tu, +defined by tu(x′)(y) ∶= u(y)(x′) for x′ ∈ X′ and y ∈ Y . We show that this map is +well-defined. Let u ∈ Lb(Y,(X′ +b)′ +b) and x′ ∈ X′. Since u ∈ Lb(Y,(X′ +b)′ +b) and {x′} is +bounded in X′, there are n ∈ N and C > 0 such that +∣tu(x′)(y)∣ = ∣u(y)(x′)∣ ≤ C∥y∥Y,n +for all y ∈ Y , yielding tu(x′) ∈ Y ′. Let B ⊂ Y be bounded. The semi-reflexivity +of X implies that for every u(y), y ∈ B, there is a unique xu(y) ∈ X such that +u(y)(x′) = x′(xu(y)) for all x′ ∈ X′. Then we get +sup +y∈B +∣tu(x′)(y)∣ = sup +y∈B +∣u(y)(x′)∣ = sup +y∈B +∣x′(xu(y))∣. +We claim that D ∶= {xu(y) ∣ y ∈ B} is a bounded set in X. Let N ⊂ X′ be finite. +Then the set M ∶= {tu(x′) ∣ x′ ∈ N} ⊂ Y ′ is finite. We have +sup +y∈B +sup +x′∈N +∣x′(xu(y))∣ = sup +y∈B +sup +x′∈N +∣tu(x′)(y)∣ = sup +y∈B +sup +y′∈M +∣y′(y)∣ < ∞ +where the last estimate follows from the fact that the bounded set B is weakly +bounded. Thus D is weakly bounded and by [131, Mackey’s theorem 23.15, p. 268] +bounded in X. Therefore it follows from +sup +y∈B +∣tu(x′)(y)∣ = sup +y∈B +∣x′(xu(y))∣ = sup +x∈D +∣x′(x)∣ + +5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE +69 +for all x′ ∈ X′ that tu ∈ L(X′ +b,Y ′ +b ), which means that t(⋅) is well-defined. +Let +u ∈ L(Y,(X′ +b)′ +b). Then we have tu ∈ Lb(X′ +b,Y ′ +b ). In addition, for all y ∈ Y and all +x′ ∈ X′ +t(tu)(y)(x′) = tu(x′)(y) = u(y)(x′) +is valid and so t(tu)(y) = u(y) for all y ∈ Y , proving the surjectivity. +The last step is to prove the continuity of t(⋅) and its inverse. Let M ⊂ Y and +B ⊂ X′ +b be bounded sets. Then +sup +y∈M +sup +x′∈B +∣tu(y)(x′)∣ = sup +y∈M +sup +x′∈B +∣u(x′)(y)∣ = sup +x′∈B +sup +y∈M +∣u(x′)(y)∣ += sup +x′∈B +sup +y∈M +∣t(tu)(x′)(y)∣ +holds for all u ∈ L(X′ +b,Y ′ +b ). Therefore, t(⋅) and its inverse are continuous. +b) Let T ∈ L(X′ +b,E). For α ∈ A there are a bounded set B ⊂ X and C > 0 such +that +pα(T(x′)) ≤ C sup +x∈B +∣x′(x)∣ ≤ C +sup +x∈acx(B) +∣x′(x)∣ +for every x′ ∈ X′ where acx(B) is the closure of the absolutely convex hull of B. +The set acx(B) is absolutely convex and compact by [89, 6.2.1 Proposition, p. 103] +and [89, 6.7.1 Proposition, p. 112] since B is bounded in the Montel space X. Hence +we gain T ∈ L(X′ +κ,E). +Let M ⊂ X′ be equicontinuous. Due to [89, 8.5.1 Theorem (a), p. 156] M is +bounded in X′ +b. Therefore, +id∶Lb(X′ +b,E) → Le(X′ +κ,E) = XεE +is continuous. +Let T ∈ L(X′ +κ,E). For α ∈ A there are an absolutely convex compact set B ⊂ X +and C > 0 such that +pα(T(x′)) ≤ C sup +x∈B +∣x′(x)∣ +for every x′ ∈ X′. Since the compact set B is bounded, we get T ∈ L(X′ +b,E). +Let M be a bounded set in X′ +b. Then M is equicontinuous by virtue of [171, +Theorem 33.2, p. 349], as X, being a Montel space, is barrelled. Thus +id∶Le(X′ +κ,E) → Lb(X′ +b,E) +is continuous. +□ +For part e) of the next theorem we need that our map SF2(Ω2)∶F2(Ω2)εE → +F2(Ω2,E) is an isomorphism and that its inverse is given as in Theorem 3.2.4, i.e. +that +Rt∶F2(Ω2,E) → F2(Ω2)εE, f ↦ J −1 ○ Rt +f, +is the inverse of SF2(Ω2) where Rt +f(f ′)(e′) = f ′(e′ ○ f) for f ′ ∈ F2(Ω2)′ and e′ ∈ E′, +and J ∶E → E′⋆ is the canonical injection in the algebraic dual E′⋆ of E′. +5.1.2. Theorem. Let E be an lcHs, F1(Ω1) and F1(Ω1,E) as well as F2(Ω2) +and F2(Ω2,E) be ε-into-compatible. Let (T E,T K) be a consistent family for (F1,E) +such that T K∶F1(Ω1) → F2(Ω2) is continuous and T E∶F1(Ω1,E) → F2(Ω2,E). +Then the following holds: +a) T E ○ SF1(Ω1) = SF2(Ω2) ○ (T KεidE). +b) If SF1(Ω1) is surjective and T K is injective, then T E is injective, continu- +ous and +T E = SF2(Ω2) ○ (T KεidE) ○ S−1 +F1(Ω1). + +70 +5. APPLICATIONS +If in addition SF2(Ω2) is surjective and T K an isomorphism, then T E is +an isomorphism with inverse +(T E)−1 = SF1(Ω1) ○ ((T K)−1εidE) ○ S−1 +F2(Ω2). +c) If SF2(Ω2) and T KεidE are surjective, then T E is surjective. +d) If SF2(Ω2) and T K are surjective, F1(Ω1), F2(Ω2) and E are Fréchet +spaces and +(i) F1(Ω1) and F2(Ω2) are nuclear, or +(ii) E is nuclear, +then T E is surjective. +e) If SF2(Ω2) is surjective with inverse Rt, T K is surjective, F1(Ω1) and +F2(Ω2) are Fréchet spaces, kerT K is nuclear and has (Ω), and +(i) F1(Ω1) and F2(Ω2) are Montel spaces, E = F ′ +b where F is a Fréchet +space satisfying (DN), or +(ii) F1(Ω1) and F2(Ω2) are Schwartz spaces, E is an ultrabornological +PLS-space satisfying (PA), +then T E is surjective. +Proof. a) Let u ∈ F1(Ω1)εE. Then +(T E ○ SF1(Ω1))(u)(x) = u(δx ○ T K) = (u ○ (T K)t)(δx) = (T KεidE)(u)(δx) += SF2(Ω2)((T KεidE)(u))(x), +x ∈ Ω2, +as (T E,T K) is consistent for (F1,E), which proves part a). +b) If SF1(Ω1) is surjective, then SF1(Ω1) is an isomorphism, because it is an +isomorphism into, and we have +T E = SF2(Ω2) ○ (T KεidE) ○ S−1 +F1(Ω1) +by part a). If T K is injective, then T KεidE is also injective by [159, Chap. I, §1, +Proposition 1, p. 20] and thus T E by the formula above as well since SF1(Ω1) is +an isomorphism and SF2(Ω2) an isomorphism into. If SF2(Ω2) is surjective and T K +an isomorphism, then SF2(Ω2) and T KεidE are isomorphisms, the latter by [159, +Chap. I, §1, Proposition 1, p. 20] and its inverse is (T K)−1εidE. The rest of part +b) follows from the formula for T E above. +c) Let f ∈ F2(Ω2,E). +Then there is g ∈ F1(Ω1)εE such that (SF2(Ω2) ○ +(T KεidE))(g) = f. Hence we obtain h ∶= SF1(Ω1)(g) ∈ F1(Ω1,E) and T E(h) = f by +part a). +d) For n = 1,2 the continuous linear injection (see (3)) +Θn∶Fn(Ωn) ⊗π E → Fn(Ωn)εE, +k +∑ +j=1 +fj ⊗ ej �→ [y ↦ +k +∑ +j=1 +y(fj)ej], +from the tensor product Fn(Ωn) ⊗π E with the projective topology extends to a con- +tinuous linear map ̂Θn∶Fn(Ωn)̂⊗πE → Fn(Ωn)εE on the completion Fn(Ωn)̂⊗πE +of Fn(Ωn) ⊗π E. +The map ̂Θn is also a topological isomorphism since Fn(Ωn) +is nuclear for n = 1,2 in case (i) resp. E is nuclear in case (ii). +Furthermore, +T K ⊗π idE∶F1(Ω1)⊗π E → F2(Ω2)⊗π E is defined by the relation Θ2 ○(T K ⊗π idE) = +(T KεidE)○Θ1. We denote by T K ̂⊗π idE the continuous linear extension of T K⊗πidE +to the completion F1(Ω1)̂⊗πE. Moreover, Fn(Ωn) for n = 1,2 and E are Fréchet +spaces, T K and idE are linear, continuous and surjective, so T K ̂⊗π idE is surjective +by [94, 10.24 Satz, p. 255]. We observe that +T KεidE = ̂Θ2 ○ (T K ̂⊗π idE) ○ ̂Θ −1 +1 + +5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE +71 +and deduce that T KεidE is surjective. Now, we apply part c), which proves part +d). +e) Throughout this proof we use the notation X′′ ∶= (X′ +b)′ +b for a locally convex +Hausdorff space X and T ∶= T K. The space F1(Ω1) is a Fréchet space and so its +closed subspace kerT as well. Further, Fn(Ωn) is a Montel space for n = 1,2 and +kerT nuclear, thus they are reflexive. The sequence +0 → kerT +i→ F1(Ω1) +T→ F2(Ω2) → 0, +(40) +where i means the inclusion, is a topologically exact sequence of Fréchet spaces +because T is surjective by assumption. Let us denote by J0∶kerT → (kerT)′′ and +Jn∶Fn(Ωn) → Fn(Ωn)′′ for n = 1,2 the canonical embeddings which are topological +isomorphisms since kerT and Fn(Ωn) are reflexive for n = 1,2. Then the exactness +of (40) implies that +0 → (kerT)′′ i0→ F1(Ω1)′′ T1→ F2(Ω2)′′ → 0, +(41) +where i0 ∶= J0 ○ i ○ J−1 +0 +and T1 ∶= J2 ○ T ○ J−1 +1 , is an exact topological sequence. This +exact sequence is topological because the (strong) bidual of a Fréchet space is again +a Fréchet space by [131, Corollary 25.10, p. 298]. +(i) Let E = F ′ +b where F is a Fréchet space with (DN). Then Ext1(F,(kerT)′′) = +0 by [174, 5.1 Theorem, p. 186] since kerT is nuclear and satisfies (Ω) and therefore +(kerT)′′ as well. Combined with the exactness of (41) this implies that the sequence +0 → L(F,(kerT)′′) +i∗ +0→ L(F,F1(Ω1)′′) +T ∗ +1→ L(F,F2(Ω2)′′) → 0 +is exact by [137, Proposition 2.1, p. 13–14] where i∗ +0(B) ∶= i0○B and T ∗ +1 (D) ∶= T1○D +for B ∈ L(F,(kerT)′′) and D ∈ L(F,F1(Ω1)′′). In particular, we obtain that +T ∗ +1 ∶L(F,F1(Ω1)′′) → L(F,F2(Ω2)′′) +(42) +is surjective. Via E = F ′ +b and Proposition 5.1.1 (X = Fn(Ωn) and Y = F) we have +the isomorphisms into +ψn ∶= SFn(Ωn) ○ t(⋅)∶L(F,Fn(Ωn)′′) → Fn(Ωn,E), +ψn(u) = (SFn(Ωn) ○ t(⋅))(u) = [x ↦ tu(δx)], +for n = 1,2 and the inverse +ψ−1 +2 (f) = (S ○ t(⋅))−1(f) = (t(⋅) ○ S−1 +F2(Ω2))(f) = t(J −1 ○ Rt +f), +f ∈ F2(Ω2,E). +Let g ∈ F2(Ω2,E). Then ψ−1 +2 (g) ∈ L(F,F2(Ω2)′′) and by the surjectivity of (42) +there is u ∈ L(F,F1(Ω1)′′) such that T ∗ +1 u = ψ−1 +2 (g). So we get ψ1(u) ∈ F1(Ω1,E). +Next, we show that T Eψ1(u) = g is valid. Let y ∈ F and x ∈ Ω2. Then +T E(ψ1(u))(x) = tu(δx ○ T) +by consistency and +T E(ψ1(u))(x)(y) = tu(δx ○ T)(y) = u(y)(δx ○ T) = ⟨δx ○ T,J−1 +1 (u(y))⟩ += ⟨δx,TJ−1 +1 (u(y))⟩ = ⟨[J2 ○ T ○ J−1 +1 ](u(y)),δx⟩ = ⟨(T1 ○ u)(y),δx⟩ += ⟨(T ∗ +1 u)(y),δx⟩ = ψ−1 +2 (g)(y)(δx) = t(J −1 ○ Rt +g)(y)(δx) += (J −1 ○ Rt +g)(δx)(y) = J −1(J (g(x))(y) = g(x)(y). +Thus T E(ψ1(u))(x) = g(x) for every x ∈ Ω2, which proves the surjectivity. +(ii) Let E be an ultrabornological PLS-space satisfying (PA). Since the nuclear +Fréchet space kerT is also a Schwartz space, its strong dual (kerT)′ +b is a DFS- +space. +By [26, Theorem 4.1, p. 577] we obtain Ext1 +P LS((kerT)′ +b,E) = 0 as the +bidual (kerT)′′ satisfies (Ω), E is a PLS-space satisfying (PA) and condition (c) +in the theorem is fulfilled because (kerT)′ +b is the strong dual of a nuclear Fréchet + +72 +5. APPLICATIONS +space. Moreover, we have Proj1 E = 0 due to [180, Corollary 3.3.10, p. 46] because +E is an ultrabornological PLS-space. +Then the exactness of the sequence (41), +[26, Theorem 3.4, p. 567] and [26, Lemma 3.3, p. 567] (in the lemma the same +condition (c) as in [26, Theorem 4.1, p. 577] is fulfilled and we choose H = (kerT)′′, +F = F1(Ω1)′′ and G = F2(Ω2)′′), imply that the sequence +0 → L(E′ +b,(kerT)′′) +i∗ +0→ L(E′ +b,F1(Ω1)′′) +T ∗ +1→ L(E′ +b,F2(Ω2)′′) → 0 +is exact. The maps i∗ +0 and T ∗ +1 are defined as in part (i). Especially, we get that +T ∗ +1 ∶L(E′ +b,F1(Ω1)′′) → L(E′ +b,F2(Ω2)′′) +(43) +is surjective. +By [54, Remark 4.4, p. 1114] we have Lb(Fn(Ωn)′ +b,E′′) ≅ Lb(E′ +b,Fn(Ωn)′′) for +n = 1,2 via taking adjoints since Fn(Ωn), being a Fréchet–Schwartz space, is a PLS- +space and hence its strong dual an LFS-space, which is regular by [180, Corollary +6.7, 10. ⇔ 11., p. 114], and E is an ultrabornological PLS-space, in particular, +reflexive by [53, Theorem 3.2, p. 58]. In addition, the map +P∶Lb(Fn(Ωn)′ +b,E′′) → Lb(Fn(Ωn)′ +b,E), +defined by P(u)(y) ∶= J −1(u(y)) for u ∈ L(Fn(Ωn)′ +b,E′′) and y ∈ Fn(Ωn)′, is an +isomorphism because E is reflexive. Due to Proposition 5.1.1 b) with X = Fn(Ωn) +we obtain the isomorphisms into +ψn ∶= S ○ J −1 ○ t(⋅)∶Lb(E′ +b,Fn(Ωn)′′) → Fn(Ωn,E), +ψn(u) = [SFn(Ωn) ○ J −1 ○ t(⋅)](u) = [x ↦ J −1(tu(δx))], +for n = 1,2 and the inverse given by +ψ−1 +2 (f) = (SF2(Ω2) ○ J −1 ○ t(⋅))−1(f) = [t(⋅) ○ J ○ S−1 +F2(Ω2)](f) = t(J ○ J −1 ○ Rt +f) += t(Rt +f) +for f ∈ F2(Ω2,E). +Let g ∈ F2(Ω2,E). +Then ψ−1 +2 (g) ∈ L(E′ +b,F2(Ω2)′′) and by the surjectivity +of (43) there exists u ∈ L(E′ +b,F1(Ω1)′′) such that T ∗ +1 u = ψ−1 +2 (g). +So we have +ψ1(u) ∈ F1(Ω1,E). The last step is to show that T Eψ1(u) = g. As in part (i) we +gain for every x ∈ Ω2 +T E(ψ1(u))(x) = J −1(tu(δx ○ T)) +by consistency and for every y ∈ E′ +tu(δx ○ T)(y) = u(y)(δx ○ T) = (T ∗ +1 u)(y)(δx) = ψ−1 +2 (g)(y)(δx) = t(Rt +g)(y)(δx) += δx(y ○ g) = y(g(x)) = J (g(x))(y). +Thus we have tu(δx ○ T) = J (g(x)) and therefore T E(ψ1(u))(x) = g(x) for all +x ∈ Ω2. +□ +Theorem 5.1.2 d) and e) are generalisations of [116, Corollary 4.3, p. 2689] and +[112, Theorem 5, p. 7–8] where T C is the Cauchy–Riemann operator ∂ on certain +weighted spaces CV∞(Ω) of smooth functions. Our next result is the well-known +application of tensor product theory and splitting theory to linear partial differential +operators we already mentioned in the introduction. +5.1.3. Corollary. Let E be a locally complete lcHs, Ω1 ⊂ Rd open and P(∂)K +be a linear partial differential operator with C∞-smooth coefficients. Then the fol- +lowing holds: +a) P(∂)E = SC∞(Ω1) ○ (P(∂)KεidE) ○ S−1 +C∞(Ω1). +b) If K = C, P(D) ∶= P(D)C ∶= P(−i∂)C has constant coefficients and is + +5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE +73 +(i) elliptic, or +(ii) hypoelliptic and Ω1 convex, +and +(iii) E is a Fréchet space, or +(iv) E = F ′ +b where F is a Fréchet space satisfying (DN), d ≥ 2, or +(v) E is an ultrabornological PLS-space satisfying (PA), d ≥ 2, +then P(D)E∶C∞(Ω1,E) → C∞(Ω1,E) is surjective. +Proof. Part a) follows from Theorem 5.1.2 a), Example 3.2.8 a) and the con- +sistency of (P(∂)E,P(∂)K) because (7) holds for u ∈ CW∞(Ω1)εE as well. +Let us turn to part b). The inverse of SC∞(Ω1) is given by Rt by Example 3.2.8 +a). The map P(D) = P(D)C∶C∞(Ω1) → C∞(Ω1) is surjective by [86, Corollary +10.6.8, p. 43] and [86, Theorem 10.6.2, p. 41] in case (ii) resp. by [86, Corollary +10.8.2, p. 51] in case (i). The space CW∞(Ω1), i.e. C∞(Ω1) with its usual topology +(see Example 3.1.9 b)), is a nuclear Fréchet space and thus its closed subspace +kerP(D)C as well. In case (i) kerP(D)C has (Ω) due to [173, Proposition 2.5 (b), +p. 173] and in case (ii) due to [140, 4.5 Corollary (a), p. 202]. Hence the surjectivity +of P(D)E follows from Theorem 5.1.2 d)+e). +□ +Recently, it was shown in [47, Theorem 4.2, p. 13] that in the case that +Ω1 is convex, kerP(D)C has property (Ω) for any P(D) with constant coeffi- +cients (so without the assumption of P(D) being hypoelliptic). +Hence we may +replace the assumption of P(D) being hypoelliptic in (ii) by the assumption that +P(D)C∶C∞(Ω1) → C∞(Ω1) is surjective. Even more recently, a necessary and suf- +ficient condition for the surjectivity of P(D)C and kerP(D)C having property (Ω) +for P(D) with constant coefficients and general open Ω1 ⊂ Rd was derived in [48, +Theorem 1.1. (a), p. 3] using shifted fundamental solutions. +Even though Corollary 5.1.3 b) is known, it is often proved without using tensor +products or splitting theory (see e.g. [94, Theorem 10.10, p. 240]) or it is phrased +as the surjectivity of P(D)Ĉ⊗π idE (see e.g. [171, Eq. (52.4), p. 541]) and the proof +of the relation +P(D)E = SC∞(Ω1) ○ (̂Θ1 ○ (P(D)C ̂⊗π idE) ○ ̂Θ −1 +1 ) ○ S−1 +C∞(Ω1) +for Fréchet spaces E is omitted (see e.g. [171, p. 545–546]), or only the surjectivity +of T ∗ +1 = P(D)∗ +1 in part e) of Theorem 5.1.2 is actually shown and it is only stated +but not proved that this implies the surjectivity of P(D)E (see e.g. the statement +of surjectivity of P(D)E in [173, p. 168] for elliptic P(D) and E = F ′ +b for a Fréchet +space F with (DN) and that it is ‘only’ shown that P(D)∗ +1 is surjective by [173, +Proposition 2.5 (b), p. 173] and [173, Theorem 2.4 (b), p. 173] where the symbol +P(D)∗ is used instead of P(D)∗ +1 in [173, p. 172] since the isomorphism J1 = J2 is +omitted). So, apart from being the probably most classical application of tensor +products or splitting theory, that is the reason why we still included Corollary 5.1.3. +Let us give another application of Theorem 5.1.2 d) and e), namely, a vector- +valued Borel–Ritt theorem. +5.1.4. Theorem. Let E be an lcHs and (xn)n∈N0 a sequence in E. If +(i) E is a Fréchet space, or +(ii) E = F ′ +b where F is a Fréchet space satisfying (DN), or +(iii) E is an ultrabornological PLS-space satisfying (PA), +then there is f ∈ C∞ +2π(R,E) such that (∂n)Ef(0) = xn for all n ∈ N0. +Proof. By the Borel–Ritt theorem [94, Satz 9.12, p. 206] the map +T K∶C∞ +2π(R) → KN0, T K(f) ∶= ((∂n)Kf(0))n∈N0, + +74 +5. APPLICATIONS +is surjective and obviously linear and continuous as well. Now, we define the map +T E∶C∞ +2π(R,E) → EN0 by replacing K by E in the definition of T K. Due to Example +4.2.1 KN0 and EN0 are ε-compatible and the inverse of SKN0 is given by Rt. In +addition, C∞ +2π(R) and C∞ +2π(R,E) are ε-compatible by Example 4.2.27 as in all three +cases E is complete. We observe that (T E,T K) is consistent by Proposition 3.1.11 +c). +The spaces KN0 and C∞ +2π(R) are nuclear Fréchet spaces. +The first by [171, +Theorem 51.1, p. 526] and the second because it is a subspace of the nuclear space +C∞(R) by [131, Examples 28.9 (1), p. 349–350] and [131, Proposition 28.6, p. 347]. +Hence in case (i) our statement follows from Theorem 5.1.2 d). Moreover, kerT K +is nuclear since C∞ +2π(R) is nuclear. By the proof of [131, Lemma 31.3, p. 392–393] +kerT K is isomorphic to s(N0). The space s(N0) has (Ω) by [131, Lemma 29.11 +(3), p. 368] and thus kerT K as well because (Ω) is a linear topological invariant +by [131, Lemma 29.11 (1), p. 368]. Therefore our statement in case (ii) and (iii) +follows from Theorem 5.1.2 e). +□ +We close this section with an application of Theorem 5.1.2 b) to the Fourier +transformation on the Beurling–Björck spaces Sµ(Rd,E) from Example 4.2.26. +5.1.5. Theorem. Let E be a locally complete lcHs over C and µ∶Rd → [0,∞) +continuous such that µ(x) = µ(−x) for all x ∈ Rd and condition (γ) is fulfilled. +(i) If E has metric ccp, or +(ii) if µ ∈ C1(Rd) and there are k ∈ N0, C > 0 such that ∣∂enµ(x)∣ ≤ Cekµ(x) +for all x ∈ Rd and 1 ≤ n ≤ d, +then FE∶Sµ(Rd,E) → Sµ(Rd,E) is an isomorphism with FE = S ○ (FCεidE) ○ S−1. +Proof. Due to Example 4.2.26 Sµ(Rd) and Sµ(Rd,E) are ε-compatible. The +Fourier transformation FC∶Sµ(Rd) → Sµ(Rd) is a well-defined isomorphism by the +definition of Sµ(Rd) and since (FC ○ FC)(f)(x) = f(−x) for all f ∈ Sµ(Rd) as well +as µ(x) = µ(−x) for all x ∈ Rd. Due to (26) with β = 0 we have that (FE,FC) is a +consistent family for (Sµ,E) and thus it follows from Theorem 5.1.2 b) that FE is +an isomorphism and FE = S ○ (FCεidE) ○ S−1, which completes the proof. +□ +5.2. Extension of vector-valued functions +We study the problem of extending vector-valued functions via the existence +of weak extensions in this section. The precise description of this problem reads +as follows. Let E be a locally convex Hausdorff space over the field K of real or +complex numbers and F(Ω) ∶= F(Ω,K) a locally convex Hausdorff space of K- +valued functions on a set Ω. Suppose that the point evaluations δx belong to the +dual F(Ω)′ for every x ∈ Ω and that there is a locally convex Hausdorff space +F(Ω,E) of E-valued functions on Ω such that the map +S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], +(44) +is an isomorphism into, i.e. F(Ω) and F(Ω,E) are ε-into-compatible. Thus F(Ω)εE +is a linearisation of a subspace of F(Ω,E). Linearisations that are based on the +Dixmier–Ng theorem were used by Bonet, Domański and Lindström in [28, Lemma +10, p. 243] resp. Laitila and Tylli in [121, Lemma 5.2, p. 14] to describe the space +of weakly holomorphic resp. harmonic functions on the unit disc Ω = D ⊂ C with +values in a (complex) Banach space E (see also [118]). +5.2.1. Question. Let Λ be a subset of Ω and G a linear subspace of E′. Let +f∶Λ → E be such that for every e′ ∈ G, the function e′ ○ f∶Λ → K has an extension +in F(Ω). When is there an extension F ∈ F(Ω,E) of f, i.e. F∣Λ = f ? + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +75 +An affirmative answer for Λ = Ω and G = E′ is called a weak-strong principle. +For weighted continuous functions on a completely regular Hausdorff space Ω with +values in a semi-Montel or Schwartz space E a weak-strong principle is given by +Bierstedt in [17, 2.10 Lemma, p. 140]. +Weak-strong principles for holomorphic +functions on open subsets Ω ⊂ C were shown by Dunford in [59, Theorem 76, p. +354] for Banach spaces E and by Grothendieck in [82, Théorème 1, p. 37–38] for +quasi-complete E. For a wider class of function spaces weak-strong principles are +due to Grothendieck, mainly, in the case that F(Ω) is nuclear and E complete (see +[83, Chap. II, §3, n○3, Théorème 13, p. 80]), which covers the case that F(Ω) is the +space C∞(Ω) of smooth functions on an open set Ω ⊂ Rd (with its usual topology). +Gramsch [77] analysed the weak-strong principles of Grothendieck and realised +that they can be used to extend functions if Λ is a set of uniqueness, i.e. from +f ∈ F(Ω) and f(x) = 0 for all x ∈ Λ follows that f = 0, and F(Ω) a semi-Montel +space, E complete and G = E′ (see [77, 0.1, p. 217]). +An extension result for +holomorphic functions where G = E′ and E is sequentially complete was shown by +Bogdanowicz in [25, Corollary 3, p. 665]. +Grosse-Erdmann proved for holomorphic functions on Λ = Ω in [79, 5.2 The- +orem, p. 35] that it is sufficient to test locally bounded functions f with values +in a locally complete space E with functionals from a weak⋆-dense subspace G of +E′. Arendt and Nikolski [7, 8] shortened his proof in the case that E is a Fréchet +space (see [7, Theorem 3.1, p. 787] and [7, Remark 3.3, p. 787]). Arendt gave an +affirmative answer in [6, Theorem 5.4, p. 74] for harmonic functions on an open +subset Λ = Ω ⊂ Rd where the range space E is a Banach space and G a weak⋆-dense +subspace of E′. +In [77] Gramsch also derived extension results for a large class of Fréchet– +Montel spaces F(Ω) in the case that Λ is a special set of uniqueness, E sequentially +complete and G strongly dense in E′ (see [77, 3.3 Satz, p. 228–229]). He applied +it to the space of holomorphic functions and Grosse-Erdmann [81] expanded this +result to the case of E being Br-complete and G only a weak⋆-dense subspace of +E′ (see [81, Theorem 2, p. 401] and [81, Remark 2 (a), p. 406]). In a series of +papers [30, 69, 70, 92, 93] these results were generalised and improved by Bonet, +Frerick, Jordá and Wengenroth who used (44) to obtain extensions for vector-valued +functions via extensions of linear operators. In [92, 93] this was done by Jordá for +holomorphic functions on a domain (i.e. open and connected) Ω ⊂ C and weighted +holomorphic functions on a domain Ω in a Banach space. In [30] this was done +by Bonet, Frerick and Jordá for closed subsheaves F(Ω) of the sheaf of smooth +functions C∞(Ω) on a domain Ω ⊂ Rd. Their results implied some consequences +on the work of Bierstedt and Holtmanns [18] as well. Further, in [69] this was +done by Frerick and Jordá for closed subsheaves F(Ω) of smooth functions on a +domain Ω ⊂ Rd which are closed in the sheaf C(Ω) of continuous functions and in +[70] by the first two authors and Wengenroth in the case that F(Ω) is the space of +bounded functions in the kernel of a hypoelliptic linear partial differential operator, +in particular, the spaces of bounded holomorphic or harmonic functions. +In this section we present a unified approach to the extension problem for a large +class of function spaces. The spaces we treat are usually of the kind that F(Ω) +belongs to the class of semi-Montel spaces, Fréchet–Schwartz spaces or Banach +spaces. +Even quite general weighted spaces F(Ω) are treated, at least, if E is +a semi-Montel space. +Our approach is based on three ideas. +First, it is based +on the representation of (a subspace of) F(Ω,E) as a space of continuous linear +operators via the map S from (44). We note that almost all our examples of such +spaces F(Ω,E) are actually of the form of a general weighted space FV(Ω,E) from +Definition 3.1.4. Second, it is based on the idea to consider a set of uniqueness Λ + +76 +5. APPLICATIONS +not necessarily as a subset of Ω but rather as a set of functionals acting on F(Ω). +In the definition of a set of uniqueness given above one may identify Λ with the set +of functionals {δx ∣ x ∈ Λ} and this shift of perspective allows us to consider certain +sets of functionals of the form T K +m,x as sets of uniqueness for F(Ω) (see Definition +5.2.2). Third, the generalised concept of consistency and strength of a family of +operators (T E +m,T K +m)m∈M acting on (F(Ω,E),F(Ω)) from Definition 4.3.1 enables +us to generalise Question 5.2.1 and affirmatively answer this generalised question. +These three ideas are used to extend the mentioned results and we always have +to balance the sets Λ from which we extend our functions and the subspaces G ⊂ E′ +with which we test. The case of ‘thin’ sets Λ and ‘thick’ subspaces G is handled in +Section 5.2.1, the converse case of ‘thick’ sets Λ and ‘thin’ subspaces G in Section +5.2.2. +5.2.1. Extension from thin sets. Using the functionals T K +m,x, we extend the +definition of a set of uniqueness and a space of restrictions given in [30, Definition +4, 5, p. 230]. This prepares the ground for a generalisation of Question 5.2.1 using +a strong, consistent family (T E +m,T K +m)m∈M. +5.2.2. Definition (set of uniqueness). Let Ω be a non-empty set, F(Ω) ⊂ KΩ +an lcHs, (ωm)m∈M be a family of non-empty sets and T K +m∶F(Ω) → Kωm be linear for +all m ∈ M. Then U ⊂ ⋃m∈M({m}×ωm) is called a set of uniqueness for (T K +m,F)m∈M +if +(i) ∀ (m,x) ∈ U ∶ T K +m,x ∈ F(Ω)′, +(ii) ∀ f ∈ F(Ω) ∶ [∀(m,x) ∈ U ∶ T K +m(f)(x) = 0] ⇒ f = 0. +We omit the index m in ωm and T K +m if M is a singleton and consider U as a subset +of ω. +If U is a set of uniqueness for (T K +m,F)m∈M, then span{T K +m,x ∣ (m,x) ∈ U} is +dense in F(Ω)′ +σ (and F(Ω)′ +κ) by the bipolar theorem. +5.2.3. Remark. Let Ω be a non-empty set and F(Ω) ⊂ KΩ an lcHs. +a) A simple set of uniqueness for (idKΩ,F) is given by U ∶= Ω if δx ∈ F(Ω)′ +for all x ∈ Ω. +b) If F(Ω) has a Schauder basis (fn)n∈N with associated sequence of coef- +ficient functionals T K ∶= (T K +n )n∈N, then U ∶= N is a set of uniqueness for +(T K,F). +An example for b) is the space of holomorphic functions on an open disc +Dr(z0) ⊂ C with radius 0 < r ≤ ∞ and center z0 ∈ C. +If we equip this space +with the topology of compact convergence, then it has the shifted monomials +((⋅ − z0)n)n∈N0 as a Schauder basis with the point evaluations (δz0 ○ ∂n +C)n∈N0 given +by (δz0 ○ ∂n +C)(f) ∶= f (n)(z0) as associated sequence of coefficient functionals. We +will explore further sets of uniqueness for concrete function spaces in the upcoming +examples and come back to b) in Section 5.7. +5.2.4. Definition (restriction space). Let G ⊂ E′ be a separating subspace +and U a set of uniqueness for (T K +m,F)m∈M. Let FG(U,E) be the space of functions +f∶U → E such that for every e′ ∈ G there is fe′ ∈ F(Ω) with T K +m(fe′)(x) = (e′ ○ +f)(m,x) for all (m,x) ∈ U. +5.2.5. Remark. Since U is a set of uniqueness, the functions fe′ are unique +and the map Rf∶E′ → F(Ω), Rf(e′) ∶= fe′, is well-defined and linear. The map Rf +resembles the map Rf defined above Lemma 3.2.1. +5.2.6. Remark. Let F(Ω) and F(Ω,E) be ε-into-compatible. Consider a set +of uniqueness U for (T K +m,F)m∈M, a separating subspace G ⊂ E′ and a strong, + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +77 +consistent family (T E +m,T K +m)m∈M for (F,E). For u ∈ F(Ω)εE set f ∶= S(u). Then +f ∈ F(Ω,E) by the ε-into-compatibility and we set ̃f∶U → E, ̃f(m,x) ∶= T E +m(f)(x). +It follows that +(e′ ○ ̃f)(m,x) = (e′ ○ T E +m(f))(x) = T K +m(e′ ○ f)(x) +for all (m,x) ∈ U and fe′ ∶= e′ ○f ∈ F(Ω) for all e′ ∈ E′ by the strength of the family. +We conclude that ̃f ∈ FG(U,E). +5.2.7. Remark. If U is a set of uniqueness for (T K +m,F)m∈M, then the existence +of operators (T E +m)m∈M such that (T E +m,T K +m)m∈M is a strong, consistent family for +(F,E) is often guaranteed by the Riesz–Markov–Kakutani representation theorems +in Section 4.3. +Under the assumptions of Remark 5.2.6 the map +RU,G∶S(F(Ω)εE) → FG(U,E), f ↦ (T E +m(f)(x))(m,x)∈U, +is well-defined. +The map RU,G is also linear since T E +m is linear for all m ∈ M. +Further, the strength of the defining family guarantees that RU,G is injective. +5.2.8. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a +separating subspace and U a set of uniqueness for (T K +m,F)m∈M. If (T E +m,T K +m)m∈M is +a strong family for (F,E), then the map +T E∶F(Ω,E) → EU, f ↦ (T E +m(f)(x))(m,x)∈U, +is injective, in particular, RU,G is injective. +Proof. Let f ∈ F(Ω,E) with T E(f) = 0. Then +0 = (e′ ○ T E(f))(m,x) = (e′ ○ T E +m(f))(x) = T K +m(e′ ○ f)(x), +(m,x) ∈ U, +and e′ ○ f ∈ F(Ω) for all e′ ∈ E′ by the strength of the family. Since U is a set of +uniqueness, we get that e′ ○ f = 0 for all e′ ∈ E′, which implies f = 0. +□ +5.2.9. Question. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a sepa- +rating subspace, (T E +m,T K +m)m∈M a strong family for (F,E) and U a set of uniqueness +for (T K +m,F)m∈M. When is the injective restriction map +RU,G∶S(F(Ω)εE) → FG(U,E), f ↦ (T E +m(f)(x))(m,x)∈U, +surjective? +The Question 5.2.1 is a special case of this question if there is a set of uniqueness +U for (T K +m,F)m∈M with {T K +m,x ∣ (m,x) ∈ U} = {δx ∣ x ∈ Λ}, Λ ⊂ Ω. We observe that +a positive answer to the surjectivity of RΩ,G results in the following weak-strong +principle. +5.2.10. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a +separating subspace such that e′ ○ f ∈ F(Ω) for all e′ ∈ G and f ∈ F(Ω,E). If +RΩ,G∶S(F(Ω)εE) → FG(Ω,E), f ↦ f, +with the set of uniqueness Ω for (idKΩ,F) is surjective, then +F(Ω)εE ≅ F(Ω,E) +via S +and +F(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)}. +Proof. From the ε-into-compatibility and the surjectivity of RΩ,G we obtain +{f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)} = FG(Ω,E) = S(F(Ω)εE) ⊂ F(Ω,E). +Further, the assumption that e′ ○ f ∈ F(Ω) for all e′ ∈ G and f ∈ F(Ω,E), implies +that F(Ω,E) is a subspace of the space on the left-hand side, which proves our +statement, in particular, the surjectivity of S. +□ + +78 +5. APPLICATIONS +To answer Question 5.2.9 for general sets of uniqueness we have to restrict to +a certain class of separating subspaces of E′. +5.2.11. Definition (determine boundedness [30, p. 230]). A linear subspace +G ⊂ E′ determines boundedness if every σ(E,G)-bounded set B ⊂ E is already +bounded in E. +In [67, p. 139] such a space G is called uniform boundedness deciding by Fer- +nández et al. and in [134, p. 63] w∗-thick by Nygaard if E is a Banach space. +5.2.12. Remark. +a) Let E be an lcHs. Then G ∶= E′ determines bound- +edness by [131, Mackey’s theorem 23.15, p. 268]. +b) Let X be a barrelled lcHs, Y an lcHs and E ∶= Lb(X,Y ). For x ∈ X and +y′ ∈ Y ′ we set δx,y′∶L(X,Y ) → K, T ↦ y′(T(x)), and G ∶= {δx,y′ ∣ x ∈ +X, y′ ∈ Y ′} ⊂ E′. Then the span of G determines boundedness (in E) by +Mackey’s theorem and the uniform boundedness principle. For Banach +spaces X,Y this is already observed in [30, Remark 11, p. 233] and, if in +addition Y = K, in [7, Remark 1.4 b), p. 781]. +c) Further examples and a characterisation of subspaces G ⊂ E′ that de- +termine boundedness can be found in [7, Remark 1.4, p. 781–782], [134, +Theorem 1.5, p. 63–64] and [134, Theorem 2.3, 2.4, p. 67–68] in the case +that E is a Banach space. +F(Ω) a semi-Montel space and E (sequentially) complete. Our next +results are in need of spaces F(Ω) such that closed graph theorems hold with +Banach spaces as domain spaces and F(Ω) as the range space. Let us formally +define this class of spaces. +5.2.13. Definition (BC-space [142, p. 395]). We call an lcHs F a BC-space +if for every Banach space X and every linear map f∶X → F with closed graph in +X × F, one has that f is continuous. +A characterisation of BC-spaces is given by Powell in [142, 6.1 Corollary, p. 400– +401]. Since every Banach space is ultrabornological and barrelled, the [131, Closed +graph theorem 24.31, p. 289] of de Wilde and the Pták–K¯omura–Adasch–Valdivia +closed graph theorem [103, §34, 9.(7), p. 46] imply that webbed spaces and Br- +complete spaces are BC-spaces. We recall that an lcHs F is said to be Br-complete +if every σ(F ′,F)-dense σf(F ′,F)-closed linear subspace of F ′ equals F ′ where +σf(F ′,F) is the finest topology coinciding with σ(F ′,F) on all equicontinuous sets +in F ′ (see [103, §34, p. 26]). An lcHs F is called B-complete if every σf(F ′,F)-closed +linear subspace of F ′ is weakly closed. In particular, B-complete spaces are Br- +complete and every Br-complete space is complete by [103, §34, 2.(1), p. 26]. These +definitions are equivalent to the original definitions of Br- and B-completeness by +Pták [143, Definition 2, 5, p. 50, 55] due to [103, §34, 2.(2), p. 26–27] and we +note that they are also called infra-Pták spaces and Pták spaces, respectively. In +particular, Fréchet spaces are B-complete by [89, 9.5.2 Krein–˘Smulian Theorem, p. +184] but we will encounter non-Fréchet B-complete spaces as well. +The following proposition is a modification of [94, Satz 10.6, p. 237] and uses +the map Rf∶e′ ↦ fe′ from Remark 5.2.5. +5.2.14. Proposition. Let U be a set of uniqueness for (T K +m,F)m∈M and F(Ω) +a BC-space. Then Rf(B○ +α) is bounded in F(Ω) for every f ∈ FE′(U,E) and α ∈ A +where Bα ∶= {x ∈ E ∣ pα(x) < 1}. In addition, if F(Ω) is a semi-Montel space, then +Rf(B○ +α) is relatively compact in F(Ω). + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +79 +Proof. Let f ∈ FE′(U,E) and α ∈ A. The polar B○ +α is compact in E′ +σ and +thus E′ +B○α is a Banach space by [131, Corollary 23.14, p. 268]. +We claim that +the restriction of Rf to E′ +B○α has closed graph. Indeed, let (e′ +τ) be a net in E′ +B○α +converging to e′ in E′ +B○α and Rf(e′ +τ) converging to g in F(Ω). For (m,x) ∈ U we +note that +T K +m,x(Rf(e′ +τ)) = T K +m(fe′τ )(x) = (e′ +τ ○ f)(m,x) → (e′ ○ f)(m,x) = T K +m(fe′)(x) += T K +m(Rf(e′))(x). +The left-hand side converges to T K +m,x(g) since T K +m,x ∈ F(Ω)′ for all (m,x) ∈ U. +Hence we have T K +m(g)(x) = T K +m(Rf(e′))(x) for all (m,x) ∈ U. From U being a +set of uniqueness follows that g = Rf(e′). Thus the restriction of Rf to E′ +B○α has +closed graph and is continuous since F(Ω) is a BC-space. This yields that Rf(B○ +α) +is bounded as B○ +α is bounded in E′ +B○α. If F(Ω) is also a semi-Montel space, then +Rf(B○ +α) is even relatively compact. +□ +Now, we are ready to prove our first extension theorem. Its proof of surjectivity +of RU,E′ is just an adaptation of the proof of surjectivity of S given in Theorem +3.2.4. Let U be a set of uniqueness for (T K +m,F)m∈M. For f ∈ FE′(U,E) we consider +the dual map +Rt +f∶F(Ω)′ → E′⋆, Rt +f(y)(e′) ∶= y(fe′), +where E′⋆ is the algebraic dual of E′. Further, we recall the notation J ∶E → E′⋆ +for the canonical injection. +5.2.15. Theorem. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E +m,T K +m)m∈M +a strong, consistent family for (F,E), F(Ω) a semi-Montel BC-space and U a set +of uniqueness for (T K +m,F)m∈M. If +(i) E is complete, or +(ii) E is sequentially complete and for every f ∈ FE′(U,E) and f ′ ∈ F(Ω)′ +there is a sequence (f ′ +n)n∈N in F(Ω)′ converging to f ′ in F(Ω)′ +κ such that +Rt +f(f ′ +n) ∈ J (E) for every n ∈ N, +then the restriction map RU,E′∶S(F(Ω)εE) → FE′(U,E) is surjective. +Proof. Let f ∈ FE′(U,E). +As in Theorem 3.2.4 we equip J (E) with the +system of seminorms given by +pB○α(J (x)) ∶= sup +e′∈B○α +∣J (x)(e′)∣ = pα(x), +x ∈ E, +(45) +for all α ∈ A where Bα ∶= {x ∈ E ∣ pα(x) < 1}. We claim Rt +f ∈ L(F(Ω)′ +κ,J (E)). +Indeed, we have for y ∈ F(Ω)′ +pB○α(Rt +f(y)) = sup +e′∈B○α +∣y(fe′)∣ = +sup +x∈Rf (B○α) +∣y(x)∣ ≤ sup +x∈Kα +∣y(x)∣ +(46) +where Kα ∶= Rf(B○α). +Due to Proposition 5.2.14 the set Rf(B○ +α) is absolutely +convex and relatively compact, implying that Kα is absolutely convex and compact +in F(Ω) by [89, 6.2.1 Proposition, p. 103]. Further, we have for all e′ ∈ E′ and +(m,x) ∈ U +Rt +f(T K +m,x)(e′) = T K +m,x(fe′) = (e′ ○ f)(m,x) = J (f(m,x))(e′) +(47) +and thus Rt +f(T K +m,x) ∈ J (E). +First, let condition (i) be satisfied, i.e. let E be complete, and f ′ ∈ F(Ω)′. The +span of {T K +m,x ∣ (m,x) ∈ U} is dense in F(Ω)′ +κ since U is a set of uniqueness for + +80 +5. APPLICATIONS +F(Ω). Thus there is a net (f ′ +τ) converging to f ′ in FV(Ω)′ +κ with Rt +f(f ′ +τ) ∈ J (E) +and +pB○α(Rt +f(f ′ +τ) − Rt +f(f ′)) ≤ +(46) sup +x∈Kα +∣(f ′ +τ − f ′)(x)∣ → 0 +(48) +for all α ∈ A. We gain that (Rt +f(f ′ +τ)) is a Cauchy net in the complete space J (E). +Hence it has a limit g ∈ J (E) which coincides with Rt +f(f ′) since +pB○α(g − Rt +f(f ′)) ≤ pB○α(g − Rt +f(f ′ +τ)) + pB○α(Rt +f(f ′ +τ) − Rt +f(f ′)) +≤ +(48)pB○α(g − Rt +f(f ′ +τ)) + sup +x∈Kα +∣(f ′ +τ − f ′)(x)∣ → 0 +for all α ∈ A. We conclude that Rt +f(f ′) ∈ J (E) for every f ′ ∈ F(Ω)′. +Second, let condition (ii) be satisfied and f ′ ∈ F(Ω)′. Then there is a sequence +(f ′ +n) in F(Ω)′ converging to f ′ in F(Ω)′ +κ such that Rt +f(f ′ +n) ∈ J (E) for every +n ∈ N. From (46) we derive that (Rt +f(f ′ +n)) is a Cauchy sequence in the sequentially +complete space J (E) converging to Rt +f(f ′) ∈ J (E). +Therefore we obtain in both cases that Rt +f ∈ L(F(Ω)′ +κ,J (E)). So we get for +all α ∈ A and y ∈ F(Ω)′ +pα((J −1 ○ Rt +f)(y)) = +(45) pB○α(J ((J −1 ○ Rt +f)(y))) = pB○α(Rt +f(y)) ≤ +(46) sup +x∈Kα +∣y(x)∣. +This implies J −1 ○ Rt +f ∈ L(F(Ω)′ +κ,E) = F(Ω)εE (as linear spaces). We set F ∶= +S(J −1 ○ Rt +f) and obtain from consistency that +T E +m(F)(x) = T E +mS(J −1 ○Rt +f)(x) = J −1(Rt +f(T K +m,x)) = +(47) J −1(J (f(m,x))) = f(m,x) +for every (m,x) ∈ U, which means RU,E′(F) = f. +□ +If E is complete and U a set of uniqueness for (T K +m,F)m∈M with {T K +m,x ∣ (m,x) ∈ +U} = {δx ∣ x ∈ Λ}, Λ ⊂ Ω, then we get [77, 0.1, p. 217] as a special case. Condition +(i) and (ii) are adaptations of Condition 3.2.3 a) and c) from FV(Ω,E) and Rf to +FE′(U,E) and Rf. We also treat an adaptation of Condition 3.2.3 e) in Theorem +5.2.52. Condition 3.2.3 b) and d) may be adapted as well but we restrict to the +ones we actually apply. First, we apply Theorem 5.2.15 to the space of bounded +zero-solutions of a hypoelliptic linear partial differential operator equipped with the +strict topology β from Proposition 4.2.24. +5.2.16. Proposition. Let Ω ⊂ Rd be open and P(∂)K a hypoelliptic linear +partial differential operator. +Then (C∞ +P (∂),b(Ω),β) is a B-complete semi-Montel +space. +Proof. Due to the proof of Proposition 4.2.24 we know that β coincides with +the mixed topology γ(τc,∥ ⋅ ∥∞). It is easy to check that the closed ∥ ⋅ ∥∞-unit ball +B∥⋅∥∞ is τc-compact in C∞ +P (∂),b(Ω). Thus [46, Section I.1, 1.13 Proposition, p. 11] +yields that (C∞ +P (∂),b(Ω),β) is a semi-Montel space. From [150, 2.9 Theorem, p. 185] +it follows that the space is B-complete. +□ +5.2.17. Corollary. Let Ω ⊂ Rd be open, E a complete lcHs, P(∂)K a hypoel- +liptic linear partial differential operator, (T E +m,T K +m)m∈M a strong, consistent family +for ((C∞ +P (∂),b(Ω),β),E) and U a set of uniqueness for (T K +m,(C∞ +P (∂),b(Ω),β))m∈M. +If f∶U → E is a function such that there is fe′ ∈ C∞ +P (∂),b(Ω) for each e′ ∈ E′ +with T K +m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U, then there is a unique F ∈ +C∞ +P (∂),b(Ω,E) with T E +m(F)(x) = f(m,x) for all (m,x) ∈ U. + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +81 +Proof. The space (C∞ +P (∂),b(Ω),β) is a semi-Montel BC-space by Proposition +5.2.16. Moreover, (C∞ +P (∂),b(Ω),β) and (C∞ +P (∂),b(Ω,E),β) are ε-compatible by Propo- +sition 4.2.24, yielding our statement by Theorem 5.2.15 (i) and Proposition 5.2.8. +□ +Especially, for any m ∈ N0 the family ((∂β)E,(∂β)K)β∈Nd +0,∣β∣≤m is strong and con- +sistent for ((C∞ +P (∂),b(Ω),β),E) by the proof of Proposition 4.2.24. It is always pos- +sible to construct a strong, consistent family (T E +m,T K +m)m∈M for ((C∞ +P (∂),b(Ω),β),E) +from a given set of uniqueness (T K +m,(C∞ +P (∂),b(Ω),β))m∈M due to Remark 4.3.11 b) +and c). +Similarly, we may apply Theorem 5.2.15 to the space E{Mp}(Ω,E) of ultradiffer- +entiable functions of class {Mp} of Roumieu-type from Example 3.1.9 f). E{Mp}(Ω) +is a projective limit of a countable sequence of DFS-spaces by [99, Theorem 2.6, +p. 44] and thus webbed because being webbed is stable under the formation of +projective and inductive limits of countable sequences by [89, 5.3.3 Corollary, p. +92]. Further, if the sequence (Mp)p∈N0 satisfies Komatsu’s conditions (M.1) and +(M.3)’, then E{Mp}(Ω) is a Montel space by [99, Theorem 5.12, p. 65–66]. The +spaces E{Mp}(Ω) and E{Mp}(Ω,E) are ε-compatible if (M.1) and (M.3)’ hold and +E is complete by Example 3.2.11 b). Hence Theorem 5.2.15 (i) is applicable. +5.2.18. Remark. We remark that Remark 5.2.6 and Theorem 5.2.15 still hold +if the map S∶F(Ω)εE → F(Ω,E) is only a linear isomorphism into, i.e. an isomor- +phism into of linear spaces, since the topological nature of ε-into-compatibility is +not used in the proof. In particular, this means that it can be applied to the space +M(Ω,E) of meromorphic functions on an open, connected set Ω ⊂ C with values +in an lcHs E over C (see [29, p. 356]). The space M(Ω) is a Montel LF-space, +thus webbed by [89, 5.3.3 Corollary (b), p. 92], due to the proof of [80, Theorem +3 (a), p. 294–295] if it is equipped with the locally convex topology τML given in +[80, p. 292]. By [29, Proposition 6, p. 357] the map S∶M(Ω)εE → M(Ω,E) is +a linear isomorphism if E is locally complete and does not contain the space CN. +Therefore we can apply Theorem 5.2.15 if E is complete and does not contain CN. +This augments [92, Theorem 12, p. 12] where E is assumed to be locally complete +with suprabarrelled strong dual and (T E,T C) = (idEΩ,idCΩ). +F(Ω) a Fréchet–Schwartz space and E locally complete. We recall the +following abstract extension result. +5.2.19. Proposition ([30, Proposition 7, p. 231]). Let E be a locally complete +lcHs, Y a Fréchet–Schwartz space, X ⊂ Y ′ +b (= Y ′ +κ) dense and A∶X → E linear. Then +the following assertions are equivalent: +a) There is a (unique) extension ̂A ∈ Y εE of A. +b) (At)−1(Y ) (= {e′ ∈ E′ ∣ e′ ○ A ∈ Y }) determines boundedness in E. +Next, we generalise [30, Theorem 9, p. 232] using the preceding proposition. +The proof of the generalisation is simply obtained by replacing the set of uniqueness +in the proof of [30, Theorem 9, p. 232] by our more general set of uniqueness. +5.2.20. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E +m,T K +m)m∈M be a strong, +consistent family for (F,E), F(Ω) a Fréchet–Schwartz space and U a set of unique- +ness for (T K +m,F)m∈M. Then the restriction map RU,G∶S(F(Ω)εE) → FG(U,E) is +surjective. + +82 +5. APPLICATIONS +Proof. Let f ∈ FG(U,E). +We choose X ∶= span{T K +m,x ∣ (m,x) ∈ U} and +Y ∶= F(Ω). Let A∶X → E be the linear map generated by A(T K +m,x) ∶= f(m,x). +The map A is well-defined since G is σ(E′,E)-dense. Let e′ ∈ G and fe′ be the +unique element in F(Ω) such that T K +m(fe′)(x) = (e′ ○ A)(T K +m,x) for all (m,x) ∈ +U. +This equation allows us to consider fe′ as a linear form on X (by setting +fe′(T K +m,x) ∶= (e′ ○ A)(T K +m,x)), which yields e′ ○ A ∈ F(Ω) for all e′ ∈ G. It follows +that G ⊂ (At)−1(Y ), implying that (At)−1(Y ) determines boundedness. Applying +Proposition 5.2.19, there is an extension ̂A ∈ F(Ω)εE of A and we set F ∶= S(̂A). +We note that +T E +m(F)(x) = T E +mS(̂A)(x) = ̂A(T K +m,x) = A(T K +m,x) = f(m,x) +for all (m,x) ∈ U by consistency, yielding RU,G(F) = f. +□ +Let us apply the preceding theorem to our weighted spaces of continuously +partially differentiable functions and its subspaces from Example 3.1.9 and Example +4.2.22. +5.2.21. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness, V∞ a directed family of weights which is locally bounded away from zero on +an open set Ω ⊂ Rd, let F(Ω) be a Fréchet–Schwartz space and U ⊂ Nd +0 × Ω a set +of uniqueness for (∂β,F)β∈Nd +0 where F stands for CV∞, CV∞ +0 , CV∞ +P (∂) or CV∞ +P (∂),0. +Then the following holds: +a) If f∶U → E is a function such that there is fe′ ∈ F(Ω) for each e′ ∈ G +with ∂βfe′(x) = (e′ ○ f)(β,x) for all (β,x) ∈ U, then there is a unique +F ∈ F(Ω,E) with (∂β)EF(x) = f(β,x) for all (β,x) ∈ U. +b) If U ⊂ Ω and f∶U → E is a function such that e′ ○ f admits an extension +fe′ ∈ F(Ω) for every e′ ∈ G, then there is a unique extension F ∈ F(Ω,E) +of f. +c) F(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)}. +Proof. The strength and consistency of ((∂β)E,∂β)β∈Nd +0 for (F,E) and the ε- +compatibility of F(Ω) and F(Ω,E) follow from Example 3.2.7 e)+f) and Example +4.2.22 c)+f). This implies that part a) and its special case part b) hold by Theorem +5.2.20 and Proposition 5.2.8. Part c) follows from part b) and Proposition 5.2.10 +since U ∶= Ω is a set of uniqueness for (idKΩ,F). +□ +5.2.22. Remark. Let V∞ be a directed family of weights which is locally +bounded away from zero on an open set Ω ⊂ Rd. +a) Then any dense set U ⊂ Ω is a set of uniqueness for (idKΩ,F) with F = +CV∞, CV∞ +0 , CV∞ +P (∂) or CV∞ +P (∂),0 due to continuity. +b) Let Ω be connected and x0 ∈ Ω. +Then U ∶= {(en,x) ∣ 1 ≤ n ≤ d, x ∈ +Ω} ∪ {(0,x0)} is a set of uniqueness for (∂β,F)β∈N0 by the mean value +theorem with F from a). +c) Let K ∶= R, d ∶= 1, Ω ∶= (a,b) ⊂ R, g∶(a,b) → N and x0 ∈ (a,b). Then +U ∶= {(g(x),x) ∣ x ∈ (a,b)} ∪ {(n,x0) ∣ n ∈ N0} is a set of uniqueness for +(∂β,F)β∈N0 with F from a). Indeed, if f ∈ F(Ω) and 0 = ∂g(x)f(x) for all +x ∈ (a,b), then f is a polynomial by [56, Chap. 11, Theorem, p. 53]. If, in +addition, 0 = ∂nf(x0) for all n ∈ N0, then the polynomial f must vanish +on the whole interval Ω. +d) Let Ω ⊂ C be connected. Then any set U ⊂ Ω with an accumulation point +in Ω is a set of uniqueness for (idCΩ,CV∞ +∂ ) by the identity theorem for +holomorphic functions. + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +83 +e) Let Ω ⊂ C be connected and z0 ∈ Ω. Then U ∶= {(n,z0) ∣ n ∈ N0} is a set +of uniqueness for (∂n +C,CV∞ +∂ )n∈N0 by local power series expansion and the +identity theorem. +f) Let Ω ⊂ Rd be connected. +Then any non-empty open set U ⊂ Ω is a +set of uniqueness for (idKΩ,CV∞ +∆) by the identity theorem for harmonic +functions (see e.g. [85, Theorem 5, p. 218]). +g) Further examples of sets of uniqueness for (idKΩ,CV∞ +∆) are given in [98]. +In part e) a special case of Remark 5.2.3 b) is used, namely, that CW∞ +∂ (Dr(z0)) +has a Schauder basis with associated coefficient functionals (δz0 ○ ∂n +C)n∈N0 where +0 < r ≤ ∞ is such that Dr(z0) ⊂ Ω. In order to obtain some sets of uniqueness which +are more sensible w.r.t. the family of weights V∞, we turn to entire and harmonic +functions fulfilling some growth conditions. For a family V ∶= (νj)j∈N of continuous +weights on Rd set V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd +0 ∣ ∣β∣ ≤ m}×Rd → [0,∞), +νj,m(β,x) ∶= νj(x). We know that CVP (∂)(Rd,E) = CV∞ +P (∂)(Rd,E) as locally convex +spaces and that CVP (∂)(Rd) is a nuclear Fréchet space for P(∂) = ∂ or P(∂) = ∆ +by Proposition 4.2.19 if E is a locally complete lcHs and V fulfils Condition 4.2.18. +In particular, Condition 4.2.18 is fulfilled if νj(x) ∶= exp(−(τ + 1 +j )∣x∣), x ∈ Rd, for +all j ∈ N and some 0 ≤ τ < ∞ and thus we can apply Corollary 5.2.21 to the +spaces Aτ +∂(C,E) = CV∂(C,E) of entire and Aτ +∆(Rd,E) = CV∆(Rd,E) of harmonic +functions of exponential type τ by Remark 4.2.20. Hence we may complement our +list in Remark 5.2.22 by some more examples for spaces of functions of exponential +type 0 ≤ τ < ∞. +5.2.23. Remark. The following sets U ⊂ C are sets of uniqueness for (idCC,Aτ +∂). +a) If τ < π, then U ∶= N0 is a set of uniqueness by [21, 9.2.1 Carlson’s theorem, +p. 153]. +b) Let δ > 0 and (λn)n∈N ⊂ (0,∞) such that λn+1 − λn > δ for all n ∈ N. Then +U ∶= (λn)n∈N is a set of uniqueness if limsupr→∞ r−2τ/πψ(r) = ∞ where +ψ(r) ∶= exp(∑λn 0, by [21, 9.5.1 Fuchs’s theorem, p. 157–158]. +The following sets U are sets of uniqueness for (∂n +C,Aτ +∂)n∈N0. +c) Let (λn)n∈N0 ⊂ C with ∣λn∣ < 1 for all n ∈ N0. If τ < ln(2), then U ∶= +{(n,λn) ∣ n ∈ N0} is a set of uniqueness by [21, 9.11.1 Theorem, p. 172]. +If τ < ln(2 + +√ +3), then U ∶= {(2n + 1,0) ∣ n ∈ N0} ∪ {(2n,λn) ∣ n ∈ N0} is a +set of uniqueness by [21, 9.11.3 Theorem, p. 173]. +d) Let (λn)n∈N0 ⊂ C with limsupn→∞ n−1 ∑n +k=1 ∣λk∣ ≤ 1. +If τ < e−1, then +U ∶= {(n,λn) ∣ n ∈ N0} is a set of uniqueness by [21, 9.11.4 Theorem, p. +173]. +The following sets U ⊂ Rd are sets of uniqueness for (idRRd ,Aτ +∆). +e) Let d ∶= 2. If there is k ∈ N with τ < π/k, then U ∶= Z ∪ (Z + ik) is a set of +uniqueness by [22, Theorem 1, p. 425]. +f) Let d ∶= 2. If τ < π and θ ∉ πQ, then U ∶= Z ∪ (eiθZ) is a set of uniqueness +by [22, Theorem 2, p. 426]. +g) If τ < π, then U ∶= {0,1} × Zd−1 is a set of uniqueness by [145, Corollary +1.8, p. 312]. +h) If τ < π and a ∈ R with ∣a∣ ≤ +√ +1/(d − 1), then U ∶= Zd−1 × {0,a} is a set of +uniqueness by [185, Theorem A, p. 335]. +i) Further examples of sets of uniqueness can be found in [10]. +The following sets U are sets of uniqueness for ((∂β)R,Aτ +∆)β∈Nd +0. +j) If τ < π, then U ∶= {(β,(x,0)) ∣ β ∈ {0,ed}, x ∈ Zd−1} is a set of uniqueness +by [185, Theorem B, p. 335]. Further examples can be found in [10]. + +84 +5. APPLICATIONS +We need the following weak-strong principle in our last section for the space +E0(E) of E-valued infinitely continuously partially differentiable functions on (0,1) +such that all derivatives can be continuously extended to the boundary and vanish +at 1. +5.2.24. Corollary. Let E be a locally complete lcHs and G ⊂ E′ determine +boundedness. Then E0(E) = {f∶(0,1) → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ E0}. +Proof. By Example 4.2.29 E0 is a Fréchet–Schwartz space and E0 and E0(E) +are ε-compatible. We derive our statement from Theorem 5.2.20 and Proposition +5.2.10 with (T E,T K) ∶= (idEU ,idKU ) and U ∶= (0,1). +□ +Fν(Ω) a Banach space and E locally complete. In this subsection we +consider function spaces F(Ω,E) with a certain structure, namely, spaces FV(Ω,E) +from Definition 3.1.4 where the family of weights V = (νj,m)j∈J,m∈M only consists +of one weight function, i.e. the sets J and M can be chosen as singletons. So for +two non-empty sets Ω and ω, a weight ν∶ω → (0,∞), a linear operator T E∶EΩ ⊃ +domT E → Eω and a linear subspace AP(Ω,E) of EΩ we consider the space +Fν(Ω,E) = {f ∈ F(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣α < ∞} +where +F(Ω,E) = AP(Ω,E) ∩ domT E +and +∣f∣α = ∣f∣Fν(Ω),α = sup +x∈ω pα(T E(f)(x))ν(x). +For instance, if Ω ∶= ω, T E ∶= idEΩ and ν ∶= 1 on Ω, then Fν(Ω,E) is the +linear subspace of F(Ω,E) consisting of bounded functions. We use the methods +developed in [70, 93] where, in particular, the special case that Fν(Ω) is the space +of bounded smooth functions on an open set Ω ⊂ Rd in the kernel of a hypoelliptic +linear partial differential operator resp. a weighted space of holomorphic functions +on an open subset Ω of a Banach space is treated. The lack of compact subsets of an +infinite dimensional Banach space Fν(Ω) is compensated in [70, 93] by equipping +F(Ω) with a locally convex Hausdorff topology such that the closed unit ball of +Fν(Ω) is compact in F(Ω). Among others, the space F(Ω,E) ∶= (O(Ω,E),τc) of +holomorphic functions on an open set Ω ⊂ C with values in a locally complete space +E equipped with topology τc of compact convergence is used in [70] and the space +Fν(Ω,E) ∶= H∞(Ω,E) of E-valued bounded holomorphic functions on Ω. +5.2.25. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E,T K) +a consistent family for (F,E) and a generator for (Fν,E) and the map i∶Fν(Ω) → +F(Ω), f ↦ f, continuous. We set +Fεν(Ω,E) ∶= S({u ∈ F(Ω)εE ∣ u(B○F (Ω)′ +Fν(Ω) ) is bounded in E}) +where B○F (Ω)′ +Fν(Ω) ∶= {y′ ∈ F(Ω)′ ∣ ∀ f ∈ BFν(Ω) ∶ ∣y′(f)∣ ≤ 1} and BFν(Ω) is the closed +unit ball of Fν(Ω). Then the following holds: +a) Fν(Ω) is a dom-space. +b) Let u ∈ F(Ω)εE. Then +sup +y′∈B○F (Ω)′ +Fν(Ω) +pα(u(y′)) = ∣S(u)∣Fν(Ω),α, +α ∈ A. +In particular, +Fεν(Ω,E) = S({u ∈ F(Ω)εE ∣ ∀ α ∈ A ∶ ∣S(u)∣Fν(Ω),α < ∞}). + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +85 +c) S(Fν(Ω)εE) ⊂ Fεν(Ω,E) ⊂ Fν(Ω,E) as linear spaces. +If F(Ω) and +F(Ω,E) are even ε-compatible, then Fεν(Ω,E) = Fν(Ω,E). +d) If Fν(Ω,E) is Hausdorff, then +(i) (T E,T K) is a consistent generator for (Fν,E). +(ii) Fν(Ω) and Fν(Ω,E) are ε-into-compatible. +(iii) (T E,T K) is a strong generator for (Fν,E) if it is a strong family for +(F,E). +Proof. Part a) follows from the continuity of the map i and the ε-into- +compatibility of F(Ω) and F(Ω,E). Let us turn to part b). As in Lemma 3.1.8 it +follows from the bipolar theorem that +B○F (Ω)′ +Fν(Ω) = acx{T K +x (⋅)ν(x) ∣ x ∈ ω}, +where acx denotes the closure w.r.t. κ(F(Ω)′,Fν(Ω)) of the absolutely convex hull +acx of the set D ∶= {T K +x (⋅)ν(x) ∣ x ∈ ω} on the right-hand side, and that +sup +y′∈B○F (Ω)′ +Fν(Ω) +pα(u(y′)) = +sup +y′∈acx(D) +pα(u(y′)) = sup +y′∈D +pα(u(y′)) = sup +x∈ω pα(u(T K +x ))ν(x) += sup +x∈ω pα(T E(S(u))(x))ν(x) = ∣S(u)∣Fν(Ω),α +by consistency, which proves part b). +Let us address part c). +The continuity of the map i implies the continuity +of the inclusion Fν(Ω)εE ↪ F(Ω)εE and thus we obtain u∣F (Ω)′ ∈ F(Ω)εE for +every u ∈ Fν(Ω)εE. If u ∈ Fν(Ω)εE and α ∈ A, then there are C0,C1 > 0 and an +absolutely convex compact set K ⊂ Fν(Ω) such that K ⊂ C1BFν(Ω) and +sup +y′∈B○F (Ω)′ +Fν(Ω) +pα(u(y′)) ≤ C0 +sup +y′∈B○F (Ω)′ +Fν(Ω) +sup +f∈K +∣y′(f)∣ ≤ C0C1, +which implies S(Fν(Ω)εE) ⊂ Fεν(Ω,E). If f ∶= S(u) ∈ Fεν(Ω,E) and α ∈ A, then +S(u) ∈ F(Ω,E) and +∣f∣Fν(Ω),α = sup +x∈ω pα(u(T K +x )ν(x)) < ∞ +by consistency, yielding Fεν(Ω,E) ⊂ Fν(Ω,E). If F(Ω) and F(Ω,E) are even +ε-compatible, then S(F(Ω)εE) = F(Ω,E), which yields Fεν(Ω,E) = Fν(Ω,E) by +part b). +Let us turn to part d). By part a) Fν(Ω) is a dom-space. Since Fν(Ω,E) is +Hausdorff, it is also a dom-space due to Remark 3.1.6 c). We have u∣F (Ω)′ ∈ F(Ω)εE +for every u ∈ Fν(Ω)εE and +SFν(Ω)(u)(x) = u(δx) = u∣F (Ω)′(δx) = SF (Ω)(u∣F (Ω)′)(x), +x ∈ Ω. +In combination with S(F(Ω)εE) ⊂ F(Ω,E) and the consistency of (T E,T K) for +(F,E) this yields that (T E,T K) is a consistent generator for (Fν,E). Thus part (i) +holds and implies part (ii) by Theorem 3.1.12. If (T E,T K) is in addition a strong +family for (F,E), then the inclusion Fν(Ω,E) ⊂ F(Ω,E) implies that e′ ○ f ∈ +F(Ω,E) and T K(e′ ○f)(x) = (e′ ○T E(f))(x) for all e′ ∈ E′, f ∈ Fν(Ω,E) and x ∈ ω. +It follows that (T E,T K) is a strong generator for (Fν,E). +□ +The canonical situation in part c) is that Fεν(Ω,E) and Fν(Ω,E) coincide +as linear spaces for locally complete E as we will encounter in the forthcoming +examples, e.g. if Fν(Ω,E) ∶= H∞(Ω,E) and F(Ω,E) ∶= (O(Ω,E),τc) for an open +set Ω ⊂ C. That all three spaces in part c) coincide is usually only guaranteed +by Corollary 3.2.5 (iii) if E is a semi-Montel space. Therefore the ‘mingle-mangle’ +space Fεν(Ω,E) is a good replacement for S(Fν(Ω)εE) for our purpose. + +86 +5. APPLICATIONS +5.2.26. Remark. Let (T E,T K) be a strong, consistent family for (F,E) and +a generator for (Fν,E). +Let F(Ω) and F(Ω,E) be ε-into-compatible and the +inclusion Fν(Ω) ↪ F(Ω) continuous. Consider a set of uniqueness U for (T K,Fν) +and a separating subspace G ⊂ E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′ +Fν(Ω) ) is bounded +in E, i.e. S(u) ∈ Fεν(Ω,E), we set f ∶= S(u). Then f ∈ F(Ω,E) by the ε-into- +compatibility and we define ̃f∶U → E, ̃f(x) ∶= T E(f)(x). This yields +(e′ ○ ̃f)(x) = (e′ ○ T E(f))(x) = T K(e′ ○ f)(x) +(49) +for all x ∈ U and fe′ ∶= e′ ○ f ∈ F(Ω) for each e′ ∈ E′ by the strength of the family. +Moreover, T K +x (⋅)ν(x) ∈ B○F (Ω)′ +Fν(Ω) for every x ∈ ω, which implies that for every e′ ∈ E′ +there are α ∈ A and C > 0 such that +∣fe′∣Fν(Ω) = sup +x∈ω +∣e′(u(T K +x (⋅)ν(x))∣ ≤ C +sup +y′∈B○F (Ω)′ +Fν(Ω) +pα(u(y′)) < ∞ +by strength and consistency. Hence fe′ ∈ Fν(Ω) for every e′ ∈ E′ and ̃f ∈ FνG(U,E). +Under the assumptions of Remark 5.2.26 the map +RU,G∶Fεν(Ω,E) → FνG(U,E), f ↦ (T E(f)(x))x∈U, +(50) +is well-defined and linear. In addition, we derive from (49) that RU,G is injective +since U is a set of uniqueness and G ⊂ E′ separating. The replacement of Question +5.2.9 reads as follows. +5.2.27. Question. Let the assumptions of Remark 5.2.26 be fulfilled. When is +the injective restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E), f ↦ (T E(f)(x))x∈U, +surjective? +Due to Proposition 5.2.25 c) the Question 5.2.1 is a special case of this question +if Λ ⊂ Ω =∶ ω and U ∶= Λ is a set of uniqueness for (idKΩ,Fν). We recall the following +extension result for continuous linear operators. +5.2.28. Proposition ([70, Proposition 2.1, p. 691]). Let E be a locally complete +lcHs, G ⊂ E′ determine boundedness, Z a Banach space whose closed unit ball BZ +is a compact subset of an lcHs Y and X ⊂ Y ′ be a σ(Y ′,Z)-dense subspace. If +A∶X → E is a σ(X,Z)-σ(E,G)-continuous linear map, then there exists a (unique) +extension ̂A ∈ Y εE of A such that ̂A(B○Y ′ +Z +) is bounded in E where B○Y ′ +Z +∶= {y′ ∈ +Y ′ ∣ ∀ z ∈ BZ ∶ ∣y′(z)∣ ≤ 1}. +Now, we are able to generalise [70, Theorem 2.2, p. 691] and [93, Theorem 10, +p. 5]. +5.2.29. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bounded- +ness and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K) be a generator for +(Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a Banach space whose +closed unit ball BFν(Ω) is a compact subset of F(Ω) and U a set of uniqueness for +(T K,Fν). Then the restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E) +is surjective. +Proof. Let f ∈ FνG(U,E). We set X ∶= span{T K +x ∣ x ∈ U}, Y ∶= F(Ω) and +Z ∶= Fν(Ω). The consistency of (T E,T K) for (F,E) yields that X ⊂ Y ′. From +U being a set of uniqueness of Z follows that X is σ(Z′,Z)-dense. Since BZ is a +compact subset of Y , it follows that Z is a linear subspace of Y and the inclusion + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +87 +Z ↪ Y is continuous, which yields y′ +∣Z ∈ Z′ for every y′ ∈ Y ′. Thus X is σ(Y ′,Z)- +dense. Let A∶X → E be the linear map determined by A(T K +x ) ∶= f(x). The map A +is well-defined since G is σ(E′,E)-dense. Due to +e′(A(T K +x )) = (e′ ○ f)(x) = T K +x (fe′) +for every e′ ∈ G and x ∈ U we have that A is σ(X,Z)-σ(E,G)-continuous. We +apply Proposition 5.2.28 and gain an extension ̂A ∈ Y εE of A such that ̂A(B○Y ′ +Z +) is +bounded in E. We set ̃F ∶= S(̂A) ∈ Fεν(Ω,E) and get for all x ∈ U that +T E( ̃F)(x) = T ES(̂A)(x) = ̂A(T K +x ) = f(x) +by consistency for (F,E), implying RU,G( ̃F) = f. +□ +Let Ω ⊂ Rd be open, E an lcHs and P(∂)E∶C∞(Ω,E) → C∞(Ω,E) a linear +partial differential operator which is hypoelliptic if E = K. We consider the weighted +space CVP (∂)(Ω,E) of zero solutions from Proposition 4.2.14 where the familiy of +weights V only consists of one continuous weight ν∶Ω → (0,∞), i.e. the space +CνP (∂)(Ω,E) = {f ∈ C∞ +P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣ν,α ∶= sup +x∈Ω +pα(f(x))ν(x) < ∞}. +5.2.30. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness, Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential operator, +ν∶Ω → (0,∞) continuous and U a set of uniqueness for (idKΩ,CνP (∂)). If f∶U → E +is a function such that e′ ○ f admits an extension fe′ ∈ CνP (∂)(Ω) for every e′ ∈ G, +then there exists a unique extension F ∈ CνP (∂)(Ω,E) of f. +Proof. We choose F(Ω) ∶= (C∞ +P (∂)(Ω),τc) and F(Ω,E) ∶= (C∞ +P (∂)(Ω,E),τc). +Then we have Fν(Ω) = CνP (∂)(Ω) and Fν(Ω,E) = CνP (∂)(Ω,E) with the generator +(T E,T K) ∶= (idEΩ,idKΩ) for (Fν,E). +We note that F(Ω) and F(Ω,E) are ε- +compatible and (T E,T K) is a strong, consistent family for (F,E) by Proposition +4.2.17. We observe that Fν(Ω) is a Banach space by Proposition 4.2.14 and for +every compact K ⊂ Ω we have +sup +x∈K +∣f(x)∣ ≤ +(22) sup +z∈K +ν(z)−1∣f∣ν ≤ sup +z∈K +ν(z)−1, +f ∈ BFν(Ω), +yielding that BFν(Ω) is bounded in F(Ω). The space F(Ω) = (C∞ +P (∂)(Ω),τc) is a +Fréchet–Schwartz space, thus a Montel space, and it is easy to check that BFν(Ω) is +τc-closed. Hence the bounded and τc-closed set BFν(Ω) is compact in F(Ω). Finally, +we remark that the ε-compatibility of F(Ω) and F(Ω,E) in combination with the +consistency of (idEΩ,idKΩ) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces +by Proposition 5.2.25 c). From Theorem 5.2.29 follows our statement. +□ +If Ω = D ⊂ C is the open unit disc, P(∂) = ∂ the Cauchy–Riemann operator +and ν = 1 on D, then CνP (∂)(Ω,E) = H∞(D,E) and a sequence U ∶= (zn)n∈N ⊂ D of +distinct elements is a set of uniqueness for (idCD,H∞) if and only if it satisfies the +Blaschke condition ∑n∈N(1 − ∣zn∣) = ∞ (see e.g. [149, 15.23 Theorem, p. 303]). +For a continuous function ν∶D → (0,∞) and a complex lcHs E we define the +Bloch type spaces +Bν(D,E) ∶= {f ∈ O(D,E) ∣ ∀ α ∈ A ∶ ∣f∣ν,α < ∞} +with +∣f∣ν,α ∶= max(pα(f(0)),sup +z∈D +pα((∂1 +C)Ef(z))ν(z)). +If E = C, we write f ′(z) ∶= (∂1 +C)Cf(z) for z ∈ D and f ∈ O(D). +5.2.31. Proposition. If ν∶D → (0,∞) is continuous, then Bν(D) is a Banach +space. + +88 +5. APPLICATIONS +Proof. Let f ∈ Bν(D). From the estimates +∣f(z)∣ ≤ ∣f(0)∣ + ∣ +z +∫ +0 +f ′(ζ)dζ∣ ≤ ∣f(0)∣ + +∣z∣ +minξ∈[0,z] ν(ξ) sup +ζ∈[0,z] +∣f ′(ζ)∣ν(ζ) +≤ 2max(1, +∣z∣ +minξ∈[0,z] ν(ξ))∣f∣ν +for every z ∈ D and +max +∣z∣≤r ∣f(z)∣ ≤ 2max(1, +r +min∣z∣≤r ν(z))∣f∣ν +(51) +for all 0 < r < 1 and f ∈ Bν(D) it follows that Bν(D) is a Banach space by using +the completeness of (O(D),τc) analogously to the proof of Proposition 4.2.14. +□ +5.2.32. Proposition. Let Ω ⊂ C be open and E a locally complete lcHs over +C. Then ((∂n +C)E,(∂n +C)C)n∈N0 is a strong, consistent family for ((O(Ω),τc),E). +Proof. We recall from (5) that the real and complex derivatives are related +by +(∂β)Ef(z) = iβ2(∂∣β∣ +C )Ef(z), +z ∈ Ω, +(52) +for every f ∈ O(Ω,E) and β = (β1,β2) ∈ N2 +0. Further, the Fréchet space (O(Ω),τc) +is barrelled. Due to Proposition 3.1.11 c) and (52) we have for all u ∈ (O(Ω),τc)εE +(∂n +C)ES(u)(z) = u(δz ○ (∂n +C)C), +n ∈ N0, z ∈ Ω, +which means that ((∂n +C)E,(∂n +C)C)n∈N0 is consistent. +Moreover, we have +(∂n +C)C(e′ ○ f)(z) = e′((∂n +C)Ef(z)), +n ∈ N0, z ∈ Ω, +for all e′ ∈ E′ and f ∈ O(Ω,E), implying the strength of ((∂n +C)E,(∂n +C)C)n∈N0. +□ +Let E be an lcHs and ν∶D → (0,∞) be continuous. We set ω ∶= {0}∪{(1,z) ∣ z ∈ +D}, define the operator T E∶O(D,E) → Eω by +T E(f)(0) ∶= f(0) +and +T E(f)(1,z) ∶= (∂1 +C)Ef(z), z ∈ D, +and the weight ν∗∶ω → (0,∞) by +ν∗(0) ∶= 1 +and +ν∗(1,z) ∶= ν(z), z ∈ D. +Then we have for every α ∈ A that +∣f∣ν,α = sup +x∈ω pα(T E(f)(x))ν∗(x), +f ∈ Bν(D,E), +and with F(D,E) ∶= O(D,E) we observe that Fν∗(D,E) = Bν(D,E) with generator +(T E,T C). +5.2.33. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness, ν∶D → (0,∞) continuous and U∗ ⊂ D have an accumulation point in D. If +f∶{0} ∪ ({1} × U∗) → E is a function such that there is fe′ ∈ Bν(D) for each e′ ∈ G +with fe′(0) = e′(f(0)) and f ′ +e′(z) = e′(f(1,z)) for all z ∈ U∗, then there exists a +unique F ∈ Bν(D,E) with F(0) = f(0) and (∂1 +C)EF(z) = f(1,z) for all z ∈ U∗. +Proof. We take F(D) ∶= (O(D),τc) and F(D,E) ∶= (O(D,E),τc). Then we +have Fν∗(D) = Bν(D) and Fν∗(Ω,E) = Bν(D,E) with the weight ν∗ and generator +(T E,T C) for (Fν∗,E) described above. +The spaces F(D) and F(D,E) are ε- +compatible by Proposition 4.2.17 in combination with (23), and the generator is +a strong, consistent family for (F,E) by Proposition 5.2.32. Due to Proposition +5.2.31 Fν∗(D) = Bν(D) is a Banach space and we deduce from (51) that BFν∗(D) is +compact in the Montel space (O(D),τc). We note that the ε-compatibility of F(Ω) + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +89 +and F(Ω,E) in combination with the consistency of (T E,T C) for (F,E) gives +Fεν∗(D,E) = Fν∗(D,E) as linear spaces by Proposition 5.2.25 c). In addition, +U ∶= {0} ∪ {(1,z) ∣ z ∈ U∗} is a set of uniqueness for (T C,Fν∗) by the identity +theorem, proving our statement by Theorem 5.2.29. +□ +E a Fréchet space. In this section we restrict to the case that E is a Fréchet +space and G ⊂ E′ is generated by a sequence that fixes the topology in E. +5.2.34. Definition ([30, Definition 12, p. 8]). Let Y be a Fréchet space. An +increasing sequence (Bn)n∈N of bounded subsets of Y ′ +b fixes the topology in Y if +(B○ +n)n∈N is a fundamental system of zero neighbourhoods of Y . +5.2.35. Remark. Let Y be a Banach space. If B ⊂ Y ′ +b is bounded, i.e. bounded +w.r.t. the operator norm, such that B fixes the topology in Y , i.e. B○ is bounded +in Y , then B is called an almost norming subset. Examples of almost norming +subspaces are given in [7, Remark 1.2, p. 780–781]. For instance, the set of point +evaluations B ∶= {δ1/n ∣ n ∈ N} is almost norming for the Y ∶= H∞(D) ∶= C∞ +∂,b(D). +5.2.36. Definition (sb-restriction space). Let E be a Fréchet space, (Bn) fix +the topology in E and G ∶= span(⋃n∈N Bn). Let FV(Ω) be a dom-space, U a set of +uniqueness for (T K +m,FV)m∈M and set +FVG(U,E)sb ∶= {f ∈ FVG(U,E) ∣ ∀ n ∈ N ∶ {fe′ ∣ e′ ∈ Bn} is bounded in FV(Ω)}. +Let E be a Fréchet space, (Bn) fix the topology in E, G ∶= span(⋃n∈N Bn), +(T E +m,T K +m)m∈M be a strong, consistent generator for (FV,E) and U a set of unique- +ness for (T K +m,FV)m∈M. For u ∈ FV(Ω)εE we have RU,G(f) ∈ FVG(U,E) with +f ∶= S(u) by Remark 5.2.6 and for j ∈ J and m ∈ M +sup +e′∈Bn +∣fe′∣j,m = sup +e′∈Bn +sup +x∈ωm +∣e′(T E +m(f)(x)νj,m(x))∣ = sup +e′∈Bn +sup +y∈Nj,m(f) +∣e′(y)∣ +with Nj,m(f) ∶= {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm}. This set is bounded in E since +sup +y∈Nj,m(f) +pα(f) = ∣f∣j,m,α < ∞ +for all α ∈ A, implying supe′∈Bn ∣fe′∣j,m < ∞ and RU,G(f) ∈ FVG(U,E)sb. Hence the +injective linear map +RU,G∶S(FV(Ω)εE) → FVG(U,E)sb, f ↦ (T E +m(f)(x))(m,x)∈U, +is well-defined. +5.2.37. Question. Let E be a Fréchet space, (Bn) fix the topology in E +and G ∶= span(⋃n∈N Bn). Let (T E +m,T K +m)m∈M be a strong, consistent generator for +(FV,E) and U a set of uniqueness for (T K +m,FV)m∈M. When is the injective restric- +tion map +RU,G∶S(FV(Ω)εE) → FVG(U,E)sb, f ↦ (T E +m(f)(x))(m,x)∈U, +surjective? +5.2.38. Remark. Let E be a Fréchet space with increasing system of seminorms +(pαn)n∈N, Bn ∶= B○ +αn where Bαn ∶= {x ∈ E ∣ pαn(x) < 1}, (T E +m,T K +m)m∈M a strong, +consistent generator for (FV,E) and U a set of uniqueness for (T K +m,FV)m∈M. If +FV(Ω) is a BC-space, then FVE′(U,E)sb = FVE′(U,E) by Proposition 5.2.14. +Hence Theorem 5.2.15 (i) answers Question 5.2.37 in this case. +Let us turn to the case where G need not coincide with E′. + +90 +5. APPLICATIONS +FV(Ω) a Fréchet–Schwartz space and E a Fréchet space. We recall the +following result. +5.2.39. Proposition ([69, Lemma 9, p. 504]). Let E be a Fréchet space, (Bn) +fix the topology in E, Y a Fréchet–Schwartz space and X ⊂ Y ′ +b (= Y ′ +κ) a dense +subspace. +Set G ∶= span(⋃n∈N Bn) and let A∶X → E be a linear map which is +σ(X,Y )-σ(E,G)-continuous and satisfies that At(Bn) is bounded in Y for each +n ∈ N. Then A has a (unique) extension ̂A ∈ Y εE. +Next, we improve [69, Theorem 1 ii), p. 501]. +5.2.40. Theorem. Let E be a Fréchet space, (Bn) fix the topology in E and G ∶= +span(⋃n∈N Bn), (T E +m,T K +m)m∈M a strong, consistent generator for (FV,E), FV(Ω) +a Fréchet–Schwartz space and U a set of uniqueness for (T K +m,FV)m∈M. Then the +restriction map RU,G∶S(FV(Ω)εE) → FVG(U,E)sb is surjective. +Proof. Let f ∈ FVG(U,E)sb. +We set X ∶= span{T K +m,x ∣ (m,x) ∈ U} and +Y ∶= FV(Ω). Let A∶X → E be the linear map determined by A(T K +m,x) ∶= f(m,x) +which is well-defined since G is σ(E′,E)-dense. From +e′(A(T K +m,x)) = (e′ ○ f)(m,x) = T K +m,x(fe′) +for every e′ ∈ G and (m,x) ∈ U it follows that A is σ(X,Y )-σ(E,G)-continuous and +sup +e′∈Bn +∣At(e′)∣j,k = sup +e′∈Bn +∣fe′∣j,k < ∞ +for all j ∈ J, k ∈ M and n ∈ N. Due to Proposition 5.2.39 there is an extension +̂A ∈ FV(Ω)εE of A. We set F ∶= S(̂A) and get for all (m,x) ∈ U that +T E +m(F)(x) = T E +mS(̂A)(x) = ̂A(T K +m,x) = f(m,x) +by consistency, which means RU,G(F) = f. +□ +5.2.41. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and +G ∶= span(⋃n∈N Bn). +Let V ∶= (νj)j∈N be an increasing family of weights which +is locally bounded away from zero on an open set Ω ⊂ Rd, P(∂)K a hypoelliptic +linear partial differential operator, CVP (∂)(Ω) a Schwartz space and U ⊂ Ω a set of +uniqueness for (idKΩ,CVP (∂)). If f∶U → E is a function such that e′ ○ f admits +an extension fe′ ∈ CVP (∂)(Ω) for each e′ ∈ G and {fe′ ∣ e′ ∈ Bn} is bounded in +CVP (∂)(Ω) for each n ∈ N, then there is a unique extension F ∈ CVP (∂)(Ω,E) of f. +Proof. CVP (∂)(Ω) is a Fréchet–Schwartz space and (idEΩ,idKΩ) a strong, +consistent generator for (CVP (∂),E) by Proposition 4.2.14 and the proof of Example +4.2.16 b). Now, Theorem 5.2.40 and Proposition 5.2.8 prove our statement. +□ +We already mentioned examples of families of weights V such that CVP (∂)(Rd) +is a nuclear Fréchet space and sets of uniqueness for (idKRd ,CVP (∂)) in Remark +4.2.20 and Remark 5.2.23 and if P(∂) = ∂ or P(∂) = ∆. Further sets of uniqueness +are given in Remark 5.2.66. If E is a Banach space, then an almost norming set +fixes the topology and examples can be found via Remark 5.2.35. +Fν(Ω) a Banach space and E a Fréchet space. Let E be a Fréchet space, +(Bn) fix the topology in E and recall the assumptions of Remark 5.2.26. +Let +(T E,T K) be a strong, consistent family for (F,E) and a generator for (Fν,E). +Let F(Ω) and F(Ω,E) be ε-into-compatible and the inclusion Fν(Ω) ↪ F(Ω) +continuous. Consider a set of uniqueness U for (T K,Fν) and G ∶= span(⋃n∈N Bn) ⊂ + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +91 +E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′ +Fν(Ω) ) is bounded in E we have RU,G(f) ∈ +FνG(U,E) with f ∶= S(u) ∈ Fεν(Ω,E) by (50). We note that +sup +e′∈Bn +∣fe′∣Fν(Ω) = sup +e′∈Bn +sup +x∈ω ∣e′(T E(f)(x)ν(x))∣ = sup +e′∈Bn +sup +y∈Nω(f) +∣e′(y)∣ +with the bounded set Nω(f) ∶= {T E(f)(x)ν(x) ∣ x ∈ ω} ⊂ E, implying RU,G(f) ∈ +FVG(U,E)sb. Thus the injective linear map +RU,G∶Fεν(Ω,E) → FνG(U,E)sb, f ↦ (T E(f)(x))x∈U, +is well-defined. +5.2.42. Question. Let the assumptions of Remark 5.2.26 be fulfilled, E be a +Fréchet space, (Bn) fix the topology in E and G ∶= span(⋃n∈N Bn). When is the +injective restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E)sb, f ↦ (T E(f)(x))x∈U, +surjective? +Now, we can generalise [70, Corollary 2.4, p. 692] and [93, Theorem 11, p. 5]. +5.2.43. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E, set +G ∶= span(⋃n∈N Bn) and let F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K) +be a generator for (Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a +Banach space whose closed unit ball BFν(Ω) is a compact subset of F(Ω) and U a +set of uniqueness for (T K,Fν). Then the restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E)sb +is surjective. +Proof. Let f ∈ FνG(U,E)sb. +Then {fe′ ∣ e′ ∈ Bn} is bounded in Fν(Ω) +for each n ∈ N. We deduce for each n ∈ N, (ak)k∈N ∈ ℓ1 and (e′ +k)k∈N ⊂ Bn that +(∑k∈N ake′ +k)○f admits the extension ∑k∈N akfe′ +k in Fν(Ω). Due to [69, Proposition +7, p. 503] the LB-space E′((Bn)n∈N) ∶= lim +←� +n∈N +E′(Bn), where +E′(Bn) ∶= {∑ +k∈N +ake′ +k ∣ (ak)k∈N ∈ ℓ1, (e′ +k)k∈N ⊂ Bn} +is endowed with its Banach space topology for n ∈ N, determines boundedness in +E. +Hence we conclude that f ∈ FνE′((Bn)n∈N)(U,E), which yields that there is +u ∈ F(Ω)εE with bounded u(B○F (Ω)′ +Fν(Ω) ) ⊂ E such that RU,G(S(u)) = f by Theorem +5.2.29. +□ +As an application we directly obtain the following two corollaries of Corol- +lary 5.2.43 since its assumptions are fulfilled by the proof of Corollary 5.2.30 and +Corollary 5.2.33, respectively. +5.2.44. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and +G ∶= span(⋃n∈N Bn), Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential +operator, ν∶Ω → (0,∞) continuous and U a set of uniqueness for (idKΩ,CνP (∂)). If +f∶U → E is a function such that e′ ○f admits an extension fe′ ∈ CνP (∂)(Ω) for each +e′ ∈ G and {fe′ ∣ e′ ∈ Bn} is bounded in CνP (∂)(Ω) for each n ∈ N, then there exists +a unique extension F ∈ CνP (∂)(Ω,E) of f. +5.2.45. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and +G ∶= span(⋃n∈N Bn), ν∶D → (0,∞) continuous and U∗ ⊂ D have an accumulation +point in D. If f∶{0} ∪ ({1} × U∗) → E is a function such that there is fe′ ∈ Bν(D) +for each e′ ∈ G with fe′(0) = e′(f(0)) and f ′ +e′(z) = e′(f(1,z)) for all z ∈ U∗ and + +92 +5. APPLICATIONS +{fe′ ∣ e′ ∈ Bn} is bounded in Bν(D) for each n ∈ N, then there exists a unique +F ∈ Bν(D,E) with F(0) = f(0) and (∂1 +C)EF(z) = f(1,z) for all z ∈ U∗. +5.2.2. Extension from thick sets. In order to obtain an affirmative answer +to Question 5.2.9 for general separating subspaces of E′ we have to restrict to the +spaces FV(Ω) from Definition 3.1.4 and a certain class of sets of uniqueness. +5.2.46. Definition (fix the topology). Let FV(Ω) be a dom-space. We say +that U ⊂ ⋃m∈M({m}×ωm) fixes the topology in FV(Ω) if for every j ∈ J and m ∈ M +there are i ∈ J, k ∈ M and C > 0 such that +∣f∣j,m ≤ C +sup +x∈ωk +(k,x)∈U +∣T K +k (f)(x)∣νi,k(x), +f ∈ FV(Ω). +In particular, U is a set of uniqueness if it fixes the topology. The present +definition of fixing the topology is a generalisation of [30, Definition 13, p. 234]. +Sets that fix the topology appear under several different notions. Rubel and Shields +call them dominating in [148, 4.10 Definition, p. 254] in the context of bounded +holomorphic functions. In the context of the space of holomorphic functions with +the topology of compact convergence studied by Grosse-Erdmann [81, p. 401] they +are said to determine locally uniform convergence. Ehrenpreis [61, p. 3,4,13] (cf. +[156, Definition 3.2, p. 166]) refers to them as sufficient sets when he considers +inductive limits of weighted spaces of entire resp. holomorphic functions, including +the case of Banach spaces. In the case of Banach spaces sufficient sets coincide +with weakly sufficient sets defined by Schneider [156, Definition 2.1, p. 163] (see e.g. +[102, §7, 1), p. 547]) and these notions are extended beyond spaces of holomorphic +functions by Korobe˘ınik [102, p. 531]. Seip [162, p. 93] uses the term sampling sets +in the context of weighted Banach spaces of holomorphic functions whereas Beurling +uses the term balayage in [14, p. 341] and [14, Definition, p. 343]. Leibowitz [122, +Exercise 4.1.4, p. 53], Stout [170, 7.1 Definition, p. 36] and Globevnik [76, p. 291– +292] call them boundaries in the context of subalgebras of the algebra C(Ω,C) of +complex-valued continuous functions on a compact Hausdorff space Ω with sup- +norm. Fixing the topology is also connected to the notion of frames used by Bonet +et al. in [31]. Let us set +ℓV(U,E) ∶= {f∶U → E ∣ ∀ j ∈ J,m ∈ M,α ∈ A ∶ ∥f∥j,m,α < ∞} +(53) +with +∥f∥j,m,α ∶= +sup +x∈ωm +(m,x)∈U +pα(f(m,x))νj,m(x) +for an lcHs E and a set U which fixes the topology in FV(Ω). If M is a singleton, +ωm = Ω = U, then ℓV(U,E) coincides with the space defined right above Example +4.2.2. If U is countable, then the inclusion ℓV(U) ↪ KU continuous where KU is +equipped with the topology of pointwise convergence and ℓV(U) contains the space +of sequences (on U) with compact support as a linear subspace, then (T K +k,x)(k,x)∈U +is an ℓV(U)-frame in the sense of [31, Definition 2.1, p. 3]. +5.2.47. Definition (lb-restriction space). Let FV(Ω) be a dom-space, U fix +the topology in FV(Ω) and G ⊂ E′ a separating subspace. We set +NU,i,k(f) ∶= {f(k,x)νi,k(x) ∣ x ∈ ωk, (k,x) ∈ U} +for i ∈ J, k ∈ M and f ∈ FVG(U,E) and +FVG(U,E)lb ∶={f ∈ FVG(U,E) ∣ ∀ i ∈ J, k ∈ M ∶ NU,i,k(f) bounded in E} +=FVG(U,E) ∩ ℓV(U,E). + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +93 +Consider a set U which fixes the topology in FV(Ω), a separating subspace +G ⊂ E′ and a strong, consistent family (T E +m,T K +m)m∈M for (FV,E). For u ∈ FV(Ω)εE +set f ∶= S(u) ∈ FV(Ω,E) by Theorem 3.1.12. Then we have RU,G(f) ∈ FVG(U,E) +with f ∶= S(u) by Remark 5.2.6 and for i ∈ J and k ∈ M +sup +y∈NU,i,k(RU,G(f)) +pα(y) = +sup +x∈ωk +(k,x)∈U +pα(T E +k (f)(x))νi,k(x) ≤ ∣f∣i,k,α < ∞ +for all α ∈ A, implying the boundedness of NU,i,k(RU,G(f)) in E. Thus RU,G(f) ∈ +FVG(U,E)lb and the injective linear map +RU,G∶S(FV(Ω)εE) → FVG(U,E)lb, f ↦ (T E +m(f)(x))(m,x)∈U, +is well-defined. +5.2.48. Question. Let G ⊂ E′ be a separating subspace, (T E +m,T K +m)m∈M a +strong, consistent generator for (FV,E) and U fix the topology in FV(Ω). When +is the injective restriction map +RU,G∶S(FV(Ω)εE) → FVG(U,E)lb, f ↦ (T E +m(f)(x))(m,x)∈U, +surjective? +If G ⊂ E′ determines boundedness and U fixes the topology in FV(Ω), then +the preceding question and Question 5.2.9 coincide. +5.2.49. Remark. Let G ⊂ E′ determine boundedness, (T E +m,T K +m)m∈M a strong, +consistent generator for (FV,E) and U fix the topology in FV(Ω). Then +FVG(U,E)lb = FVG(U,E). +Proof. We only need to show that the inclusion ‘⊃’ holds. Let f ∈ FVG(U,E). +Then there is fe′ ∈ FV(Ω) with T K +m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U and +sup +y∈NU,i,k(f) +∣e′(y)∣ = +sup +x∈ωk +(k,x)∈U +∣(e′ ○ f)(k,x)∣νi,k(x) ≤ ∣fe′∣i,k < ∞ +for each e′ ∈ G, i ∈ J and k ∈ M. Since G ⊂ E′ determines boundedness, this means +that NU,i,k(f) is bounded in E and hence f ∈ FVG(U,E)lb. +□ +FV(Ω) arbitrary and E a semi-Montel space. +5.2.50. Definition (generalised Schwartz space). We call an lcHs E a gener- +alised Schwartz space if every bounded set in E is already precompact. +In particular, semi-Montel spaces and Schwartz spaces are generalised Schwartz +spaces by [89, 10.4.3 Corollary, p. 202]. Conversely, a generalised Schwartz space is +a Schwartz space if it is quasi-normable by [89, 10.7.3 Corollary, p. 215]. Moreover, +looking at the proof of Lemma 3.2.2 b), we see that this lemma not only holds for +semi-Montel or Schwartz spaces but for all generalised Schwartz spaces. +5.2.51. Proposition. Let E be an lcHs, FV(Ω) a dom-space and U fix the +topology in FV(Ω). Then Rf ∈ L(E′ +b,FV(Ω)) and Rf(B○ +α) is bounded in FV(Ω) +for every f ∈ FVE′(U,E)lb and α ∈ A where Bα ∶= {x ∈ E ∣ pα(x) < 1} and Rf is +the map from Remark 5.2.5. In addition, if E is a generalised Schwartz space, then +Rf ∈ L(E′ +γ,FV(Ω)) and Rf(B○ +α) is relatively compact in FV(Ω). +Proof. Let f ∈ FVE′(U,E)lb, j ∈ J and m ∈ M. Then there are i ∈ J, k ∈ M +and C > 0 such that for every e′ ∈ E′ +∣Rf(e′)∣j,m = ∣fe′∣j,m ≤ C +sup +x∈ωk +(k,x)∈U +∣T K +k (fe′)(x)∣νi,k(x) + +94 +5. APPLICATIONS += C +sup +x∈ωk +(k,x)∈U +∣(e′ ○ f)(k,x)∣νi,k(x) = C +sup +y∈NU,i,k(f) +∣e′(y)∣, +which proves the first part because NU,i,k(f) is bounded in E. Let us consider the +second part. The bounded set NU,i,k(f) is already precompact in E because E is +a generalised Schwartz space. Therefore we have Rf ∈ L(E′ +γ,FV(Ω)). The polar +B○ +α is relatively compact in E′ +γ for every α ∈ A by the Alaoğlu–Bourbaki theorem +and thus Rf(B○ +α) in FV(Ω) as well. +□ +5.2.52. Theorem. Let E be a semi-Montel space, (T E +m,T K +m)m∈M a strong, con- +sistent generator for (FV,E) and U fix the topology in FV(Ω). Then the restriction +map RU,E′∶S(FV(Ω)εE) → FVE′(U,E)lb is surjective. +Proof. Let f ∈ FVE′(Ω,E)lb and e′ ∈ E′. For every f ′ ∈ FV(Ω)′ there are +j ∈ J, m ∈ M and C0 > 0 with +∣Rt +f(f ′)(e′)∣ = ∣f ′(fe′)∣ ≤ C0∣fe′∣j,m. +By the proof of Proposition 5.2.51 there are i ∈ J, k ∈ M and C > 0 such that +∣Rt +f(f ′)(e′)∣ ≤ C0C +sup +y∈NU,i,k(f) +∣e′(y)∣ ≤ C0C +sup +y∈acx(NU,i,k(f)) +∣e′(y)∣. +The set acx(NU,i,k(f)) is absolutely convex and compact by [89, 6.2.1 Proposi- +tion, p. 103] and [89, 6.7.1 Proposition, p. 112] because E is a semi-Montel space. +Therefore Rt +f(f ′) ∈ (E′ +κ)′ = J (E) by the Mackey–Arens theorem. As in Theorem +5.2.15 we obtain J −1○Rt +f ∈ FV(Ω)εE by (45), (46) and Proposition 5.2.51. Setting +F ∶= S(J −1 ○ Rt +f), we conclude T E +m(F)(x) = f(m,x) for all (m,x) ∈ U by (47) and +so RU,E′(F) = f. +□ +5.2.53. Remark. Let E be a Fréchet space with increasing system of seminorms +(pαn)n∈N, Bn ∶= B○ +αn where Bαn ∶= {x ∈ E ∣ pαn(x) < 1}, (T E +m,T K +m)m∈M a strong, +consistent generator for (FV,E) and U a set of uniqueness for (T K +m,FV)m∈M. If U +fixes the topology of FV(Ω), then FVE′(U,E)sb = FVE′(U,E) by Remark 5.2.49 +and Proposition 5.2.51. Hence Theorem 5.2.52 answers Question 5.2.37 if E is a +Fréchet–Montel space. +Our first application of Theorem 5.2.52 concerns the space Cbu(Ω,E) of bounded +uniformly continuous functions from a metric space Ω to an lcHs E from Example +4.2.7. +5.2.54. Corollary. Let Ω be a metric space, U ⊂ Ω a dense subset and E a +semi-Montel space. If f∶U → E is a function such that e′ ○ f admits an extension +fe′ ∈ Cbu(Ω) for each e′ ∈ E′, then there is a unique extension F ∈ Cbu(Ω,E) of f. +In particular, +Cbu(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ Cbu(Ω)}. +Proof. (idEΩ,idKΩ) is a strong, consistent generator for (Cbu,E) and we have +Cbu(Ω)εE ≅ Cbu(Ω,E) via S by Example 4.2.7. Due to Theorem 5.2.52, Proposition +5.2.8 and Remark 5.2.49 with G = E′ the extension F exists and is unique because +the dense set U ⊂ Ω fixes the topology in Cbu(Ω). The rest follows from Proposition +5.2.10. +□ +Next, we consider the space A(Ω,E) of continuous functions from Ω to an lcHs +E over C which are holomorphic on an open and bounded set Ω ⊂ C from Example +4.2.13. + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +95 +5.2.55. Corollary. Let Ω ⊂ C be open and bounded, U ⊂ Ω fix the topology in +A(Ω) and E a semi-Montel space over C. If f∶U → E is a function such that e′ ○f +admits an extension fe′ ∈ A(Ω) for each e′ ∈ E′, then there is a unique extension +F ∈ A(Ω,E) of f. In particular, +A(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ A(Ω)}. +Proof. (idEΩ,idCΩ) is a strong, consistent generator for (A,E) and A(Ω)εE ≅ +A(Ω,E) via S by Example 4.2.13. Due to Theorem 5.2.52, Proposition 5.2.8 and +Remark 5.2.49 with G = E′ the extension F exists and is unique. The remaining +part follows from Proposition 5.2.10. +□ +If Ω ⊂ C is connected, then the boundary ∂Ω of Ω fixes the topology in A(Ω) +by the maximum principle. If Ω = D, then ∂D is the intersection of all sets that fix +the topology in A(D) by [170, 7.7 Example, p. 39]. +If E is a generalised Schwartz space which is not a semi-Montel space, we do not +know whether the extension results in Corollary 5.2.54 and Corollary 5.2.55 hold +but we still have a weak-strong principle due to the following observation which is +based on [87, Chap. 3, §9, Proposition 2, p. 231] with σ(E,E′) replaced by σ(E,G). +5.2.56. Proposition. If +(i) E is a semi-Montel space and G ⊂ E′ a separating subspace, or +(ii) E is a generalised Schwartz space and G ⊂ ̂E′ a separating subspace, i.e. +separates the points of the completion ̂E, +then the initial topology of E and the topology σ(E,G) coincide on the bounded sets +of E. +Proof. (i) Let B ⊂ E be a bounded set. If E is a semi-Montel space, then the +closure B is compact in E. The topology induced by σ(E,G) on B is Hausdorff and +weaker than the initial topology induced by E. Thus the two topologies coincide +on B and so on B by the remarks above [87, Chap. 3, §9, Proposition 2, p. 231]. +(ii) Let B ⊂ E be a bounded set. If E is a generalised Schwartz space, then B is +precompact in E and relatively compact in the completion ̂E by [89, 3.5.1 Theorem, +p. 64]. Hence the closure B is compact in ̂E. The topology induced by σ( ̂E,G) on +B is Hausdorff and weaker than the initial topology induced by ̂E, implying that +the two topologies coincide on B as in part (i). This yields that σ(E,G) and the +initial topology of E coincide on B because σ(E,G) = σ( ̂E,G) on B and the initial +topologies of E and ̂E coincide on B as well. +□ +Concerning (ii), we note that a separating subspace G ⊂ E′ of E need not +separate the points of ̂E by [79, 5.4 Example, p. 36] (even though E′ = ̂E′ by [89, +3.4.2 Theorem, p. 61–62]). Next, we apply Proposition 5.2.56 to the space A(Ω,E). +5.2.57. Remark. Let E be an lcHs over C and Ω ⊂ C open and bounded. If +(i) E is a semi-Montel space and G ⊂ E′ determines boundedness, or +(ii) E is a generalised Schwartz space and G ⊂ ̂E′ a separating subspace which +determines boundedness in E, +then +A(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ A(Ω)}. +Indeed, let us denote the right-hand side by A(Ω,E)σ and set Eσ ∶= (E,σ(E,G)). +Then A(Ω,E)σ = A(Ω,Eσ) and f(Ω) is bounded for every f ∈ A(Ω,E)σ as G +determines boundedness in E. The initial topology of E and σ(E,G) coincide on +the bounded range f(Ω) of f ∈ A(Ω,E)σ by Proposition 5.2.56. Hence we deduce +that +A(Ω,E)σ = A(Ω,Eσ) = A(Ω,E). + +96 +5. APPLICATIONS +In this way Bierstedt proves his weak-strong principles for weighted continuous +functions in [17, 2.10 Lemma, p. 140] with G = E′ = ̂E′. +FV(Ω) a Fréchet–Schwartz space and E locally complete. +5.2.58. Definition (chain-structured). Let FV(Ω) be a dom-space. We say +that U ⊂ ⋃m∈N({m} × ωm) is chain-structured if +(i) (k,x) ∈ U +⇒ ∀ m ∈ N, m ≥ k ∶ (m,x) ∈ U, +(ii) ∀ (k,x) ∈ U, m ∈ N, m ≥ k, f ∈ FV(Ω) ∶ T K +k (f)(x) = T K +m(f)(x). +5.2.59. Remark. Let Ω ⊂ Rd be open and V∞ a directed family of weights. +Concerning the operators (T K +m)m∈N0 of CV∞(Ω) from Example 3.1.9 a) where ωm ∶= +{β ∈ Nd +0 ∣ ∣β∣ ≤ m} × Ω resp. ωm ∶= Nd +0 × Ω, we have for all k ∈ N0 and f ∈ CV∞(Ω) +that +T K +k (f)(β,x) = ∂βf(x) = T K +m(f)(β,x), +β ∈ Nd +0, ∣β∣ ≤ k, x ∈ Ω, +for all m ∈ N0, m ≥ k. Hence condition (ii) of Definition 5.2.58 is fulfilled for any +U ⊂ ⋃m∈N0({m} × ωm) in this case. Condition (i) says that once a ‘link’ (k,β,x) +belongs to U for some order k, then the ‘link’ (m,β,x) belongs to U for any higher +order m as well. +5.2.60. Definition (diagonally dominated, increasing). We say that a family +V ∶= (νj,m)j,m∈N of weights on Ω is diagonally dominated and increasing if ωm ⊂ +ωm+1 for all m ∈ N and νj,m ≤ νmax(j,m),max(j,m) on ωmin(j,m) for all j,m ∈ N as well +as νj,j ≤ νj+1,j+1 on ωj for all j ∈ N. +5.2.61. Remark. Let FV(Ω) be a dom-space, U ⊂ ⋃m∈N({m} × ωm) chain- +structured, G ⊂ E′ a separating subspace and V diagonally dominated and increas- +ing. +a) If U fixes the topology in FV(Ω), then +FVG(U,E)lb = {f ∈ FVG(U,E) ∣ ∀ i ∈ N ∶ NU,i(f) bounded in E} +with NU,i(f) ∶= NU,i,i(f). +b) Let FV(Ω) be a Fréchet space. We set Um ∶= {(m,x) ∈ U ∣ x ∈ ωm} and +Bj ∶= ⋃j +m=1{T K +m,x(⋅)νm,m(x) ∣ (m,x) ∈ Um} ⊂ FV(Ω)′ for j ∈ N. Then +U fixes the topology in FV(Ω) in the sense of Definition 5.2.46 if and +only if the sequence (Bj)j∈N fixes the topology in FV(Ω) in the sense of +Definition 5.2.34. +Proof. Let us begin with a). We only need to show that the inclusion ‘⊃’ holds. +Let f be an element of the right-hand side and i,k ∈ N. We set m ∶= max(i,k) and +observe that for (k,x) ∈ U we have (m,x) ∈ U by (i) and +(e′ ○ f)(k,x) = T K +k (fe′)(x) = +(ii) T K +m(fe′)(x) = (e′ ○ f)(m,x) +for each e′ ∈ G with (i) and (ii) from the definition of U being chain-structured. +Since G is separating, it follows that f(k,x) = f(m,x). Hence we get for all α ∈ A +sup +y∈NU,i,k(f) +pα(y) = +sup +x∈ωk +(k,x)∈U +pα(f(k,x))νi,k(x) ≤ +(i) +sup +x∈ωm +(m,x)∈U +pα(f(k,x))νm,m(x) += +sup +x∈ωm +(m,x)∈U +pα(f(m,x))νm,m(x) < ∞ +using that ωk ⊂ ωm and V is diagonally dominated. +Let us turn to part b). ‘⇒’: Let j ∈ N and A ⊂ FV(Ω) be bounded. Then +sup +y∈Bj +sup +f∈A +∣y(f)∣ = +sup +1≤m≤j +(m,x)∈Um +sup +f∈A +∣T K +m(f)(x)∣νm,m(x) ≤ sup +f∈A +sup +1≤m≤j +∣f∣m,m < ∞ + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +97 +since A is bounded, implying that Bj is bounded in FV(Ω)′ +b. Further, (Bj) is +increasing by definition. Additionally, for all j ∈ N +B○ +j = +j +⋂ +m=1 +{f ∈ FV(Ω) ∣ +sup +x∈ωm +(m,x)∈U +∣T K +m(f)(x)∣νm,m(x) ≤ 1} += {f ∈ FV(Ω) ∣ +sup +x∈ωj +(j,x)∈U +∣T K +j (f)(x)∣νj,j(x) ≤ 1} +because U is chain-structured and V increasing. Thus (B○ +j) is a fundamental system +of zero neighbourhoods of FV(Ω) if U fixes the topology. +‘⇐’: Let j,m ∈ N. Then there are i ∈ N and ε > 0 such that +εB○ +i ⊂ {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} =∶ Dj,m +which follows from fixing the topology in the sense of Definition 5.2.34. Let f ∈ Dj,m +and set +∣f∣Ui ∶= +sup +(i,x)∈Ui +∣T K +i (f)(x)∣νi,i(x). +If ∣f∣Ui = 0, then tf ∈ εB○ +i for all t > 0 and hence t∣f∣j,m = ∣tf∣j,m ≤ 1 for all t > 0, +which yields ∣f∣j,m = 0 = ∣f∣Ui. If ∣f∣Ui ≠ 0, then +f +∣f∣Ui ∈ B○ +i and thus ε +f +∣f∣Ui ∈ Dj,m, +implying +∣f∣j,m = 1 +ε∣f∣Ui∣ε f +∣f∣Ui +∣j,m ≤ 1 +ε∣f∣Ui. +The inequality ∣f∣j,m ≤ 1 +ε∣f∣Ui still holds if ∣f∣Ui = 0. +□ +5.2.62. Theorem ([30, Theorem 16, p. 236]). Let Y be a Fréchet–Schwartz +space, (Bj)j∈N fix the topology in Y and A∶X ∶= span(⋃j∈N Bj) → E be a linear map +which is bounded on each Bj. If +a) (At)−1(Y ) is dense in E′ +b and E locally complete, or +b) (At)−1(Y ) is dense in E′ +σ and E is Br-complete, +then A has a (unique) extension ̂A ∈ Y εE. +Now, we generalise [30, Theorem 17, p. 237]. +5.2.63. Theorem. Let E be an lcHs and G ⊂ E′ a separating subspace. Let +(T E +m,T K +m)m∈M be a strong, consistent generator for (FV,E), FV(Ω) a Fréchet– +Schwartz space, V diagonally dominated and increasing and U be chain-structured +and fix the topology in FV(Ω). If +a) G is dense in E′ +b and E locally complete, or +b) E is Br-complete, +then the restriction map RU,G∶S(FV(Ω)εE) → FVG(U,E)lb is surjective. +Proof. Let f ∈ FVG(U,E)lb. We set X ∶= span(⋃j∈N Bj) with Bj from Re- +mark 5.2.61 b) and Y ∶= FV(Ω). Let A∶X → E be the linear map determined +by +A(T K +m,x(⋅)νm,m(x)) ∶= f(m,x)νm,m(x) +for 1 ≤ m ≤ j and (m,x) ∈ Um with Um from Remark 5.2.61 b). The map A is +well-defined since G is σ(E′,E)-dense, and bounded on each Bj because A(Bj) = +⋃j +m=1 NU,m(f). +Let e′ ∈ G and fe′ be the unique element in FV(Ω) such that +T K +m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U, which implies T K +m(fe′)(x)νm,m(x) = +(e′○A)(T K +m,x(⋅)νm,m(x)) for all (m,x) ∈ Um. This equation allows us to consider fe′ +as a linear form on X (by fe′(T K +m,x(⋅)νm,m(x)) ∶= (e′ ○ A)(T K +m,x(⋅)νm,m(x))), which +yields e′ ○ A ∈ FV(Ω) for all e′ ∈ G. It follows that G ⊂ (At)−1(Y ). Noting that G +is σ(E′,E)-dense, we apply Theorem 5.2.62 and obtain an extension ̂A ∈ FV(Ω)εE + +98 +5. APPLICATIONS +of A. We set F ∶= S(̂A) and observe that for all (m,x) ∈ U there is j ∈ N, j ≥ m, +such that (j,x) ∈ Uj and νj,j(x) > 0 by (6) and because U is chain-structured and +V diagonally dominated and increasing. Due to the proof of Remark 5.2.61 a) we +have f(j,x) = f(m,x) and thus +T E +m(F)(x) = T E +mS(̂A)(x) = ̂A(T K +m,x) = +1 +νj,j(x) +̂A(T K +m,x(⋅)νj,j(x)) += +1 +νj,j(x) +̂A(T K +j,x(⋅)νj,j(x)) = f(j,x) = f(m,x) +by consistency, yielding RU,G(F) = f. +□ +In particular, condition a) is fulfilled if E is semi-reflexive. Indeed, if E is semi- +reflexive, then E is quasi-complete by [153, Chap. IV, 5.5, Corollary 1, p. 144] and +G +b(E′,E) = G +τ(E′,E) = E′ by [89, 11.4.1 Proposition, p. 227] and the bipolar theorem. +For instance, condition b) is satisfied if E is a Fréchet space or E = (C∞ +∂,b(D),β) +which is a Br-complete space by Proposition 5.2.16 and is not a Fréchet space by +Remark 4.2.23. +As stated, our preceding theorem generalises [30, Theorem 17, p. 237] where +FV(Ω) is a closed subspace of CW∞(Ω) for open, connected Ω ⊂ Rd. A characteri- +sation of sets that fix the topology in the space CW∞ +∂ (Ω) of holomorphic functions +on an open, connected set Ω ⊂ C is given in [30, Remark 14, p. 235]. The characteri- +sation given in [30, Remark 14 (b), p. 235] is still valid and applied in [30, Corollary +18, p. 238] for closed subspaces of CW∞ +P (∂)(Ω) where P(∂)K is a hypoelliptic linear +partial differential operator which satisfies the maximum principle, namely, that +U ⊂ Ω fixes the topology if and only if there is a sequence (Ωn)n∈N of relatively +compact, open subsets of Ω with ⋃n∈N Ωn = Ω such that ∂Ωn ⊂ U ∩ Ωn+1 for all +n ∈ N. Among the hypoelliptic operators P(∂)K satisfying the maximum principle +are the Cauchy–Riemann operator ∂ and the Laplacian ∆. Further examples can +be found in [74, Corollary 3.2, p. 33]. The statement of [30, Corollary 18, p. 238] +for the space of holomorphic functions is itself a generalisation of [81, Theorem 2, +p. 401] with [81, Remark 2 (a), p. 406] where E is Br-complete and of [92, Theo- +rem 6, p. 10] where E is semi-reflexive. The case that G is dense in E′ +b and E is +sequentially complete is covered by [77, 3.3 Satz, p. 228–229], not only for spaces +of holomorphic functions, but for several classes of function spaces. +Let us turn to other families of weights than W∞. Due to Proposition 4.2.19 +we already know that U ∶= {0} × C fixes the topology in CV∞ +∂ (C) = CV∂(C) and +U ∶= {0} × Rd in CV∞ +∆(Rd) = CV∆(Rd) if V ∶= (νj)j∈N fulfils Condition 4.2.18 and +V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd +0 ∣ ∣β∣ ≤ m} × Rd → [0,∞), νj,m(β,x) ∶= +νj(x). +5.2.64. Corollary. Let E be an lcHs, G ⊂ E′ a separating subspace, V ∶= +(νj)j∈N an increasing family of weights which is locally bounded away from zero +on an open set Ω ⊂ Rd, P(∂)K a hypoelliptic linear partial differential operator, +CVP (∂)(Ω) a Schwartz space and U ⊂ Ω fix the topology of CVP (∂)(Ω). If +a) G is dense in E′ +b and E locally complete, or +b) E is Br-complete, +and f∶U → E is a function in ℓV(U) such that e′ ○ f admits an extension fe′ ∈ +CVP (∂)(Ω) for each e′ ∈ G, then there is a unique extension F ∈ CVP (∂)(Ω,E) of f. +Proof. The existence of F follows from Proposition 4.2.14, Example 4.2.16 b) +and Theorem 5.2.63 with (T E +m,T K +m)m∈M ∶= (idEΩ,idKΩ). The uniqueness of F is a +result of Proposition 5.2.8. +□ + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +99 +We have the following sufficient conditions on a family of weights V which +guarantee the existence of a countable set U ⊂ C that fixes the topology of CV∂(C) +due to Abanin and Varziev [2]. +5.2.65. Proposition. Let V ∶= (νj)j∈N where νj(z) ∶= exp(ajµ(z)−ϕ(z)), z ∈ C, +with some continuous, subharmonic function µ∶C → [0,∞), a continuous function +ϕ∶C → R and a strictly increasing, positive sequence (aj)j∈N with a ∶= limj→∞ aj ∈ +(0,∞]. Let there be +(i) s ≥ 0 and C > 0 such that ∣ϕ(z) − ϕ(ζ)∣ ≤ C and ∣µ(z) − µ(ζ)∣ ≤ C for all +z,ζ ∈ C with ∣z − ζ∣ ≤ (1 + ∣z∣)−s, +(ii) max(ϕ(z),µ(z)) ≤ ∣z∣q + C0 for some q,C0 > 0 and +(iii) ln(∣z∣) = O(µ(z)) as ∣z∣ → ∞ if a = ∞, or ln(∣z∣) = o(µ(z)) as ∣z∣ → ∞ if +0 < a < ∞. +Let (λk)k∈N be the sequence of simple zeros of a function L ∈ C̃V∂(C) having no +other zeros where ̃V ∶= (ν2 +j /νmj)j∈N for some sequence (mj)j∈N in N. Suppose that +there are j0 ∈ N and a sequence of circles {z ∈ C ∣ ∣z∣ = Rm} with Rm ↗ ∞ such that +∣L(z)∣νj0(z) ≥ Cm, +m ∈ N, z ∈ C, ∣z∣ = Rm, +for some Cm ↗ ∞ and +∣L′(λk)∣νj0(λk) ≥ 1 +for all sufficiently large k ∈ N. +Then CV∂(C) is a nuclear Fréchet space for all a ∈ (0,∞] and U ∶= (λk)k∈N fixes the +topology of CV∂(C) if a = ∞. If µ is a radial function, i.e. µ(z) = µ(∣z∣), z ∈ C, with +µ(2z) ∼ µ(z) as ∣z∣ → ∞, then U fixes the topology of CV∂(C) for all a ∈ (0,∞]. +Proof. First, we check that Condition 4.2.18 is satisfied, which implies that +CV∂(C) is a nuclear Fréchet space by Proposition 4.2.19. We set k ∶= max(s,2) and +observe that (i) is also fulfilled with k instead of s. Let z ∈ C and ∥ζ∥∞,∥η∥∞ ≤ +(1/ +√ +2)(1 + ∣z∣)−k =∶ r(z). From ∣ ⋅ ∣ ≤ +√ +2∥ ⋅ ∥∞ and (i) it follows +∣µ(z + ζ) − µ(z + η)∣ ≤ ∣µ(z + ζ) − µ(z)∣ + ∣µ(z) − µ(z + η)∣ ≤ 2C +and thus µ(z+ζ) ≤ 2C+µ(z+η). In the same way we obtain −ϕ(z+ζ) ≤ 2C−ϕ(z+η). +Hence we have +ajµ(z + ζ) − ϕ(z + ζ) ≤ 2C(aj + 1) + ajµ(z + η) − ϕ(z + η) +for j ∈ N, implying +νj(z + ζ) ≤ e2C(aj+1)νj(z + η), +which means that (α.1) of Condition 4.2.18 holds. By (iii) there are ε > 0 and +R > 0 such that ln(∣z∣) ≤ εµ(z) for all z ∈ C with ∣z∣ ≥ R if a = ∞. This yields for all +∣z∣ ≥ max(2,R) that +ajµ(z) + k ln(1 + ∣z∣) ≤ ajµ(z) + k ln(∣z∣2) = ajµ(z) + 2k ln(∣z∣) ≤ ajµ(z) + 2kεµ(z). +Since a = ∞, there is n ∈ N such that an ≥ aj + 2kε, resulting in +ajµ(z) + k ln(1 + ∣z∣) ≤ anµ(z) +for all ∣z∣ ≥ max(2,R). Therefore we derive +ajµ(z) + k ln(1 + ∣z∣) ≤ anµ(z) + k ln(1 + max(2,R)) +(54) +for all z ∈ C, which means that (α.2) and (α.3) hold with ψj(z) ∶= r(z). If 0 < a < ∞, +for every ε > 0 there is R > 0 such that ln(∣z∣) ≤ εµ(z) for all z ∈ C with ∣z∣ ≥ R by +(iii). Thus we may choose ε > 0 such that aj+1 −aj ≥ 2kε > 0 because (aj) is strictly +increasing. We deduce that (54) with n ∶= j + 1 holds in this case as well and (α.2) +and (α.3), too. +Observing that the condition that U = (λk)k∈N is the sequence of simple zeros +of a function L ∈ C̃V∂(C) means that L ∈ L (Φa +ϕ,µ;U) and (i) that ϕ and µ vary + +100 +5. APPLICATIONS +slowly w.r.t. r(z) ∶= (1 + ∣z∣)−s in the notation of [2, Definition, p. 579, 584] and [2, +p. 585], respectively, the statement that U fixes the topology is a consequence of +[2, Theorem 2, p. 585–586]. +□ +5.2.66. Remark. +a) Let D ⊂ C be convex, bounded and open with 0 ∈ D. +Let ϕ(z) ∶= HD(z) ∶= supζ∈D Re(zζ), z ∈ C, be the supporting function of +D, µ(z) ∶= ln(1 + ∣z∣), z ∈ C, and aj ∶= j, j ∈ N. Then ϕ and µ fulfil the +conditions of Proposition 5.2.65 with a = ∞ by [2, p. 586] and the existence +of an entire function L which fulfils the conditions of Proposition 5.2.65 +is guaranteed by [3, Theorem 1.6, p. 1537]. Thus there is a countable +set U ∶= (λk)k∈N ⊂ C which fixes the topology in A−∞ +D +∶= CV∂(C) with +V ∶= (exp(ajµ − ϕ))j∈N. +b) An explicit construction of a set U ∶= (λk)k∈N ⊂ C which fixes the topology +in A−∞ +D +is given in [1, Algorithm 3.2, p. 3629]. This construction does not +rely on the entire function L. In particular (see [31, p. 15]), for D ∶= D +we have ϕ(z) = ∣z∣, for each k ∈ N we may take lk ∈ N, lk > 2πk2, and set +λk,j ∶= krk,j, 1 ≤ j ≤ lk, where rk,j denote the lk-roots of unity. Ordering +λk,j in a sequence of one index appropriately, we obtain a sequence which +fixes the topology of A−∞ +D . +c) Let µ∶C → [0,∞) be a continuous, subharmonic, radial function which +increases with ∣z∣ and satisfies +(i) supζ∈C,∥ζ∥∞≤r(z) µ(z + ζ) ≤ infζ∈C,∥ζ∥∞≤r(z) µ(z + ζ) + C for some con- +tinuous function r∶C → (0,1] and C > 0, +(ii) ln(1 + ∣z∣2) = o(µ(∣z∣)) as ∣z∣ → ∞, +(iii) µ(2∣z∣) = O(µ(∣z∣)) as ∣z∣ → ∞. +Then V ∶= (exp(−(1/j)µ))j∈N fulfils Condition 4.2.18 where (α.1) follows +from (i) and (α.2), (α.3) as in the proof of Proposition 5.2.65. +Thus +CV∂(C) is a nuclear Fréchet space by Proposition 4.2.19. If µ(∣z∣) = o(∣z∣2) +as ∣z∣ → ∞ or µ(∣z∣) = ∣z∣2, z ∈ C, then U ∶= {αn + iβm ∣ n,m ∈ Z} fixes the +topology in the space A0 +µ ∶= CV∂(C) for any α,β > 0 by [31, Corollary 4.6, +p. 20] and [31, Proposition 4.7, p. 20], respectively. +d) For instance, the conditions on µ in c) are fulfilled for µ(z) ∶= ∣z∣γ, z ∈ +C, with 0 < γ ≤ 2 by [130, 1.5 Examples (3), p. 205]. +If γ = 1, then +A0 +µ = A0 +∂(C) is the space of entire functions of exponential type zero (see +Remark 4.2.20). +e) More general characterisations of countable sets that fix the topology of +CV∂(C) can be found in [2, Theorem 1, p. 580] and [31, Theorem 4.5, p. +17]. +The spaces A0 +µ from c) are known as Hörmander algebras and the space A−∞ +D +considered in a) is isomorphic to the strong dual of the Korenblum space A−∞(D) +via Laplace transform by [132, Proposition 4, p. 580]. +Fν(Ω) a Banach space and E locally complete. For a dom-space Fν(Ω), +a set U that fixes the topology in Fν(Ω) and a separating subspace G ⊂ E′ we have +FνG(U,E)lb ={f ∈ FνG(U,E) ∣ NU(f) bounded in E} +=FνG(U,E) ∩ ℓν(U,E) +where NU(f) ∶= {f(x)ν(x) ∣ x ∈ U}. Let us recall the assumptions of Remark 5.2.26 +but now U fixes the topology. Let (T E,T K) be a strong, consistent family for (F,E) +and a generator for (Fν,E). Let F(Ω) and F(Ω,E) be ε-into-compatible and the +inclusion Fν(Ω) ↪ F(Ω) continuous. Consider a set U which fixes the topology in +Fν(Ω) and a separating subspace G ⊂ E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′ +Fν(Ω) ) is + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +101 +bounded in E we have RU,G(f) ∈ FνG(U,E) with f ∶= S(u) ∈ Fεν(Ω,E) by (50). +Further, T K +x (⋅)ν(x) ∈ B○F (Ω)′ +Fν(Ω) for every x ∈ ω, which implies that +sup +x∈U +pα(RU,G(f)(x))ν(x) = sup +x∈U +pα(u(T K +x (⋅)ν(x))) ≤ +sup +y′∈B○F (Ω)′ +Fν(Ω) +pα(u(y′)) < ∞ +for all α ∈ A by consistency. Hence RU,G(f) ∈ FνG(U,E)lb. Therefore the injective +linear map +RU,G∶Fεν(Ω,E) → FνG(U,E)lb, f ↦ (T E(f)(x))x∈U, +is well-defined and the question we want to answer is: +5.2.67. Question. Let the assumptions of Remark 5.2.26 be fulfilled and U fix +the topology in Fν(Ω). When is the injective restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E)lb, f ↦ (T E(f)(x))x∈U, +surjective? +5.2.68. Proposition ([70, Proposition 3.1, p. 692]). Let E be a locally complete +lcHs, G ⊂ E′ a separating subspace and Z a Banach space whose closed unit ball +BZ is a compact subset of an lcHs Y . Let B1 ⊂ B○Y ′ +Z +such that B○Z +1 +∶= {z ∈ Z ∣ ∀ y′ ∈ +B1 ∶ ∣y′(z)∣ ≤ 1} is bounded in Z. If A∶X ∶= spanB1 → E is a linear map which is +bounded on B1 such that there is a σ(E′,E)-dense subspace G ⊂ E′ with e′ ○ A ∈ Z +for all e′ ∈ G, then there exists a (unique) extension ̂A ∈ Y εE of A such that ̂A(B○Y ′ +Z +) +is bounded in E. +The following theorem is a generalisation of [70, Theorem 3.2, p. 693] and [93, +Theorem 12, p. 5]. +5.2.69. Theorem. Let E be a locally complete lcHs, G ⊂ E′ a separating sub- +space and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K) be a generator +for (Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a Banach space +whose closed unit ball BFν(Ω) is a compact subset of F(Ω) and U fix the topology +in Fν(Ω). Then the restriction map +RU,G∶Fεν(Ω,E) → FνG(U,E)lb +is surjective. +Proof. Let f ∈ FνG(U,E)lb. We set B1 ∶= {T K +x (⋅)ν(x) ∣ x ∈ U}, X ∶= spanB1, +Y ∶= F(Ω) and Z ∶= Fν(Ω). We have B1 ⊂ Y ′ since (T E,T K) is a consistent family +for (F,E). If f ∈ BZ, then +∣T K +x (f)ν(x)∣ ≤ ∣f∣Fν(Ω) ≤ 1 +for all x ∈ U and thus B1 ⊂ B○Y ′ +Z +. Further on, there is C > 0 such that for all f ∈ B○Z +1 +∣f∣Fν(Ω) ≤ C sup +x∈U +∣T K +x (f)∣ν(x) ≤ C +as U fixes the topology in Z, implying the boundedness of B○Z +1 +in Z. Let A∶X → E +be the linear map determined by +A(T K +x (⋅)ν(x)) ∶= f(x)ν(x). +The map A is well-defined since G is σ(E′,E)-dense, and bounded on B1 be- +cause A(B1) = NU(f). +Let e′ ∈ G and fe′ be the unique element in Fν(Ω) +such that T K(fe′)(x) = (e′ ○ f)(x) for all x ∈ U, which implies T K(fe′)(x)ν(x) = +(e′○A)(T K +x (⋅)ν(x)). Again, this equation allows us to consider fe′ as a linear form on +X (by setting fe′(T K +x (⋅)ν(x)) ∶= (e′○A)(T K +x (⋅)ν(x))), which yields e′○A ∈ Fν(Ω) = Z +for all e′ ∈ G. +Hence we can apply Proposition 5.2.68 and obtain an extension + +102 +5. APPLICATIONS +̂A ∈ Y εE of A such that ̂A(B○Y ′ +Z +) is bounded in E. We set ̃F ∶= S(̂A) ∈ Fεν(Ω,E) +and get for all x ∈ U that +T E( ̃F)(x) = T ES(̂A)(x) = ̂A(T K +x ) = +1 +ν(x)A(T K +x (⋅)ν(x)) = f(x) +by consistency for (F,E), yielding RU,G( ̃F) = f. +□ +5.2.70. Corollary. Let E be a locally complete lcHs, G ⊂ E′ a separating +subspace, Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential operator, +ν∶Ω → (0,∞) continuous and U fix the topology in CνP (∂)(Ω). If f∶U → E is a +function in ℓν(U,E) such that e′ ○ f admits an extension fe′ ∈ CνP (∂)(Ω) for every +e′ ∈ G, then there exists a unique extension F ∈ CνP (∂)(Ω,E) of f. +Proof. Observing that f ∈ FνG(U,E)lb with Fν(Ω) = CνP (∂)(Ω), our state- +ment follows directly from Theorem 5.2.69 whose conditions are fulfilled by the +proof of Corollary 5.2.30. +□ +Sets that fix the topology in CνP (∂)(Ω) for different weights ν are well-studied +if P(∂) = ∂ is the Cauchy–Riemann operator. If Ω ⊂ C is open, P(∂) = ∂ and ν = 1, +then CνP (∂)(Ω) = H∞(Ω) is the space of bounded holomorphic functions on Ω. +Brown, Shields and Zeller characterise the countable discrete sets U ∶= (zn)n∈N ⊂ Ω +that fix the topology in H∞(Ω) with C = 1 and equality in Definition 5.2.46 for +Jordan domains Ω in [36, Theorem 3, p. 167]. In particular, they prove for Ω = D +that a discrete U = (zn)n∈N fixes the topology in H∞(D) if and only if almost every +boundary point is a non-tangential limit of a sequence in U. Bonsall obtains the +same characterisation for bounded harmonic functions, i.e. P(∂) = ∆ and ν = 1, on +Ω = D by [32, Theorem 2, p. 473]. An example of such a set U = (zn)n∈N ⊂ D is +constructed in [36, Remark 6, p. 172]. Probably the first example of a countable +discrete set U ⊂ D that fixes the topology in H∞(D) is given by Wolff in [183, p. +1327] (cf. [81, Theorem (Wolff), p. 402]). In [148, 4.14 Theorem, p. 255] Rubel and +Shields give a charaterisation of sets U ⊂ Ω that fix the topology in H∞(Ω) by means +of bounded complex measures where Ω ⊂ C is open and connected. The existence +of a countable U fixing the topology in H∞(Ω) is shown in [148, 4.15 Proposition, +p. 256]. In the case of several complex variables the existence of such a countable +U is treated by Sibony in [167, Remarques 4 b), p. 209] and by Massaneda and +Thomas in [128, Theorem 2, p. 838]. +If Ω = C and P(∂) = ∂, then CνP (∂)(Ω) =∶ F ∞ +ν (C) is a generalised L∞-version +of the Bargmann–Fock space. In the case that ν(z) = exp(−α∣z∣2/2), z ∈ C, for +some α > 0, Seip and Wallstén show in [162, Theorem 2.3, p. 93] that a countable +discrete set U ⊂ C fixes the topology in F ∞ +ν (C) if and only if U contains a uniformly +discrete subset U ′ with lower uniform density D−(U ′) > α/π (the proof of sufficiency +is given in [165] and the result was announced in [161, Theorem 1.3, p. 324]). A +generalisation of this result using lower angular densities is given by Lyubarski˘ı and +Seip in [124, Theorem 2.2, p. 162] to weights of the form ν(z) = exp(−φ(arg z)∣z∣2/2), +z ∈ C, with a 2π-periodic 2-trigonometrically convex function φ such that φ ∈ +C2([0,2π]) and φ(θ) + (1/4)φ′′(θ) > 0 for all θ ∈ [0,2π]. An extension of the results +in [162] to weights of the form ν(z) = exp(−φ(z)), z ∈ C, with a subharmonic +function φ such that ∆φ(z) ∼ 1 is given in [135, Theorem 1, p. 249] by Ortega- +Cerdà and Seip. +Here, f(x) ∼ g(x) for two functions f,g∶Ω → R means that +there are C1,C2 > 0 such that C1g(x) ≤ f(x) ≤ C2g(x) for all x ∈ Ω. +Marco, +Massaneda and Ortega-Cerdà describe sets that fix the topology in F ∞ +ν (C) with +ν(z) = exp(−φ(z)), z ∈ C, for some subharmonic function φ whose Laplacian ∆φ is +a doubling measure (see [126, Definition 5, p. 868]), e.g. φ(z) = ∣z∣β for some β > 0, + +5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS +103 +in [126, Theorem A, p. 865]. The case of several complex variables is handled by +Ortega-Cerdà, Schuster and Varolin in [136, Theorem 2, p. 81]. +If Ω = D and P(∂) = ∂, then CνP (∂)(Ω) =∶ A∞ +ν (D) is a generalised L∞-version +of the weighted Bergman space (and of H∞(D)). For ν(z) = (1 − ∣z∣2)n, z ∈ D, for +some n ∈ N, Seip proves that a countable discrete set U ⊂ D fixes the topology in +A∞ +ν (D) if and only if U contains a uniformly discrete subset U ′ with lower uniform +density D−(U ′) > n by [163, Theorem 1.1, p. 23], and gives a typical example +in [163, p. 23]. +Later on, this is extended by Seip in [164, Theorem 2, p. 718] +to weights ν(z) = exp(−φ(z)), z ∈ D, with a subharmonic function φ such that +∆φ(z) ∼ (1 − ∣z∣2)−2, e.g. φ(z) = −β ln(1 − ∣z∣2), z ∈ D, for some β > 0. Domański +and Lindström give necessary resp. sufficient conditions for fixing the topology in +A∞ +ν (D) in the case that ν is an essential weight on D, i.e. there is C > 0 with +ν(z) ≤ ̃ν(z) ≤ Cν(z) for each z ∈ D where ̃ν(z) ∶= (sup{∣f(z)∣ ∣ f ∈ BA∞ +ν (D)})−1 is +the associated weight. In [55, Theorem 29, p. 260] they describe necessary resp. +sufficient conditions for fixing the topology if the upper index Uν is finite (see [55, +p. 242]), and necessary and sufficient conditions in [55, Corollary 31, p. 261] if +0 < Lν = Uν < ∞ holds where Lν is the lower index (see [55, p. 243]), which for +example can be applied to ν(z) = (1 − ∣z∣2)n(ln( +e +1−∣z∣))β, z ∈ D, for some n > 0 and +β ∈ R. The case of simply connected open Ω ⊂ C is considered in [55, Corollary 32, +p. 261–262]. +Borichev, Dhuez and Kellay treat A∞ +ν (D) and F ∞ +ν (C) simultaneously. +Let +ΩR ∶= D, if R = 1, and ΩR ∶= C if R = ∞. They take ν(z) = exp(−φ(z)), z ∈ ΩR, +where φ∶[0,R) → [0,∞) is an increasing function such that φ(0) = 0, limr→R φ(r) = +∞, φ is extended to ΩR by φ(z) ∶= φ(∣z∣), φ ∈ C2(ΩR), and, in addition ∆φ(z) ≥ 1 +if R = ∞ (see [33, p. 564–565]). Then they set ρ∶[0,R) → R, ρ(r) ∶= [∆φ(r)]−1/2, +and suppose that ρ decreases to 0 near R, ρ′(r) → 0, r → R, and either (ID) the +function r ↦ ρ(r)(1 − r)−C increases for some C ∈ R and for r close to 1, resp. (IC) +the function r ↦ ρ(r)rC increases for some C ∈ R and for large r, or (IIΩR) that +ρ′(r)ln(1/ρ(r)) → 0, r → R (see [33, p. 567–569]). Typical examples for (ID) are +φ(r) = ln(ln( 1 +1−r))ln( 1 +1−r) +or +φ(r) = +1 +1−r, +a typical example for (IID) is φ(r) = exp( 1 +1−r), for (IC) +φ(r) = r2 ln(ln(r)) +or +φ(r) = rp, for some p > 2, +and a typical example for (IIC) is φ(r) = exp(r). Sets that fix the topology in +A∞ +ν (D) are described by densities in [33, Theorem 2.1, p. 568] and sets that fix the +topology in F ∞ +ν (C) in [33, Theorem 2.5, p. 569]. +Wolf uses sets that fix the topology in A∞ +ν (D) for the characterisation of +weighted composition operators on A∞ +ν (D) with closed range in [182, Theorem +1, p. 36] for bounded ν. +5.2.71. Corollary. Let E be a locally complete lcHs, G ⊂ E′ a separating +subspace, ν∶D → (0,∞) continuous and U ∶= {0} ∪ ({1} × U∗) fix the topology in +Bν(D) with U∗ ⊂ D. If f∶U → E is a function in ℓν∗(U,E) such that there is fe′ ∈ +Bν(D) for each e′ ∈ G with fe′(0) = e′(f(0)) and f ′ +e′(z) = e′(f(1,z)) for all z ∈ U∗, +then there exists a unique F ∈ Bν(D,E) with F(0) = f(0) and (∂1 +C)EF(z) = f(1,z) +for all z ∈ U∗. +Proof. As in Corollary 5.2.70 but with Fν∗(D) = Bν(D) and Corollary 5.2.33 +instead of Corollary 5.2.30. +□ +Sets that fix the topology in Bν(D) play an important role in the characteri- +sation of composition operators on Bν(D) with closed range. Chen and Gauthier +give a characterisation in [42] for weights of the form ν(z) = (1 − ∣z∣2)α, z ∈ D, + +104 +5. APPLICATIONS +for some α ≥ 1. We recall the following definitions which are needed to phrase +this characterisation. For a continuous function ν∶D → (0,∞) and a non-constant +holomorphic function φ∶D → D we set +τ ν +φ(z) ∶= ν(z)∣φ′(z)∣ +ν(φ(z)) , z ∈ D, +and +Ων +ε ∶= {z ∈ D ∣ τ ν +φ(z) ≥ ε}, ε > 0, +and define the pseudohyperbolic distance +ρ(z,w) ∶= ∣ z − w +1 − zw∣, z,w ∈ D. +For 0 < r < 1 a set B ⊂ D is called a pseudo r-net if for every w ∈ D there is z ∈ B +with ρ(z,w) ≤ r (see [42, p. 195]). A set U⋆ ⊂ D is a sampling set for Bν(D) with ν +as above in the sense of [42, p. 198] if and only if {0}∪({1}×U⋆) fixes the topology +in Bν(D) (see the definitions above Corollary 5.2.33). +5.2.72. Theorem ([42, Theorem 3.1, p. 199, Theorem 4.3, p. 202]). Let φ∶D → +D be a non-constant holomorphic function and ν(z) = (1 − ∣z∣2)α, z ∈ D, for some +α ≥ 1. Then the following statements are equivalent. +(i) The composition operator Cφ∶Bν(D) → Bν(D), Cφ(f) ∶= f ○ φ, is bounded +below (i.e. has closed range). +(ii) There is ε > 0 such that {0} ∪ ({1} × φ(Ων +ε)) fixes the topology in Bν(D). +(iii) There are ε > 0 and 0 < r < 1 such that φ(Ων +ε) is a pseudo r-net. +This theorem has some predecessors. The implications (i)⇒(iii) and (iii), r < +1/4 ⇒(i) for α = 1 are due to Ghatage, Yan and Zheng by [72, Proposition 1, +p. 2040] and [72, Theorem 2, p. 2043]. This was improved by Chen to (i)⇔(iii) +for α = 1 by removing the restriction r < 1/4 in [41, Theorem 1, p. 840]. +The +proof of the equivalence (i)⇔(ii) given in [73, Theorem 1, p. 1372] for α = 1 is +due to Ghatage, Zheng and Zorboska. A non-trivial example of a sampling set +for α = 1 can be found in [73, Example 2, p. 1376] (cf. [42, p. 203]). In the case +of several complex variables a characterisation corresponding to Theorem 5.2.72 +is given by Chen in [41, Theorem 2, p. 844] and Deng, Jiang and Ouyang in [50, +Theorem 1–3, p. 1031–1032, 1034] where Ω is the unit ball of Cd. Giménez, Malavé +and Ramos-Fernández extend Theorem 5.2.72 by [75, Theorem 3, p. 112] and [75, +Corollary 1, p. 113] to more general weights of the form ν(z) = µ(1−∣z∣2) with some +continuous function µ∶(0,1] → (0,∞) such that µ(r) → 0, r → 0+, which can be +extended to a holomorphic function µ0 on D1(1) without zeros in D1(1) and fulfilling +µ(1−∣1−z∣) ≤ C∣µ0(z)∣ for all z ∈ D1(1) and some C > 0 (see [75, p. 109]). Examples +of such functions µ are µ1(r) ∶= rα, α > 0, µ2 ∶= r ln(2/r) and µ3(r) ∶= rβ ln(1 − r), +β > 1, for r ∈ (0,1] (see [75, p. 110]) and with ν(z) = µ1(1 − ∣z∣2) = (1 − ∣z∣2)α, z ∈ D, +one gets Theorem 5.2.72 back for α ≥ 1. For 0 < α < 1 and ν(z) = µ1(1−∣z∣2), z ∈ D, +the equivalence (i)⇔(ii) is given in [184, Proposition 4.4, p. 14] of Yoneda as well and +a sufficient condition implying (ii) in [184, Corollary 4.5, p. 15]. Ramos-Fernández +generalises the results given in [75] to bounded essential weights ν on D by [144, +Theorem 4.3, p. 85] and [144, Remark 4.2, p. 84]. In [141, Theorem 2.4, p. 3106] +Pirasteh, Eghbali and Sanatpour use sets that fix the topology in Bν(D) for radial +essential ν to characterise Li–Stević integral-type operators on Bν(D) with closed +range instead of composition operators. The composition operator on the harmonic +variant of the Bloch type space Bν(D) with ν(z) = (1 −∣z∣2)α, z ∈ D, for some α > 0 +is considered by Esmaeili, Estaremi and Ebadian, who give a corresponding result +in [64, Theorem 2.8, p. 542]. +5.3. Weak-strong principles for differentiability of finite order +This section is dedicated to Ck-weak-strong principles for differentiable func- +tions. So the question is: + +5.3. WEAK-STRONG PRINCIPLES FOR DIFFERENTIABILITY OF FINITE ORDER +105 +5.3.1. Question. Let E be an lcHs, G ⊂ E′ a separating subspace, Ω ⊂ Rd +open and k ∈ N0 ∪ {∞}. If f∶Ω → E is such that e′ ○ f ∈ Ck(Ω) for each e′ ∈ G, does +f ∈ Ck(Ω,E) hold? +An affirmative answer to the preceding question is called a Ck-weak-strong +principle. It is a result of Bierstedt [17, 2.10 Lemma, p. 140] that for k = 0 the +C0-weak-strong principle holds if Ω ⊂ Rd is open (or more general a Hausdorff kR- +space), G = E′ and E is such that every bounded set is already precompact in E, +i.e. E is a generalised Schwartz space (see Definition 5.2.50 and Remark 5.2.57). +For instance, the last condition is fulfilled if E is a semi-Montel or Schwartz space. +The C0-weak-strong principle does not hold for general E by [94, Beispiel, p. 232]. +Grothendieck sketches in a footnote [82, p. 39] (cf. [84, Chap. 3, Sect. 8, Corol- +lary 1, p. 134]) the proof that for k < ∞ a weakly-Ck+1 function f∶Ω → E on an +open set Ω ⊂ Rd with values in a quasi-complete lcHs E is already Ck, i.e. that +from e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′ it follows f ∈ Ck(Ω,E). A detailed proof of this +statement is given by Schwartz in [158], simultaneously weakening the condition +from quasi-completeness of E to sequential completeness and from weakly-Ck+1 to +weakly-Ck,1 +loc . +5.3.2. Theorem ([158, Appendice, Lemme II, Remarques 10), p. 146–147]). +Let E be a sequentially complete lcHs, Ω ⊂ Rd open and k ∈ N0. +a) If f∶Ω → E is such that e′ ○ f ∈ Ck,1 +loc (Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E). +b) If f∶Ω → E is such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E). +Here Ck,1 +loc (Ω) denotes the space of functions in Ck(Ω) whose partial derivatives +of order k are locally Lipschitz continuous. +Part b) clearly implies a C∞-weak- +strong principle for open Ω ⊂ Rd, G = E′ and sequentially complete E. This can +be generalised to locally complete E. Waelbroeck has shown in [177, Proposition +2, p. 411] and [176, Definition 1, p. 393] that the C∞-weak-strong principle holds if +Ω is a manifold, G = E′ and E is locally complete. It is a result of Bonet, Frerick +and Jordá that the C∞-weak-strong principle still holds by [30, Theorem 9, p. 232] +if Ω ⊂ Rd is open, G ⊂ E′ determines boundedness and E is locally complete. Due +to [104, 2.14 Theorem, p. 20] of Kriegl and Michor an lcHs E is locally complete if +and only if the C∞-weak-strong principle holds for Ω = R and G = E′. +One of the goals of this section is to improve Theorem 5.3.2. We recall the +following definition from Example 4.2.28. For k ∈ N0 the space of k-times con- +tinuously partially differentiable E-valued functions on an open set Ω ⊂ Rd whose +partial derivatives up to order k are continuously extendable to the boundary of Ω +is +Ck(Ω,E) = {f ∈ Ck(Ω,E) ∣ (∂β)Ef cont. extendable on Ω for all β ∈ Nd +0, ∣β∣ ≤ k} +equipped with the system of seminorms given by +∣f∣Ck(Ω),α = +sup +x∈Ω +β∈Nd +0,∣β∣≤k +pα((∂β)Ef(x)), +f ∈ Ck(Ω,E), α ∈ A. +The space of functions in Ck(Ω,E) such that all its k-th partial derivatives are +γ-Hölder continuous with 0 < γ ≤ 1 is given by +Ck,γ(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣Ck,γ(Ω),α < ∞} +where +∣f∣Ck,γ(Ω),α ∶= max(∣f∣Ck(Ω),α, +sup +β∈Nd +0,∣β∣=k +∣(∂β)Ef∣C0,γ(Ω),α) + +106 +5. APPLICATIONS +with +∣f∣C0,γ(Ω),α ∶= sup +x,y∈Ω +x≠y +pα(f(x) − f(y)) +∣x − y∣γ +. +We set Ck,γ(Ω) ∶= Ck,γ(Ω,K) and +ω1 ∶= {β ∈ Nd +0 ∣ ∣β∣ ≤ k} × Ω +and +ω2 ∶= {β ∈ Nd +0 ∣ ∣β∣ = k} × (Ω2 ∖ {(x,x) ∣ x ∈ Ω}) +as well as ω ∶= ω1 ∪ ω2. We define the operator T E∶Ck(Ω,E) → Eω by +T E(f)(β,x) ∶=(∂β)E(f)(x) +, (β,x) ∈ ω1, +T E(f)(β,(x,y)) ∶=(∂β)E(f)(x) − (∂β)E(f)(y) +, (β,(x,y)) ∈ ω2. +and the weight ν∶ω → (0,∞) by +ν(β,x) ∶= 1, (β,x) ∈ ω1, +and +ν(β,(x,y)) ∶= +1 +∣x − y∣γ , (β,(x,y)) ∈ ω2. +By setting F(Ω,E) ∶= Ck(Ω,E) and observing that +∣f∣Ck,γ(Ω),α = sup +x∈ω pα(T E(f)(x))ν(x), +f ∈ Ck,γ(Ω,E), α ∈ A, +we have Fν(Ω,E) = Ck,γ(Ω,E) with generator (T E,T K). +5.3.3. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness, Ω ⊂ Rd open and bounded, k ∈ N0 and 0 < γ ≤ 1. In the case k ≥ 1, assume +additionally that Ω has Lipschitz boundary. If f∶Ω → E is such that e′ ○f ∈ Ck,γ(Ω) +for all e′ ∈ G, then f ∈ Ck,γ(Ω,E). +Proof. We take F(Ω) ∶= Ck(Ω) and F(Ω,E) ∶= Ck(Ω,E) and have Fν(Ω) = +Ck,γ(Ω) and Fν(Ω,E) = Ck,γ(Ω,E) with the weight ν and generator (T E,T K) for +(Fν,E) described above. Due to the proof of Example 4.2.28 and Theorem 3.1.12 +the spaces F(Ω) and F(Ω,E) are ε-into-compatible for any lcHs E (the condi- +tion that E has metric ccp in Example 4.2.28 is only needed for ε-compatibility). +Another consequence of Example 4.2.28 is that +T E(S(u))(β,x) = (∂β)E(S(u))(x) = u(δx ○ (∂β)K) = u(T K +β,x), +(β,x) ∈ ω1, +holds for all u ∈ F(Ω)εE, implying +T E(S(u))(β,(x,y)) = T E(S(u))(β,x) − T E(S(u))(β,y) = u(T K +β,x) − u(T K +β,y) += u(T K +β,(x,y)), +(β,(x,y)) ∈ ω2. +Thus (T E,T K) is a consistent family for (F,E) and its strength is easily seen. In +addition, Fν(Ω) = Ck,γ(Ω) is a Banach space by [57, Theorem 9.8, p. 110] (cf. [4, +1.7 Hölderstetige Funktionen, p. 46]) whose closed unit ball is compact in F(Ω) = +Ck(Ω) by [4, 8.6 Einbettungssatz in Hölder-Räumen, p. 338]. Moreover, the ε-into- +compatibility of F(Ω) and F(Ω,E) in combination with the consistency of (T E,T K) +for (F,E) implies Fεν(Ω,E) ⊂ Fν(Ω,E) as linear spaces by Proposition 5.2.25 +c). Hence our statement follows from Theorem 5.2.29 with the set of uniqueness +U ∶= {0} × Ω for (T K,Fν). +□ +5.3.4. Remark. We point out that Corollary 5.3.3 corrects our result [117, +Corollary 5.3, p. 16] by adding the missing assumption that Ω should additionally +have Lipschitz boundary in the case k ≥ 1. This is needed to deduce that the closed +unit ball of Ck,γ(Ω) is compact in Ck(Ω) by [4, 8.6 Einbettungssatz in Hölder- +Räumen, p. 338] (in the notation of [74] Ω having Lipschitz boundary means that +it is a C0,1 domain, see [74, Lemma 6.36, p. 136] and the comments below and +above this lemma). This additional assumption is missing in [57, Theorem 14.32, +p. 232], which is our main reference in [117] for the compact embedding, but it is + +5.3. WEAK-STRONG PRINCIPLES FOR DIFFERENTIABILITY OF FINITE ORDER +107 +needed due to [4, U8.1 Gegenbeispiel zu Einbettungssätzen, p. 365] (cf. [74, p. 53]). +However, this only affects the result [117, Corollary 6.3, p. 21–22] where we have to +add this missing assumption as well (see Corollary 5.4.4 for this). The other results +of [117] derived from [117, Corollary 5.3, p. 16] are not affected by this missing +assumption since they are all a consequence of [117, Corollary 5.4, p. 17] and [117, +Corollary 6.4, p. 22], whose proofs can be adjusted without additional assumptions +(see Corollary 5.3.5 and Corollary 5.4.5 for this). +Next, we use the preceding corollary to generalise the theorem of Grothendieck +and Schwartz on weakly Ck+1-functions. For k ∈ N0 and 0 < γ ≤ 1 we define the +space of k-times continuously partially differentiable E-valued functions with locally +γ-Hölder continuous partial derivatives of k-th order on an open set Ω ⊂ Rd by +Ck,γ +loc (Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ K ⊂ Ω compact, α ∈ A ∶ ∣f∣K,α < ∞} +where +∣f∣K,α ∶= max(∣f∣Ck(K),α, +sup +β∈Nd +0,∣β∣=k +∣(∂β)Ef∣C0,γ(K),α) +with +∣f∣Ck(K),α ∶= +sup +x∈K +β∈Nd +0,∣β∣≤k +pα((∂β)Ef(x)) +and +∣f∣C0,γ(K),α ∶= sup +x,y∈K +x≠y +pα(f(x) − f(y)) +∣x − y∣γ +. +Further, we set Ck,γ +loc (Ω) ∶= Ck,γ +loc (Ω,K). Using Corollary 5.3.3, we are able to +improve Theorem 5.3.2 to the following form. +5.3.5. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness, Ω ⊂ Rd open, k ∈ N0 and 0 < γ ≤ 1. +a) If f∶Ω → E is such that e′ ○f ∈ Ck,γ +loc (Ω) for all e′ ∈ G, then f ∈ Ck,γ +loc (Ω,E). +b) If f∶Ω → E is such that e′ ○f ∈ Ck+1(Ω) for all e′ ∈ G, then f ∈ Ck,1 +loc (Ω,E). +Proof. Let us start with a). Let f∶Ω → E be such that e′ ○ f ∈ Ck,γ +loc (Ω) for +all e′ ∈ G. Let (Ωn)n∈N be an exhaustion of Ω with open, relatively compact sets +Ωn ⊂ Ω with Lipschitz boundaries ∂Ωn (e.g. choose each Ωn as the interior of a +finite union of closed axis-parallel cubes, see the proof of [168, Theorem 1.4, p. 7] +for the construction) that satisfies Ωn ⊂ Ωn+1 for all n ∈ N. Then the restriction of +e′ ○ f to Ωn is an element of Ck,γ(Ωn) for every e′ ∈ G and n ∈ N. Due to Corollary +5.3.3 we obtain that f ∈ Ck,γ(Ωn,E) for every n ∈ N. Thus f ∈ Ck,γ +loc (Ω,E) since +differentiability is a local property and for each compact K ⊂ Ω there is n ∈ N such +that K ⊂ Ωn. +Let us turn to b), i.e. let f∶Ω → E be such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ G. +Since Ω ⊂ Rd is open, for every x ∈ Ω there is εx > 0 such that Bεx(x) ⊂ Ω. For all +e′ ∈ G, β ∈ Nd +0 with ∣β∣ = k and w,y ∈ Bεx(x), w ≠ y, it holds that +∣(∂β)K(e′ ○ f)(w) − (∂β)K(e′ ○ f)(y)∣ +∣w − y∣ +≤ Cd max +1≤n≤d +max +z∈Bεx(x) +∣(∂β+en)K(e′ ○ f)(z)∣ +by the mean value theorem applied to the real and imaginary part where Cd ∶= +√ +d +if K = R, and Cd ∶= 2 +√ +d if K = C. +Thus e′ ○ f ∈ Ck,1 +loc (Ω) for all e′ ∈ G since +for each compact set K ⊂ Ω there are m ∈ N and xi ∈ Ω, 1 ≤ i ≤ m, such that +K ⊂ ⋃m +i=1 Bεxi (xi). It follows from part a) that f ∈ Ck,1 +loc (Ω,E). +□ + +108 +5. APPLICATIONS +If Ω = R, γ = 1 and G = E′, then part a) of Corollary 5.3.5 is already known +by [104, 2.3 Corollary, p. 15]. A ‘full’ Ck-weak-strong principle for k < ∞, i.e. the +conditions of part b) imply f ∈ Ck+1(Ω,E), does not hold in general (see [104, p. +11–12]) but it holds if we restrict the class of admissible lcHs E. +5.3.6. Theorem. Let E be a semi-Montel space, G ⊂ E′ determine boundedness, +Ω ⊂ Rd open and k ∈ N. If f∶Ω → E is such that e′ ○ f ∈ Ck(Ω) for all e′ ∈ G, then +f ∈ Ck(Ω,E). +Proof. Let f∶Ω → E be such that e′○f ∈ Ck(Ω) for all e′ ∈ G. Due to Corollary +5.3.5 b) we already know that f ∈ Ck−1,1 +loc +(Ω,E) since semi-Montel spaces are quasi- +complete and thus locally complete. Now, let x ∈ Ω, εx > 0 such that Bεx(x) ⊂ Ω, +β ∈ Nd +0 with ∣β∣ = k − 1 and n ∈ N with 1 ≤ n ≤ d. The set +B ∶= {(∂βf)E(x + hen) − (∂βf)Ef(x) +h +∣ h ∈ R, 0 < h ≤ εx} +is bounded in E because f ∈ Ck−1,1 +loc +(Ω,E). As E is a semi-Montel space, the closure +B is compact in E. Let (hm)m∈N be a sequence in R such that 0 < hm ≤ εx for all +m ∈ N. From the compactness of B we deduce that there is a subnet (hmι)ι∈I of +(hm)m∈N and yx ∈ B with +yx = lim +ι∈I +(∂βf)E(x + hmιen) − (∂βf)Ef(x) +hmι +=∶ lim +ι∈I yι. +Further, we note that the limit +(∂β+en)K(e′ ○ f)(x) = +lim +h→0 +h∈R,h≠0 +∂β(e′ ○ f)(x + hen) − ∂β(e′ ○ f)(x) +h +(55) +exists for all e′ ∈ G and that (e′(yι))ι∈I is a subnet of the net of difference quotients +on the right-hand side of (55) as (∂β)K(e′ ○ f) = e′ ○ (∂β)Ef. Therefore +(∂β+en)K(e′ ○ f)(x) = +lim +h→0 +h∈R,h≠0 +e′((∂β)Ef(x + hen) − (∂β)Ef(x) +h +) += +lim +h→0 +h∈R,0 1? +(iii) For every ε > 0 does there exist a function g ∈ Ck(R,E) such that λ({x ∈ +Ω ∣ f(x) ≠ g(x)}) < ε in Corollary 5.3.8 where λ is the one-dimensional +Lebesgue measure. In the case that E = Rn this is true by [66, Theorem +3.1.15, p. 227]. + +110 +5. APPLICATIONS +(iv) Is there a ‘Radon–Nikodým type’ characterisation of generalised Gelfand +spaces as in the Banach case? +5.4. Vector-valued Blaschke theorems +In this section we prove several convergence theorems for Banach-valued func- +tions in the spirit of Blaschke’s convergence theorem [38, Theorem 7.4, p. 219] +as it is done in [7, Theorem 2.4, p. 786] and [7, Corollary 2.5, p. 786–787] for +bounded holomorphic functions and more general in [70, Corollary 4.2, p. 695] for +bounded functions in the kernel of a hypoelliptic linear partial differential operator. +Blaschke’s convergence theorem says that if (zn)n∈N ⊂ D is a sequence of distinct +elements with ∑n∈N(1 − ∣zn∣) = ∞ and if (fk)k∈N is a bounded sequence in H∞(D) +such that (fk(zn))k converges in C for each n ∈ N, then there is f ∈ H∞(D) such +that (fk)k converges uniformly to f on the compact subsets of D, i.e. w.r.t. to τc. +5.4.1. Proposition ([70, Proposition 4.1, p. 695]). Let (E,∥ ⋅ ∥) be a Banach +space, Z a Banach space whose closed unit ball BZ is a compact subset of an lcHs +Y and let (Aι)ι∈I be a net in Y εE such that +sup +ι∈I +{∥Aι(y)∥ ∣ y ∈ B○Y ′ +Z +} < ∞. +Assume further that there exists a σ(Y ′,Z)-dense subspace X ⊂ Y ′ such that +limι Aι(x) exists for each x ∈ X. Then there is A ∈ Y εE with A(B○Y ′ +Z +) bounded +and limι Aι = A uniformly on the equicontinuous subsets of Y ′, i.e. for all equicon- +tinuous B ⊂ Y ′ and ε > 0 there exists ς ∈ I such that +sup +y∈B +∥Aι(y) − A(y)∥ < ε +for each ι ≥ ς. +Next, we generalise [70, Corollary 4.2, p. 695]. +5.4.2. Corollary. Let (E,∥ ⋅ ∥) be a Banach space and F(Ω) and F(Ω,E) be +ε-into-compatible. Let (T E,T K) be a generator for (Fν,E) and a strong, consistent +family for (F,E), Fν(Ω) a Banach space whose closed unit ball BFν(Ω) is a compact +subset of F(Ω) and U a set of uniqueness for (T K,Fν). +If (fι)ι∈I ⊂ Fεν(Ω,E) is a bounded net in Fν(Ω,E) such that limι T E(fι)(x) +exists for all x ∈ U, then there is f ∈ Fεν(Ω,E) such that (fι)ι∈I converges to f in +F(Ω,E). +Proof. We set X ∶= span{T K +x ∣ x ∈ U}, Y ∶= F(Ω) and Z ∶= Fν(Ω). As in +the proof of Theorem 5.2.29 we observe that X is σ(Y ′,Z)-dense in Y ′. +From +(fι)ι∈I ⊂ Fεν(Ω,E) follows that there are Aι ∈ F(Ω)εE with S(Aι) = fι for all ι ∈ I. +Since (fι)ι∈I is a bounded net in Fν(Ω,E), we note that +sup +ι∈I +sup +x∈ω ∥Aι(T K +x (⋅)ν(x))∥ = sup +ι∈I +sup +x∈ω ∥T ES(Aι)(x)∥ν(x) = sup +ι∈I +sup +x∈ω ∥T Efι(x)∥ν(x) += sup +ι∈I +∣fι∣Fν(Ω,E) < ∞ +by consistency. Further, limι S(Aι)(T K +x ) = limι T E(fι)(x) exists for each x ∈ U, +implying the existence of limι S(Aι)(x) for each x ∈ X by linearity. +We apply +Proposition 5.4.1 and obtain f ∶= S(A) ∈ Fεν(Ω,E) such that (Aι)ι∈I converges +to A in F(Ω)εE. From F(Ω) and F(Ω,E) being ε-into-compatible it follows that +(fι)ι∈I converges to f in F(Ω,E). +□ + +5.4. VECTOR-VALUED BLASCHKE THEOREMS +111 +First, we apply the preceding corollary to the space C[γ] +z (Ω,E) of γ-Hölder +continuous functions on Ω that vanish at a fixed point z ∈ Ω from Example 4.2.9 +a). We recall that for a metric space (Ω,d), z ∈ Ω, an lcHs E and 0 < γ ≤ 1 we have +C[γ] +z (Ω,E) = {f ∈ EΩ ∣ f(z) = 0 and ∀ α ∈ A ∶ ∣f∣C0,γ(Ω),α < ∞}. +Further, we set ω ∶= Ω2 ∖ {(x,x) ∣ x ∈ Ω}, F(Ω,E) ∶= {f ∈ C(Ω,E) ∣ f(z) = 0} and +T E∶F(Ω,E) → Eω, T E(f)(x,y) ∶= f(x) − f(y), and +ν∶ω → [0,∞), ν(x,y) ∶= +1 +d(x,y)γ . +Then we have for every α ∈ A that +∣f∣C0,γ(Ω),α = sup +x∈ω pα(T E(f)(x))ν(x), +f ∈ C[γ] +z (Ω,E), +and observe that Fν(Ω,E) = C[γ] +z (Ω,E) with generator (T E,T K). +5.4.3. Corollary. Let E be a Banach space, (Ω,d) a metric space, z ∈ Ω and +0 < γ ≤ 1. If (fι)ι∈I is a bounded net in C[γ] +z (Ω,E) such that limι fι(x) exists for all +x in a dense subset U ⊂ Ω, then there is f ∈ C[γ] +z (Ω,E) such that (fι)ι∈I converges +to f in C(Ω,E) uniformly on compact subsets of Ω. +Proof. We choose F(Ω) ∶= {f ∈ C(Ω) ∣ f(z) = 0} and F(Ω,E) ∶= {f ∈ +C(Ω,E) ∣ f(z) = 0}. Then we have Fν(Ω) = C[γ] +z (Ω) and Fν(Ω,E) = C[γ] +z (Ω,E) +with the weight ν and generator (T E,T K) for (Fν,E) described above. Due to [17, +3.1 Bemerkung, p. 141] the spaces F(Ω) and F(Ω,E), equipped with the topology +τc of compact convergence, are ε-compatible. Obviously, (T E,T K) is a strong, con- +sistent family for (F,E). In addition, Fν(Ω) = C[γ] +z (Ω) is a Banach space by [179, +Proposition 1.6.2, p. 20]. For all f from the closed unit ball BFν(Ω) of Fν(Ω) we +have +∣f(x) − f(y)∣ ≤ d(x,y)γ, +x,y ∈ Ω, +and +∣f(x)∣ = ∣f(x) − f(z)∣ ≤ d(x,z)γ, +x ∈ Ω. +It follows that BFν(Ω) is (uniformly) equicontinuous and {f(x) ∣ f ∈ BFν(Ω)} is +bounded in K for all x ∈ Ω. Ascoli’s theorem (see e.g. [133, Theorem 47.1, p. 290]) +implies the compactness of BFν(Ω) in F(Ω) (see also [118, 3.7 Theorem (a), p. 10]). +Furthermore, the ε-compatibility of F(Ω) and F(Ω,E) in combination with the +consistency of (T E,T K) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces by +Proposition 5.2.25 c). We note that limι fι(x) = limι T E(fι)(x,z) for all x in U, +proving our claim by Corollary 5.4.2. +□ +The space C[γ] +z (Ω) is named Lip0(Ωγ) in [179] (see [179, Definition 1.6.1 (b), p. +19] and [179, Definition 1.1.2, p. 2]). Corollary 5.4.3 generalises [179, Proposition +2.1.7, p. 38] (in combination with [179, Proposition 1.2.4, p. 5]) where Ω is compact, +U = Ω and E = K. +5.4.4. Corollary. Let E be a Banach space, Ω ⊂ Rd open and bounded, k ∈ N0 +and 0 < γ ≤ 1. In the case k ≥ 1, assume additionally that Ω has Lipschitz boundary. +If (fι)ι∈I is a bounded net in Ck,γ(Ω,E) such that +(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if +(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω +is connected and there is x0 ∈ Ω such that limι fι(x0) exists and k ≥ 1, +then there is f ∈ Ck,γ(Ω,E) such that (fι)ι∈I converges to f in Ck(Ω,E). + +112 +5. APPLICATIONS +Proof. As in Corollary 5.3.3 we take F(Ω) ∶= Ck(Ω) and F(Ω,E) ∶= Ck(Ω,E) +as well as Fν(Ω) ∶= Ck,γ(Ω) and Fν(Ω,E) ∶= Ck,γ(Ω,E) with the weight ν and +generator (T E,T K) for (Fν,E) described above of Corollary 5.3.3. By the proof +of Corollary 5.3.3 all conditions of Corollary 5.4.2 are satisfied, which implies our +statement. +□ +We recall that CWk(Ω,E) is the space Ck(Ω,E) equipped with its usual topol- +ogy for an open set Ω ⊂ Rd, k ∈ N∞ ∪ {0} and an lcHs E (see Example 3.1.9 b) for +k ∈ N∞ and the definition above Proposition 3.1.11 for k = 0). +5.4.5. Corollary. Let E be a Banach space, Ω ⊂ Rd open, k ∈ N0 and 0 < γ ≤ 1. +If (fι)ι∈I is a bounded net in Ck,γ +loc (Ω,E) such that +(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if +(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω +is connected and there is x0 ∈ Ω such that limι fι(x0) exists and k ≥ 1, +then there is f ∈ Ck,γ +loc (Ω,E) such that (fι)ι∈I converges to f in CWk(Ω,E). +Proof. Let (Ωn)n∈N be an exhaustion of Ω with open, relatively compact sets +Ωn ⊂ Ω such that Ωn has Lipschitz boundary, Ωn ⊂ Ωn+1 for all n ∈ N and, in +addition, x0 ∈ Ω1 and Ωn is connected for each n ∈ N in case (ii) (see the proof of +Corollary 5.3.5). The restriction of (fι)ι∈I to Ωn is a bounded net in Ck,γ(Ωn,E) +for each n ∈ N. By Corollary 5.4.4 there is Fn ∈ Ck,γ(Ωn,E) for each n ∈ N such +that the restriction of (fι)ι∈I to Ωn converges to Fn in Ck(Ωn,E) since U ∩ Ωn is +dense in Ωn due to Ωn being open and x0 being an element of the connected set +Ωn in case (ii). The limits Fn+1 and Fn coincide on Ωn for each n ∈ N. Thus the +definition f ∶= Fn on Ωn for each n ∈ N gives a well-defined function f ∈ Ck,γ +loc (Ω,E), +which is a limit of (fι)ι∈I in CWk(Ω,E). +□ +5.4.6. Corollary. Let E be a Banach space, Ω ⊂ Rd open and k ∈ N0. If +(fι)ι∈I is a bounded net in Ck+1(Ω,E) such that +(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if +(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω +is connected and there is x0 ∈ Ω such that limι fι(x0) exists, +then there is f ∈ Ck,1 +loc (Ω,E) such that (fι)ι∈I converges to f in CWk(Ω,E). +Proof. By Corollary 5.3.5 b) (fι)ι∈I is a bounded net in Ck,1 +loc (Ω,E). Hence +our statement is a consequence of Corollary 5.4.5. +□ +The preceding result directly implies a C∞-smooth version. +5.4.7. Corollary. Let E be a Banach space and Ω ⊂ Rd open. If (fι)ι∈I is a +bounded net in C∞(Ω,E) such that +(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if +(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω +is connected and there is x0 ∈ Ω such that limι fι(x0) exists, +then there is f ∈ C∞(Ω,E) such that (fι)ι∈I converges to f in CW∞(Ω,E). +Now, we turn to weighted kernels of hypoelliptic linear partial differential op- +erators. +5.4.8. Corollary. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoelliptic +linear partial differential operator, ν∶Ω → (0,∞) continuous and U ⊂ Ω a set of +uniqueness for (idKΩ,CνP (∂)). If (fι)ι∈I is a bounded net in (CνP (∂)(Ω,E),∣ ⋅ ∣ν) +such that limι fι(x) exists for all x ∈ U, then there is f ∈ CνP (∂)(Ω,E) such that +(fι)ι∈I converges to f in (C∞ +P (∂)(Ω,E),τc). + +5.4. VECTOR-VALUED BLASCHKE THEOREMS +113 +Proof. Our statement follows from Corollary 5.4.2 since by the proof of Corol- +lary 5.2.30 all conditions needed are fulfilled. +□ +For ν = 1 on Ω the preceding corollary is included in [70, Corollary 4.2, p. +695] but then an even better result is available, whose proof we prepare next. +We recall the definition of the space (C∞ +P (∂),b(Ω,E),β) with the strict topology β +from Proposition 4.2.24. For an open set Ω ⊂ Rd, an lcHs E and a linear partial +differential operator P(∂)E which is hypoelliptic if E = K the space of bounded +zero solutions is +C∞ +P (∂),b(Ω,E) = {f ∈ C∞ +P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∥f∥∞,α = sup +x∈Ω +pα(f(x)) < ∞}. +We equip this space with strict topology β induced by the seminorms +∣f∣̃ν,α ∶= sup +x∈Ω +pα(f(x))∣̃ν(x)∣, +f ∈ C∞ +P (∂),b(Ω,E), +for ̃ν ∈ C0(Ω). Now, we phrase for C∞ +P (∂),b(Ω,E) = CνP (∂)(Ω,E) with ν = 1 on Ω +the improved version of Corollary 5.4.8. +5.4.9. Corollary. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoelliptic +linear partial differential operator and U ⊂ Ω a set of uniqueness for (idKΩ,C∞ +P (∂),b). +If (fι)ι∈I is a bounded net in (C∞ +P (∂),b(Ω,E),∥ ⋅ ∥∞) such that limι fι(x) exists +for all x ∈ U, then there is f ∈ C∞ +P (∂),b(Ω,E) such that (fι)ι∈I converges to f in +(C∞ +P (∂),b(Ω,E),β). +Proof. We take F(Ω) ∶= (C∞ +P (∂),b(Ω),β) and F(Ω,E) ∶= (C∞ +P (∂),b(Ω,E),β) as +well as Fν(Ω) ∶= (C∞ +P (∂),b(Ω),∥ ⋅ ∥∞) and Fν(Ω,E) ∶= (C∞ +P (∂),b(Ω,E),∥ ⋅ ∥∞) with +the weight ν(x) ∶= 1, x ∈ Ω, and generator (idEΩ,idΩK) for (Fν,E). The generator +is strong and consistent for (F,E) and F(Ω) and F(Ω,E) are ε-compatible by +Proposition 4.2.24. The space Fν(Ω) is a Banach space as a closed subspace of +the Banach space (Cb(Ω),∥ ⋅ ∥∞). Its closed unit ball BFν(Ω) is τc-compact because +(C∞ +P (∂)(Ω),τc) is a Fréchet–Schwartz space, in particular, a Montel space. Thus +BFν(Ω) is ∥ ⋅ ∥∞-bounded and τc-compact, which implies that it is also β-compact +by [45, Proposition 1 (viii), p. 586] and [45, Proposition 3, p. 590]. In addition, +the ε-compatibility of F(Ω) and F(Ω,E) in combination with the consistency of +(idEΩ,idKΩ) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces by Proposition +5.2.25 c), verifying our statement by Corollary 5.4.2. +□ +A direct consequence of Corollary 5.4.9 is the following remark. +5.4.10. Remark. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoel- +liptic linear partial differential operator and (fι)ι∈I a bounded net in the space +(C∞ +P (∂),b(Ω,E),∥ ⋅ ∥∞). Then the following statements are equivalent: +(i) (fι) converges pointwise, +(ii) (fι) converges uniformly on compact subsets of Ω, +(iii) (fι) is β-convergent. +In the case of complex-valued bounded holomorphic functions of one variable, +i.e. E = C, Ω ⊂ C is open and P(∂) = ∂ is the Cauchy–Riemann operator, conver- +gence w.r.t. β is known as bounded convergence (see [147, p. 13–14, 16]) and the +preceding remark is included in [148, 3.7 Theorem, p. 246] for connected sets Ω. +A similar improvement of Corollary 5.4.3 for the space C[γ] +z (Ω,E) of γ-Hölder +continuous functions on a metric space (Ω,d) that vanish at a given point z ∈ Ω is + +114 +5. APPLICATIONS +possible, using the strict topology β on C[γ] +z (Ω) given by the seminorms +∣f∣ν ∶= sup +x,y∈Ω +x≠y +∣f(x) − f(y)∣ +∣x − y∣γ +∣ν(x,y)∣, +f ∈ C[γ] +z (Ω), +for ν ∈ C0(ω) with ω = Ω2 ∖ {(x,x) ∣ x ∈ Ω}. If Ω is compact and E a Banach +space, this follows as in Corollary 5.4.9 from the observation that β is the mixed +topology γ(∣ ⋅ ∣C0,γ(Ω),τc) by [90, Theorem 3.3, p. 645], that a set is β-compact if +and only if it is ∣⋅∣C0,γ(Ω)-bounded and τc-compact by [90, Theorem 2.1 (6), p. 642], +the ε-compatibility (C[γ] +z (Ω),β)εE ≅ (C[γ] +z (Ω,E),γτγ) by [90, Theorem 4.4, p. 648] +where the topology γτγ is described in [90, Definition 4.1, p. 647] and coincides +with β if E = K by [90, Proposition 4.3 (i), p. 647]. +Let us turn to Bloch type spaces. The result corresponding to Corollary 5.4.8 +for Bloch type spaces reads as follows. +5.4.11. Corollary. Let E be a Banach space, ν∶D → (0,∞) continuous and +U∗ ⊂ D have an accumulation point in D. If (fι)ι∈I is a bounded net in Bν(D,E) +such that limι fι(0) and limι(∂1 +C)Efι(z) exist for all z ∈ U∗, then there is f ∈ +Bν(D,E) such that (fι)ι∈I converges to f in (O(D,E),τc). +Proof. Due to the proof of Corollary 5.2.33 all conditions needed to apply +Corollary 5.4.2 are fulfilled, which proves our statement. +□ +5.5. Wolff type results +The following theorem gives us a Wolff type description of the dual of F(Ω) and +generalises [70, Theorem 3.3, p. 693] and [70, Corollary 3.4, p. 694] whose proofs +only need a bit of adaptation. Wolff’s theorem [183, p. 1327] (cf. [81, Theorem +(Wolff), p. 402]) phrased in a functional analytic way (see [70, p. 240]) says: if +Ω ⊂ C is a domain (i.e. open and connected), then for each µ ∈ O(Ω)′ there are +a sequence (zn)n∈N which is relatively compact in Ω and a sequence (an)n∈N in ℓ1 +such that µ = ∑∞ +n=1 anδzn. +5.5.1. Theorem. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E,T K) be a +generator for (Fν,E) and a strong, consistent family for (F,E) for every Banach +space E. Let F(Ω) be a nuclear Fréchet space and Fν(Ω) a Banach space whose +closed unit ball BFν(Ω) is a compact subset of F(Ω) and (xn)n∈N fixes the topology +in Fν(Ω). +a) Then there is 0 < λ ∈ ℓ1, i.e. λ ∈ ℓ1 and λn > 0 for all n ∈ N, such that for +every bounded B ⊂ F(Ω)′ +b there is C ≥ 1 with +{µ∣Fν(Ω) ∣ µ ∈ B} ⊂ { +∞ +∑ +n=1 +anν(xn)T K +xn ∈ Fν(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}. +b) Let (∥ ⋅ ∥k)k∈N denote the system of seminorms generating the topology of +F(Ω). Then there is a decreasing zero sequence (εn)n∈N such that for all +k ∈ N there is C ≥ 1 with +∥f∥k ≤ C sup +n∈N +∣T K(f)(xn)∣ν(xn)εn, +f ∈ Fν(Ω). +Proof. We start with part a). +Let B1 ∶= {T K +xn(⋅)ν(xn) ∣ n ∈ N} ⊂ F(Ω)′, +X ∶= spanB1, Y ∶= F(Ω), Z ∶= Fν(Ω) and E1 ∶= {∑∞ +n=1 anν(xn)T K +xn ∣ a ∈ ℓ1}. From +∣j1(a)(f)∣ ∶= ∣ +∞ +∑ +n=1 +anν(xn)T K +xn(f)∣ ≤ sup +n∈N +∣T K(f)(xn)∣ν(xn)∥a∥ℓ1 ≤ ∣f∣Fν(Ω)∥a∥ℓ1 +for all f ∈ Fν(Ω) and a ∈ ℓ1 it follows that E1 is a linear subspace of Fν(Ω)′ +and the continuity of the map j1∶ℓ1 → Fν(Ω)′ where Fν(Ω)′ is equipped with the + +5.5. WOLFF TYPE RESULTS +115 +operator norm. In addition, we deduce that the linear map j∶ℓ1/kerj1 → Fν(Ω)′, +j([a]) ∶= j1(a), where [a] denotes the equivalence class of a ∈ ℓ1 in the quotient +space ℓ1/kerj1, is continuous w.r.t. the quotient norm since +∥j([a])∥Fν(Ω)′ ≤ +inf +b∈ℓ1,[b]=[a]∥b∥ℓ1 = ∥[a]∥ℓ1/ ker j1. +By setting E ∶= j(ℓ1/kerj1) and ∥j([a])∥E ∶= ∥[a]∥ℓ1/ ker j1, a ∈ ℓ1, and observing +that ℓ1/kerj1 is a Banach space, we obtain that E is also a Banach space, which is +continuously embedded in Fν(Ω)′. +We denote by A∶X → E the restriction to Z = Fν(Ω) determined by +A(T K +xn(⋅)ν(xn)) ∶= T K +xn(⋅)∣Fν(Ω)ν(xn) = j([en]) +where en is the n-th unit sequence in ℓ1. We consider Fν(Ω) as a subspace of E′ +via +f(j([a])) ∶= j([a])(f) = +∞ +∑ +n=1 +anν(xn)T K(f)(xn), +a ∈ ℓ1, +for f ∈ Fν(Ω). The space G ∶= Fν(Ω) clearly separates the points of E, thus is +σ(E′,E)-dense and +(f ○ A)(T K +xn(⋅)ν(xn)) = A(T K +xn(⋅)ν(xn))(f) = j([en])(f) = f(j([en])) +for all n ∈ N. Hence we may consider f ○ A by identification with f as an element +of Z = Fν(Ω) for all f ∈ G = Fν(Ω). It follows from Proposition 5.2.68 that there +is a unique extension ̂A ∈ F(Ω)εE of A such that S(̂A) ∈ Fεν(Ω,E). +For each e′ ∈ E′ there are C0,C1 > 0 and an absolutely convex compact set +K ⊂ F(Ω) such that +∣(e′ ○ ̂A)(µ)∣ ≤ C0∥̂A(µ)∥E ≤ C0C1 sup +f∈K +∣µ(f)∣ +for all µ ∈ F(Ω)′, implying e′ ○ ̂A ∈ (F(Ω)′ +b)′. Due to the reflexivity of the nuclear +Fréchet space F(Ω) we obtain e′ ○ ̂A ∈ F(Ω) for each e′ ∈ E′. Further, for each +e′ ∈ E′ we have +∥e′ ○ ̂A∥Fν(Ω) = sup +x∈ω ∣T K(e′ ○ ̂A)(x)∣ν(x) = sup +x∈ω ∣(e′ ○ ̂A)(T K +x (⋅)ν(x))∣ +≤ C0 sup +x∈ω ∥̂A(T K +x (⋅)ν(x))∥E < ∞ +since ̂A(B○F (Ω)′ +Fν(Ω) ) is bounded in E. This yields e′ ○ ̂A ∈ Fν(Ω) for each e′ ∈ E′. In +particular, we get that ̂A is σ(F(Ω)′,Fν(Ω))-σ(E,E′) continuous. The restriction +r∶F(Ω)′ → Fν(Ω)′, r(µ) ∶= µ∣Fν(Ω), is σ(F(Ω)′,Fν(Ω))-σ(Fν(Ω)′,Fν(Ω)) contin- +uous and coincides with ̂A on the σ(F(Ω)′,Fν(Ω))-dense subspace X = spanB1 ⊂ +F(Ω)′. Therefore ̂A(µ) = r(µ) = µ∣Fν(Ω) for all µ ∈ F(Ω)′. +Let B be an absolutely convex, closed and bounded subset of F(Ω)′ +b. +We +endow W ∶= spanB with the Minkowski functional of B. Due to the nuclearity +of F(Ω), there are an absolutely convex, closed and bounded subset V ⊂ F(Ω)′ +b, +(w′ +k)k∈N ⊂ BW ′, (µk)k∈N ⊂ V and 0 ≤ γ ∈ ℓ1, i.e. γ ∈ ℓ1 and γn ≥ 0 for all n ∈ N, such +that +µ = +∞ +∑ +k=1 +γkw′ +k(µ)µk, +µ ∈ B, +by [24, 2.9.1 Theorem, p. 134, 2.9.2 Definition, p. 135]. The boundedness of ̂A(V ) +in E and the definition of E give us a bounded sequence ([β(k)])k∈N ⊂ E with +µk∣Fν(Ω) = ̂A(µk) = +∞ +∑ +n=1 +β(k) +n ν(xn)T K +xn + +116 +5. APPLICATIONS +for all k ∈ N. The sequence (β(k))k∈N ⊂ ℓ1 is also bounded by [131, Remark 5.11, p. +36] and we set ρn ∶= ∑∞ +k=1 γk∣β(k) +n ∣ for n ∈ N. With ρ ∶= (ρn)n∈N we have +∥ρ∥ℓ1 = +∞ +∑ +n=1 +∞ +∑ +k=1 +γk∣β(k) +n ∣ ≤ +∞ +∑ +n=1 +sup +l∈N +∣β(l) +n ∣ +∞ +∑ +k=1 +γk = sup +l∈N +∥β(l)∥ℓ1∥γ∥ℓ1 < ∞, +which means that ρ ∈ ℓ1. For every µ ∈ B we set an ∶= ∑∞ +k=1 γkw′ +k(µ)β(k) +n , n ∈ N, +and conclude that a ∈ ℓ1 with ∣an∣ ≤ ρn for all n ∈ N and +µ∣Fν(Ω) = +∞ +∑ +n=1 +anν(xn)T K +xn. +(57) +The strong dual F(Ω)′ +b of the Fréchet–Schwartz space F(Ω) is a DFS-space and +thus there is a fundamental sequence of bounded (closed, absolutely convex) sets +(Bl)l∈N in F(Ω)′ +b by [131, Proposition 25.19, p. 303]. Due to our preceding results +there is ρ(l) ∈ ℓ1 with (57) for each l ∈ N. Finally, part a) follows from choosing +0 < λ ∈ ℓ1 such that each ρ(l) is componentwise smaller than a multiple of λ, i.e. +we choose λ in a way that for each l ∈ N there is Cl ≥ 1 with ρ(l) +n +≤ Clλn for all +n ∈ N (w.l.o.g. we may assume (the worst case) that limn→∞ ρ(l+1) +n +/ρ(l) +n = ∞ for each +l ∈ N. Then the construction of a suitable 0 < λ ∈ ℓ1 is given in [97, Chap. IX, §41, +7., p. 301–302]: set c(l) +n ∶= ρ(l) +n +for all l,n ∈ N and define λn ∶= cn + +1 +n2 for all n ∈ N +with the (cn) ∈ ℓ1 constructed there. Then set C1 ∶= 1 and Cl ∶= (max{c(l) +n ∣ 1 ≤ n ≤ +nl−1}/min{λn ∣ 1 ≤ n ≤ nl−1}) + 1, l ≥ 2, for the sequence of indices (nl)l∈N from the +construction of (cn).). +Let us turn to part b). We choose λ ∈ ℓ1 from part a) and a decreasing zero +sequence (εn)n∈N such that ( λn +εn )n∈N still belongs to ℓ1 (e.g. take εn ∶= (∑∞ +k=n λk)1/2 +for n ∈ N by [97, Chap. IX, §39, Theorem of Dini, p. 293]). For k ∈ N we set +̃Bk ∶= {f ∈ F(Ω) ∣ ∥f∥k ≤ 1} +and note that the polar ̃B○ +k is bounded in F(Ω)′ +b. Due to part a) there exists C ≥ 1 +such that +̂A( ̃B○ +k) ⊂ { +∞ +∑ +n=1 +anν(xn)T K +xn ∈ Fν(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}. +By [131, Proposition 22.14, p. 256] the formula +∥f∥k = sup +y′∈ ̃ +B○ +k +∣y′(f)∣, +f ∈ F(Ω), +is valid and hence +∥f∥k = sup +y′∈ ̃ +B○ +k +∣r(y′)(f)∣ = sup +y′∈ ̃ +B○ +k +∣̂A(y′)(f)∣ ≤ C sup +a∈ℓ1 +∣an∣≤λn +∣ +∞ +∑ +n=1 +anν(xn)T K(f)(xn)∣ +≤ C∥(λn +εn +) +n∥ +ℓ1 sup +n∈N +∣T K(f)(xn)∣ν(xn)εn +for all f ∈ Fν(Ω). +□ +5.5.2. Remark. The proof of Theorem 5.5.1 shows it is not needed that the +assumption that F(Ω) and F(Ω,E) are ε-into-compatible, (T E,T K) is a generator +for (Fν,E) and a strong, consistent family for (F,E) is fulfilled for every Banach +space E. It is sufficient that it is fulfilled for the Banach space E ∶= j(ℓ1/kerj1). +We recall from (53) that for a positive sequence ν ∶= (νn)n∈N and an lcHs E we +have +ℓν(N,E) = {x = (xn)n∈N ∈ EN ∣ ∀ α ∈ A ∶ ∥x∥α = sup +n∈N +pα(xn)νn < ∞}. + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +117 +Further, we equip the space EN of all sequences in E from Example 4.2.1 with the +topology of pointwise convergence, i.e. the topology generated by the seminorms +∣x∣k,α ∶= sup +1≤n≤k +pα(xn), +x = (xn)n∈N ∈ EN, +for k ∈ N and α ∈ A. +5.5.3. Corollary. Let ν ∶= (νn)n∈N be a positive sequence. +a) Then there is 0 < λ ∈ ℓ1 such that for every bounded B ⊂ (KN)′ +b there is +C ≥ 1 with +{µ∣ℓν(N) ∣ µ ∈ B} ⊂ { +∞ +∑ +n=1 +anνnδn ∈ ℓν(N)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}. +b) Then there is a decreasing zero sequence (εn)n∈N such that for all k ∈ N +there is C ≥ 1 with +sup +1≤n≤k +∣xn∣ ≤ C sup +n∈N +∣xn∣νnεn, +x = (xn)n∈N ∈ ℓν(N). +Proof. We take F(N) ∶= KN and F(N,E) ∶= EN as well as Fν(N) ∶= ℓν(N) and +Fν(N,E) ∶= ℓν(N,E) where (T E,T K) ∶= (idEN,idKN) is the generator for (Fν,E). +We remark that F(N) and F(N,E) are ε-compatible and (T E,T K) is a strong, +consistent family for (F,E) by Example 4.2.1 for every Banach space E. Moreover, +Fν(N) = ℓν(N) is a Banach space by [131, Lemma 27.1, p. 326] since ℓν(N) = λ∞(A) +with the Köthe matrix A ∶= (an,j)n,j∈N given by an,j ∶= νn for all n,j ∈ N. +In +addition, we have for every k ∈ N +sup +1≤n≤k +∣xn∣ ≤ sup +1≤n≤k +ν−1 +n ∣x∣ν ≤ sup +1≤n≤k +ν−1 +n , +x = (xn)n∈N ∈ BFν(N), +which means that BFν(N) is bounded in F(N). The space F(N) = KN is a nuclear +Fréchet space and BFν(N) is obviously closed in KN. Thus the bounded and closed +set BFν(N) is compact in F(N), implying our statement by Theorem 5.5.1. +□ +5.5.4. Corollary. Let Ω ⊂ Rd be open, P(∂)K a hypoelliptic linear partial +differential operator, ν∶Ω → (0,∞) continuous and (xn)n∈N fix the topology in +CνP (∂)(Ω). +a) Then there is 0 < λ ∈ ℓ1 such that for every bounded B ⊂ (C∞ +P (∂)(Ω),τc)′ +b +there is C ≥ 1 with +{µ∣CνP (∂)(Ω) ∣ µ ∈ B} ⊂ { +∞ +∑ +n=1 +anν(xn)δxn ∈ CνP (∂)(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}. +b) Then there is a decreasing zero sequence (εn)n∈N such that for all compact +K ⊂ Ω there is C ≥ 1 with +sup +x∈K +∣f(x)∣ ≤ C sup +n∈N +∣f(xn)∣ν(xn)εn, +f ∈ CνP (∂)(Ω). +Proof. Due to the proof of Corollary 5.2.30 and the observation that the space +F(Ω) = (C∞ +P (∂)(Ω),τc) is a nuclear Fréchet space all conditions of Theorem 5.5.1 +are fulfilled, which yields our statement. +□ +5.6. Series representation of vector-valued functions via Schauder +decompositions +The purpose of this section is to lift series representations known from scalar- +valued functions to vector-valued functions and its underlying idea was derived from +the classical example of the (local) power series representation of a holomorphic +function. We recall that a C-valued function f on the open disc Dr(0) around zero + +118 +5. APPLICATIONS +with radius r > 0 belongs to the space O(Dr(0)) of holomorphic functions on Dr(0) +if the limit +f (1)(z) ∶= +lim +h→0 +h∈C,h≠0 +f(z + h) − f(z) +h +, +z ∈ Dr(0), +(58) +exists in C. It is well-known that every f ∈ O(Dr(0)) can be written as +f(z) = +∞ +∑ +n=0 +f (n)(0) +n! +zn, +z ∈ Dr(0), +where the power series on the right-hand side converges uniformly on every com- +pact subset of Dr(0) and f (n)(0) is the n-th complex derivative of f at 0 which +is defined from (58) by the recursion f (0) ∶= f and f (n) ∶= (f (n−1))(1) for n ∈ N. +By [79, 2.1 Theorem and Definition, p. 17–18] and [79, 5.2 Theorem, p. 35], this +series representation remains valid if f is a holomorphic function on Dr(0) with +values in a locally complete locally convex Hausdorff space E over C where holo- +morphy means that the limit (58) exists in E and the higher complex derivatives +are defined recursively as well. Analysing this example, we observe that O(Dr(0)), +equipped with the topology τc of uniform convergence on compact subsets of Dr(0), +is a Fréchet space, in particular, barrelled, with a Schauder basis formed by the +monomials z ↦ zn. Further, the formulas for the complex derivatives of a C-valued +resp. an E-valued function f on Dr(0) are built up in the same way by (58) (see +Chapter 2). +Our goal is to derive a mechanism which uses these observations and transfers +known series representations for other spaces of scalar-valued functions to their +vector-valued counterparts. Let us describe the general setting. We recall from [89, +14.2, p. 292] that a sequence (fn) in a locally convex Hausdorff space F over a field +K is called a topological basis, or simply a basis, if for every f ∈ F there is a unique +sequence of coefficients (λK +n(f)) in K such that +f = +∞ +∑ +n=1 +λK +n(f)fn +(59) +where the series converges in F. +Due to the uniqueness of the coefficients the +map λK +n∶f ↦ λK +n(f) is well-defined, linear and called the n-th coefficient functional +associated to (fn). Further, for each k ∈ N the map +Pk∶F → F, Pk(f) ∶= +k +∑ +n=1 +λK +n(f)fn, +is a linear projection whose range is span{f1,...,fn} and it is called the k-th ex- +pansion operator associated to (fn). A basis (fn) of F is called equicontinuous if +the expansion operators Pk form an equicontinuous sequence in the linear space +L(F,F) of continuous linear maps from F to F (see [89, 14.3, p. 296]). A basis +(fn) of F is called a Schauder basis if the coefficient functionals are continuous, i.e. +λK +n ∈ F ′ for each n ∈ N. In particular, this is already fulfilled if F is a Fréchet space +by [131, Corollary 28.11, p. 351]. If F is barrelled, then a Schauder basis of F is +already equicontinuous and F has the (bounded) approximation property by the +uniform boundedness principle. +The starting point for our approach is equation (59). Let F and E be non- +trivial locally convex Hausdorff spaces over a field K where F has an equicontinuous +Schauder basis (fn) with associated coefficient functionals (λK +n). The expansion +operators (Pk) form a so-called Schauder decomposition of F (see [27, p. 77]), i.e. +they are continuous projections on F such that +(i) PkPj = Pmin(j,k) for all j,k ∈ N, +(ii) Pk ≠ Pj for k ≠ j, + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +119 +(iii) (Pkf) converges to f for each f ∈ F. +This operator theoretic definition of a Schauder decomposition is equivalent to the +usual definition in terms of closed subspaces of F given in [96, p. 377] (see [123, +p. 219]). In our main Theorem 5.6.1 of this section we prove that (PkεidE) is a +Schauder decomposition of Schwartz’ ε-product FεE and each u ∈ FεE has the +series representation +u(f ′) = +∞ +∑ +n=1 +u(λK +n)f ′(fn), +f ′ ∈ F ′. +Now, suppose that F = F(Ω) is a space of K-valued functions on a set Ω with a +topology such that the point-evaluation functionals δx, x ∈ Ω, belong to F(Ω)′ and +that there is a locally convex Hausdorff space F(Ω,E) of functions from Ω to E +such that the map +S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], +is an isomorphism, i.e. suppose that F(Ω) and F(Ω,E) are ε-compatible. Assuming +that for each n ∈ N and u ∈ F(Ω)εE there is λE +n (S(u)) ∈ E with +λE +n (S(u)) = u(λK +n), +(60) +i.e. (λE,λK) is consistent, we obtain in Corollary 5.6.5 that (S ○ (PkεidE) ○ S−1)k +is a Schauder decomposition of F(Ω,E) and +f = lim +k→∞(S ○ (PkεidE) ○ S−1)(f) = +∞ +∑ +n=1 +λE +n (f)fn, +f ∈ F(Ω,E), +which is the desired series representation in F(Ω,E). In particular, the consis- +tency condition (60) guarantees in the case of E-valued holomorphic functions on +Dr(0) that the complex derivatives at 0 appear in the Schauder decomposition of +O(Dr(0),E) since (∂n +C)ES(u)(0) = u(δ0 ○(∂n +C)C) for all u ∈ O(Dr(0))εE and n ∈ N0 +by Proposition 5.2.32 if E is locally complete. We apply our result to sequence +spaces, spaces of continuously differentiable functions on a compact interval, the +space of holomorphic functions, the Schwartz space and the space of smooth func- +tions which are 2π-periodic in each variable. +As a byproduct of Theorem 5.6.1 we obtain that every element of the completion +F ̂⊗εE of the injective tensor product F ⊗ε E has a series representation as well if +F is a complete space with an equicontinuous Schauder basis and E is complete. +Concerning series representation in F ̂⊗εE, little seems to be known whereas for +the completion F ̂⊗πE of the projective tensor product F ⊗π E of two metrisable +locally convex spaces F and E it is well-known that every f ∈ F ̂⊗πE has a series +representation +f = +∞ +∑ +n=1 +anfn ⊗ en +where (an) ∈ ℓ1, i.e. (an) is absolutely summable, and (fn) and (en) are null +sequences in F and E, respectively (see e.g. [83, Chap. I, §2 , n○1, Théorème 1, p. +51] or [89, 15.6.4 Corollary, p. 334]). If F and E are metrisable and one of them is +nuclear, then the isomorphism F ̂⊗πE ≅ F ̂⊗εE holds and we trivially have a series +representation of the elements of F ̂⊗εE as well. Other conditions on the existence +of series representations of the elements of F ̂⊗εE can be found in [151, Proposition +4.25, p. 88], where F and E are Banach spaces and both of them have a Schauder +basis, and in [91, Theorem 2, p. 283], where F and E are locally convex Hausdorff +spaces and both of them have an equicontinuous Schauder basis. + +120 +5. APPLICATIONS +5.6.1. Schauder decomposition. Let us start with our main theorem on +Schauder decompositions of ε-products. We recall from (3) that we consider the +tensor product F ⊗ E as a linear subspace of FεE for two locally convex Hausdorff +spaces F and E by means of the linear injection +Θ∶F ⊗ E → FεE, +k +∑ +n=1 +fn ⊗ en �→ [y ↦ +k +∑ +n=1 +y(fn)en]. +The next theorem is essentially due to José Bonet, improving a previous version +of us which became Corollary 5.6.5. +5.6.1. Theorem. Let F and E be lcHs, (fn)n∈N an equicontinuous Schauder +basis of F with associated coefficient functionals (λn)n∈N and set Qn∶F → F, +Qn(f) ∶= λn(f)fn for every n ∈ N. Then the following holds: +a) The sequence (Pk)k∈N given by Pk ∶= (∑k +n=1 Qn)εidE is a Schauder decom- +position of FεE. +b) Each u ∈ FεE has the series representation +u(f ′) = +∞ +∑ +n=1 +u(λn)f ′(fn), +f ′ ∈ F ′. +c) F ⊗ E is sequentially dense in FεE. +Proof. Since (fn) is a Schauder basis of F, the sequence (∑k +n=1 Qn) converges +to idF in Lσ(F,F). Thus we deduce from the equicontinuity of (fn) that (∑k +n=1 Qn) +converges to idF in Lκ(F,F) by [89, Theorem 8.5.1 (b), p. 156]. For f ′ ∈ F ′ and +f ∈ F it holds +(Qt +n ○ Qt +m)(f ′)(f) = Qt +m(f ′)(Qn(f)) = Qt +m(f ′)(λn(f)fn) = f ′(λm(λn(f)fn)fm) += λm(fn)λn(f)f ′(fm) = +⎧⎪⎪⎨⎪⎪⎩ +λn(f)f ′(fn) +, m = n, +0 +, m ≠ n, +due to the uniqueness of the coefficient functionals (λn) (see [89, 14.2.1 Proposition, +p. 292]) and it follows for k,j ∈ N that +( +j +∑ +n=1 +Qt +n ○ +k +∑ +m=1 +Qt +m)(f ′)(f) = +min(j,k) +∑ +n=1 +λn(f)f ′(fn) = +min(j,k) +∑ +n=1 +Qt +n(f ′)(f). +This implies that +(PkPj)(u) = u ○ +j +∑ +n=1 +Qt +n ○ +k +∑ +m=1 +Qt +m = u ○ +min(j,k) +∑ +n=1 +Qt +n = Pmin(j,k)(u) +for all u ∈ FεE. If k ≠ j, w.l.o.g. k > j, we choose x ∈ E, x ≠ 0,3 and consider fk ⊗ x +as an element of FεE via the map Θ. Then +(Pk − Pj)(fk ⊗ x) = +k +∑ +n=j+1 +(fk ⊗ x) ○ Qt +n = fk ⊗ x ≠ 0 +since +((fk⊗x)○Qt +n)(f ′) = (fk⊗x)(λn(⋅)f ′(fn)) = λn(fk)f ′(fn)x = +⎧⎪⎪⎨⎪⎪⎩ +(fk ⊗ x)(f ′) , n = k, +0 +, n ≠ k. +It remains to prove that for each u ∈ FεE +lim +k→∞Pk(u) = u +3The lcHs E is non-trivial by our assumptions in Chapter 2. + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +121 +in FεE. Let (qβ)β∈B denote the system of seminorms inducing the locally convex +topology of F. Let u ∈ FεE and α ∈ A. Due to the continuity of u there are an +absolutely convex compact set K = K(u,α) ⊂ F and C0 = C0(u,α) > 0 such that +for each f ′ ∈ F ′ we have +pα((Pk(u) − u)(f ′)) = pα(u(( +k +∑ +n=1 +Qt +n − idF ′)(f ′))) ≤ C0 sup +f∈K +∣( +k +∑ +n=1 +Qt +n − idF ′)(f ′)(f)∣ += C0 sup +f∈K +∣f ′( +k +∑ +n=1 +Qnf − f)∣. +Let V be an absolutely convex zero neighbourhood in F. As a consequence of the +equicontinuity of the polar V ○ there are C1 > 0 and β ∈ B such that +sup +f ′∈V ○ pα((Pk(u) − u)(f ′)) ≤ C0C1 sup +f∈K +qβ( +k +∑ +n=1 +Qnf − f). +In combination with the convergence of (∑k +n=1 Qn) to idF in Lκ(F,F) this yields +the convergence of (Pk(u)) to u in FεE and settles part a). +Let us turn to b) and c). Since +Pk(u)(f ′) = u( +k +∑ +n=1 +Qt +n(f ′)) = +k +∑ +n=1 +u(λn)f ′(fn) +for every f ′ ∈ F ′, we note that the range of Pk(u) is contained in span{u(λn) ∣ 1 ≤ +n ≤ k} for each u ∈ FεE and k ∈ N. Hence Pk(u) has finite rank and thus belongs +to F ⊗ E, implying the sequential density of F ⊗ E in FεE and the desired series +representation by part a). +□ +The index set of the equicontinuous Schauder basis of F in Theorem 5.6.1 need +not be N (or N0) but may be any other countable index set as long as the equicon- +tinuous Schauder basis is unconditional which is, for instance, always fulfilled if F +is nuclear by [89, 21.10.1 Dynin-Mitiagin Theorem, p. 510]. +5.6.2. Remark. If F and E are complete, we have under the assumption of +Theorem 5.6.1 that F ̂⊗εE ≅ FεE by c) since FεE is complete by [94, Satz 10.3, p. +234] and F ̂⊗εE is the closure of F ⊗ E in FεE. Thus each element of F ̂⊗εE has +a series representation. +Let us apply the preceding theorem to spaces of Lebesgue integrable functions. +We consider the measure space ([0,1],L ([0,1]),λ) of Lebesgue measurable sets +and use the notation Lp[0,1] for the space of (equivalence classes) of Lebesgue +p-integrable functions on [0,1]. The Haar system hn∶[0,1] → R, n ∈ N, given by +h1(x) ∶= 1 for all x ∈ [0,1] and +h2k+j(x) ∶= +⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ +1 +,(2j − 2)/2k+1 ≤ x < (2j − 1)/2k+1, +−1 +,(2j − 1)/2k+1 ≤ x < 2j/2k+1, +0 +,else, +for k ∈ N0 and 1 ≤ j ≤ 2k forms a Schauder basis of Lp[0,1] for every 1 ≤ p < ∞ and +the associated coefficient functionals are +λn(f) ∶= ∫ +[0,1] +f(x)hn(x)dλ(x), +f ∈ Lp[0,1], n ∈ N, +(see [154, Satz I, p. 317]). Because Lp[0,1] is Banach space and thus barrelled, its +Schauder basis (hn) is equicontinuous and we directly obtain from Theorem 5.6.1 +the following corollary. + +122 +5. APPLICATIONS +5.6.3. Corollary. Let E be an lcHs and 1 ≤ p < ∞. (∑k +n=1 λn(⋅)hnεidE)k∈N is +a Schauder decomposition of Lp[0,1]εE and for each u ∈ Lp[0,1]εE it holds +u(f ′) = +∞ +∑ +n=1 +u(λn)f ′(hn), +f ′ ∈ Lp[0,1]′. +Defining Lp([0,1],E) ∶= Lp[0,1]εE, we can read the corollary above as a state- +ment on series representations in the vector-valued version of Lp[0,1]. However, +in many cases of spaces F(Ω) of scalar-valued functions there is a more natural +way to define the vector-valued version F(Ω,E) of F(Ω), namely, that F(Ω) and +F(Ω,E) are ε-compatible. +5.6.4. Remark. If F(Ω) and F(Ω,E) are ε-into-compatible, then we get by +identification of isomorphic subspaces +F(Ω) ⊗ε E ⊂ F(Ω)εE ⊂ F(Ω,E) +and the embedding F(Ω) ⊗ E ↪ F(Ω,E) is given by f ⊗ e �→ [x ↦ f(x)e]. +Proof. The inclusions obviously hold and F(Ω)εE and F(Ω,E) induce the +same topology on F(Ω) ⊗ E. Further, we have +f ⊗ e +Θ +�→ [y ↦ y(f)e] +S +�→ [x �→ [y ↦ y(f)e](δx)] = [x ↦ f(x)e]. +□ +5.6.5. Corollary. Let F(Ω) and F(Ω,E) be ε-compatible, (fn)n∈N an equi- +continuous Schauder basis of F(Ω) with associated coefficient functionals λK ∶= +(λK +n)n∈N. Let there be λE∶F(Ω,E) → EN such that (λE,λK) is a consistent family +for (F,E), and set QE +n ∶F(Ω,E) → F(Ω,E), QE +n (f) ∶= λE +n (f)fn for every n ∈ N. +Then the following holds: +a) The sequence (P E +k )k∈N given by P E +k ∶= ∑k +n=1 QE +n is a Schauder decomposi- +tion of F(Ω,E). +b) Each f ∈ F(Ω,E) has the series representation +f = +∞ +∑ +n=1 +λE +n (f)fn. +c) F(Ω) ⊗ E is sequentially dense in F(Ω,E). +Proof. For each u ∈ F(Ω)εE and x ∈ Ω we note that with Pk from Theorem +5.6.1 it holds +(S ○ Pk)(u)(x) = u( +k +∑ +n=1 +Qt +n(δx)) = u( +k +∑ +n=1 +λK +n(⋅)fn(x)) = +k +∑ +n=1 +u(λK +n)fn(x) += +k +∑ +n=1 +λE +n (S(u))fn(x) = (P E +k ○ S)(u)(x), +which means that S ○ Pk = P E +k ○ S. This implies part a) and b) by Theorem 5.6.1 +a) since S is an isomorphism. Part c) is a direct consequence of Theorem 5.6.1 c) +and the isomorphism F(Ω)εE ≅ F(Ω,E). +□ +In the preceding corollary we used the isomorphism S to obtain a Schauder +decomposition. On the other hand, if S is an isomorphism into, which is often the +case (see Theorem 3.1.12), we can use a Schauder decomposition of F(Ω,E) to +prove the surjectivity of S. +5.6.6. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible. Let there be +(fn)n∈N in F(Ω) and for every f ∈ F(Ω,E) a sequence (λE +n (f))n∈N in E such that +f = +∞ +∑ +n=1 +λE +n (f)fn, +f ∈ F(Ω,E). +Then the following holds: + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +123 +a) F(Ω) ⊗ E is sequentially dense in F(Ω,E). +b) If F(Ω) and E are sequentially complete, then +F(Ω,E) ≅ F(Ω)εE. +c) If F(Ω) and E are complete, then +F(Ω,E) ≅ F(Ω)εE ≅ F(Ω)̂⊗εE. +Proof. Let f ∈ F(Ω,E) and observe that +P E +k (f) ∶= +k +∑ +n=1 +λE +n (f)fn = +k +∑ +n=1 +fn ⊗ λE +n (f) ∈ F(Ω) ⊗ E +for every k ∈ N by Remark 5.6.4. Due to our assumption we have the convergence +P E +k (f) → f in F(Ω,E). Thus F(Ω) ⊗ E is sequentially dense in F(Ω,E). +Let us turn to part b). If F(Ω) and E are sequentially complete, then F(Ω)εE +is sequentially complete by [94, Satz 10.3, p. 234]. Since S is an isomorphism into +and +S(Θ( +k +∑ +n=q +fn ⊗ λE +n (f))) = +k +∑ +n=q +λE +n (f)fn +for all k,q ∈ N, k > q, we get that (Θ(∑k +n=1 fn ⊗ λE +n (f)) is a Cauchy sequence in +F(Ω)εE and thus convergent. Hence we deduce that +S( lim +k→∞Θ( +k +∑ +n=1 +fn ⊗ λE +n (f))) = lim +k→∞ +k +∑ +n=1 +(S ○ Θ)(fn ⊗ λE +n (f)) = +∞ +∑ +n=1 +λE +n (f)fn = f, +which proves the surjectivity of S. +If F(Ω) and E are complete, then F(Ω)̂⊗εE is the closure of F(Ω)⊗ε E in the +complete space F(Ω)εE by [94, Satz 10.3, p. 234]. As limk→∞ Θ(∑k +n=1 fn ⊗ λE +n (f)) +is an element of the closure, we obtain part c). +□ +5.6.2. Examples of Schauder decompositions. +Sequence spaces. For our first application we recall the definition of some +sequence spaces. For an lcHs E and a Köthe matrix A ∶= (ak,j)k,j∈N we define the +topological subspace of λ∞(A,E) from Corollary 4.2.3 a) by +c0(A,E) ∶= {x = (xk) ∈ EN ∣ ∀ j ∈ N ∶ lim +k→∞xkak,j = 0}. +In particular, the space c0(N,E) of null-sequences in E is obtained as c0(N,E) = +c0(A,E) with ak,j ∶= 1 for all k,j ∈ N. The space of convergent sequences in E is +defined by +c(N,E) ∶= {x ∈ EN ∣ x = (xk) converges in E} +and equipped with the system of seminorms +∣x∣α ∶= sup +k∈N +pα(xk), +x ∈ c(N,E), +for α ∈ A. Further, we set c0(A) ∶= c0(A,K), c0(N) ∶= c0(N,K) and c(N) ∶= c(N,K). +Furthermore, we equip the space EN with the system of seminorms given by +∥x∥l,α ∶= sup +k∈N +pα(xk)χ{1,...,l}(k), +x = (xk) ∈ EN, +for l ∈ N and α ∈ A. For a non-empty set Ω we define for n ∈ Ω the n-th unit +function by +ϕn,Ω∶Ω → K, ϕn,Ω(k) ∶= +⎧⎪⎪⎨⎪⎪⎩ +1 +, k = n, +0 +, else, +and we simply write ϕn instead of ϕn,Ω if no confusion seems to be likely. Further, +we set ϕ∞∶N → K, ϕ∞(k) ∶= 1, and x∞ ∶= δ∞(x) ∶= limk→∞ xk for x ∈ c(N,E). +For series representations of the elements in these sequence spaces we do not need + +124 +5. APPLICATIONS +Corollary 5.6.5 due to the subsequent proposition but we can use the representation +to obtain the surjectivity of S for sequentially complete E. +5.6.7. Proposition. Let E be an lcHs and ℓ(Ω,E) one of the spaces c0(A,E), +EN, s(Nd,E), s(Nd +0,E) or s(Zd,E). +a) Then (∑n∈Ω,∣n∣≤k δnϕn)k∈N is a Schauder decomposition of ℓ(Ω,E) and +x = ∑ +n∈Ω +xnϕn, +x ∈ ℓ(Ω,E). +b) Then (δ∞ϕ∞+∑k +n=1(δn−δ∞)ϕn)k∈N is a Schauder decomposition of c(N,E) +and +x = x∞ϕ∞ + +∞ +∑ +n=1 +(xn − x∞)ϕn, +x ∈ c(N,E). +Proof. Let us begin with a). First, we note that (ϕn)n∈Ω is an unconditional +equicontinuous Schauder basis of s(Ω), Ω = Nd, Nd +0, Zd, since s(Ω) is a nuclear +Fréchet space. Now, for x = (xn) ∈ ℓ(Ω,E) let (P E +k ) be the sequence in ℓ(Ω,E) +given by P E +k (x) ∶= ∑∣n∣≤k xnϕn. It is easy to see that P E +k is a continuous projection +on ℓ(Ω,E), P E +k P E +j += P E +min(k,j) for all k,j ∈ N and P E +k ≠ P E +j +for k ≠ j. Let ε > 0, +α ∈ A and j ∈ N. For x ∈ c0(A,E) there is N0 ∈ N such that pα(xnan,j) < ε for all +n ≥ N0. Hence we have for x ∈ c0(A,E) +∣x − P E +k (x)∣j,α = sup +n>k +pα(xn)an,j ≤ sup +n≥N0 +pα(xn)an,j ≤ ε +for all k ≥ N0. For x ∈ EN and l ∈ N we have +∥x − P E +k (x)∥l,α = 0 < ε +for all k ≥ l. For x ∈ s(Ω,E), Ω = Nd, Nd +0, Zd, we notice that there is N1 ∈ N such +that for all n ∈ Ω with ∣n∣ ≥ N1 we have +(1 + ∣n∣2)j/2 +(1 + ∣n∣2)j += (1 + ∣n∣2)−j/2 < ε. +Thus we deduce for ∣n∣ ≥ N1 +pα(xn)(1 + ∣n∣2)j/2 < εpα(xn)(1 + ∣n∣2)j ≤ ε∣x∣2j,α +and hence +∣x − P E +k (x)∣j,α = sup +∣n∣>k +pα(xn)(1 + ∣n∣2)j/2 ≤ sup +∣n∣≥N1 +pα(xn)(1 + ∣n∣2)j/2 ≤ ε∣x∣2j,α +for all k ≥ N1. Therefore (P E +k (x)) converges to x in ℓ(Ω,E) and +x = lim +k→∞P E +k (x) = ∑ +n∈Ω +xnϕn. +Now, we turn to b). For x = (xn) ∈ c(N,E) let ( ̃P E +k (x)) be the sequence in +c(N,E) given by ̃P E +k (x) ∶= x∞ϕ∞ + ∑k +n=1(xn − x∞)ϕn. +Again, it is easy to see +that ̃P E +k is a continuous projection on c(N,E), ̃P E +k ̃P E +j += ̃P E +min(k,j) for all k,j ∈ N +and ̃P E +k ≠ ̃P E +j +for k ≠ j. Let ε > 0 and α ∈ A. Then there is N2 ∈ N such that +pα(xn − x∞) < ε for all n ≥ N2. Thus we obtain +∣x − ̃P E +k (x)∣α = sup +n>k +pα(xn − x∞) ≤ sup +n≥N2 +pα(xn − x∞) ≤ ε +for all k ≥ N2, implying that ( ̃P E +k (x)) converges to x in c(N,E) and +x = lim +k→∞ +̃P E +k (x) = x∞ϕ∞ + +∞ +∑ +n=1 +(xn − x∞)ϕn. +□ + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +125 +5.6.8. Theorem. Let E be a sequentially complete lcHs and ℓ(Ω,E) one of the +spaces c0(A,E), EN, s(Nd,E), s(Nd +0,E) or s(Zd,E). Then +(i) ℓ(Ω,E) ≅ ℓ(Ω)εE, +(ii) c(N,E) ≅ c(N)εE. +Proof. The map Sℓ(Ω) is an isomorphism into by Theorem 3.1.12 and, in +addition, by Proposition 4.1.9 (i) if ℓ(Ω,E) = c0(A,E). Considering c(N,E), we +observe that for x ∈ c(N) +δn(x) = xn → x∞ = δ∞(x), +which implies the convergence δn → δ∞ in c(N)′ +γ by the Banach–Steinhaus theorem +since c(N) is a Banach space. Hence we get +u(δ∞) = lim +n→∞u(δn) = lim +n→∞S(u)(n) = δ∞(S(u)) +for every u ∈ c(N)εE, which implies that Sc(N) is an isomorphism into by Theorem +3.1.12. From Proposition 5.6.7 and Proposition 5.6.6 we deduce our statement. +□ +More general, we note that Theorem 5.6.8 holds for any lcHs E if ℓ(Ω,E) = EN +by Example 4.2.1, for E with metric ccp if ℓ(Ω,E) = c0(A,E) by Example 4.2.11 (ii), +and for locally complete E if ℓ(Ω,E) = s(Ω,E) with Ω = Nd, Nd +0, Zd by Corollary +4.2.3 b). +Continuous and differentiable functions on a compact interval. We +start with continuous functions on compact sets. Let E be an lcHs and Ω ⊂ Rd +compact. We equip the space C(Ω,E) of continuous functions on Ω with values in +E with the system of seminorms given by +∣f∣α ∶= sup +x∈Ω +pα(f(x)), +f ∈ C(Ω,E), +for α ∈ A. We want to apply our preceding results to intervals. Let −∞ < a < b < ∞ +and T ∶= (tj)0≤j≤n be a partition of the interval [a,b], i.e. a = t0 < t1 < ... < tn = b. +The hat functions hT +tj∶[a,b] → R for the partition T are given by +hT +tj(x) ∶= +⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ +x−tj +tj−tj−1 +,tj−1 ≤ x ≤ tj, +tj+1−x +tj+1−tj +,tj < x ≤ tj+1, +0 +,else, +for 2 ≤ j ≤ n − 1 and +hT +a (x) ∶= +⎧⎪⎪⎨⎪⎪⎩ +t1−x +t1−a +,a ≤ x ≤ t1, +0 +,else, +hT +b (x) ∶= +⎧⎪⎪⎨⎪⎪⎩ +x−tn−1 +b−tn−1 +,tn−1 ≤ x ≤ b, +0 +,else. +Let T ∶= (tn)n∈N0 be a dense sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ +m. For T n ∶= {t0,...,tn} there is a (unique) enumeration σ∶{0,...,n} → {0,...,n} +of T n such that Tn ∶= (tσ(j))0≤j≤n is a partition of [a,b] with T n = {tσ(1),...,tσ(n)}. +The functions ϕT +0 ∶= hT1 +t0 , ϕT +1 ∶= hT1 +t1 and ϕT +n ∶= hTn +tσ(j) with j = σ−1(n) for n ≥ 2 +are called Schauder hat functions for the sequence T and form a Schauder basis of +C([a,b]) with associated coefficient functionals given by λK +0 (f) ∶= f(t0), λK +1 (f) ∶= +f(t1) and +λK +n+1(f) ∶= f(tn+1) − +n +∑ +k=0 +λK +k (f)ϕT +k (tn+1), +f ∈ C([a,b]), n ≥ 1, +by [166, 2.3.5 Proposition, p. 29]. Looking at the coefficient functionals, we see that +the right-hand sides even make sense if f ∈ C([a,b],E) and thus we define λE +n on +C([a,b],E) for n ∈ N0 accordingly. + +126 +5. APPLICATIONS +5.6.9. Theorem. Let E be an lcHs with metric ccp and T ∶= (tn)n∈N0 a dense +sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m. Then (∑k +n=0 λE +k ϕT +n )k∈N0 +is a Schauder decomposition of C([a,b],E) and +f = +∞ +∑ +n=0 +λE +n (f)ϕT +n , +f ∈ C([a,b],E). +Proof. The spaces C([a,b]) and C([a,b],E) are ε-compatible by Example +4.2.12 if E has metric ccp. C([a,b]) is a Banach space and thus barrelled, implying +that its Schauder basis (ϕT +n ) is equicontinuous. We note that for all u ∈ C([a,b])εE +and x ∈ [a,b] +λE +n (S(u))(x) = u(δtn) = u(λK +n), +n ∈ {0,1}, +and by induction +λE +n+1(S(u))(x) = u(δtn+1) − +n +∑ +k=0 +λE +k (S(u))ϕT +k (tn+1) = u(δtn+1) − +n +∑ +k=0 +u(λK +k )ϕT +k (tn+1) += u(λK +n+1), +n ≥ 1. +Thus (λE,λK) is consistent, proving our claim by Corollary 5.6.5. +□ +If a = 0, b = 1 and T is the sequence of dyadic numbers given in [166, 2.1.1 +Definitions, p. 21], then (ϕT +n ) is the so-called Faber–Schauder system. Using the +Schauder basis and coefficient functionals of the space C0(R) of continuous functions +vanishing at infinity given in [166, 2.7.1, p. 41–42] and [166, 2.7.4 Corollary, p. 43] +and that SC0(R) is an isomorphism by Example 4.2.11 (ii) if E has metric ccp, +the corresponding result for the E-valued counterpart C0(R,E) holds as well by a +similar reasoning. Another corresponding result holds for the space C[γ] +0,0 ([0,1],E), +0 < γ < 1, of γ-Hölder continuous functions on [0,1] with values in E that vanish +at zero and at infinity if one uses the Schauder basis and coefficient functionals of +C[γ] +0,0 ([0,1]) from [44, Theorem 2, p. 220] and [43, Theorem 3, p. 230]. This result is +a bit weaker since Example 4.2.9 only guarantees that SC[γ] +0,0([0,1]) is an isomorphism +if E is quasi-complete. +Now, we turn to the spaces Ck([a,b],E) of continuously differentiable functions +on an interval (a,b) with values in an lcHs E such that all derivatives can be +continuously extended to the boundary from Example 4.2.28. We set f (k)(x) ∶= +(∂k)Kf(x) for x ∈ (a,b) and f ∈ Ck([a,b]). From the Schauder hat functions (ϕT +n ) +for a dense sequence T ∶= (tn)n∈N0 in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m +and the associated coefficient functionals λK +n we can easily get a Schauder basis for +the space Ck([a,b]), k ∈ N, by applying ∫ +(⋅) +a +k-times to the series representation +f (k) = +∞ +∑ +n=0 +λK +n(f (k))ϕT +n , +f ∈ Ck([a,b]), +where we identified f (k) with its continuous extension. The resulting Schauder basis +f T +n ∶[a,b] → R and associated coefficient functionals µK +n∶Ck([a,b]) → K, n ∈ N0, are +f T +n (x) = 1 +n!(x − a)n, +µK +n(f) = f (n)(a), +0 ≤ n ≤ k − 1, +f T +n (x) = +x +∫ +a +sk−1 +∫ +a +⋯ +s2 +∫ +a +s1 +∫ +a +ϕT +n−kdsds1 ...dsk−1, +µK +n(f) = λK +n−k(f (k)), +n ≥ k, +for x ∈ [a,b] and f ∈ Ck([a,b]) (see e.g. [157, p. 586–587], [166, 2.3.7, p. 29]). Again, +the mapping rule for the coefficient functionals still makes sense if f ∈ Ck([a,b],E) +and so we define µE +n on Ck([a,b],E) for n ∈ N0 accordingly. + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +127 +5.6.10. Theorem. Let E be an lcHs with metric ccp, k ∈ N, T ∶= (tn)n∈N0 +a dense sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m. +Then +(∑l +n=0 µE +n f T +n )l∈N0 is a Schauder decomposition of Ck([a,b],E) and +f = +∞ +∑ +n=0 +µE +n (f)f T +n , +f ∈ Ck([a,b],E). +Proof. The spaces Ck([a,b]) and Ck([a,b],E) are ε-compatible by Example +4.2.28 if E has metric ccp. The Banach space Ck([a,b]) is barrelled giving the +equicontinuity of its Schauder basis. Due to Proposition 3.1.11 c) we have for all +u ∈ Ck([a,b])εE, β ∈ N0, β ≤ k, and x ∈ (a,b) +(∂β)ES(u)(x) = u(δx ○ (∂β)K). +Further, for every sequence (xn) in (a,b) converging to t ∈ {a,b} we obtain by +Proposition 4.1.7 in combination with Lemma 4.1.8 applied to T ∶= (∂β)K +lim +n→∞(∂β)ES(u)(xn) = u( lim +n→∞δxn ○ (∂β)K). +From these observations we deduce that µE +n (S(u)) = u(µK +n) for all n ∈ N0, i.e. +(µE,µK) is consistent. Therefore our statement is a consequence of Corollary 5.6.5. +□ +Holomorphic functions. In this short subsection we show how to get the +result on power series expansion of holomorphic functions from the introduction. +Let E be an lcHs over C, z0 ∈ C, r ∈ (0,∞] and equip O(Dr(z0),E) with the +topology τc of compact convergence. +5.6.11. Theorem. Let E be a locally complete lcHs over C, z0 ∈ C and r ∈ +(0,∞]. Then (f ↦ ∑k +n=0 +(∂n +C )Ef(z0) +n! +(⋅ − z0)n)k∈N0 is a Schauder decomposition of +O(Dr(z0),E) and +f = +∞ +∑ +n=0 +(∂n +C)Ef(z0) +n! +(⋅ − z0)n, +f ∈ O(Dr(z0),E). +Proof. The spaces O(Dr(z0)) and O(Dr(z0),E) are ε-compatible by Propo- +sition 4.2.17 and (23) (cf. [30, Theorem 9, p. 232]) if E is locally complete. Further, +the Schauder basis ((⋅−z0)n) of O(Dr(z0)) is equicontinuous since the Fréchet space +O(Dr(z0)) is barrelled. Due to Proposition 5.2.32 we have for all u ∈ O(Dr(z0))εE +(∂n +C)ES(u)(z) = u(δz ○ (∂n +C)C), +n ∈ N0, z ∈ Dr(z0), +which yields that (λE,λC) is consistent where λE∶O(Dr(z0),E) → EN0 is given +by λE +n (f) ∶= (∂n +C )Ef(z0) +n! +for n ∈ N0 (and analogously for E replaced by C). Hence +Corollary 5.6.5 implies our statement. +□ +Theorem 5.6.11 holds for holomorphic functions in several variables as well (see +[113, Theorem 5.7, p. 264]). +Fourier expansions. In this subsection we turn our attention to Fourier ex- +pansions in the Schwartz space S(Rd,E) and in the space C∞ +2π(Rd,E) of smooth +functions that are 2π-periodic in each variable. +We recall the definition of the Hermite functions. For n ∈ N0 we set +hn∶R → R, hn(x) ∶= (2nn!√π)−1/2(x − d +dx) +n +e−x2/2 = (2nn!√π)−1/2Hn(x)e−x2/2, +with the Hermite polynomials Hn of degree n which can be computed recursively +by +H0(x) = 1, Hn+1(x) = 2xHn(x) − H′ +n(x) and H′ +n(x) = 2nHn−1(x), +x ∈ R, n ∈ N0. + +128 +5. APPLICATIONS +For n = (nk) ∈ Nd +0 we define the n-th Hermite function by +hn∶Rd → R, hn(x) ∶= +d +∏ +k=1 +hnk(xk), +and +Hn∶Rd → R, Hn(x) ∶= +d +∏ +k=1 +Hnk(xk). +5.6.12. Proposition. Let E be a locally complete lcHs, f ∈ S(Rd,E) and n ∈ +Nd +0. Then fhn is Pettis-integrable on Rd. +Proof. First, we set ψ∶Rd → R, ψ(x) ∶= e−∣x∣2/2, as well as g∶Rd → [0,∞), +g(x) ∶= e∣x∣2/2. Then ψ ∈ L1(Rd,λ) and ψg = 1. Moreover, let u∶Rd → E, u(x) ∶= +f(x)hn(x)g(x), and note that +(∂ej)Eu(x) = (∂ej)Ef(x)hn(x)g(x) + f(x)g(x)∂ejhn(x) + f(x)hn(x)g(x)xj +where +∂ejhn(x) = (2njnj!√π)−1/2(H′ +nj(xj)e−x2 +j/2 − Hnj(xj)xje−x2 +j/2) +d +∏ +k=1,k≠j +hnk(xk) += (2njnj!√π)−1/2(2njHnj−1(xj) − xjHnj(xj))e−x2 +j/2 +d +∏ +k=1,k≠j +hnk(xk) +for all x = (xk) ∈ Rd and 1 ≤ j ≤ d. We set Cn ∶= (∏d +i=1 2nini!√π)−1/2 and observe +that +g(x)∂ejhn(x) = e∣x∣2/2∂ejhn(x) = Cn(2njHnj−1(xj) − xjHnj(xj)) +d +∏ +k=1,k≠j +Hnk(xk) +is a polynomial in d variables. The functions given by +hn(x)g(x) = e∣x∣2/2hn(x) = CnHn(x) +and +hn(x)g(x)xj = CnHn(x)xj +are polynomials in d variables as well. Thus there are m ∈ N and C > 0 such that +max(∣hn(x)g(x)∣,∣g(x)∂ejhn(x)∣,∣hn(x)g(x)xj∣) ≤ C(1 + ∣x∣2)m/2 +for all x ∈ Rd and 1 ≤ j ≤ d, which implies +pα((∂ej)Eu(x)) ≤ C(pα((∂ej)Ef(x))(1 + ∣x∣2)m/2 + 2pα(f(x))(1 + ∣x∣2)m/2) +for all α ∈ A and hence +sup +x∈Rd +β∈Nd +0,∣β∣≤1 +pα((∂β)Eu(x)) ≤ 3C∣f∣S(Rd),m,α. +Therefore u = fhng is (weakly) C1 +b , which yields u ∈ C[1] +b (Rd,E) by Proposition +A.1.5. Further, we set h∶Rd → (0,∞), h(x) ∶= 1 + ∣x∣2, and observe that +sup +x∈Rd pα(u(x)h(x)) ≤ sup +x∈Rd pα(f(x))∣hn(x)g(x)h(x)∣ ≤ C∣f∣m+2,α < ∞ +for all α ∈ A. In addition, we remark that for every ε > 0 there is r > 0 such that +1 ≤ εh(x) for all x ∉ Br(0) =∶ K. We deduce from Proposition A.2.7 (iii) that fhn +is Pettis-integrable on Rd. +□ +Due to the previous proposition we can define the n-th Fourier coefficient of +f ∈ S(Rd,E) by +̂f(n) ∶= F E +n (f) ∶= ∫ +Rd +f(x)hn(x)dx = ∫ +Rd +f(x)hn(x)dx, +n ∈ Nd +0, +if E is locally complete. We know that the map +F K∶S(Rd) → s(Nd +0), F K(f) ∶= ( ̂f(n))n∈Nd +0, + +5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS +129 +is an isomorphism (see e.g. [94, Satz 3.7, p. 66]). We improve this result to locally +complete E and derive a Schauder decomposition of S(Rd,E) as well. +5.6.13. Theorem. Let E be a locally complete lcHs. Then the following holds: +a) (∑n∈Nd +0,∣n∣≤k F E +n hn)k∈N is a Schauder decomposition of S(Rd,E) and +f = ∑ +n∈Nd +0 +̂f(n)hn, +f ∈ S(Rd,E). +b) The map +F E∶S(Rd,E) → s(Nd +0,E), F E(f) ∶= ( ̂f(n))n∈Nd +0, +is an isomorphism and +F E = Ss(Nd +0) ○ (F KεidE) ○ S−1 +S(Rd). +Proof. Let us begin with part a). Due to Corollary 3.2.10 the spaces S(Rd) +and S(Rd,E) are ε-compatible and the inverse of the isomorphism S∶S(Rd)εE → +S(Rd,E) is given by the map Rt∶S(Rd,E) → S(Rd)εE, f ↦ J −1 ○ Rt +f, according +to Theorem 3.2.4. Moreover, S(Rd) is a nuclear Fréchet space, thus barrelled, and +hence its Schauder basis (hn) is equicontinuous and unconditional. From the Pettis- +integrability of fhn by Proposition 5.6.12 and Proposition 4.3.3 with (T E +0 ,T K +0 ) ∶= +(hn idERd ,hn idKRd ) we obtain that (F E,F K) is consistent. Hence we conclude our +statement from Corollary 5.6.5. +Let us turn to part b). First, we show that the map F E is well-defined. Let +f ∈ S(Rd,E). Then e′ ○ f ∈ S(Rd) and +⟨e′,F E(f)n⟩ = ⟨e′, ̂f(n)⟩ = ̂ +e′ ○ f(n) = F K(e′ ○ f)n +for every n ∈ Nd +0 and e′ ∈ E′. Thus we have F K(e′ ○ f) ∈ s(Nd +0) for every e′ ∈ E′, +which implies by [131, Mackey’s theorem 23.15, p. 268] that F E(f) ∈ s(Nd +0,E) and +that F E is well-defined. Due to Corollary 3.2.10 and Corollary 4.2.3 the maps +SS(Rd) and Ss(Nd +0) are isomorphisms, which implies that F E is also an isomorphism +with F E = Ss(Nd +0) ○ (F KεidE) ○ S−1 +S(Rd) by Theorem 5.1.2 b). +□ +Our last example of this subsection is devoted to Fourier expansions in the +space C∞ +2π(Rd,E). We recall that C∞ +2π(Rd,E) denotes the topological subspace of +CW∞(Rd,E) consisting of the functions which are 2π-periodic in each variable. Due +to Lemma A.2.2 we are able to define the n-th Fourier coefficient of f ∈ C∞ +2π(Rd,E) +by +̂f(n) ∶= FE +n (f) ∶= (2π)−d +∫ +[−π,π]d +f(x)e−i⟨n,x⟩dx, +n ∈ Zd, +where ⟨⋅,⋅⟩ is the usual scalar product on Rd, if E is locally complete. We know +that the map +FC∶C∞ +2π(Rd) → s(Zd), FC(f) ∶= ( ̂f(n))n∈Zd, +is an isomorphism (see e.g. [94, Satz 1.7, p. 18]), which we lift to the E-valued case. +5.6.14. Theorem. Let E be a locally complete lcHs over C. +a) Then (∑n∈Zd,∣n∣≤k FE +n ei⟨n,⋅⟩)k∈N is a Schauder decomposition of C∞ +2π(Rd,E) +and +f = ∑ +n∈Zd +̂f(n)ei⟨n,⋅⟩, +f ∈ C∞ +2π(Rd,E). + +130 +5. APPLICATIONS +b) The map +FE∶C∞ +2π(Rd,E) → s(Zd,E), FE(f) ∶= ( ̂f(n))n∈Zd, +is an isomorphism and +FE = Ss(Zd) ○ (FCεidE) ○ S−1 +C∞ +2π(Rd). +Proof. The spaces C∞ +2π(Rd) and C∞ +2π(Rd,E) are ε-compatible by Example +4.2.27. +The space C∞ +2π(Rd) is barrelled since it is a nuclear Fréchet space and thus +its Schauder basis (ei⟨n,⋅⟩) is equicontinuous and unconditional. By Theorem 3.2.4 +the inverse of SC∞ +2π(Rd) is given by Rt∶C∞ +2π(Rd,E) → C∞ +2π(Rd)εE, f ↦ J −1 ○ Rt +f. +From the Pettis-integrability of fe−i⟨n,⋅⟩ and Proposition 4.3.3 with (T E +0 ,T K +0 ) ∶= +(e−i⟨n,⋅⟩ idERd ,e−i⟨n,⋅⟩ idCRd ) we obtain that (FE,FC) is consistent. Hence we con- +clude part a) from Corollary 5.6.5. +Let us turn to part b). As in Theorem 5.6.13 it follows from [131, Mackey’s +theorem 23.15, p. 268] that the map FE is well-defined. Due to Corollary 4.2.3 and +Example 4.2.27 the maps Ss(Zd) and SC∞ +2π(Rd) are isomorphisms, which implies that +FE is an isomorphism as well with FE = Ss(Zd) ○ (F CεidE) ○ S−1 +C∞ +2π(Rd) by Theorem +5.1.2 b). +□ +For quasi-complete E Theorem 5.6.14 is already known by [94, Satz 10.8, p. +239]. +5.7. Representation by sequence spaces +Our last section is dedicated to the representation of weighted spaces of E- +valued functions by weighted spaces of E-valued sequences if there is a counterpart +of this representation in the scalar-valued case involving the coefficient functionals +associated to a Schauder basis (see Remark 5.2.3 b)). We only touched upon this +problem in Section 5.6 for special cases like S(Rd,E) and C∞ +2π(Rd,E) in Theorem +5.6.13 b) and Theorem 5.6.14 b). We solve this problem in a different way by an +application of our extension results from Section 5.2. As an example we treat the +space O(DR(0),E) of holomorphic functions and the multiplier space OM(R,E) of +the Schwartz space (see Corollary 5.7.3). +5.7.1. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bound- +edness and F(Ω) and F(Ω,E) resp. ℓ(N) and ℓ(N,E) be ε-into-compatible with +e′ ○ g ∈ ℓ(N) for all e′ ∈ E′ and g ∈ ℓ(N,E). Let (fn)n∈N be an equicontinuous +Schauder basis of F(Ω) with associated coefficient functionals (T K +n )n∈N such that +T K∶F(Ω) → ℓ(N), T K(f) ∶= (T K +n (f))n∈N, +is an isomorphism and let there be T E∶F(Ω,E) → EN such that (T E,T K) is a +strong, consistent family for (F,E). If +(i) F(Ω) is a Fréchet–Schwartz space, or +(ii) E is sequentially complete, G = E′ and F(Ω) is a semi-Montel BC-space, +then the following holds: +a) FG(N,E) = ℓ(N,E). +b) ℓ(N) and ℓ(N,E) are ε-compatible, in particular, ℓ(N)εE ≅ ℓ(N,E). +c) The map +T E∶F(Ω,E) → ℓ(N,E), T E(f) ∶= (T E +n (f))n∈N, +is a well-defined isomorphism, F(Ω) and F(Ω,E) are ε-compatible, in +particular, F(Ω)εE ≅ F(Ω,E), and T E = Sℓ(N) ○ (T KεidE) ○ S−1 +F(Ω). + +5.7. REPRESENTATION BY SEQUENCE SPACES +131 +Proof. a)(1) First, we remark that N is a set of uniqueness for (T K,F). Let +u ∈ F(Ω)εE and n ∈ N. Then +RN,G(SF(Ω)(u))(n) = (T E ○ SF(Ω))(u)(n) = T E +n (SF(Ω)(u)) = u(T K +n ) = u(δn ○ T K) += (u ○ (T K)t)(δn) = (T KεidE)(u)(δn) += (Sℓ(N) ○ (T KεidE))(u)(n) +(61) +by consistency and the ε-into-compatibility, yielding FG(N,E) ⊂ ℓ(N,E) once we +have shown that RN,G is surjective, which we postpone to part b). +a)(2) Let g ∈ ℓ(N,E). Then e′ ○g ∈ ℓ(N) for all e′ ∈ E′ and ge′ ∶= (T K)−1(e′ ○g) ∈ +F(Ω). We note that T K +n (ge′) = (e′ ○ g)(n) for all n ∈ N, which implies ℓ(N,E) ⊂ +FG(N,E). +b) We only need to show that Sℓ(N) is surjective. Let g ∈ ℓ(N,E), which implies +g ∈ FG(N,E) by part a)(2). +We claim that RN,G is surjective. In case (i) this follows directly from Theorem +5.2.20. Let us turn to case (ii) and denote by (fn)n∈N the equicontinuous Schauder +basis of F(Ω) associated to (T K +n )n∈N. +We check that condition (ii) of Theorem +5.2.15 is fulfilled. Let f ′ ∈ F(Ω)′ and set +f ′ +k∶F(Ω) → K, f ′ +k(f) ∶= +k +∑ +n=1 +T K +n (f)f ′(fn), +for k ∈ N. Then f ′ +k ∈ F(Ω)′ for every k ∈ N and (f ′ +k) converges to f ′ in F(Ω)′ +σ since +(∑k +n=1 T K +n (f)fn) converges to f in F(Ω). From the equicontinuity of the Schauder +basis we deduce that (f ′ +k) converges to f ′ in F(Ω)′ +κ by [89, 8.5.1 Theorem (b), p. +156]. Let f ∈ FE′(N,E). For each e′ ∈ E′ and k ∈ N we have +Rt +f(f ′ +k)(e′) = f ′ +k(fe′) = +k +∑ +n=1 +T K +n (fe′)f ′(fn) = e′( +k +∑ +n=1 +f(n)f ′(fn)) +since f ∈ FE′(N,E), implying Rt +f(f ′ +k) ∈ J (E). Hence we can apply Theorem 5.2.15 +(ii) and obtain that RN,E′ is surjective, finishing the proof of part a)(1). +Thus there is u ∈ F(Ω)εE such that RN,E′(SF(Ω)(u)) = g in both cases. Then +(T KεidE)(u) ∈ ℓ(N)εE and from (61) we derive +Sℓ(N)((T KεidE)(u)) = RN,G(SF(Ω)(u)) = g, +proving the surjectivity of Sℓ(N). +c) First, we note that the map T E is well-defined. Indeed, we have (e′○T E)(f) = +T K(e′ ○f) ∈ ℓ(N) for all f ∈ F(Ω,E) and e′ ∈ E′ by the strength of the family. Part +a) implies that T E(f) ∈ FG(N,E) = ℓ(N,E) and thus the map T E is well-defined +and its linearity follows from the linearity of the T E +n for n ∈ N. Next, we prove that +T E is surjective. Let g ∈ ℓ(N,E). Since T KεidE is an isomorphism and Sℓ(N) by +part b) as well, we obtain that u ∶= ((T KεidE)−1 ○ S−1 +ℓ(N))(g) ∈ F(Ω)εE. Therefore +SF(Ω)(u) ∈ F(Ω,E) and from (61) we get +T E(SF(Ω)(u)) = (T E ○ SF(Ω))(u) = (Sℓ(N) ○ (T KεidE))(u) = g, +which means that T E is surjective. The injectivity of T E by Proposition 5.2.8, +implies that +SF(Ω) = (T E)−1 ○ (Sℓ(N) ○ (T KεidE)), +yielding the surjectivity of SF(Ω) and thus the ε-compatibility of F(Ω) and F(Ω,E). +Furthermore, we have T E = Sℓ(N) ○ (T KεidE) ○ S−1 +F(Ω), resulting in T E being an +isomorphism. +□ + +132 +5. APPLICATIONS +We note that one should not confuse the coefficient space ℓ(N) of the Schauder +series expansion of functions from F(Ω) in the theorem above with the space +ℓ1 = ℓ1(N) of absolutely summable sequences. +We remark again (see Theorem +5.6.1) that the index set of the equicontinuous Schauder basis of F(Ω) in Theorem +5.7.1 need not be N (or N0) but may be any other countable index set as long as +the equicontinuous Schauder basis is unconditional which is, for instance, always +fulfilled if F(Ω) is nuclear by [89, 21.10.1 Dynin-Mitiagin Theorem, p. 510]. +Theorem 5.7.1 (i) gives another proof of Theorem 5.6.13 b) and Theorem 5.6.14 +b). +Let us demonstrate an application of the preceding theorem which relates +the space of O(DR(0),E), 0 < R ≤ ∞, of holomorphic functions on DR(0) with +values in a complex locally complete lcHs E (see Theorem 5.6.11) and the Köthe +space λ∞(AR,E) with Köthe matrix AR ∶= (rk +j )k∈N0,j∈N for some strictly increasing +sequence (rj)j∈N in (0,R) converging to R (see Corollary 4.2.3), using the sequence +of Taylor coefficients of a holomorphic function. +5.7.2. Corollary. Let E be a locally complete lcHs over C, 0 < R ≤ ∞ and +define the Köthe matrix AR ∶= (rk +j )k∈N0,j∈N for some strictly increasing sequence +(rj)j∈N in (0,R) converging to R. Then λ∞(AR)εE ≅ λ∞(AR,E) and +λE∶O(DR(0),E) → λ∞(AR,E), λE(f) ∶= ((∂k +C)Ef(0) +k! +) +k∈N0, +is an isomorphism with λE = Sλ∞(AR) ○ (λCεidE) ○ S−1 +O(DR(0)). +Proof. By Proposition 4.2.17 and (23) the spaces O(DR(0)) and O(DR(0),E) +are ε-compatible. Moreover, λ∞(AR) and λ∞(AR,E) are ε-compatible by Corollary +4.2.3 as limk→∞( rj +rj+1 )k = 0 for any j ∈ N. Clearly, we have e′ ○ x ∈ λ∞(AR) for all +e′ ∈ E′ and x ∈ λ∞(AR,E). The space O(DR(0)) with the topology τc of compact +convergence is a nuclear Fréchet space and thus a Fréchet–Schwartz space. +In +particular, this space is barrelled and its Schauder basis of monomials (z ↦ zk)k∈N0 +is equicontinuous. The corresponding coefficient functionals are given by λC +k and +the map λC is an isomorphism by [131, Example 27.27, p. 341–342]. By the proof +of Theorem 5.6.11 the family (λE,λC) is consistent for (O,E) and its strength +follows from Proposition 5.2.32. Now, we can apply Theorem 5.7.1 (i), yielding our +statement. +□ +Let us present another application of Theorem 5.7.1 to the space OM(Rd,E) of +multipliers for the Schwartz space from Example 3.1.9 d). For simplicity we restrict +to the case d = 1. Fix a compactly supported test function ϕ ∈ C∞ +c (R) with ϕ(x) = 1 +for x ∈ [0, 1 +4] and ϕ(x) = 0 for x ≥ 1 +2. For f ∈ C∞(R,E) we set +fj(x) ∶= f(x + j) − +∞ +∑ +k=0 +akϕ(−2k(x − 1))f(−2k(x − 1 + j) + 1), x ∈ [0,1], j ∈ Z, +where +ak ∶= +∞ +∏ +j=0,j≠k +1 + 2j +2j − 2k , k ∈ N0. +Fixing x ∈ [0,1), we observe that fj(x) is well-defined for each j ∈ Z since there are +only finitely many summands due to the compact support of ϕ and −2k(x−1) → ∞ +for k → ∞. For x = 1 we have fj(1) = 0 for each j and the convergence of the series +in E follows from the uniform continuity of f on [0,1], f(0) = 0 and ∑∞ +k=0 ak = 1 by +the case n = 0 in [160, Lemma (iii), p. 625]. For each e′ ∈ E′ and j ∈ Z we note that +e′(fj(x)) = (e′ ○f)(x+j)− +∞ +∑ +k=0 +akϕ(−2k(x−1))(e′ ○f)(−2k(x−1+j)+1), x ∈ [0,1], + +5.7. REPRESENTATION BY SEQUENCE SPACES +133 +which implies that e′ ○fj ∈ E0 by [11, Proposition 3.2, p. 15]. Using the weak-strong +principle Corollary 5.2.24, we obtain that fj ∈ E0(E) for all j ∈ Z if E is locally +complete. Setting +ρ∶R → [0,1], ρ(x) ∶= 1 − cos(arctan(x)) = 1 − +1 +√ +1 + x2 , +we deduce from the proof and with the notation of [12, Proposition 2.2, p. 1494] +that e′○fj ○ρ = (Φ−1 +2 ○Φ1)(e′○fj) is an element of the Schwartz space S(R) for each +e′ ∈ E′. The weak-strong principle Corollary 5.2.21 c) yields that fj ○ ρ ∈ S(R,E) +if E is locally complete. Hence (fj ○ ρ) ⋅ h2n is Pettis-integrable on R for every +j ∈ Z and n ∈ N0 by Proposition 5.6.12 if E is locally complete where hn is the n-th +Hermite function. Therefore the Pettis-integral +bn,j(f) ∶= ⟨fj ○ ρ,h2n⟩L2 ∶= ∫ +R +fj(ρ(x))h2n(x)dx, j ∈ Z, n ∈ N0, +is a well-defined element of E by Proposition 5.6.12 if E is locally complete. By +[12, Theorem 2.1, p. 1496–1497] (cf. [172, Theorem 3, p. 478]) the map +ΦK∶OM(R) → s(N)′ +b̂⊗πs(N), ΦK(f) ∶= (bσ(n,j)(f))(n,j)∈N2, +is an isomorphism where σ∶N2 → N0 × Z is the enumeration given by σ(n,j) ∶= +(n − 1,(j − 1)/2) if j is odd, and σ(n,j) ∶= (n − 1,−j/2) if j is even. Here, we have +to interpret ΦK(f) as an element of s(N)′ +b̂⊗πs(N) by identification of isomorphic +spaces. Namely, +s(N)′ +b̂⊗πs(N) ≅ s(N)̂⊗πs(N)′ +b ≅ s(N)εs(N)′ +b ≅ s(N,s(N)′ +b) +holds where the first isomorphism is due to the commutativity of ̂⊗π, the second +due to the nuclearity of s(N) and the last due to Corollary 4.2.3 b) via Ss(N). Then +we interpret ΦK(f) as an element of s(N,s(N)′ +b) by means of +j ∈ N �→ [a ∈ s(N) ↦ ∑ +n∈N +anbσ(n,j)] +(see also (62) below). +5.7.3. Corollary. If E is a sequentially complete lcHs, then the map +ΦE∶OM(R,E) → s(N,Lb(s(N),E)), ΦE(f) ∶= (bσ(n,j)(f))(n,j)∈N2, +is an isomorphism where we interpret ΦE(f) as an element of s(N,Lb(s(N),E)). +Proof. The spaces OM(R) and OM(R,E) are ε-compatible by Corollary +3.2.10 with the inverse of SOM(R) given by the map Rt∶OM(R,E) → OM(R)εE, +f ↦ J −1 ○ Rt +f, according to Theorem 3.2.4. The barrelled nuclear space OM(R) +has the equicontinuous unconditional Schauder basis (ψσ(n,j))(n,j)∈N2 with asso- +ciated coefficient functionals δn,j ○ ΦK = bσ(n,j) given in [12, Proposition 3.2, p. +1499]. Next, we show that (ΦE,ΦK) is a strong, consistent family for (OM,E). Let +f ∈ OM(R,E). For each e′ ∈ E′ and (n,j) ∈ N2 we have +δn,j ○ ΦK(e′ ○ f) = bσ(n,j)(e′ ○ f) = ∫ +R +(e′ ○ f)(j−1)/2(ρ(x))h2(n−1)(x)dx += ⟨e′,∫ +R +f(j−1)/2(ρ(x))h2(n−1)(x)dx⟩ = ⟨e′,δn,j ○ ΦE(f)⟩ += e′(bσ(n,j)(f)) +if j is odd since (f(j−1)/2 ○ ρ) ⋅ h2(n−1) is Pettis-integrable on R. The analogous +result holds for even j as well. This implies the strength of the family. Due to + +134 +5. APPLICATIONS +Proposition 4.3.3 with (T E +0 ,T K +0 ) given by T E +0 (f) ∶= (fj ○ ρ)h2n, f ∈ OM(R,E), and +T K +0 (f) ∶= (fj ○ ρ)h2n, f ∈ OM(R), the family (ΦE,ΦK) is consistent. +In order to apply Theorem 5.7.1 we need spaces ℓV(N2) and ℓV(N2,E) of +sequences with values in K and E, respectively. +In addition, the space ℓV(N2) +has to be isomorphic to s(N,s(N)′ +b) so that ΦK∶OM(R) → s(N,s(N)′ +b) ≅ ℓV(N2) +becomes the isomorphism we need for Theorem 5.7.1. We set +ℓV(N2,E) ∶= {x = (xn,j) ∈ EN2 ∣ ∀ k ∈ N, B ⊂ s(N) bounded, α ∈ A ∶ ∥x∥k,B,α < ∞} +where +∥x∥k,B,α ∶= +sup +(j,a)∈ωB +pα(T E(x)(j,a))νk,B(j,a) +with ωB ∶= N × B and νk,B∶ωB → [0,∞), νk,B(j,a) ∶= (1 + j2)k/2, and +T E(x)(j,a) ∶= ∑ +n∈N +anxn,j. +We claim that the map +T E∶ℓV(N2,E) → s(N,Lb(s(N),E)), x ↦ (T E(x)(j,⋅))j∈N, +(62) +is an isomorphism. We remark for each k ∈ N, bounded B ⊂ s(N) and α ∈ A that +∣T E(x)∣s(N),k,(B,α) = sup +j∈N +sup +a∈B +pα(T E(x)(j,a))(1 + j2)k/2 = ∥x∥k,B,α +for all x ∈ ℓV(N2,E), implying that T E is an isomorphism into. Let y ∶= (yj) ∈ +s(N,Lb(s(N),E)). Then yj ∈ Lb(s(N),E) for j ∈ N and we set xn,j ∶= yj(en) for n ∈ +N where en is the n-th unit sequence in s(N). We note that with x ∶= (xn,j)(n,j)∈N2 +T E(x)(j,a) = ∑ +n∈N +anxn,j = ∑ +n∈N +anyj(en) = yj(∑ +n∈N +anen) = yj(a) +holds for all j ∈ N and a ∶= (an) ∈ s(N) since (en) is a Schauder basis of s(N) with +associated coefficient functionals a ↦ an. It follows that x ∈ ℓV(N2,E) and the +surjectivity of T E. +The next step is to prove that ℓV(N2) and ℓV(N2,E) are ε-into-compatible. Due +to Theorem 3.1.12 we only need to show that (T E,T K) is a consistent generator +for (ℓV,E). Let u ∈ ℓV(N2)εE. Then +m +∑ +n=1 +anSℓV(N2)(u)(j,n) = +m +∑ +n=1 +anu(δj,n) = u( +m +∑ +n=1 +anδj,n) +(63) +for all m ∈ N and a ∶= (an) ∈ s(N). Since +( +m +∑ +n=1 +anδj,n)(x) = +m +∑ +n=1 +anxj,n → T K(x)(j,a) = T K +(j,a)(x), +m → ∞, +for all x ∈ ℓV(N2), we deduce that (∑m +n=1 anδj,n)m converges to T K +(j,a)(x) in ℓV(N2)′ +κ +by the Banach–Steinhaus theorem, which is applicable as ℓV(N2) ≅ s(N,s(N)′ +b) ≅ +OM(R) is barrelled. We conclude that +u(T K +(j,a)) = lim +m→∞u( +m +∑ +n=1 +anδj,n) = +(63) +∞ +∑ +n=1 +anSℓV(N2)(u)(j,n) = T ESℓV(N2)(u)(j,a) +and thus the consistency of (T E,T K) for (ℓV,E). +Furthermore, we clearly have e′ ○ x ∈ ℓV(N2) for all x ∈ ℓV(N2,E) and the map +Φ∶OM(R) → s(N)′ +b̂⊗πs(N) ≅ ℓV(N2) is an isomorphism by [12, Theorem 2.1, p. +1496–1497] and (62). Due to [83, Chap. II, §4, n○4, Théorème 16, p. 131] the dual +OM(R)′ +b is an LF-space and thus OM(R) ≅ (OM(R)′ +b)′ +b is the strong dual of an +LF-space by reflexivity and therefore webbed by [94, Satz 7.25, p. 165]. Finally, we +can apply Theorem 5.7.1 (ii), yielding our statement. +□ + +5.7. REPRESENTATION BY SEQUENCE SPACES +135 +5.7.4. Remark. The actual isomorphism in Corollary 5.7.3 (without the inter- +pretation) is given by ̃ΦE ∶= T E ○ ΦE with T E from (62) and we have +̃ΦE = T E ○ ΦE = T E ○ SℓV(N2) ○ (ΦKεidE) ○ S−1 +OM(R). +Furthermore, Corollary 5.7.3 is valid for locally complete E as well. Indeed, similar +to Example 4.2.2 we may show that ℓV(N2,E) ≅ ℓV(N2)εE for locally complete E. +In combination with Corollary 3.2.10 and Theorem 5.1.2 b) this proves Corollary +5.7.3 for locally complete E as in Theorem 5.6.13 b). + + +Appendices +137 + + +APPENDIX A +Compactness of closed absolutely convex hulls and +Pettis-integrals +A.1. Compactness of closed absolutely convex hulls +In this section of the appendix we treat the question for which functions f∶Ω → +E, subsets K ⊂ Ω and lcHs E sets like acx(f(K)) are compact or sets like +Nj,m(f) ∶= {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm}, +j ∈ J, m ∈ M, +for f ∈ FV(Ω,E) are contained in an absolutely convex compact set. This is useful +in connection with ε-compatibility due to Corollary 3.2.5 (iv) and also relevant +in connection with the Pettis-integrability of a vector-valued function due to the +Mackey–Arens theorem. +We recall that the space of càdlàg functions on a set Ω ⊂ R with values in an +lcHs E is defined by +D(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ Ω ∶ +lim +w→x+f(w) = f(x) and f(x−) ∶= lim +w→x−f(w) exists}.4 +A.1.1. Proposition. Let Ω ⊂ R, K ⊂ Ω be compact and E an lcHs. Then f(K) +is precompact for every f ∈ D(Ω,E). If E is quasi-complete, then acx(f(K)) is +compact. +Proof. Let f ∈ D(Ω,E), α ∈ A and ε > 0. We recall and define +Br(x) = {w ∈ R ∣ ∣w − x∣ < r} +and +Bε,α(y) ∶= {w ∈ E ∣ pα(w − y) < ε} +for every x ∈ Ω, y ∈ E and r > 0. Let x ∈ Ω. Then there is rx− > 0 such that +pα(f(w) − f(x−)) < ε for all w ∈ Brx−(x) ∩ (−∞,x) ∩ Ω if x is an accumulation +point of (−∞,x] ∩ Ω. Further, there is rx+ > 0 such that pα(f(w) − f(x)) < ε for +all w ∈ Brx+(x) ∩ [x,∞) ∩ Ω if x is an accumulation point of [x,∞) ∩ Ω. If x is an +accumulation point of (−∞,x]∩Ω and [x,∞)∩Ω, we choose rx ∶= min(rx−,rx+). If x +is an accumulation point of (−∞,x]∩Ω but not of [x,∞)∩Ω, we choose rx ∶= rx−. If +x is an accumulation point of [x,∞)∩Ω but not of (−∞,x]∩Ω, we choose rx ∶= rx+. +If x is neither an accumulation point of (−∞,x] ∩ Ω nor of [x,∞) ∩ Ω, then there is +rx > 0 such that Brx(x) ∩ Ω = {x}. +Setting Vx ∶= Brx(x)∩Ω, we note that the sets Vx are open in Ω with respect to +the topology induced by R and K ⊂ ⋃x∈K Vx. Since K is compact, there are n ∈ N +and x1,...,xn ∈ K such that K ⊂ ⋃n +i=1 Vxi. W.l.o.g. each xi is an accumulation point +of (−∞,xi]∩Ω and [xi,∞)∩Ω. Then we have f(w) ∈ (Bε,α(f(xi−))∪Bε,α(f(xi))) +for all w ∈ Vxi and get +f(K) ⊂ +n +⋃ +i=1 +f(Vxi) ⊂ +n +⋃ +i=1 +(Bε,α(f(xi−)) ∪ Bε,α(f(xi))), +which means that f(K) is precompact. +4We recall that for x ∈ Ω we only demand limw→x+ f(w) = f(x) if x is an accumulation +point of [x, ∞) ∩ Ω, and the existence of the limit limw→x− f(w) if x is an accumulation point of +(−∞, x] ∩ Ω. +139 + +140 +A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS +If E is quasi-complete, then the precompact set f(K) is relatively compact +by [89, 3.5.3 Proposition, p. 65]. Hence acx(f(K)) is compact as quasi-complete +spaces have ccp. +□ +For f ∈ D(Ω,E) we define the jump function ∆∗f(x) ∶= f(x) − f(x−), x ∈ Ω, +where we set f(x−) ∶= 0 if x is not an accumulation point of (−∞,x] ∩ Ω. +A.1.2. Proposition. Let Ω ⊂ R, K ⊂ Ω be compact and E an lcHs. Then +∆∗f(K) is precompact for every f ∈ D(Ω,E). If E is quasi-complete, then the set +acx(∆∗f(K)) is compact. +Proof. If K is a finite set, then ∆∗f(K) is finite, thus compact, and we are +done. So let us assume that K is not finite. Let α ∈ A and ε > 0. We define +∆ε,α ∶= {x ∈ K ∣ pα(∆∗f(x)) ≥ ε} and claim that ∆α,ε is a finite set. +Let us +assume the contrary. Then there is an infinite sequence (xn) in ∆ε,α ⊂ K. Due +to the compactness of K there is a subsequence of (xn) which converges to some +x ∈ K. W.l.o.g. this subsequence is strictly increasing and we call this subsequence +again (xn). Since f has left limits (in left-accumulation points), for every n ∈ N, +n ≥ 2, there is wn ∈ (xn−1,xn) such that pα(f(xn−) − f(wn)) ≤ ε/2 (if xn is not an +accumulation point of (−∞,xn] ∩ Ω, then there is wn ∈ (xn−1,xn) with wn ∉ Ω and +we set f(wn) ∶= 0). Hence we have +pα(f(xn) − f(wn)) ≥ pα(f(xn) − f(xn−)) − pα(f(xn−) − f(wn)) += pα(∆∗f(xn)) − pα(f(xn−) − f(wn)) ≥ ε/2 +for all n ≥ 2. But this is a contradiction because +lim +n→∞f(xn) = lim +n→∞f(wn) = f(x−), +which proves our claim. +Next, we note that +∆∗f(K) ⊂ (Bε,α(0) ∪ ∆∗f(∆ε,α)) ⊂ +⋃ +z∈{0}∪∆∗f(∆ε,α) +z + Bε,α(0), +which implies that ∆∗f(K) is precompact as {0} ∪ ∆∗f(∆ε,α) is finite. +If E is quasi-complete, then the precompact set ∆∗f(K) is relatively compact +by [89, 3.5.3 Proposition, p. 65]. Hence acx(∆∗f(K)) is compact as quasi-complete +spaces have ccp. +□ +Proposition A.1.1 and Proposition A.1.2 are known in the case that Ω = [0,1] +and E = K (see the comments after [19, Chap. 3, Sect. 14, Lemma 1, p. 110]) since +precompactness is equivalent to boundedness if E = K. +A.1.3. Proposition. Let Ω be a locally compact topological Hausdorff space +and f ∈ C0(Ω,E). If +(i) E is an lcHs with ccp, or +(ii) E is an lcHs with metric ccp and Ω second-countable, +then acx(f(Ω)) is compact. +Proof. Let Ω be compact, then f(Ω) is compact in E as f is continuous. If +Ω is even second-countable, then Ω is metrisable by [58, Chap. XI, 4.1 Theorem, p. +233] and thus f(Ω) as well by [34, Chap. IX, §2.10, Proposition 17, p. 159]. This +yields that acx(f(Ω)) is compact in both cases. +Let Ω be non-compact and Ω∗ denote the one-point compactification of Ω. +Since f ∈ C0(Ω,E), it has a unique continuous extension ̂f to Ω∗ with ̂f(∞) = 0. +Hence K ∶= ̂f(Ω∗) is a compact set in E as Ω∗ is compact and ̂f continuous. If +Ω is even second-countable, then Ω∗ is metrisable by [58, Chap. XI, 8.6 Theorem, + +A.1. COMPACTNESS OF CLOSED ABSOLUTELY CONVEX HULLS +141 +p. 247] and thus K as well by [34, Chap. IX, §2.10, Proposition 17, p. 159]. This +yields that acx(K) is compact in both cases and thus the closed subset acx(f(Ω)), +too. +□ +We note that C0(Ω,E) = C(Ω,E) if Ω is compact. For our next proposition we +define the space of bounded γ-Hölder continuous functions, 0 < γ ≤ 1, from a metric +space (Ω,d) to an lcHs E by +C[γ] +b +(Ω,E) ∶= {f ∈ EΩ ∣ ∀α ∈ A ∶ sup +x∈Ω +pα(f(x)) < ∞ and sup +x,y∈Ω +x≠y +pα(f(x) − f(y)) +d(x,y)γ +< ∞}. +A.1.4. Proposition. Let (Ω,d) be a metric space, E a locally complete lcHs +and f ∈ C[γ] +b +(Ω,E) for some 0 < γ ≤ 1. If there is h∶Ω → (0,∞) such that fh is +bounded on Ω and with N ∶= {x ∈ Ω ∣ f(x) = 0} it holds that +∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ (K ∪ N) ∶ 1 ≤ εh(x), +then acx(f(Ω)) is compact. +Proof. Since f ∈ C[γ] +b +(Ω,E), the sets f(Ω) and +B1 ∶= {f(z) − f(t) +d(z,t)γ +∣ z,t ∈ Ω, z ≠ t} +are bounded in E. Further, the range (fh)(Ω) is bounded in E by assumption. +Thus B ∶= acx(B1 ∪ f(Ω) ∪ (fh)(Ω)) is a closed disk and EB a Banach space with +the norm ∥x∥B ∶= inf{r > 0 ∣ x ∈ rB}, x ∈ EB, as E is locally complete. Next, +we show that f(Ω) is precompact in EB. Let V be a zero neighbourhood in EB. +Then there is ε > 0 such that Uε ∶= {x ∈ EB ∣ ∥x∥B ≤ ε} ⊂ V . Moreover, there is a +compact set K ⊂ Ω such that 1 ≤ εh(x) for all x ∈ Ω∖(K ∪N). The map f∶Ω → EB +is well-defined and uniformly continuous because ∥f(z) − f(t)∥B ≤ d(z,t)γ for all +z,t ∈ Ω, which follows from B1 ⊂ B. We deduce that f(K) is compact in EB. We +note that +f(x) = f(x)h(x) +1 +h(x), +x ∈ Ω ∖ N, +which implies that ∥f(x)∥B ≤ +1 +h(x) as (fh)(Ω) ⊂ B. Hence we have +∥f(x)∥B ≤ +1 +h(x) ≤ ε, +x ∈ Ω ∖ (K ∪ N), +and the estimate 0 = ∥f(x)∥B ≤ ε is still valid for x ∈ N, yielding f(Ω ∖ K) ⊂ Uε. +Since f(K) is compact in EB, it is also precompact and so there is a finite set +P ⊂ EB such that f(K) ⊂ P + V . We derive that +f(Ω) = (f(K) ∪ f(Ω ∖ K)) ⊂ ((P + V ) ∪ Uε) ⊂ ((P ∪ {0}) + V ), +which means that f(Ω) is precompact in EB and thus acx(f(Ω)) as well by [89, +6.7.1 Proposition, p. 112]. Therefore the set acx(f(Ω)) is compact in the Banach +space EB and also compact in the weaker topology of E. +□ +The underlying idea of Proposition A.1.4 is taken from [29, Lemma 1, Propo- +sition 2, p. 354]. +A.1.5. Proposition. Let Ω ⊂ Rd be an open convex set, E an lcHs over K and +f∶Ω → E weakly C1 +b , i.e. e′ ○ f ∈ C1 +b (Ω) for each e′ ∈ E′. Then f ∈ C[1] +b (Ω,E). +Proof. Let z,t ∈ Ω, z ≠ t. By the mean value theorem we have +∣(e′ ○ f)(z) − (e′ ○ f)(t)∣ +∣z − t∣ +≤ Cd max +1≤n≤dsup +x∈Ω +∣(∂en)K(e′ ○ f)(x)∣ ≤ Cd∣e′ ○ f∣C1 +b (Ω) < ∞ + +142 +A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS +for all e′ ∈ E′ where Cd ∶= +√ +d if K = R, and Cd ∶= 2 +√ +d if K = C. It follows from +[131, Mackey’s theorem 23.15, p. 268] that f is Lipschitz continuous and bounded +as well, thus f ∈ C[1] +b (Ω,E). +□ +A.1.6. Proposition. Let FV(Ω,E) be a dom-space, let there be a set X, a +family K of sets and a map π∶⋃m∈M ωm → X such that ⋃K∈K K ⊂ X. +If f ∈ +FV(Ω,E) fulfils +∀ ε > 0, j ∈ J, m ∈ M, α ∈ A ∃ K ∈ K ∶ +(i) +sup +x∈ωm, +π(x)∉K +pα(T E +m(f)(x))νj,m(x) < ε, +(ii) Nπ⊂K,j,m(f) ∶= {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm, π(x) ∈ K} is precompact in E, +then the set Nj,m(f) is precompact in E for every j ∈ J and m ∈ M. +If E is +quasi-complete, then acx(Nj,m(f)) is compact. +Proof. Let V be a zero neighbourhood in E. Then there are α ∈ A and ε > 0 +such that Bε,α ⊂ V where Bε,α ∶= {x ∈ E ∣ pα(x) < ε}. Let j ∈ J and m ∈ M. Due to +(i) there is K ∈ K such that the set +Nπ⊄K,j,m(f) ∶= {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm, π(x) ∉ K} +is contained in Bε,α. Further, the precompactness of Nπ⊂K,j,m(f) by (ii) implies +that there exists a finite set P ⊂ E such that Nπ⊂K,j,m(f) ⊂ P + V . Hence we +conclude +Nj,m(f) = (Nπ⊄K,j,m(f) ∪ Nπ⊂K,j,m(f)) +⊂ (Bε,α ∪ (P + V )) ⊂ (V ∪ (P + V )) = (P ∪ {0}) + V, +which means that Nj,m(f) is precompact. +The second part of the statement follows from the fact that a precompact set +in a quasi-complete space is relatively compact by [89, 3.5.3 Proposition, p. 65] and +that quasi-complete spaces have ccp. +□ +The most common case is that K consists of the compact subsets of Ω and π +is a projection on X ∶= Ω (see e.g. Example 4.2.11, Example 4.2.16 and Example +4.2.22). +A.2. The Pettis-integral +We start with the definition of the Pettis-integral which we use to define Fourier +transformations of vector-valued functions (see Proposition 4.2.25, Theorem 5.6.13 +and Theorem 5.6.14) and for Riesz–Markov–Kakutani theorems in Section 4.3. +Let Σ be a σ-algebra on a set X. +A function µ∶Σ → K is called K-valued +measure if µ(∅) = 0 and µ is countably additive, i.e. for any sequence (An)n∈N of +pairwise disjoint sets in Σ it holds that +µ( ⋃ +n∈N +An) = ∑ +n∈N +µ(An) ∈ K. +If K = R, µ is also called a signed measure, and if K = C a complex measure. If K is +replaced by [0,∞], we say that µ is a positive measure. For a K-valued measure µ +its total variation ∣µ∣ given by +∣µ∣(A) ∶= sup{∑ +n∈N +∣µ(An)∣ ∣ An ∈ Σ,Am ∩ An = ∅ if m ≠ n,A = ⋃ +n∈N +An}, +A ∈ Σ, + +A.2. THE PETTIS-INTEGRAL +143 +is a well-defined positive measure by [149, 6.2 Theorem, p. 117] and it is finite by +[149, 6.4 Theorem, p. 118], i.e. ∣µ∣(X) < ∞. Obviously, a K-valued measure µ is +positive if and only if ∣µ∣ = µ. For a positive measure µ on X and 1 ≤ p < ∞ let +Lp(X,µ) ∶= {f∶X → K measurable ∣ qp(f) ∶= ∫ +X +∣f(x)∣pdµ(x) < ∞} +and define the quotient space of p-integrable functions by Lp(X,µ) ∶= Lp(X,µ)/{f ∈ +Lp(X,µ) ∣ qp(f) = 0}, which becomes a Banach space if it is equipped with the norm +∥f∥p ∶= ∥f∥Lp ∶= qp(F)1/p, f = [F] ∈ Lp(X,µ). From now on we do not distinguish +between equivalence classes and their representatives anymore. +For a K-valued measure µ there is a unique h ∈ L1(X,∣µ∣) with dµ = hd∣µ∣ by +the Radon–Nikodým theorem (see [149, 6.12 Theorem, p. 124]) and h can be chosen +such that ∣h∣ = 1, i.e. has a representative with modulus equal to 1. Now, we say +that f ∈ Lp(X,µ) if f ⋅ h ∈ Lp(X,∣µ∣). For f ∈ L1(X,µ) we define the integral of f +on X w.r.t. µ by +∫ +X +f(x)dµ(x) ∶= ∫ +X +f(x)h(x)d∣µ∣(x). +For a measure space (X,Σ,µ) and f∶X → K we say that f is integrable on +Λ ∈ Σ and write f ∈ L1(Λ,µ) if χΛf ∈ L1(X,µ). Then we set +∫ +Λ +f(x)dµ(x) ∶= ∫ +X +χΛ(x)f(x)dµ(x). +A.2.1. Definition (Pettis-integral). Let (X,Σ,µ) be a measure space and E +an lcHs. A function f∶X → E is called weakly measurable if the function e′ ○f∶X → +K, (e′ ○ f)(x) ∶= ⟨e′,f(x)⟩ ∶= e′(f(x)), is measurable for all e′ ∈ E′. +A weakly +measurable function is said to be weakly integrable if e′ ○ f ∈ L1(X,µ). A function +f∶X → E is called Pettis-integrable on Λ ∈ Σ if it is weakly integrable on Λ and +∃ eΛ ∈ E ∀e′ ∈ E′ ∶ ⟨e′,eΛ⟩ = ∫ +Λ +⟨e′,f(x)⟩dµ(x). +In this case eΛ is unique due to E being Hausdorff and we set the Pettis-integral +∫ +Λ +f(x)dµ(x) ∶= eΛ. +If we consider the measure space (X,L (X),λ) of Lebesgue measurable sets +for X ⊂ Rd, we just write dx ∶= dλ(x). +A.2.2. Lemma. Let E be a locally complete lcHs, Ω ⊂ Rd open and f∶Ω → E. +If f is weakly C1, i.e. e′ ○ f ∈ C1(Ω) for every e′ ∈ E′, then f is Pettis-integrable on +every compact subset K ⊂ Ω with respect to any locally finite positive measure µ on +Ω and +pα(∫ +K +f(x)dµ(x)) ≤ µ(K)sup +x∈K +pα(f(x)), +α ∈ A. +Proof. Let K ⊂ Ω be compact and (Ω,Σ,µ) a measure space with locally +finite measure µ, i.e. Σ contains the Borel σ-algebra B(Ω) on Ω and for every +x ∈ Ω there is a neighbourhood Ux ⊂ Ω of x such that µ(Ux) < ∞. Since the map +e′ ○ f is differentiable for every e′ ∈ E′, thus Borel-measurable, and B(Ω) ⊂ Σ, it is +measurable. We deduce that e′ ○f ∈ L1(K,µ) for every e′ ∈ E′ because locally finite +measures are finite on compact sets. Therefore the map +I∶E′ → K, I(e′) ∶= ∫ +K +⟨e′,f(x)⟩dµ(x) + +144 +A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS +is well-defined and linear. We estimate +∣I(e′)∣ ≤ ∣µ(K)∣ sup +x∈f(K) +∣e′(x)∣ ≤ µ(K) +sup +x∈acx(f(K)) +∣e′(x)∣, +e′ ∈ E′. +Due to f being weakly C1 and [29, Proposition 2, p. 354] the absolutely convex set +acx(f(K)) is compact, yielding I ∈ (E′ +κ)′ ≅ E by the theorem of Mackey–Arens, +which means that there is eK ∈ E such that +⟨e′,eK⟩ = I(e′) = ∫ +K +⟨e′,f(x)⟩dµ(x), +e′ ∈ E′. +Hence f is Pettis-integrable on K w.r.t. µ. For α ∈ A we set Bα ∶= {x ∈ E ∣ pα(x) < 1} +and observe that +pα(∫ +K +f(x)dµ(x)) = sup +e′∈B○α +∣⟨e′,∫ +K +f(x)dµ(x)⟩∣ = sup +e′∈B○α +∣∫ +K +e′(f(x))dµ(x)∣ +≤ µ(K) sup +e′∈B○α +sup +x∈K +∣e′(f(x))∣ = µ(K)sup +x∈K +pα(f(x)) +where we used [131, Proposition 22.14, p. 256] in the first and last equation to get +from pα to supe′∈B○α and back. +□ +A.2.3. Lemma. Let E be a sequentially complete lcHs, Ω ⊂ Rd open, (Ω,Σ,µ) +a measure space with locally finite positive measure µ and f∶Ω → E. If f is weakly +C1 and there are ψ ∈ L1(Ω,µ) and g∶Ω → [0,∞) measurable such that ψg ≥ 1 and +fg is bounded on Ω, then f is Pettis-integrable on Ω and +pα(∫ +Ω +f(x)dµ(x)) ≤ ∥ψ∥1 sup +x∈Ω +pα(f(x)g(x)), +α ∈ A. +Proof. Let (Kn)n∈N be a compact exhaustion of Ω. Due to Lemma A.2.2 the +Pettis-integral +en ∶= ∫ +Kn +f(x)dµ(x) +is a well-defined element of E for every n ∈ N. +Next, we show that (en) is a +Cauchy sequence in E. +Let α ∈ A, m ∈ N0 and k,n ∈ N with k > n. +We set +Bα ∶= {x ∈ E ∣ pα(x) < 1} and Qk,n ∶= Kk ∖ Kn and note that +pα(ek − en) = sup +e′∈B○α +∣e′(ek − en)∣ = sup +e′∈B○α +∣ ∫ +Qk,n +e′(f(x))dµ(x)∣ +≤ ∫ +Qk,n +∣ψ(x)∣dµ(x) sup +e′∈B○α +sup +x∈Ω +∣e′(f(x)g(x))∣ += ∫ +Qk,n +∣ψ(x)∣dµ(x)sup +x∈Ω +pα(f(x)g(x)) +(64) +where we used [131, Proposition 22.14, p. 256] to switch from pα to supe′∈B○α and +back. Since ψ ∈ L1(Ω,µ), we have that (en) is a Cauchy sequence in the sequentially +complete space E. Thus eΩ ∶= limn→∞ en exists in E and the dominated convergence +theorem implies +e′(eΩ) = lim +n→∞e′(en) = lim +n→∞∫ +Kn +e′(f(x))dµ(x) = ∫ +Ω +e′(f(x))dµ(x), +e′ ∈ E′. +Hence f is Pettis-integrable on Ω with ∫Ω f(x)dµ(x) = eΩ. As in (64) we have +pα(en) ≤ ∫ +Kn +∣ψ(x)∣dµ(x)sup +x∈Ω +pα(f(x)g(x)) ≤ ∥ψ∥1 sup +x∈Ω +pα(f(x)g(x)) + +A.2. THE PETTIS-INTEGRAL +145 +for every n ∈ N. Letting n → ∞, we derive the estimate in our statement. +□ +A.2.4. Remark. Let µ be a K-valued measure and Σ contain B(Ω). +Then +Lemma A.2.2 is still valid with µ(K) replaced by ∣µ∣(K) due to the definition of +the integral w.r.t. a K-valued measure and as ∣µ∣(K) ≤ ∣µ∣(Ω) < ∞. Thus Lemma +A.2.3 holds in this case as well. +The following definition is analogous to the definition of the Pettis-integral. +A.2.5. Definition (Pettis-summable). Let I be a non-empty set and E an +lcHs. A family (fi)i∈I in E is called weakly summable if (⟨e′,fi⟩)i∈I ∈ ℓ1(I,K) for +all e′ ∈ E′. A family (fi)i∈I in E is called Pettis-summable if it is weakly summable +and +∃ eI ∈ E ∀e′ ∈ E′ ∶ ⟨e′,eI⟩ = ∑ +i∈I +⟨e′,fi⟩. +In this case eI is unique due to E being Hausdorff and we set +∑ +i∈I +fi ∶= eI. +For the elements f of the space D([0,1],E) of E-valued càdlàg functions on +[0,1] and their jump functions ∆∗f we have the following result. +A.2.6. Proposition. Let E be a quasi-complete lcHs, µ a K-valued Borel mea- +sure on [0,1] and ψ ∈ ℓ1([0,1],K). Then f ∈ D([0,1],E) is Pettis-integrable on +[0,1] and +pα( ∫ +[0,1] +f(x)dµ(x)) ≤ ∣µ∣([0,1]) sup +x∈[0,1] +pα(f(x)), +α ∈ A, +and (∆∗f)ψ is Pettis-summable on [0,1] and +pα( ∑ +x∈[0,1] +(∆∗f)(x)ψ(x)) ≤ ∥ψ∥ℓ1 sup +x∈[0,1] +pα(∆∗f(x)), +α ∈ A. +Proof. By [19, Chap. 3, Sect. 14, Lemma 1, p. 110] e′ ○ f is Borel measurable +for every e′ ∈ E′ and integrable due to its boundedness on [0,1]. Thus the map +I∶E′ → K, I(e′) ∶= ∫ +[0,1] +e′(f(x))dµ(x), +is well-defined and linear. It follows from Proposition A.1.1 that acx(f([0,1])) is +absolutely convex and compact in E. In combination with the estimate +∣I(e′)∣ ≤ ∣µ∣([0,1]) +sup +x∈f([0,1]) +∣e′(x)∣ ≤ ∣µ∣([0,1]) +sup +x∈acx(f([0,1])) +∣e′(x)∣ +for every e′ ∈ E′ we deduce that I ∈ (E′ +κ)′ ≅ E by the theorem of Mackey–Arens, +which implies that there is e[0,1] ∈ E such that +⟨e′,e[0,1]⟩ = I(e′) = ∫ +[0,1] +e′(f(x))dµ(x), +e′ ∈ E′. +Thus f is Pettis-integrable on [0,1]. +Since ψ ∈ ℓ1([0,1]) and e′ ○ ∆∗f bounded on [0,1] for every e′ ∈ E′, the map +I0∶E′ → K, I0(e′) ∶= +∑ +x∈[0,1] +e′((∆∗f)(x)ψ(x)), +is well-defined and linear. Moreover, the set acx(∆∗f([0,1])) is absolutely convex +and compact by Proposition A.1.2. Again, the estimate +∣I0(e′)∣ ≤ +∑ +x∈[0,1] +∣ψ(x)∣ +sup +x∈∆∗f([0,1]) +∣e′(x)∣ ≤ ∥ψ∥ℓ1 +sup +x∈acx(∆∗f([0,1])) +∣e′(x)∣ + +146 +A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS +for every e′ ∈ E′, implies our statement. +The remaining estimates are deduced +analogously to Lemma A.2.2. +□ +A.2.7. Proposition. Let E be an lcHs, Ω a topological Hausdorff space and +(Ω,Σ,µ) a measure space. +If f∶Ω → E is weakly integrable and there are ψ ∈ +L1(Ω,µ) and g∶Ω → [0,∞) measurable such that ψg ≥ 1 and +(i) E has ccp, Ω is locally compact and fg ∈ C0(Ω,E), or +(ii) E has metric ccp, Ω is locally compact and second-countable, and fg ∈ +C0(Ω,E), or +(iii) E is locally complete, Ω a metric space, fg ∈ C[γ] +b +(Ω,E) for some 0 < γ ≤ 1 +and there is h∶Ω → (0,∞) such that fgh is bounded on Ω and with N ∶= +{x ∈ Ω ∣ f(x)g(x) = 0} it holds that +∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ (K ∪ N) ∶ 1 ≤ εh(x), +then f is Pettis-integrable on Ω and +pα(∫ +Ω +f(x)dµ(x)) ≤ ∥ψ∥1 sup +x∈Ω +pα(f(x)g(x)), +α ∈ A. +Proof. Since f is weakly integrable, the map +I∶E′ → K, I(e′) ∶= ∫ +Ω +e′(f(x))dµ(x), +is well-defined and linear. It follows from Proposition A.1.3 in case (i)-(ii) and from +Proposition A.1.4 in case (iii) that acx(fg(Ω)) is absolutely convex and compact +in E. If µ is a positive measure, i.e. [0,∞]-valued, we observe that +∣I(e′)∣ ≤ ∫ +Ω +∣ψ(x)∣dµ(x) +sup +x∈fg(Ω) +∣e′(x)∣ ≤ ∥ψ∥1 +sup +x∈acx(fg(Ω)) +∣e′(x)∣ +for every e′ ∈ E′. If µ is a K-valued measure, then the same estimate holds with µ +replaced by ∣µ∣. We deduce from this estimate that I ∈ (E′ +κ)′ ≅ E by the theorem +of Mackey–Arens, which implies that there is eΩ ∈ E such that +⟨e′,eΩ⟩ = I(e′) = ∫ +Ω +e′(f(x))dµ(x), +e′ ∈ E′. +Thus f is Pettis-integrable on Ω. The remaining estimate is deduced analogously +to Lemma A.2.2. +□ +The idea how to prove Proposition A.2.7 (ii) for Ω = Rd is due to an anonymous +reviewer of [110] but did not make it into [110] because of page limits. + +List of Symbols +Sets and systems of sets +acx(M) +absolutely convex hull of the set M 17 +acx(M) +closure of the absolutely convex hull of the set M 17 +B○F (Ω)′ +Fν(Ω) +the polar {y′ ∈ F(Ω)′ ∣ ∀ f ∈ BFν(Ω) ∶ ∣y′(f)∣ ≤ 1} 84 +Br(x) +ball {w ∈ Rd ∣ ∣w − x∣ < r} around x ∈ Rd with radius r > 0 17 +ch(M) +circled hull of the set M 17 +cx(M) +convex hull of the set M 17 +Dr(z) +disc {w ∈ C ∣ ∣w − z∣ < r} around z ∈ C with radius r > 0 17 +D +open unit disc D1(0) 19 +G○ +the polar set of G 17 +Mm +the set {β ∈ Nd +0 ∣ ∣β∣ ≤ min(m,k)} for m ∈ N0 and k ∈ N∞ 24 +M +closure of the set M 17 +M +t +closure of the set M w.r.t. the topology t 17 +M +X +closure of the set M in the topological space X 17 +∂M +boundary of the set M 17 +N∞ +the set N ∪ {∞} 19 +σ(E,G) +weak topology induced on E by a separating subspace G ⊂ E′ 78 +τc +topology of compact convergence 21 +Locally convex Hausdorff spaces & spaces of continuous linear operators +E′⋆ +algebraic dual of the dual E′ 29 +E +locally convex Hausdorff space 17 +ED +space ⋃n∈N nD for a disk D ⊂ E 18 +F ′ +dual space of F 17 +t(F ′,F) +topology on F ′ where t = b, γ, κ, σ or τ 17 +t(F) +bornology on F that induces the topology t(F ′,F) 17 +F ≅ E +locally convex Hausdorff spaces F and E are isomorphic 17 +FεE +space Le(F ′ +κ,E) where L(F ′ +κ,E) is equipped with the topology of +uniform convergence on the equicontinuous subsets of F ′ 17 +F ⊗ E +tensor product of F and E 18 +F ⊗ε E +F ⊗ E equipped with the topology induced by FεE +18 +F ̂⊗εE +completion of F ⊗ε E 18 +F ⊗π E +F ⊗ E equipped with the projective topology 70 +F ̂⊗πE +completion of F ⊗π E 70 +L(F,E) +space of continuous linear operators from F to E 17 +Lt(F,E) +space L(F,E) equipped with the topology t 17 +b +topology t = b on L(F,E) of uniform convergence on +the bounded subsets of F 17 +147 + +148 +List of Symbols +γ +topology t = γ on L(F,E) of uniform convergence on +the precompact (totally bounded) subsets of F 17 +κ +topology t = κ on L(F,E) of uniform convergence on +the absolutely convex, compact subsets of F 17 +σ +topology t = σ on L(F,E) of uniform convergence on +the finite subsets of F 17 +τ +topology t = τ on L(F,E) of uniform convergence on +the absolutely convex, σ(F,F ′)-compact subsets of F +17 +(pα)α∈A +directed system of seminorms inducing the locally convex Hausdorff +topology on E 17 +Spaces of functions +Aτ +∂(C,E) +space of holomorphic functions f∶C → E of exponential type τ 51 +Aτ +∆(Rd,E) +space of harmonic functions f∶Rd → E of exponential type τ 51 +A(Ω,E) +space of continuous functions f∶Ω → E such that f is holomorphic +on Ω 47 +Bν(D,E) +Bloch type space 87 +C(Ω,X) +space of continuous functions f∶Ω → X 17 +C0(Ω,X) +space of continuous functions f∶Ω → X that vanish at infinity 17 +c0(A,E) +space of elements (xk) in the Köthe space λ∞(A,E) such that +(xkak,j) converges to 0 in E for all j ∈ N 123 +c(N,E) +space of convergent sequences in E 123 +Cb(Ω,E) +space of bounded continuous functions f∶Ω → E 46 +C[γ] +b +(Ω,E) +space of bounded γ-Hölder continuous functions f∶Ω → E 140 +C1 +b (Ω,E) +space of continuously partially differentiable functions f∶Ω → E +such that (∂β)Ef is bounded on Ω for all ∣β∣ ≤ 1 20 +C∞ +P (∂)(Ω,E) +kernel of the linear partial differential operator P(∂)E in C∞(Ω,E) +48 +C∞ +P (∂),b(Ω,E) +space of bounded functions f in C∞ +P (∂)(Ω,E) 53 +Cu(Ω,E) +space of uniformly continuous functions f∶Ω → E 39 +Cbu(Ω,E) +space of bounded uniformly continuous functions f∶Ω → E 45 +CC(Ω,E) +space of Cauchy continuous functions f∶Ω → E 38 +Cext(Ω,E) +space of continuous functions f∶Ω → E which have a continuous +extension to Ω 39 +C[γ] +z (Ω,E) +space of γ-Hölder continuous functions f∶Ω → E such that f(z) = 0 +45 +C[γ] +z,0(Ω,E) +space of functions f in C[γ] +z (Ω,E) that vanish at infinity 45 +Ck(Ω,E) +space of k-times continuously partially differentiable functions +f∶Ω → E 19 +C∞ +2π(Rd,E) +space of functions f in C∞(Rd,E) which are 2π-periodic in each +variable 58 +Ck(Ω,E) +space of functions f in Ck(Ω,E) such that all partial derivatives +(∂β)Ef up to order k are continuously extendable on Ω 58 +Ck,γ(Ω,E) +space of functions f in Ck(Ω,E) such that all partial derivatives +(∂β)Ef of order k are γ-Hölder continuous 105 +Ck,γ +loc (Ω,E) +space of functions f in Ck(Ω,E) such that all partial derivatives +(∂β)Ef of order k are locally γ-Hölder continuous 107 +CVk(Ω,E) +space of functions f in Ck(Ω,E) s.t. (β,x) ↦ (∂β)Ef(x)νj,m(β,x) +is bounded on ωm for all j ∈ J and m ∈ N0 24 + +List of Symbols +149 +CWk(Ω,E) +space Ck(Ω,E) equipped with the topology of uniform convergence +of partial derivatives up to order k on compact subsets of Ω 25 +CVk +P (∂)(Ω,E) +kernel of the linear partial differential operator P(∂)E in CVk(Ω,E) +26 +CVk +0(Ω,E) +space of functions f in CVk(Ω,E) that vanish with all their deriva- +tives when weighted at infinity 51 +CVk +0,P (∂)(Ω,E) kernel of the linear partial differential operator P(∂)E in CVk +0(Ω,E) +52 +CV(Ω,E) +space of continuous functions f∶Ω → E such that fν is bounded on +Ω for all ν ∈ V 46 +CV0(Ω,E) +space of functions f in CV(Ω,E) such that fν vanishes at infinity +for all ν ∈ V 46 +CVP (∂)(Ω,E) +kernel of the linear partial differential operator P(∂)E in CV(Ω,E) +48 +CV0,P (∂)(Ω,E) kernel of the linear partial differential operator P(∂)E in CV0(Ω,E) +48 +D(Ω,E) +space of càdlàg functions f∶Ω → E 44 +E0(E) +space of functions f in C∞([0,1],E) such that (∂k)Ef(1) = 0 59 +E{Mp}(Ω,E) +space of ultradifferentiable functions of class {Mp} of Roumieu-type +25 +E(Mp)(Ω,E) +space of ultradifferentiable functions of class (Mp) of Beurling-type +25 +FG(U,E) +space of functions f∶U → E such that for every e′ ∈ G there is +fe′ ∈ F(Ω) with T K +m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U 76 +FVG(U,E)lb +lb-restriction space 92 +FVG(U,E)sb +sb-restriction space 89 +FV(Ω,E)σ +space of functions f∶Ω → E such that e′ ○ f ∈ FV(Ω) for all e′ ∈ E′ +28 +FV(Ω,E)κ +space of functions f in FV(Ω,E)σ such that Rf(B○ +α) is relatively +compact in FV(Ω) for all α ∈ A 29 +FV(Ω,E) +space of functions in F(Ω,E) with a weighted graph topology in- +duced by the family of weights V 22 +AP(Ω,E) +subspace of functions with additional properties in +EΩ 22 +APFV(Ω,E) the space AP(Ω,E) with an emphasis on the depen- +dence on FV(Ω) 22 +∣f∣j,m,α +seminorms applied to f inducing the weighted graph +topology on FV(Ω,E) 22 +∣f∣FV(Ω),j,m,αthe seminorm ∣f∣j,m,α applied to f with an emphasis +on the dependence on FV(Ω) 22 +F(Ω,E) +the intersection AP(Ω,E) ∩ (⋂m∈M domT E +m) 22 +F(Ω) +the space F(Ω,K) 22 +FV(Ω) +the space FV(Ω,K) 22 +Nj,m(f) +the set {T E +m(f)(x)νj,m(x) ∣ x ∈ ωm} 22 +Fν(Ω,E) +the space FV(Ω,E) with V = (ν) 84 +Fεν(Ω,E) +the space of all functions S(u) s.t. u ∈ F(Ω)εE and u(B○F (Ω)′ +Fν(Ω) ) is +bounded in E 84 +H∞(Ω,E) +space of bounded holomorphic functions f∶Ω → E 84 +Lp(X,µ) +space of equivalence classes of p-integrable functions f∶X → K w.r.t. +the measure µ 142 +λ∞(A,E) +Köthe space 42 + +150 +List of Symbols +ℓV(Ω,E) +space of functions f in EΩ such that fν is bounded on Ω for all +ν ∈ V 42 +M(Ω,E) +space of meromorphic functions f∶Ω → E 81 +OM(Rd,E) +multiplier space for the Schwartz space 25 +O(Ω,E) +space of holomorphic functions f∶Ω → E 20 +s(Ω,E) +space of sequences (xk) in E such that the sequence (xk(1+∣k∣2)j/2) +is bounded in E for all j ∈ N 43 +Sµ(Rd,E) +Beurling–Björck space 55 +(γ) +property of µ 55 +S(Rd,E) +Schwartz space 25 +XΩ +space of maps f∶Ω → X 17 +Maps +χK +characteristic function of a set K ⊂ Ω 17 +∂ +Ef +Cauchy–Riemann operator applied to an E-valued function f 20 +∆∗f +the jump function of a càdlàg function f 139 +δx +point evaluation functional f ↦ f(x) 21 +Hn +n-th Hermite polynomial 128 +hn +n-th Hermite function 128 +J +the canonical injection E → E′⋆, x �→ [e′ ↦ e′(x)] 30 +∣µ∣ +total variation of a K-valued measure µ 142 +(∂β)Ef +β-th partial derivative of an E-valued function f 19 +∂βf +β-th partial derivative (∂β)Kf of a K-valued function +f 20 +(∂n +C)Ef +n-th complex derivative (∂n +C)Ef of an E-valued function f 20 +f (n) +n-th complex derivative (∂n +C)Cf of a C-valued func- +tion f 20 +Rf +the map E′ → F(Ω), e′ ↦ fe′, for given f ∈ FG(U,E) 76 +Rf +the map E′ → FV(Ω), e′ ↦ e′ ○ f, for given f ∈ FV(Ω,E)σ 29 +Rt +f +the map FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], for given f ∈ +FV(Ω,E)σ 29 +S +the map F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)] 21 +SF(Ω) +the map S with an emphasis on the dependence on +F(Ω) 21 +T1εT2 +ε-product of the continuous linear operators T1 and T2 18 +T E +m,x +the map f ↦ T E +m(f)(x) 23 +Θ +the linear injection F ⊗ E → FεE 18 +V +family of weight functions 22 +(V∞) +vanishing at infinity condition on the family of weight +functions 34 +Miscellaneous +(DN) +property of a Fréchet space 67 +(Ω) +property of a Fréchet space 67 +(PA) +property of a PLS-space 67 + +Index +almost norming 89 +B-complete 78 +BC-space 78 +Beurling–Björck space 55 +Blaschke’s convergence theorem 110 +Bloch type space 87 +Borel–Ritt theorem 73 +Br-complete 78 +càdlàg function 44 +Cauchy continuous 38 +Cauchy–Riemann operator 20 +coefficient functional 118 +completely regular 37 +consistent family 60 +continuously partially differentiable 19 +convex compactness property (ccp) 18 +metric (metric ccp) 18 +determine boundedness 78 +disk 18 +dom-space 23 +ε-compatible 21 +ε-into-compatible 21 +ε-product 17 +equicontinuous basis 118 +E-valued weak FV-function 28 +expansion operator 118 +exponential type 51 +family of weight functions 22 +C1-controlled 34 +directed 23 +locally bounded 52 +locally bounded away from zero 26, 48 +Favard space 43 +fix the topology 89, 92 +Fourier expansion 127 +Fourier transformation 55, 74 +generalised Gelfand space 109 +generalised Schwartz space 93 +generator 23 +consistent 23 +strong 23 +hat function 125 +Hermite function 128 +Hermite polynomial 127 +Hölder continuous 45 +holomorphic 20 +infra-exponential type 51 +injective tensor product 18 +jump function 139 +Köthe matrix 42 +Köthe space 42 +kR-space 37 +k-space 37 +K-valued measure 142 +lb-restriction space 92 +lcHs 17 +local closure 18 +local limit point 18 +locally closed 18 +locally complete 18 +multiplier space 25 +Nachbin-family 46 +Pettis-integrable 143 +Pettis-integral 143 +Pettis-summable 144 +PLS-space 67 +positive measure 142 +projective tensor product 70 +restriction space 76 +Riesz–Markov–Kakutani theorem 59 +sb-restriction space 89 +Schauder basis 118 +Schauder decomposition 118 +151 + +152 +Index +Schauder hat function 125 +Schwartz space 25 +semiflow 43 +separating subspace 17 +set of uniqueness 76 +strict topology 53, 114 +strong family 60 +topological basis 118 +total variation 142 +ultradifferentiable 25 +uniformly discrete metric space 42 +vanish at infinity in the weighted topol- +ogy w.r.t. 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Soc., 61(2):335–340, 1976. doi:10.1090/S0002- +9939-1976-0425144-6. + diff --git a/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/load_file.txt b/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aee4fda1aef178040bd2f5cb74efcbd86ea87555 --- /dev/null +++ b/4dFRT4oBgHgl3EQfpDe0/content/tmp_files/load_file.txt @@ -0,0 +1,8879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf,len=8878 +page_content='On vector-valued functions and the ε-product Habilitationsschrift vorgelegt am 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2022 der Technischen Universität Hamburg von Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' rer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Karsten Kruse, geboren am 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1984 in Papenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Die Habilitationsschrift wurde in der Zeit von Juli 2020 bis Januar 2022 im Institut für Mathematik der Technischen Universität Hamburg angefertigt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13612v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='FA] 31 Jan 2023 Gutachter: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' José Bonet Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Leonhard Frerick Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thomas Kalmes PD Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Christian Seifert eingereicht: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Januar 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' überarbeitet: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Januar 2023 Tag des Habilitationskolloquiums: 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Juli 2022 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='15480/882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4898 ORCID: 0000-0003-1864-4915 Creative Commons Lizenz: Diese Arbeit steht unter der Creative Commons Lizenz Namensnennung 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0 (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Das bedeutet, dass sie vervielfältigt, verbreitet und öffentlich zugänglich gemacht werden darf, auch kommerziell, sofern dabei stets der Urheber, die Quelle des Textes und o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lizenz genannt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Die genaue Formulierung der Lizenz kann unter https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0/legalcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='de auf- gerufen werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Acknowledgement It is quite hard to express how grateful I am to the people who helped, in one way or the other, to finish this thesis which spans a part of my work between 2016 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' But I will give it a try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' First of all, I am deeply indebted to Marko Lindner and Christian Seifert who always supported and encouraged me since I joined the TUHH in 2014 and gave me a home so that I could work on the kind of mathematics I love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' I know how lucky I was to meet you both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Second, I would like to thank the two people who taught me probably the most I know about complex analysis and functional analysis, namely, Andreas Defant and Michael Langenbruch (Oldenburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Third, I am utterly grateful to José Bonet and Enrique Jordá (Valencia) for many helpful suggestions and comments, improving some of the papers this thesis is based on, as well as enduring the quite abstract setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let us come to the honorable mentions, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' just kidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In 2015 I was lucky again because Jan Meichsner joined the TUHH as a PhD student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Despite him being a physicist and a dispraiser of green cabbage, it was a real pleasure to share an office, do mathematics or just spend time with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, let me thank Dennis Gallaun with whom I spent a lot of effort and gaffer tape setting up the mobile e-assessment center at the TUHH between 2018 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Apart from the people mentioned above I would like to thank my other mathe- matical co-authors Hans Daduna, Ruslan Krenzler, Felix Schwenninger and Lin Xie whose work is not physically present in this thesis but whose mathematical influence or spirit probably is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, I am thankful to the whole Institute of Mathematics of the TUHH, in particular, the group of Applied Analysis for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, I am grateful to the reviewers of this thesis Leonhard Frerick (Trier) and Thomas Kalmes (Chemnitz) besides José Bonet and Christian Seifert, and the anonymous reviewers of the papers it is based on for their work, helpful comments and corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Finally, I would like to thank my family for their continuous support and my love Sonja for sharing my mathematical interests, bearing my kind of humour and so much more which I cannot put into words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' These words only roughly express my gratitude and I hope that the minimum that remains after reading the acknowledgement is the thought ‘At least, he gave it a try.’ and a smile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3 Abstract This habilitation thesis centres on linearisation of vector-valued functions which means that vector-valued functions are represented by continuous linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The first question we face is which vector-valued functions may be represented by continuous linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We study this problem in the framework of ε- products and give sufficient conditions in Chapter 3 and 4 when a space F(Ω,E) of vector-valued functions on a set Ω coincides (up to an isomorphism) with the ε-product F(Ω)εE of a corresponding space of scalar-valued functions F(Ω) and the codomain E which is usually an infinite-dimensional locally convex Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The ε-product F(Ω)εE is a space of continuous linear operators from the dual space F(Ω)′ to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Once we have a representation of a space F(Ω,E) of vector-valued functions by an ε-product F(Ω)εE, we have access to the rich theory of continuous linear operators which allows us to lift results that are known for the scalar-valued case to the vector-valued case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The whole Chapter 5, which spans more than half of this thesis, is dedicated to this lifting mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' But we should point out that this is not only about transferring results from the scalar-valued to the vector-valued case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The results in the vector-valued case encode additional information for the scalar-valued case as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' we may deduce from the solvability of a linear partial differential equation in the vector-valued case affirmative answers on the parameter dependence of solutions in the scalar-valued case (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 we give a unified approach to handle the problem of extending functions with values in E, which have weak extensions in F(Ω), to functions in the vector-valued counterpart F(Ω,E) of F(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We present different extension the- orems depending on the topological properties of the spaces F(Ω) and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' These theorems also cover weak-strong principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, we study weak-strong principles for continuously partially differentiable functions of finite order in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 and improve the well-known weak-strong principles of Grothendieck and Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We use our results on the extension of vector-valued functions to de- rive Blaschke’s convergence theorem for several spaces of vector-valued functions and Wolff’s theorem for the description of dual spaces of several function spaces F(Ω) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Starting from the observation that every scalar-valued holomorphic function has a local power series expansion and that this is still true for holomorphic functions with values in E if E is locally complete, we develop a machinery which is based on linearisation and Schauder decomposition to transfer known series expansions from scalar-valued to vector-valued functions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Especially, we apply this machinery to derive Fourier expansions for E-valued Schwartz functions and C∞-smooth functions on Rd that are 2π-periodic in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The last section of Chapter 5 is devoted to the representation of spaces F(Ω,E) of vector-valued functions by sequence spaces, which can be used to iden- tify the coefficient spaces of the series expansions from the preceding section, if one knows the coefficient space in the scalar-valued case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, we give several new conditions on the Pettis-integrability of vector-valued functions in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, which are, for instance, needed for the Fourier expansions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 5 Kurzfassung Im Mittelpunkt dieser Habilitationsschrift steht die Linearisierung vektorwer- tiger Funktionen, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' vektorwertige Funktionen sollen durch stetige lineare Opera- toren dargestellt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Die erste Frage, der man sich stellen muss, ist, welche vektorwertigen Funktionen durch stetige lineare Operatoren dargestellt werden kön- nen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Wir untersuchen dieses Problem im Rahmen von ε-Produkten und geben hinreichende Bedingungen in Kapitel 3 und 4 an, wann ein Raum F(Ω,E) von vektorwertigen Funktionen auf einer Menge Ω mit dem ε-Produkt F(Ω)εE eines entsprechenden Raums skalarwertiger Funktionen F(Ω) und des Wertebereichs E (bis auf Isomorphie) übereinstimmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hierbei ist E üblicherweise ein unendlich- dimensionaler lokalkonvexer Hausdorff Raum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Das ε-Produkt F(Ω)εE ist ein Raum stetiger linearer Operatoren, die vom Dualraum F(Ω)′ nach E abbilden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Sobald wir eine Darstellung eines Raums F(Ω,E) von vektorwertigen Funk- tionen durch ein ε-Produkt F(Ω)εE gewonnen haben, ist es uns möglich die reich- haltige Theorie der stetigen linearen Operatoren zu nutzen, die es uns erlaubt, Ergebnisse, die für den skalarwertigen Fall bekannt sind, auf den vektorwertigen Fall zu übertragen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Das gesamte Kapitel 5, das mehr als die Hälfte dieser Arbeit einnimmt, widmet sich diesem Übertragungsmechanismus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Es sei jedoch darauf hingewiesen, dass es hier nicht nur um die Übertragung von Ergebnissen aus dem skalarwertigen auf den vektorwertigen Fall geht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Die Ergebnisse im vektorwertigen Fall beinhalten auch zusätzliche Informationen für den skalarwertigen Fall, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' können wir aus der Lösbarkeit einer linearen partiellen Differentialgleichung im vektorwertigen Fall Antworten auf die Frage nach der Parameterabhängigkeit der Lösungen im skalarwertigen Fall ableiten (siehe Abschnitt 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Abschnitt 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 stellen wir einen einheitlichen Ansatz zur Lösung des Fort- setzungsproblems von Funktionen mit Werten in E, die schwache Fortsetzungen in F(Ω) haben, zu Funktionen im vektorwertigen Gegenstück F(Ω,E) von F(Ω) vor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Wir präsentieren verschiedene Fortsetzungssätze in Abhängigkeit von den topologi- schen Eigenschaften der Räume F(Ω) und E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Diese Sätze decken auch schwach- stark Prinzipien ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Insbesondere untersuchen wir schwach-stark Prinzipien für endlich oft stetig partiell differenzierbare Funktionen in Abschnitt 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 und verbes- sern die bekannten schwach-starken Prinzipien von Grothendieck und Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Zudem leiten wir von unseren Ergebnissen zur Fortsetzung vektorwertiger Funktio- nen den Konvergenzsatz von Blaschke für diverse Räume vektorwertiger Funktionen ab und übertragen den Satz von Wolff auf Dualräume mehrerer Funktionenräume F(Ω) in den Abschnitten 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 und 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Ausgehend von der Beobachtung, dass jede skalarwertige holomorphe Funktion eine lokale Potenzreihenentwicklung hat und dass dies auch für holomorphe Funktionen mit Werten in E gilt, wenn E lokal vollständig ist, entwickeln wir einen Mechanismus, der auf Linearisierung und Schauder-Zerlegung basiert, um in Abschnitt 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 bekannte Reihenentwicklungen von skalarwertigen auf vektorwertige Funktionen zu erweitern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Insbesondere wen- den wir diesen Mechanismus an, um Fourier-Entwicklungen für E-wertige Schwartz- Funktionen und C∞-glatte Funktionen auf Rd, die 2π-periodisch in jeder Variablen sind, zu erhalten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Der letzte Abschnitt von Kapitel 5 ist der Darstellung von 7 8 KURZFASSUNG Räumen F(Ω,E) vektorwertiger Funktionen durch Folgenräume gewidmet, was man dazu nutzen kann, die Koeffizientenräume der Reihenentwicklungen aus dem vorangegangenen Abschnitt zu bestimmen, sofern man den Koeffizientenraum im skalarwertigen Fall kennt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Außerdem legen wir mehrere neue Bedingungen für die Pettis-Integrierbarkeit von vektorwertigen Funktionen in Anhang A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 dar, die z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' für die Fourier-Entwicklungen in Abschnitt 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 benötigt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Contents Acknowledgement 3 Abstract 5 Kurzfassung 7 Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Introduction 11 Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Notation and preliminaries 17 Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The ε-product for weighted function spaces 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-into-compatibility 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-compatibility 28 Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Consistency 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The spaces AP(Ω,E) and consistency 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further examples of ε-products 41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Riesz–Markov–Kakutani representation theorems 59 Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Applications 67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lifting the properties of maps from the scalar-valued case 67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Extension of vector-valued functions 74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Extension from thin sets 76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Extension from thick sets 92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Weak-strong principles for differentiability of finite order 104 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Vector-valued Blaschke theorems 110 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Wolff type results 114 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Series representation of vector-valued functions via Schauder decompositions 117 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Schauder decomposition 120 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Examples of Schauder decompositions 123 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Representation by sequence spaces 130 Appendices Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Compactness of closed absolutely convex hulls and Pettis- integrals 139 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Compactness of closed absolutely convex hulls 139 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The Pettis-integral 142 List of Symbols 147 Index 151 Bibliography 153 9 CHAPTER 1 Introduction This work is dedicated to a classical topic, namely, the linearisation of weighted spaces of vector-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The setting we are interested in is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let F(Ω) be a locally convex Hausdorff space of functions from a non-empty set Ω to a field K and E be a locally convex Hausdorff space over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The ε-product of F(Ω) and E is defined as the space of linear continuous operators F(Ω)εE ∶= Le(F(Ω)′ κ,E) equipped with the topology of uniform convergence on equicontinuous subsets of the dual F(Ω)′ which itself is equipped with the topology of uniform convergence on absolutely convex compact subsets of F(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Suppose that the point-evaluation functionals δx, x ∈ Ω, belong to F(Ω)′ and that there is a locally convex Hausdorff space F(Ω,E) of E-valued functions on Ω such that the map S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], (1) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The main question we want to answer reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' When is F(Ω)εE a linearisation of F(Ω,E), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' when is S an isomorphism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In [15, 16, 17] Bierstedt treats the space CV(Ω,E) of continuous functions on a completely regular Hausdorff space Ω weighted with a Nachbin-family V and its topological subspace CV0(Ω,E) of functions that vanish at infinity in the weighted topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' He derives sufficient conditions on Ω, V and E such that the answer to our question is affirmative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' S is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Schwartz answers this question for several weighted spaces of k-times continuously partially differentiable functions on Ω = Rd like the Schwartz space in [158, 159] for quasi-complete E with regard to vector-valued distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Grothendieck treats the question in [83], mainly for nuclear F(Ω) and complete E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In [99, 100, 101] Komatsu gives a positive answer for ultradifferentiable functions of Beurling or Roumieu type and sequentially complete E with regard to vector-valued ultradistributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For the space of k-times continuously partially differentiable functions on open subsets Ω of infinite dimensional spaces equipped with the topology of uniform convergence of all partial derivatives up to order k on compact subsets of Ω sufficient conditions for an affirmative answer are deduced by Meise in [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For holomorphic functions on open subsets of infinite dimensional spaces a positive answer is given in [52] by Dineen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Bonet, Frerick and Jordá show in [30] that S is an isomorphism for certain closed subsheaves of the sheaf C∞(Ω,E) of smooth functions on an open subset Ω ⊂ Rd with the topology of uniform convergence of all partial derivatives on compact subsets of Ω and locally complete E which, in particular, covers the spaces of harmonic and holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' An important application of linearisation is within the field of partial differen- tial equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be a linear space of functions on a set U and P(∂)∶C∞(Ω) → C∞(Ω) a linear partial differential operator with C∞-smooth coefficients where C∞(Ω) ∶= C∞(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call the elements of U parameters and say that a family (fλ)λ∈U in C∞(Ω) depends on a parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' E if the map λ ↦ fλ(x) is an element of E for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The question of parameter dependence is whether for 11 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' INTRODUCTION every family (fλ)λ∈U in C∞(Ω) depending on a parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' E there is a family (uλ)λ∈U in C∞(Ω) with the same kind of parameter dependence which solves the partial differential equation P(∂)uλ = fλ, λ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, it is the question of Ck-smooth (holomorphic, distributional, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=') parameter dependence if E is the space Ck(U) of k-times continuously partially differentiable functions on an open set U ⊂ Rd (the space O(U) of holomorphic functions on an open set U ⊂ C, the space of distributions D(V )′ on an open set V ⊂ Rd where U = D(V ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The question of parameter dependence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' E has an affirmative answer for several locally convex Hausdorff spaces E due to tensor product techniques and splitting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Indeed, the answer is affirmative if the topology of E is stronger than the topology of pointwise convergence on U and P(∂)E∶C∞(Ω,E) → C∞(Ω,E) is surjective where P(∂)E is the version of P(∂) for E-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The oper- ator P(∂)E is surjective if its version P(∂) for scalar-valued functions is surjective, for instance, if P(∂) is elliptic, and E is a Fréchet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is a consequence of Grothendieck’s theory of tensor products [83], the nuclearity of C∞(Ω) and the isomorphism C∞(Ω,E) ≅ C∞(Ω)εE for locally complete E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thanks to the splitting theory of Vogt for Fréchet spaces [173] and of Bonet and Domański for PLS-spaces [54] we even have in case of an elliptic P(∂) that P(∂)E for d > 1 is surjective if E ∶= F ′ b where F is a Fréchet space satisfying the condition (DN) or if E is an ultra- bornological PLS-space having the property (PA) since kerP(∂) has the property (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, these three results cover the cases that E = Ck(U), O(U) or D(V )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Of course, this technique to answer the question of parameter dependence is not restricted to linear partial differential operators or the space C∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Another application of linearisation lies in the problem of extending a vector- valued function f∶Λ → E from a subset Λ ⊂ Ω to a locally convex Hausdorff space E if the scalar-valued functions e′ ○ f are extendable for each continuous linear functional e′ from certain linear subspaces G of E′ under the constraint of preserving the properties, like holomorphy, of the scalar-valued extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This problem was considered, among others, by Grothendieck [82, 83], Bierstedt [17], Gramsch [77], Grosse-Erdmann [79, 81], Arendt and Nikolski [6, 7, 8], Bonet, Frerick, Jordá and Wengenroth [30, 69, 70, 92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Even the simple case Λ = Ω and G = E′ is interesting and an affirmative answer is called a weak-strong principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Our goal is to give a unified and flexible approach to linearisation which is able to handle new examples and covers the already known examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Organisation of this thesis After fixing some notions and preliminaries on locally convex Hausdorff spaces, continuous linear operators and continuously partially differentiable functions in Chapter 2, we study the problem of linearisation in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 we introduce our standard example of spaces F(Ω,E) that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Namely, spaces of functions FV(Ω,E) from Ω to E which are subspaces of sections of domains of linear operators T E on EΩ, and whose topology is generated by a family of weight functions V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' These spaces cover many examples of classical spaces of functions appearing in analysis like the mentioned ones and an example of the operators T E are the partial derivative operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then we exploit the structure of our spaces to describe a sufficient condition, which we call consistency, on the interplay of the pairs of operators (T E,T K) and the map S such that S becomes an isomorphism into, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' an isomorphism to its range (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ORGANISATION OF THIS THESIS 13 In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 we tackle the problem of surjectivity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In our main Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and its Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 we give several sufficient conditions on the pairs of operators (T E,T K) and the spaces involved such that S∶FV(Ω)εE → FV(Ω,E) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Looking at the pair of partial differential operators (P(∂)E,P(∂)) considered above, these conditions allow us to express P(∂)E as P(∂)E = S ○ (P(∂)εidE) ○ S−1 where P(∂)εidE is the ε-product of P(∂) and the identity idE on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence it becomes obvious that the surjectivity of P(∂)E is equivalent to the surjectivity of P(∂)εidE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is used in [105, 109, 112, 116, 119] in the case of the Cauchy–Riemann operator P(∂) = ∂ on spaces of smooth functions with exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Chapter 4 we take a closer look at the notion of consistency of (T E,T K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 we characterise several properties of the functions S(u) for u ∈ FV(Ω)εE that are inherited from the elements of FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 is devoted to several concrete examples of spaces of vector-valued functions that may be linearised by S and which we use for our applications in the forthcoming sections and chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 we answer in several cases the question whether given a con- tinuous linear functional T K on F(Ω) there is always a continuous linear operator T E on F(Ω,E) such that (T E,T K) is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is closely related to Riesz– Markov–Kakutani theorems for T K, which we transfer to the vector-valued case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Chapter 5 is dedicated to applications of linearisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 we come back to our problem of parameter dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We show in our main Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 of this section how to use linearisations to transfer properties like injectivity, sur- jectivity or bijectivity from a map T K∶F1(Ω1) → F2(Ω2) to the corresponding map T E∶F2(Ω1,E) → F2(Ω2,E) if the pair (T E,T K) is consistent under suitable as- sumptions on the spaces involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Besides the problem of parameter dependence for (hypo)elliptic linear partial differential operators (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3), we de- duce a vector-valued version of the Borel–Ritt theorem (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4) from this main theorem and give sufficient conditions under which the Fourier transfor- mation FC∶Sµ(Rd) → Sµ(Rd) on the Beurling–Björck space is still an isomorphism in the vector-valued case and may be decomposed as FE = S ○ (FCεidE) ○ S−1 (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 we present a general approach to the extension problem consid- ered above for a large class of function spaces F(Ω,E) if the map S is an isomor- phism into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The spaces we treat are of the kind that F(Ω) belongs to the class of semi-Montel, Fréchet–Schwartz or Banach spaces, or that E is a semi-Montel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Apart from linearisation and consistency, the main ingredient of this approach is to view the set Λ ⊂ Ω from which we want to extend our functions as a set of function- als {δx ∣ x ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This view allows us to generalise the extension problem in Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 by swapping this set of functionals by other functionals, which opens up new possibilities in applications that we explore in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In the extension problem we always have to balance the sets Λ from which we extend our functions and the subspaces G ⊂ E′ with which we test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The case of ‘thin’ sets Λ and ‘thick’ subspaces G is handled in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 with main theorems Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='15, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='20 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='29, the converse case of ‘thick’ sets Λ and ‘thin’ subspaces G is handled in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 with main theorems Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='52, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='63 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 we consider weak-strong principles for continuously partially differentiable functions of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For locally complete E it is well-known that a function f belongs to C∞(Ω,E) if and only if e′ ○ f ∈ C∞(Ω) for all e′ ∈ E′ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' [30, Theorem 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 232]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If k ∈ N0, then it is still true that f ∈ Ck(Ω,E) implies e′ ○ f ∈ Ck(Ω) for all e′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' But the converse is not true anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Only a weaker 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' INTRODUCTION version of this weak-strong principle holds which is due to Grothendieck [82] and Schwartz [158] (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Namely, if k ∈ N0, E is sequentially complete and f∶Ω → E is such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Using the results from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, we improve this weaker version of the weak-strong principle by allowing E to be locally complete, only testing with less functionals from certain linear subspaces G ⊂ E′ and getting that f does not only belong to Ck(Ω,E) but that all partial derivatives of order k are actually locally Lipschitz continuous (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If we restrict to semi-Montel spaces E, then even a ‘full’ weak-strong principle Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 holds as in the C∞-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 we derive vector-valued Blaschke theorems like Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 for several function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This generalises results of Arendt and Nikolski [7] for bounded holomorphic functions and Frerick, Jordá and Wengenroth [70] for bounded functions in the kernel of a hypoelliptic linear partial differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' These are results of the form: given a bounded net (fι)ι∈I in some space F1(Ω,E) of Banach-valued functions which converges pointwise on a certain subset of Ω there is a limit f ∈ F1(Ω,E) of this net w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a weaker topology of a linear superspace F2(Ω,E) of F1(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Blaschke’s classical convergence theorem [38, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 219] we have E = C, F1(Ω,E) is the space of bounded holomorphic functions on the open unit disc D ⊂ C, F2(Ω,E) is the space of holomorphic functions on D and the weaker topology is the topology of compact convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 we present Wolff type descriptions of the dual space of several function spaces F(Ω) using linearisation (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Wolff’s theorem [183, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 1327] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' [81, Theorem (Wolff), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 402]) phrased in a functional analytic way (see [70, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 240]) says: if Ω ⊂ C is a domain, then for each µ ∈ O(Ω)′ there are a sequence (zn)n∈N which is relatively compact in Ω and a sequence (an)n∈N in the space ℓ1 of absolutely summable sequences such that µ = ∑∞ n=1 anδzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 we derive a general result for Schauder decompositions of the ε-product FεE for locally convex Hausdorff spaces F and E if F has an equicon- tinuous Schauder basis (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In combination with linearisation and consistency this can be used for F = F(Ω) to lift series representations like the power series expansion of holomorphic functions from scalar-valued functions to vector-valued functions (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We present several examples in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, for instance, Fourier expansions in the Schwartz space S(Rd,E) and in the space C∞ 2π(Rd,E) of functions in C∞(Rd,E) that are 2π-periodic in each vari- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, we combine these expansions for locally complete E with the results from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 to identify the coefficient spaces of the Fourier expansions in S(Rd,E) and C∞ 2π(Rd,E) (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 an application of our extension results from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 is given to represent function spaces F(Ω,E) by sequence spaces if one knows such a represen- tation for F(Ω) (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As examples we treat the space O(DR(0),E) of E-valued holomorphic functions on the disc DR(0) ⊂ C around 0 with radius 0 < R ≤ ∞ and the multiplier space OM(R,E) of the Schwartz space for locally complete E (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The first section Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 of the Appendix A is devoted to the question when the closure of an absolutely convex hull of a set is compact in a locally convex Hausdorff space E and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 to the related question of Pettis-integrability of an E-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CONCERNING ORIGINALITY 15 Concerning originality We note that some parts of chapters or sections are based on our papers and preprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Chapter 3, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 are based on our paper Weighted spaces of vector-valued functions and the ε-product [110] and its extended preprint [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 contains results from Sections 3 and 6 of our accepted preprint Extension of weighted vector-valued func- tions and sequence space representation [115] and our paper Extension of weighted vector-valued functions and weak–strong principles for differen- tiable functions of finite order [117] and its extended preprint [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 generalises some results of our papers Surjectivity of the ∂- operator between weighted spaces of smooth vector-valued functions [116] and Parameter dependence of solutions of the Cauchy–Riemann equation on weighted spaces of smooth functions [112] and its extended preprint [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 are based on our accepted preprint [115] and our paper [117] (and its extended preprint [120]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 is based on our paper Series representations in spaces of vector- valued functions via Schauder decompositions [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, the introduction Chapter 1 and Chapter 2 on notation and pre- liminaries are based on the corresponding sections in our papers and preprints [106, 110, 112, 114, 115, 116, 117, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' However, not all of the results given in this thesis are already contained in our preprints or papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Chapter 3 the new, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' not contained in our preprints or papers, results are Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (ii), Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 e)+f), Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 the new examples and results are Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='16, Ex- ample 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='22 which extends [107, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='17 a), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 244] of our paper The approximation property for weighted spaces of differentiable function [107], Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='25 which extends [114, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 370] from sequentially complete E to locally complete E, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='26 and Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='28 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' All the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 are new except for Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 which is [115, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Definition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4] (and also not a result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The main theorem of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, is new even though special cases appeared in [112, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 are new as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 extends [120, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Corollary, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 22] from metric spaces with finite diameter to arbitrary metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14 b) extend [114, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 371–372] and [114, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 375] from sequentially complete E to locally complete E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 is new in the sense that there is only a sketch how to prove it in [115, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The results of Appendix A are also new except for Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, Propo- sition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6, which are contained in [106, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 24] and [106, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13 Lemma d), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 10], and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 which is [114, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 369].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CHAPTER 2 Notation and preliminaries Basics of topology We equip the spaces Rd, d ∈ N, and C with the usual Euclidean norm ∣ ⋅ ∣, denote by Br(x) ∶= {w ∈ Rd ∣ ∣w − x∣ < r} the ball around x ∈ Rd and by Dr(z) ∶= {w ∈ C ∣ ∣w − z∣ < r} the disc around z ∈ C with radius r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, for a subset M of a topological space (X,t) we denote the closure of M by M and the boundary of M by ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If we want to emphasize that we take the closure in X resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the topology t, then we write M X resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' M t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For a subset M of a vector space X we denote by ch(M) the circled hull, by cx(M) the convex hull and by acx(M) the absolutely convex hull of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If X is a topological vector space, we write acx(M) for the closure of acx(M) in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Locally convex Hausdorff spaces and continuous linear operators By E we always denote a non-trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' E ≠ {0}, locally convex Hausdorff space over the field K = R or C equipped with a directed fundamental system of seminorms (pα)α∈A and, in short, we write that E is an lcHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E = K, then we set (pα)α∈A ∶= {∣ ⋅ ∣}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' By XΩ we denote the set of maps from a non-empty set Ω to a non-empty set X, by χK we mean the characteristic function of K ⊂ Ω, by C(Ω,X) the space of continuous functions from a topological space Ω to a topological space X, and by C0(Ω,X) its subspace of continuous functions that vanish at infinity if X is a locally convex Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We denote by L(F,E) the space of continuous linear operators from F to E where F and E are locally convex Hausdorff spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E = K, we just write F ′ ∶= L(F,K) for the dual space and G○ for the polar set of G ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If F and E are linearly topologically isomorphic, we just write that F and E are isomorphic, in symbols F ≅ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We denote by Lt(F,E) the space L(F,E) equipped with the locally convex topology t of uniform convergence on the finite subsets of F if t = σ, on the absolutely convex, compact subsets of F if t = κ, on the absolutely convex, σ(F,F ′)-compact subsets of F if t = τ, on the precompact (totally bounded) subsets of F if t = γ and on the bounded subsets of F if t = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We use the symbols t(F ′,F) for the corresponding topology on F ′ and t(F) for the corresponding bornology on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We say that a subspace G ⊂ F ′ is separating (the points of F) if for every x ∈ F it follows from y(x) = 0 for all y ∈ G that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Clearly, this is equivalent to G being σ(F ′,F)-dense in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For details and notions on the theory of locally convex spaces not explained in this thesis see [68, 89, 131, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-products and tensor products The so-called ε-product of Schwartz is defined by FεE ∶= Le(F ′ κ,E) (2) where L(F ′ κ,E) is equipped with the topology of uniform convergence on equicon- tinuous subsets of F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This definition of the ε-product coincides with the original 17 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' NOTATION AND PRELIMINARIES one by Schwartz [159, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' I, §1, Définition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It is symmetric which means that FεE ≅ EεF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In the literature the definition of the ε-product is sometimes done the other way around, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' EεF is defined by the right-hand side of (2) but due to the symmetry these definitions are equivalent and for our purpose the given definition is more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If we replace F ′ κ by F ′ γ, we obtain Grothendieck’s definition of the ε-product and we remark that the two ε-products coincide if F is quasi-complete because then F ′ γ = F ′ κ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' However, we stick to Schwartz’ definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For locally convex Hausdorff spaces Fi, Ei and Ti ∈ L(Fi,Ei), i = 1,2, we define the ε-product T1εT2 ∈ L(F1εF2,E1εE2) of the operators T1 and T2 by (T1εT2)(u) ∶= T2 ○ u ○ T t 1, u ∈ F1εF2, where T t 1∶E′ 1 → F ′ 1, e′ ↦ e′ ○ T1, is the dual map of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If T1 is an isomorphism and F2 = E2, then T1εidE2 is also an isomorphism with inverse T −1 1 εidE2 by [159, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' I, §1, Proposition 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 20] (or [89, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 347] if the Fi are complete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As usual we consider the tensor product F ⊗E as a linear subspace of FεE for two locally convex Hausdorff spaces F and E by means of the linear injection Θ∶F ⊗ E → FεE, k ∑ n=1 fn ⊗ en �→ [y ↦ k ∑ n=1 y(fn)en].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (3) Via Θ the space F ⊗ E is identified with the space of operators with finite rank in FεE and a locally convex topology is induced on F ⊗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We write F ⊗ε E for F ⊗ E equipped with this topology and F ̂⊗εE for the completion of the injective tensor product F ⊗εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For more information on the theory of ε-products and tensor products see [49, 89, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Several degrees of completeness The sufficient conditions for surjectivity of the map S∶F(Ω)εE → F(Ω,E) from the introduction, which we derive in the forthcoming, depend on assumptions on different types of completeness of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For this purpose we recapitulate some definitions which are connected to completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We start with local completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For a disk D ⊂ E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a bounded, absolutely convex set, the linear space ED ∶= ⋃n∈N nD becomes a normed space if it is equipped with the gauge functional of D as a norm (see [89, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space E is called locally complete if ED is a Banach space for every closed disk D ⊂ E (see [89, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 197]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call a non-empty subset A of an lcHs E locally closed if every local limit point of A belongs to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Here, a point x ∈ E is called a local limit point of A if there is a sequence (xn)n∈N in A that converges locally to x (see [138, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 154–155]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' there is a disk D ⊂ E such that (xn) converges to x in ED (see [138, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The local closure of a subset A of E is defined as the smallest locally closed subset of E which contains A (see [138, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 155]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, we note that every locally complete linear subspace of E is locally closed and a locally closed linear subspace of a locally complete space is locally complete by [138, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='20 (i), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, a locally convex Hausdorff space is locally complete if and only if it is convenient by [104, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14 Theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, every complete locally convex Hausdorff space is quasi-complete, every quasi-complete space is sequentially complete and every sequentially complete space is locally complete and all these implications are strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The first two by [89, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 58] and the third by [138, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8 Corollary, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 153] and [138, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12 Example, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Now, let us recall the following definition from [181, 9-2-8 Definition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 134] and [175, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A locally convex Hausdorff space is said to have the [metric] convex compactness property ([metric] ccp) if the closure of the absolutely convex hull of VECTOR-VALUED CONTINUOUSLY PARTIALLY DIFFERENTIABLE FUNCTIONS 19 every [metrisable] compact set is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Sometimes this condition is phrased with the term convex hull instead of absolutely convex hull but these definitions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Indeed, the first definition implies the second since every convex hull of a set A ⊂ E is contained in its absolutely convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' On the other hand, we have acx(A) = cx(ch(A)) by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 103] and the circled hull ch(A) of a [metrisable] compact set A is compact by [153, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' I, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 26] [and metrisable by [34, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' IX, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, Proposition 17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 159] since D × A is metrisable and ch(A) = ME(D × A) where ME∶K × E → E is the continuous scalar multiplication and D ∶= D1(0) the open unit disc], which yields the other implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, every locally convex Hausdorff space with ccp has obviously met- ric ccp, every quasi-complete locally convex Hausdorff space has ccp by [181, 9-2-10 Example, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 134], every sequentially complete locally convex Hausdorff space has metric ccp by [23, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 Proposition (ii), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 364] and every locally convex Hausdorff space with metric ccp is locally complete by [175, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' All these implications are strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The second by [181, 9-2-10 Example, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 134] and the others by [175, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For more details on the [metric] convex compactness property and local completeness see [29, 175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In addition, we remark that every semi-Montel space is semi-reflexive by [89, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 230] and every semi-reflexive locally convex Hausdorff space is quasi-complete by [153, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' IV, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, Corollary 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 144] and these implications are strict as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Summarizing, we have the following diagram of strict implications: semi-Montel ⇒ semi-reflexive ⇓ complete ⇒ quasi-complete ⇒ sequentially complete ⇒ locally complete ⇓ ⇓ �⇒ ccp ⇒ metric ccp Vector-valued continuously partially differentiable functions Since weighted spaces of continuously partially differentiable resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' holomorphic vector-valued functions will serve as our standard examples, we recall the definition of the spaces Ck(Ω,E) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' O(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A function f∶Ω → E on an open set Ω ⊂ Rd to an lcHs E is called continuously partially differentiable (f is C1) if for the n-th unit vector en ∈ Rd the limit (∂en)Ef(x) ∶= lim h→0 h∈R,h≠0 f(x + hen) − f(x) h exists in E for every x ∈ Ω and (∂en)Ef is continuous on Ω ((∂en)Ef is C0) for every 1 ≤ n ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For k ∈ N a function f is said to be k-times continuously partially differentiable (f is Ck) if f is C1 and all its first partial derivatives are Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A function f is called infinitely continuously partially differentiable (f is C∞) if f is Ck for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For k ∈ N∞ ∶= N∪{∞} the functions f∶Ω → E which are Ck form a linear space which is denoted by Ck(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For β ∈ Nd 0 with ∣β∣ ∶= ∑d n=1 βn ≤ k and a function f∶Ω → E on an open set Ω ⊂ Rd to an lcHs E we set (∂βn)Ef ∶= f if βn = 0, and (∂βn)Ef(x) ∶= (∂en)E⋯(∂en)E ������������������������������������������������������������������������������������������ βn-times f(x) if βn ≠ 0 and the right-hand side exists in E for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we define (∂β)Ef(x) ∶= ((∂β1)E⋯(∂βd)E)f(x) 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' NOTATION AND PRELIMINARIES if the right-hand side exists in E for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E = K, we often just write ∂βf ∶= (∂β)Kf for β ∈ Nd 0, ∣β∣ ≤ k, and f ∈ Ck(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, we define the space of bounded continuously partially differentiable functions by C1 b (Ω,E) ∶= {f ∈ C1(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣C1 b (Ω),α ∶= sup x∈Ω β∈Nd 0,∣β∣≤1 pα((∂β)Ef(x)) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Vector-valued holomorphic functions A function f∶Ω → E on an open set Ω ⊂ C to an lcHs E over C is called holomorphic if the limit (∂1 C)Ef(z) ∶= lim h→0 h∈C,h≠0 f(z + h) − f(z) h , z ∈ Ω, exists in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We denote by O(Ω,E) the linear space of holomorphic functions f∶Ω → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Defining the vector-valued version of the Cauchy–Riemann operator by ∂ Ef ∶= 1 2((∂e1)E + i(∂e2)E)f for f ∈ C(Ω,E) such that the partial derivatives (∂en)Ef, n = 1,2, exist in E, we remark that O(Ω,E) = {f ∈ C(Ω,E) ∣ f ∈ ker∂ E} = {f ∈ C∞(Ω,E) ∣ f ∈ ker∂ E} (4) by [113, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 267] if E is locally complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we set (∂0 C)Ef ∶= f and note that the (n + 1)-th complex derivative (∂n+1 C )Ef ∶= (∂1 C)E((∂n C)Ef) exists for all n ∈ N0 and f ∈ O(Ω,E) by [79, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Theorem and Definition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 17–18] and [79, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 35] if E is locally complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E = C, we often just write f (n) ∶= (∂n C)Cf for n ∈ N0 and f ∈ O(Ω) ∶= O(Ω,C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that the real and complex derivatives are related by (∂β)Ef(z) = iβ2(∂∣β∣ C )Ef(z), z ∈ Ω, (5) for every f ∈ O(Ω,E) and β = (β1,β2) ∈ N2 0 by [113, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 270] if E is locally complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CHAPTER 3 The ε-product for weighted function spaces 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-into-compatibility In the introduction we already mentioned that linearisations of spaces of vector- valued functions by means of ε-products are essential for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Here, one of the important questions is which spaces of vector-valued functions can be represented by ε-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a non-empty set and E an lcHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If F(Ω) ⊂ KΩ is an lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω, then the map S∶F(Ω)εE → EΩ, u �→ [x ↦ u(δx)], is well-defined and linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Definition (ε-into-compatible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a non-empty set and E an lcHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let F(Ω) ⊂ KΩ and F(Ω,E) ⊂ EΩ be lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call the spaces F(Ω) and F(Ω,E) ε-into-compatible if the map S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)], is a well-defined isomorphism into, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' to its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call F(Ω) and F(Ω,E) ε-compatible if S is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We write SF(Ω) if we want to emphasise the dependency on F(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In this section we introduce the weighted space FV(Ω,E) of E-valued functions on Ω as a subspace of sections of domains in EΩ of linear operators T E m equipped with a generalised version of a weighted graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This space is the role model for many function spaces and an example for these operators are the partial derivative operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then we treat the question whether FV(Ω,E) and FV(Ω)εE are ε-into-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is deeply connected with the interplay of the pair of operators (T E m,T K m) with the map S (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In our main theorem of this section we give sufficient conditions such that S∶FV(Ω)εE → FV(Ω,E) is an isomorphism into (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In the next section we provide conditions such that S becomes surjective (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We start with the well-known example Ck(Ω,E) of k-times continuously partially differentiable E-valued func- tions to motivate our definition of FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let k ∈ N∞ and Ω ⊂ Rd be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Consider the space C(Ω,E) of continuous functions f∶Ω → E with the topology τc of compact convergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the topology given by the seminorms ∥f∥K,α ∶= sup x∈K pα(f(x)), f ∈ C(Ω,E), for compact K ⊂ Ω and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The usual topology on the space Ck(Ω,E) of k-times continuously partially differentiable functions is the graph topology generated by the partial derivative operators (∂β)E∶Ck(Ω,E) → C(Ω,E) for β ∈ Nd 0, ∣β∣ ≤ k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the topology given by the seminorms ∥f∥K,β,α ∶= max(∥f∥K,α,∥(∂β)Ef∥K,α), f ∈ Ck(Ω,E), 21 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES for compact K ⊂ Ω, β ∈ Nd 0, ∣β∣ ≤ k, and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The same topology is induced by the directed system of seminorms given by ∣f∣K,m,α ∶= sup β∈Nd 0,∣β∣≤m ∥f∥K,β,α = sup x∈K β∈Nd 0,∣β∣≤m pα((∂β)Ef(x)), f ∈ Ck(Ω,E), for compact K ⊂ Ω, m ∈ N0, m ≤ k, and α ∈ A and may also be seen as a weighted topology induced by the family (χK) of characteristic functions of the compact sets K ⊂ Ω by writing ∣f∣K,m,α = sup x∈Ω β∈Nd 0,∣β∣≤m pα((∂β)Ef(x))χK(x), f ∈ Ck(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This topology is inherited by linear subspaces of functions having additional prop- erties like being holomorphic or harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We turn to the weight functions which we use to define a kind of weighted graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Definition (weight function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let J be a non-empty set and (ωm)m∈M a family of non-empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call V ∶= (νj,m)j∈J,m∈M a family of weight functions on (ωm)m∈M if it fulfils νj,m∶ωm → [0,∞) for all j ∈ J, m ∈ M and ∀ m ∈ M, x ∈ ωm ∃ j ∈ J ∶ 0 < νj,m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (6) From the structure of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 we arrive at the following definition of the weighted spaces of vector-valued functions we want to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a non-empty set, V ∶= (νj,m)j∈J,m∈M a family of weight functions on (ωm)m∈M and T E m∶EΩ ⊃ domT E m → Eωm a linear map for every m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let AP(Ω,E) be a linear subspace of EΩ and define the space of intersections F(Ω,E) ∶= AP(Ω,E) ∩ ( ⋂ m∈M domT E m) as well as FV(Ω,E) ∶= {f ∈ F(Ω,E) ∣ ∀ j ∈ J, m ∈ M, α ∈ A ∶ ∣f∣j,m,α < ∞} where ∣f∣j,m,α ∶= sup x∈ωm pα(T E m(f)(x))νj,m(x) = sup e∈Nj,m(f) pα(e) with Nj,m(f) ∶= {T E m(f)(x)νj,m(x) ∣ x ∈ ωm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we write F(Ω) ∶= F(Ω,K) and FV(Ω) ∶= FV(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If we want to empha- sise dependencies, we write M(E) instead of M, APFV(Ω,E) instead of AP(Ω,E) and ∣f∣FV(Ω),j,m,α instead of ∣f∣j,m,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If J, M or A are singletons, we omit the index j, m resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' α in ∣f∣j,m,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Note that ωm need not be a subset of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space AP(Ω,E) is a placeholder where we collect additional properties (AP) of our functions not being reflected by the operators T E m which we integrated in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' However, these additional properties might come from being in the domain or kernel of additional operators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' harmonicity means being in the kernel of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' But often AP(Ω,E) can be chosen as EΩ or C(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space FV(Ω,E) is locally convex but need not be Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since it is easier to work with Hausdorff spaces and a directed family of seminorms plus the point evaluation functionals δx∶FV(Ω) → K, f ↦ f(x), for x ∈ Ω and their continuity play a big role, we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-INTO-COMPATIBILITY 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Definition (dom-space and T E m,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We call FV(Ω,E) a dom-space if it is a locally convex Hausdorff space, the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A is directed and, in addition, δx ∈ FV(Ω)′ for every x ∈ Ω if E = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We define the point evaluation of T E m by T E m,x∶domT E m → E, T E m,x(f) ∶= T E m(f)(x), for m ∈ M and x ∈ ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) It is easy to see that FV(Ω,E) is Hausdorff if there is m ∈ M such that ωm = Ω and T E m = idEΩ since E is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) If E = K, then T K m,x ∈ FV(Ω)′ for every m ∈ M and x ∈ ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Indeed, for m ∈ M and x ∈ ωm there exists j ∈ J such that νj,m(x) > 0 by (6), implying for every f ∈ FV(Ω) that ∣T K m,x(f)∣ = 1 νj,m(x)∣T K m(f)(x)∣νj,m(x) ≤ 1 νj,m(x)∣f∣j,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, this implies δx ∈ FV(Ω)′ for all x ∈ Ω if there is m ∈ M such that ωm = Ω and T K m = idKΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) Let the family of weight functions V be directed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ∀ j1,j2 ∈ J,m1,m2 ∈ M ∃ j3 ∈ J, m3 ∈ M, C > 0 ∀ i ∈ {1,2} ∶ (ωm1 ∪ ωm2) ⊂ ωm3 and νji,mi ≤ Cνj3,m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A is directed if V is directed and additionally it holds with mi, i ∈ {1,2,3}, from above that ∀ f ∈ FV(Ω,E), i ∈ {1,2}, x ∈ ωmi ∶ T E mi(f)(x) = T E m3(f)(x), since the system (pα)α∈A of E is already directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We point out that the additional condition in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 c) is missing in [110, Remark 5 c), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 1516] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' [106, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 Remark, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 6]), which we correct here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For the lcHs E over K we want to define a natural E-valued version of a dom- space FV(Ω) = FV(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The natural E-valued version of FV(Ω) should be a dom-space FV(Ω,E) such that there is a canonical relation between the families (T K m) and (T E m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This canonical relation will be explained in terms of their interplay with the map S∶FV(Ω)εE → EΩ, u �→ [x ↦ u(δx)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, the elements of our E-valued version FV(Ω,E) of FV(Ω) should be com- patible with a weak definition in the sense that e′ ○f ∈ FV(Ω) should hold for every e′ ∈ E′ and f ∈ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Definition (generator, consistent, strong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let FV(Ω) and FV(Ω,E) be dom-spaces such that M ∶= M(K) = M(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) We call (T E m,T K m)m∈M a generator for (FV(Ω),E), in short, (FV,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) We call (T E m,T K m)m∈M consistent if we have for all u ∈ FV(Ω)εE that S(u) ∈ F(Ω,E) and ∀ m ∈ M, x ∈ ωm ∶ (T E mS(u))(x) = u(T K m,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) We call (T E m,T K m)m∈M strong if we have for all e′ ∈ E′, f ∈ FV(Ω,E) that e′ ○ f ∈ F(Ω) and ∀ m ∈ M, x ∈ ωm ∶ T K m(e′ ○ f)(x) = (e′ ○ T E m(f))(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' More precisely, T K m,x in b) means the restriction of T K m,x to FV(Ω) and the term u(T K m,x) is well-defined by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Consistency will guarantee that the map S∶FV(Ω)εE → FV(Ω,E) is a well-defined isomorphism into, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-into- compatibility, and strength will help us to prove its surjectivity under some ad- ditional assumptions on FV(Ω) and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let us come to a lemma which describes 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES the topology of FV(Ω)εE in terms of the operators T K m with m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It was the motivation for the definition of consistency and allows us to consider FV(Ω)εE as a topological subspace of FV(Ω,E) via S, assuming consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let FV(Ω) be a dom-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then the topology of FV(Ω)εE is given by the system of seminorms defined by ∥u∥j,m,α ∶= sup x∈ωm pα(u(T K m,x))νj,m(x), u ∈ FV(Ω)εE, for j ∈ J, m ∈ M and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We define the sets Dj,m ∶= {T K m,x(⋅)νj,m(x) ∣ x ∈ ωm} and Bj,m ∶= {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} for every j ∈ J and m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We claim that acx(Dj,m) is dense in the polar B○ j,m with respect to κ(FV(Ω)′,FV(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The observation D○ j,m = {T K m,x(⋅)νj,m(x) ∣ x ∈ ωm}○ = {f ∈ FV(Ω) ∣ ∀x ∈ ωm ∶ ∣T K m(f)(x)∣νj,m(x) ≤ 1} = {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} = Bj,m yields acx(Dj,m)κ(FV(Ω)′,FV(Ω)) = (Dj,m)○○ = B○ j,m by the bipolar theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' By [89, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 152, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 156–157] the system of seminorms defined by qj,m,α(u) ∶= sup y∈B○ j,m pα(u(y)), u ∈ FV(Ω)εE, for j ∈ J, m ∈ M and α ∈ A gives the topology on FV(Ω)εE (here it is used that the system of seminorms (∣ ⋅ ∣j,m) of FV(Ω) is directed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As every u ∈ FV(Ω)εE is continuous on B○ j,m, we may replace B○ j,m by a κ(FV(Ω)′,FV(Ω))-dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore we obtain qj,m,α(u) = sup{pα(u(y)) ∣ y ∈ acx(Dj,m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For y ∈ acx(Dj,m) there are n ∈ N, λk ∈ K, xk ∈ ωm, 1 ≤ k ≤ n, with ∑n k=1 ∣λk∣ ≤ 1 such that y = ∑n k=1 λkT K m,xk(⋅)νj,m(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then we have for every u ∈ FV(Ω)εE pα(u(y)) ≤ n ∑ k=1 ∣λk∣pα(u(T K m,xk))νj,m(xk) ≤ ∥u∥j,m,α, thus qj,m,α(u) ≤ ∥u∥j,m,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' On the other hand, we derive qj,m,α(u) ≥ sup y∈Dj,m pα(u(y)) = sup x∈ωm pα(u(T K m,x))νj,m(x) = ∥u∥j,m,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Let us turn to a more general version of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, namely, to weighted spaces of k-times continuously partially differentiable functions and kernels of linear partial differential operators in these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let k ∈ N∞ and Ω ⊂ Rd be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We consider the cases (i) ωm ∶= Mm × Ω with Mm ∶= {β ∈ Nd 0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0, or (ii) ωm ∶= Nd 0 × Ω for all m ∈ N0 and k = ∞, and let Vk ∶= (νj,m)j∈J,m∈N0 be a directed family of weights on (ωm)m∈N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) We define the weighted space of k-times continuously partially differentiable functions with values in an lcHs E as CVk(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ j ∈ J, m ∈ N0, α ∈ A ∶ ∣f∣j,m,α < ∞} where ∣f∣j,m,α ∶= sup (β,x)∈ωm pα((∂β)Ef(x))νj,m(β,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-INTO-COMPATIBILITY 25 Setting domT E m ∶= Ck(Ω,E) and T E m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)], as well as AP(Ω,E) ∶= EΩ, we observe that CVk(Ω,E) is a dom-space by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 and ∣f∣j,m,α = sup x∈ωm pα(T E mf(x))νj,m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) The space Ck(Ω,E) with its usual topology given in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 is a special case of a)(i) with J ∶= {K ⊂ Ω ∣ K compact}, νK,m(β,x) ∶= χK(x), (β,x) ∈ ωm, for all m ∈ N0 and K ∈ J where χK is the characteristic function of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In this case we write Wk ∶= Vk for the family of weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) The Schwartz space is defined by S(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ m ∈ N0, α ∈ A ∶ ∣f∣m,α < ∞} where ∣f∣m,α ∶= sup x∈Rd β∈Nd 0,∣β∣≤m pα((∂β)Ef(x))(1 + ∣x∣2)m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶= {1} and ν1,m(β,x) ∶= (1 + ∣x∣2)m/2, (β,x) ∈ ωm, for all m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' d) The multiplier space for the Schwartz space is defined by OM(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ g ∈ S(Rd), m ∈ N0, α ∈ A ∶ ∥f∥g,m,α < ∞} where ∥f∥g,m,α ∶= sup x∈Rd β∈Nd 0,∣β∣≤m pα((∂β)Ef(x))∣g(x)∣ (see [158, 40), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶= {j ⊂ S(Rd) ∣ j finite} and νj,1,m(β,x) ∶= maxg∈j ∣g(x)∣, (β,x) ∈ ωm, for all m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This choice of J guarantees that the family V∞ is directed and does not change the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' e) Let K ∶= {K ⊂ Ω ∣ K compact} and (Mp)p∈N0 be a sequence of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space E(Mp)(Ω,E) of ultradifferentiable functions of class (Mp) of Beurling-type is defined as E(Mp)(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ K ∈ K, h > 0, α ∈ A ∶ ∣f∣(K,h),α < ∞} where ∣f∣(K,h),α ∶= sup x∈K β∈Nd 0 pα((∂β)Ef(x)) 1 h∣β∣M∣β∣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is a special case of a)(ii) with J ∶= K × R>0 and ν(K,h),m(β,x) ∶= χK(x) 1 h∣β∣M∣β∣ , (β,x) ∈ ωm, for all (K,h) ∈ J and m ∈ N0 where R>0 ∶= (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' f) Let K and (Mp)p∈N0 be as in e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space E{Mp}(Ω,E) of ultradifferentiable functions of class {Mp} of Roumieu-type is defined as E{Mp}(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ (K,H) ∈ J, α ∈ A ∶ ∣f∣(K,H),α < ∞} where J ∶= K × {H = (Hn)n∈N ∣ ∃ (hk)k∈N, hk > 0, hk ↗ ∞ ∀ n ∈ N ∶ Hn = h1 ⋅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ⋅ hn} and ∣f∣(K,H),α ∶= sup x∈K β∈Nd 0 pα((∂β)Ef(x)) 1 H∣β∣M∣β∣ 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES (see [101, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 675]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Again, this is a special case of a)(ii) with ν(K,H),m(β,x) ∶= χK(x) 1 H∣β∣M∣β∣ , (β,x) ∈ ωm, for all (K,H) ∈ J and m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' g) Let n ∈ N, βi ∈ Nd 0 with ∣βi∣ ≤ k and ai∶Ω → K for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We set P(∂)E∶Ck(Ω,E) → EΩ, P(∂)E(f)(x) ∶= n ∑ i=1 ai(x)(∂βi)E(f)(x) and obtain the (topological) subspace of CVk(Ω,E) given by CVk P (∂)(Ω,E) ∶= {f ∈ CVk(Ω,E) ∣ f ∈ kerP(∂)E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Choosing AP(Ω,E) ∶= kerP(∂)E, we see that this is also a dom-space by a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If P(∂)E is the Cauchy–Riemann operator (and E locally complete) or the Laplacian, we obtain the weighted space of holomorphic resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let us show that the generators of these spaces are strong and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In order to obtain consistency for their generators we have to restrict to directed families of weights which are locally bounded away from zero on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ∀ K ⊂ Ω compact, m ∈ N0 ∃ j ∈ J ∀ β ∈ Nd 0, ∣β∣ ≤ min(m,k) ∶ inf x∈K νj,m(β,x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This condition on Vk guarantees that the map I∶CVk(Ω) → CWk(Ω), f ↦ f, is continuous which is needed for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator of (CVk,E) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (CVk P (∂),E) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 is strong and consistent if CVk(Ω) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CVk P (∂)(Ω) is barrelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We recall the definitions from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have ωm ∶= Mm × Ω with Mm ∶= {β ∈ Nd 0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0 or ωm ∶= Nd 0 × Ω for all m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, APCVk(Ω,E) = EΩ, APCVk P (∂)(Ω,E) = kerP(∂)E, domT E m ∶= Ck(Ω,E) and T E m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)], for all m ∈ N0 and the same with K instead of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The family (T E m,T K m)m∈N0 is a strong generator for (CVk,E) because (∂β)K(e′ ○ f)(x) = e′((∂β)Ef(x)), (β,x) ∈ ωm, for all e′ ∈ E′, f ∈ CVk(Ω,E) and m ∈ N0 due to the linearity and continuity of e′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In addition, e′ ○ f ∈ kerP(∂)K for all e′ ∈ E′ and f ∈ CVk P (∂)(Ω,E), which implies that (T E m,T K m)m∈N0 is also a strong generator for (CVk P (∂),E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For consistency we need to prove that (∂β)ES(u)(x) = u(δx ○ (∂β)K), (β,x) ∈ ωm, for all u ∈ CVk(Ω)εE resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' u ∈ CVk P (∂)(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This follows from the subsequent Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11 b) since FV(Ω) = CVk(Ω) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' FV(Ω) = CVk P (∂)(Ω) is barrelled and Vk locally bounded away from zero on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus (T E m,T K m)m∈N0 is a consistent generator for (CVk,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In addition, we have with P(∂)E from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 g) that P(∂)E(S(u))(x) = n ∑ i=1 ai(x)(∂βi)E(S(u))(x) = u( n ∑ i=1 ai(x)(δx ○ (∂βi)K)) = u(δx ○ P(∂)K) = 0, x ∈ Ω, (7) for every u ∈ CVk P (∂)(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This yields S(u) ∈ kerP(∂)E for all u ∈ CVk P (∂)(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore (T E m,T K m)m∈N0 is a consistent generator for (CVk P (∂),E) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-INTO-COMPATIBILITY 27 Let us turn to the postponed part in the proof of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We denote by CW(Ω) the space of scalar-valued continuous functions on an open set Ω ⊂ Rd with the topology of uniform convergence on compact subsets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the weighted topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and we set δ(x) ∶= δx for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω ⊂ Rd be open, k ∈ N∞ and FV(Ω) a dom-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) If T ∈ L(FV(Ω),CW(Ω)), then δ ○ T ∈ C(Ω,FV(Ω)′ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) If T ∈ L(FV(Ω),CW1(Ω)) and FV(Ω) is barrelled, then (∂en)FV(Ω)′ κ(δ ○ T)(x) = lim h→0 δx+hen ○ T − δx ○ T h = δx ○ (∂en)K ○ T, x ∈ Ω, 1 ≤ n ≤ d, and δ ○ T ∈ C1(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) If the inclusion I∶FV(Ω) → CWk(Ω), f ↦ f, is continuous and FV(Ω) barrelled, then S(u) ∈ Ck(Ω,E) and (∂β)ES(u)(x) = u(δx ○ (∂β)K), β ∈ Nd 0, ∣β∣ ≤ k, x ∈ Ω, for all u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we observe that (δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f) for every f ∈ FV(Ω) as T(f) is continuous on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Second, let K ⊂ Ω be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there are j ∈ J, m ∈ M and C > 0 such that sup x∈K ∣(δx ○ T)(f)∣ = sup x∈K ∣T(f)(x)∣ ≤ C∣f∣j,m for every f ∈ FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This means that {δx ○ T ∣ x ∈ K} is equicontinuous in FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The topologies σ(FV(Ω)′,FV(Ω)) and γ(FV(Ω)′,FV(Ω)) coincide on equicontinuous subsets of FV(Ω)′, implying that the restriction (δ ○ T)∣K∶K → FV(Ω)′ γ is continuous by our first observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As δ ○ T is continuous on every compact subset of the open set Ω ⊂ Rd, it follows that δ ○ T∶Ω → FV(Ω)′ γ is well- defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) Let x ∈ Ω and 1 ≤ n ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there is ε > 0 such that x + hen ∈ Ω for all h ∈ R with 0 < ∣h∣ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that δ ○ T ∈ C(Ω,FV(Ω)′ κ) by part a), which implies δx+hen○T −δx○T h ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For every f ∈ FV(Ω) we have lim h→0 δx+hen ○ T − δx ○ T h (f) = lim h→0 T(f)(x + hen) − T(f)(x) h = (∂en)KT(f)(x) in K as T(f) ∈ C1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore 1 h(δx+hen ○ T − δx ○ T) converges to δx ○ (∂en)K ○ T in FV(Ω)′ σ and thus in FV(Ω)′ κ by the Banach–Steinhaus theorem as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, we obtain δx ○ (∂en)K ○ T = lim h→0 δx+hen ○ T − δx ○ T h = (∂en)FV(Ω)′ κ(δ ○ T)(x) in FV(Ω)′ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, δ ○ (∂en)K ○ T ∈ C(Ω,FV(Ω)′ κ) by part a) as (∂en)K ○ T ∈ L(FV(Ω),CW(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we deduce that δ ○ T ∈ C1(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) We prove our claim by induction on the order of differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For β ∈ Nd 0 with ∣β∣ = 0 we get S(u) = u ○ δ ∈ C(Ω,E) from part a) with T = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, (∂β)ES(u)(x) = S(u)(x) = u(δx) = u(δx ○ (∂β)K), x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES Let m ∈ N0, m < k, such that S(u) ∈ Cm(Ω,E) and (∂β)ES(u)(x) = u(δx ○ (∂β)K), x ∈ Ω, (8) for all β ∈ Nd 0 with ∣β∣ ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let β ∈ Nd 0 with ∣β∣ = m + 1 ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there is 1 ≤ n ≤ d and ̃β ∈ Nd 0 with ∣̃β∣ = m such that β = en + ̃β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The barrelledness of FV(Ω) yields that 1 h(δx+hen ○ (∂ ̃β)K − δx ○ (∂ ̃β)K) converges to δx ○ (∂en)K ○ (∂ ̃β)K in FV(Ω)′ κ for every x ∈ Ω by part b) with T ∶= (∂ ̃β)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore we derive from δx ○ (∂en)K ○ (∂ ̃β)K = δx ○ (∂β)K by Schwarz’ theorem that u(δx ○ (∂β)K) = lim h→0 1 h(u(δx+hen ○ (∂ ̃β)K) − u(δx ○ (∂ ̃β)K)) = (8) lim h→0 1 h((∂ ̃β)ES(u)(x + hen) − (∂ ̃β)ES(u)(x)) = (∂en)E(∂ ̃β)ES(u)(x) for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, δ ○ (∂β)K = (∂en)FV(Ω)′ κ(δ ○ T) ∈ C(Ω,FV(Ω)′ κ) for T = (∂ ̃β)K by part b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we have S(u) ∈ Cm+1(Ω,E) and it follows from Schwarz’ theorem again that u(δx ○ (∂β)K) = (∂en)E(∂ ̃β)ES(u)(x) = (∂β)ES(u)(x), x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Part a) of the preceding proposition is just a modification of [16, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Lemma, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 198], where FV(Ω) = CV(Ω) is the Nachbin-weighted space of continuous functions and T = id, and holds more general for kR-spaces Ω (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (T E m,T K m)m∈M be a consistent generator for (FV,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then the map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism into, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the spaces FV(Ω) and FV(Ω,E) are ε-into-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' First, we show that S(FV(Ω)εE) ⊂ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to the consistency of (T E m,T K m)m∈M we have S(u) ∈ AP(Ω,E) ∩ domT E m and (T E mS(u))(x) = u(T K m,x), m ∈ M x ∈ ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, we get by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8 for every j ∈ J, m ∈ M and α ∈ A ∣S(u)∣j,m,α = sup x∈ωm pα(T E m(S(u))(x))νj,m(x) = ∥u∥j,m,α < ∞, (9) implying S(u) ∈ FV(Ω,E) and the continuity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, we deduce from (9) that S is injective and that the inverse of S on the range of S is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If J, M and A are countable, then S is an isometry with respect to the induced metrics on FV(Ω,E) and FV(Ω)εE by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The basic idea for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12 was derived from analysing the proof of an analogous statement for Bierstedt’s weighted spaces CV(Ω,E) and CV0(Ω,E) of continuous functions already mentioned in the introduction (see [16, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Lemma, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Folgerung, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 199–200] and [17, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Satz, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 137]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-compatibility Now, we try to answer the natural question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' When is S surjective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The strength of a generator and a weaker concept to define a natural E-valued version of FV(Ω) come into play to answer the question on the surjectivity of our key map S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let FV(Ω) be a dom-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We define the linear space of E-valued weak FV-functions by FV(Ω,E)σ ∶= {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ FV(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-COMPATIBILITY 29 Moreover, for f ∈ FV(Ω,E)σ we define the linear map Rf∶E′ → FV(Ω), Rf(e′) ∶= e′ ○ f, and the dual map Rt f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], where E′⋆ is the algebraic dual of E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, we set FV(Ω,E)κ ∶= {f ∈ FV(Ω,E)σ ∣ ∀ α ∈ A ∶ Rf(B○ α) relatively compact in FV(Ω)} where Bα ∶= {x ∈ E ∣ pα(x) < 1} for α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Next, we give a sufficient condition for the inclusion FV(Ω,E) ⊂ FV(Ω,E)σ by means of the family (T E m,T K m)m∈M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If (T E m,T K m)m∈M is a strong generator for (FV,E), then we have FV(Ω,E) ⊂ FV(Ω,E)σ and sup e′∈B○α ∣Rf(e′)∣j,m = ∣f∣j,m,α (10) for every f ∈ FV(Ω,E), j ∈ J, m ∈ M and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have e′ ○ f ∈ F(Ω) for every e′ ∈ E′ since (T E m,T K m)m∈M is a strong generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, we have ∣Rf(e′)∣j,m = ∣e′ ○ f∣j,m = sup x∈ωm ∣T K m(e′ ○ f)(x)∣νj,m(x) = sup x∈ωm ∣e′(T E m(f)(x))∣νj,m(x) = sup x∈Nj,m(f) ∣e′(x)∣ (11) for every j ∈ J and m ∈ M with the set Nj,m(f) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that Nj,m(f) is bounded in E by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and thus weakly bounded, im- plying that the right-hand side of (11) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we conclude f ∈ FV(Ω,E)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we observe that sup e′∈B○α ∣Rf(e′)∣j,m = ∣f∣j,m,α for every j ∈ J, m ∈ M and α ∈ A due to [131, Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Now, we phrase some sufficient conditions for FV(Ω,E) ⊂ FV(Ω,E)κ to hold which is one of the key points regarding the surjectivity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If (T E m,T K m)m∈M is a strong generator for (FV,E) and one of the following conditions is fulfilled, then FV(Ω,E) ⊂ FV(Ω,E)κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) FV(Ω) is a semi-Montel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) E is a semi-Montel or Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ γ(E) ∶ Nj,m(f) ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' By virtue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 we already have f ∈ FV(Ω,E)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) For every j ∈ J, m ∈ M and α ∈ A we derive from sup e′∈B○α ∣Rf(e′)∣j,m = (10) ∣f∣j,m,α < ∞ that Rf(B○ α) is bounded and thus relatively compact in the semi-Montel space FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) It follows from (11) that Rf ∈ L(E′ γ,FV(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, the polar B○ α is relatively compact in E′ γ for every α ∈ A by the Alaoğlu–Bourbaki theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The continuity of Rf implies that Rf(B○ α) is relatively compact as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) Let j ∈ J and m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set K ∶= Nj,m(f) is bounded in E by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We deduce that K is already precompact in E by [89, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Corollary, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 202] if E is a Schwartz space resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' since it is relatively compact if E is a semi-Montel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence the statement follows from c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES Let us turn to sufficient conditions for FV(Ω,E) ≅ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For the lcHs E we denote by J ∶E → E′⋆, x �→ [e′ ↦ e′(x)], the canonical injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (T E m,T K m)m∈M be a strong generator for (FV,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Define the following conditions: a) E is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) E is quasi-complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there is a bounded net (f ′ τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′ κ such that Rt f(f ′ τ) ∈ J (E) for every τ ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) E is sequentially complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there is a sequence (f ′ n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′ κ such that Rt f(f ′ n) ∈ J (E) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' d) E is locally complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there is a sequence (f ′ n)n∈N in FV(Ω)′ locally converging to f ′ in FV(Ω)′ κ such that Rt f(f ′ n) ∈ J (E) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' e) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ τ(E) ∶ Nj,m(f) ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (T E m,T K m)m∈M be a consistent generator for (FV,E) and let FV(Ω,E) ⊂ FV(Ω,E)κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If one of the Conditions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 is fulfilled, then the map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' FV(Ω) and FV(Ω,E) are ε-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The inverse of S is given by the map Rt∶FV(Ω,E) → FV(Ω)εE, f ↦ J −1 ○ Rt f, where J ∶E → E′⋆ is the canonical injection and Rt f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], with Rf(e′) = e′ ○ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12 we only have to show that S is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We equip J (E) with the system of seminorms given by pB○α(J (x)) ∶= sup e′∈B○α ∣J (x)(e′)∣ = pα(x), x ∈ E, (12) for every α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We consider the dual map Rt f and claim that Rt f ∈ L(FV(Ω)′ κ,J (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Indeed, we have pB○α(Rt f(y)) = sup e′∈B○α ∣y(Rf(e′))∣ = sup x∈Rf (B○α) ∣y(x)∣ ≤ sup x∈Kα ∣y(x)∣ (13) for all y ∈ FV(Ω)′ where Kα ∶= Rf(B○α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since FV(Ω,E) ⊂ FV(Ω,E)κ, the set Rf(B○ α) is absolutely convex and relatively compact, implying that Kα is absolutely convex and compact in FV(Ω) by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we have for all e′ ∈ E′ and x ∈ Ω Rt f(δx)(e′) = δx(e′ ○ f) = e′(f(x)) = J (f(x))(e′) (14) and thus Rt f(δx) ∈ J (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) Let E be complete and f ′ ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since the span of {δx ∣ x ∈ Ω} is dense in F(Ω)′ κ by the bipolar theorem, there is a net (f ′ τ) converging to f ′ in FV(Ω)′ κ with Rt f(f ′ τ) ∈ J (E) by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As pB○α(Rt f(f ′ τ) − Rt f(f ′)) ≤ (13) sup x∈Kα ∣(f ′ τ − f ′)(x)∣ → 0, (15) for all α ∈ A, we gain that (Rt f(f ′ τ)) is a Cauchy net in the complete space J (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence it has a limit g ∈ J (E) which coincides with Rt f(f ′) since pB○α(g − Rt f(f ′)) ≤ pB○α(g − Rt f(f ′ τ)) + pB○α(Rt f(f ′ τ) − Rt f(f ′)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-COMPATIBILITY 31 ≤ (15)pB○α(g − Rt f(f ′ τ)) + sup x∈Kα ∣(f ′ τ − f ′)(x)∣ → 0 for all α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We conclude that Rt f(f ′) ∈ J (E) for every f ′ ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) Let Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 b) hold and f ′ ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there is a bounded net (f ′ τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′ κ such that Rt f(f ′ τ) ∈ J (E) for every τ ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to (13) we obtain that (Rt f(f ′ τ)) is a bounded Cauchy net in the quasi-complete space J (E) converging to Rt f(f ′) ∈ J (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) Let Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 c) hold and f ′ ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there is a sequence (f ′ n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′ κ such that Rt f(f ′ n) ∈ J (E) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Again (13) implies that (Rt f(f ′ n)) is a Cauchy sequence in the sequentially complete space J (E) which converges to Rt f(f ′) ∈ J (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' d) Let Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 d) hold and f ′ ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there is an absolutely convex, bounded subset D ⊂ FV(Ω)′ κ and a sequence (f ′ n)n∈N in FV(Ω)′ converging to f ′ in (FV(Ω)′ κ)D such that Rt f(fn) ∈ J (E) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let r > 0 and f ′ n − f ′ k ∈ rD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then Rt f(f ′ n − f ′ k) ∈ r(Rt f(D) ∩ J (E)), implying {r > 0 ∣ f ′ n − f ′ k ∈ rD} ⊂ {r > 0 ∣ Rt f(f ′ n − f ′ k) ∈ r(Rt f(D) ∩ J (E)) J (E)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Setting B ∶= Rt f(D) ∩ J (E) J (E), we derive qB(Rt f(f ′ n − f ′ k)) ≤ qD(f ′ n − f ′ k) where qB and qD are the gauge functionals of B resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set Rt f(D)∩J (E) is absolutely convex as the intersection of two absolutely convex sets and it is bounded by (13) and the boundedness of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' So B, being the closure of a disk, is a disk as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since (f ′ n) is a Cauchy sequence in (FV(Ω)′ κ)D, we conclude that (R′ f(f ′ n)) is a Cauchy sequence in J (E)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set B is a closed disk in the locally complete space J (E) and hence a Banach disk by [89, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus J (E)B is a Banach space and (Rt f(f ′ n)) has a limit g ∈ J (E)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The continuity of the canonical injection J (E)B ↪ J (E) implies that (Rt f(f ′ n)) converges to g in J (E) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As in a) we obtain that Rt f(f ′) = g ∈ J (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' e) Let Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 e) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ FV(Ω,E) and e′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For every f ′ ∈ FV(Ω)′ there are j ∈ J, m ∈ M and C > 0 such that ∣Rt f(f ′)(e′)∣ ≤ C∣Rf(e′)∣j,m = (11) C sup x∈Nj,m(f) ∣e′(x)∣ because (T E m,T K m)m∈M is a strong generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since there is K ∈ τ(E) such that Nj,m(f) ⊂ K, we have ∣Rt f(f ′)(e′)∣ ≤ C sup x∈K ∣e′(x)∣, implying Rt f(f ′) ∈ (E′ τ)′ = J (E) by the Mackey–Arens theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore we obtain that Rt f ∈ L(FV(Ω)′ κ,J (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' So we get for all α ∈ A and y ∈ F(Ω)′ pα((J −1 ○ Rt f)(y)) = (12) pB○α(J ((J −1 ○ Rt f)(y))) = pB○α(Rt f(y)) ≤ (13) sup x∈Kα ∣y(x)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This implies J −1 ○ Rt f ∈ L(FV(Ω)′ κ,E) = FV(Ω)εE (as linear spaces) and we gain S(J −1 ○ Rt f)(x) = J −1(Rt f(δx)) = (14) J −1(J (f(x))) = f(x) for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus S(J −1 ○ Rt f) = f, proving the surjectivity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Further sufficient conditions for S being a topological isomorphism can be found in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, we get the following corollary as a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (T E m,T K m)m∈M be a strong, consistent generator for (FV,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If (i) FV(Ω) is a semi-Montel space and E complete, or (ii) FV(Ω) is a Fréchet–Schwartz space and E locally complete, or (iii) E is a semi-Montel space, or (iv) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ κ(E) ∶ Nj,m(f) ⊂ K, then FV(Ω) and FV(Ω,E) are ε-compatible, in particluar, FV(Ω,E) ≅ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (i) Follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 a) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 with Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (ii) If FV(Ω) is a Fréchet–Schwartz space, then we have span{δx ∣ x ∈ Ω} lc = span{δx ∣ x ∈ Ω} FV(Ω)′ b = span{δx ∣ x ∈ Ω} FV(Ω)′ κ = FV(Ω)′ by [30, Lemma 6 (b), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 231] and the bipolar theorem where span{δx ∣ x ∈ Ω} lc is the local closure of span{δx ∣ x ∈ Ω} in FV(Ω)′ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence for every f ′ ∈ FV(Ω)′ there is a sequence (f ′ n) in the span of {δx ∣ x ∈ Ω} which converges locally to f ′ in FV(Ω)′ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to (14) we know that Rt f(f ′ n) ∈ J (E) for every f ∈ FV(Ω,E) and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since Fréchet–Schwartz spaces are also semi-Montel spaces, the statement follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 a) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 with Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (iv) Follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 c) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 with Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (iii) Is a special case of (iv) since the set K ∶= acx(Nj,m(f)) is absolutely convex and compact in the semi-Montel space E by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 103] and [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 112] for every f ∈ FV(Ω,E), j ∈ J and m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Linearisations of spaces FV(Ω,E)σ of weak E-valued func- tions, where FV(Ω) need not be a dom-space, are treated in [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let us apply our preceding results to our weighted spaces of k-times continu- ously partially differentiable functions on an open set Ω ⊂ Rd with k ∈ N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) CVk(Ω,E) ≅ CVk(Ω)εE if E is a semi-Montel space and CVk(Ω) barrelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) CVk P (∂)(Ω,E) ≅ CVk P (∂)(Ω)εE if E is a semi-Montel space and CVk P (∂)(Ω) barrelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) CVk(Ω,E) ≅ CVk(Ω)εE if E is complete and CVk(Ω) a Montel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' d) CVk P (∂)(Ω,E) ≅ CVk P (∂)(Ω)εE if E is complete and CVk P (∂)(Ω) a Montel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' e) CVk(Ω,E) ≅ CVk(Ω)εE if E is locally complete and CVk(Ω) a Fréchet– Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' f) CVk P (∂)(Ω,E) ≅ CVk P (∂)(Ω)εE if E is locally complete and CVk P (∂)(Ω) a Fréchet–Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator of (CVk,E) and (CVk P (∂),E) is strong and consistent by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iii) we deduce part a) and b), from (i) part c) and d) and from (ii) part e) and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Closed subspaces of Fréchet–Schwartz spaces are also Fréchet–Schwartz spaces by [131, Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The space CV∞ P (∂)(Ω) is closed in CV∞(Ω) if there is an lcHs Y such that P(∂)∣CV∞(Ω)∶CV∞(Ω) → Y is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For example, this is fulfilled if the coefficients of P(∂) belong to C(Ω), in particular, if P(∂) ∶= ∆ or ∂, with Y ∶= (C(Ω),τc) due to V∞ being locally bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The spaces CVk(Ω) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 a)(i) with ωm ∶= Mm × Ω for all m ∈ N0, where Mm ∶= {β ∈ Nd 0 ∣ ∣β∣ ≤ min(m,k)}, are Fréchet spaces and thus barrelled if the J 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-COMPATIBILITY 33 in Vk ∶= (νj,m)j∈J,m∈N0 is countable by [107, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 240].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Sufficient conditions on the weights that guarantee that CV∞(Ω) is a nuclear Fréchet space and hence a Schwartz space as well can be found in [111, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For the case ωm = Nd 0 × Ω see the references given in [111, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If Vk = Wk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Ck(Ω,E) is equipped with its usual topology of uniform conver- gence of all partial derivatives up to order k on compact subsets of Ω, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 c)+d) can be improved to quasi-complete E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For Ω = Rd this can be found in [158, Proposition 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 108, Théorème 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 111] and for general open Ω ⊂ Rd it is already mentioned in [94, (9), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 236] (without a proof) that CWk(Ω,E) ≅ CWk(Ω)εE for k ∈ N∞ and quasi-complete E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For k = ∞ we even have CW∞(Ω,E) ≅ CW∞(Ω)εE for locally complete E by [30, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Our technique allows us to generalise the first result and to get back the second result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If k < ∞ and E has metric ccp, or if k = ∞ and E is locally complete, then a) CWk(Ω,E) ≅ CWk(Ω)εE, and b) CWk P (∂)(Ω,E) ≅ CWk P (∂)(Ω)εE if CWk P (∂)(Ω) is closed in CWk(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We recall from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 b) that Wk is the family of weights given by νK,m(β,x) ∶= χK(x), (β,x) ∈ Mm × Ω, for all m ∈ N0 and compact K ⊂ Ω where Mm ∶= {β ∈ Nd 0 ∣ ∣β∣ ≤ min(m,k)} and χK is the characteristic function of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We already know that the generator for (CWk,E) and (CWk P (∂),E) is strong and consistent by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10 because Wk is locally bounded away from zero on Ω, and CWk(Ω) and its closed subspace CWk P (∂)(Ω) are Fréchet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ CWk(Ω,E), K ⊂ Ω be compact, m ∈ N0 and consider NK,m(f) = {(∂β)Ef(x)νK,m(β,x) ∣ x ∈ Ω, β ∈ Mm} = {0} ∪ ⋃ β∈Mm (∂β)Ef(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' NK,m(f) is compact since it is a finite union of compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, the compact sets {0} and (∂β)Ef(K) are metrisable by [34, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' IX, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, Proposi- tion 17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 159] and thus their finite union NK,m(f) is metrisable as well by [169, Theorem 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 361] since the compact set NK,m(f) is collectionwise normal and locally countably compact by [63, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='18 Theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 305].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E has metric ccp, then the set acx(NK,m(f)) is absolutely convex and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv) settles the case for k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If k = ∞ and E is locally complete, we observe that Kβ ∶= acx((∂β)Ef(K)) for f ∈ CW∞(Ω,E) is absolutely convex and compact by [29, Proposition 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then we have NK,m(f) ⊂ acx( ⋃ β∈Mm Kβ) and the set on the right-hand side is absolutely convex and compact by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Again, the statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The statement above for k = ∞ follows from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 e)+f) as well because CW∞(Ω) and its closed subspaces are Fréchet–Schwartz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In the context of differentiability on infinite dimensional spaces the preceding example a) remains true for an open subset Ω of a Fréchet space or DFM-space and quasi-complete E by [129, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Corollary, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 286].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Like here this can be generalised to E with [metric] ccp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A special case of example b) is already known to be a consequence of [30, Theorem 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 232], namely, if k = ∞ and P(∂)K is hypoelliptic with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, this covers the space of holomorphic functions and the space of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Holomorphy on infinite dimensional spaces is treated in [52, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='35, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 332–333] where V = W0, Ω is an open subset of a locally 34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES convex Hausdorff k-space and E a quasi-complete locally convex Hausdorff space, both over C, which can be generalised to E with [metric] ccp in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For a second improvement of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 for k = ∞ to locally complete E without the condition that CV∞(Ω) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CV∞ P (∂)(Ω) is a Fréchet–Schwartz space we introduce the following conditions on the family V∞ on (Mm × Ω)m∈N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We say that a family V∞ of weights on (Mm × Ω)m∈N0 is C1-controlled if (i) ∀ j ∈ J, m ∈ N0, β ∈ Mm ∶ νj,m(β,⋅) ∈ C1(Ω), (ii) ∀ j ∈ J, m ∈ N0, β,γ ∈ Mm,x ∈ Ω ∶ νj,m(β,x) = νj,m(γ,x), (iii) ∀ j ∈ J, m ∈ N0 ∃ i ∈ J, k ∈ N0, k ≥ m, C > 0 ∀ β ∈ Mm, x ∈ Ω, 1 ≤ n ≤ d ∶ ∣∂enνj,m(β,⋅)∣(x) ≤ Cνi,k(β,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We say that family Vk, k ∈ N∞, fulfils condition (V∞) if ∀ m ∈ N0, j ∈ J ∃ n ∈ N≥m, i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ β ∈ Mm, x ∈ Ω ∖ K ∶ νj,m(β,x) ≤ ενi,n(β,x) where N≥m ∶= {n ∈ N0 ∣ n ≥ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Here (V∞) stands for vanishing at infinity and the condition was introduced in [107, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 239] and for k = 0 in [16, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Bemerkung, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs and V∞ a directed C1-controlled family of weights on an open convex set Ω ⊂ Rd which fulfils (V∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is locally complete, then a) CV∞(Ω,E) ≅ CV∞(Ω)εE if CV∞(Ω) is barrelled, and b) CV∞ P (∂)(Ω,E) ≅ CV∞ P (∂)(Ω)εE if CV∞ P (∂)(Ω) is barrelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We already know that the generator for (CV∞,E) and (CV∞ P (∂),E) is strong and consistent by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10 because V∞ is locally bounded away from zero on Ω as νj,m(β,⋅) is continuous for all j ∈ J, m ∈ N0 and β ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ CV∞(Ω,E), j ∈ J, m ∈ N0 and β ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We set g∶Ω → E, g(x) ∶= (∂β)Ef(x)νj,m(β,x), and note that (∂en)Eg(x) = (∂β+en)Ef(x)νj,m(β,x) + (∂β)Ef(x)((∂en)Rνj,m(β,⋅))(x), x ∈ Ω, for all 1 ≤ n ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since V∞ is directed and C1-controlled there are i1,i2 ∈ J, k1,k2 ∈ N0, k1 > m, k2 ≥ m, and C1,C2 > 0 such that pα((∂en)Eg(x)) ≤ pα((∂β+en)Ef(x))νj,m(β,x) + pα((∂β)Ef(x))∣(∂en)Rνj,m(β,⋅)∣(x) ≤ C1pα((∂β+en)Ef(x))νi1,k1(β,x) + C2pα((∂β)Ef(x))νi2,k2(β,x) = C1pα((∂β+en)Ef(x))νi1,k1(β + en,x) + C2pα((∂β)Ef(x))νi2,k2(β,x) for all 1 ≤ n ≤ d and α ∈ A, which implies sup x∈Ω γ∈Nd 0,∣γ∣≤1 pα((∂γ)Eg(x)) ≤ ∣f∣j,m,α + C1∣f∣i1,k1,α + C2∣f∣i2,k2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus g is (weakly) C1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to (V∞) there are n ∈ N≥m and i ∈ J such that for all ε > 0 there is a compact set K ⊂ Ω such that for all β ∈ Mm and x ∈ Ω ∖ K we have νj,m(β,x) ≤ ενi,n(β,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since V∞ is directed, we may assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' that νj,m(β,x) ≤ νi,n(β,x) for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This implies that the zeros of νi,n(β,⋅) are zeros of νj,m(β,⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We define h∶Ω → [0,∞) by h(x) ∶= νi,n(β,x)/νj,m(β,x) for x ∈ Ω with νj,m(β,x) ≠ 0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ε-COMPATIBILITY 35 h(x) ∶= 1 if νj,m(β,x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that h(x) > 0 for all x ∈ Ω as the zeros of νi,n(β,⋅) are contained in the zeros of νj,m(β,⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows that (∂β)Ef(x)νj,m(β,x)h(x) = (∂β)Ef(x)νi,n(β,x) for x ∈ Ω with νj,m(β,x) ≠ 0 and (∂β)Ef(x)νj,m(β,x)h(x) = 0 for x ∈ Ω with νj,m(β,x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore (∂β)Efνj,m(β,⋅)h is bounded on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, εh(x) = ενi,n(β,x)/νj,m(β,x) ≥ 1 for x ∈ Ω ∖ K with νj,m(β,x) ≠ 0 because (V∞) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, the zeros of νj,m(β,⋅) are contained in N ∶= {x ∈ Ω ∣ (∂β)Ef(x)νj,m(β,x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This yields that Kβ ∶= acx((∂β)Efνj,m(β,⋅)(Ω)) is absolutely convex and compact by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, Nj,m(f) = {(∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} ⊂ acx( ⋃ β∈Mm Kβ) and the set on the right-hand side is absolutely convex and compact by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Finally, our statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ For the Schwartz space S(Rd,E) and the multiplier space OM(Rd,E) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 c) and d) an improvement of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 c) to quasi-complete E is already known, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' [158, Proposition 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 108, Théorème 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' However, due to Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 it is even allowed that E is only locally complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is a locally complete lcHs, then S(Rd,E) ≅ S(Rd)εE and OM(Rd,E) ≅ OM(Rd)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We start with the Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 a) and the barrelledness of the Fréchet space S(Rd) we only need to check that its directed family V∞ ∶= (ν1,m)m∈N0 of weights given by ν1,m(β,x) ∶= (1 + ∣x∣2)m/2, x ∈ Rd, for m ∈ N0 and β ∈ Mm is C1-controlled and fulfils (V∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Obviously, condition (i) and (ii) are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since ∣∂enν1,m(β,⋅)∣(x) = (m/2)(1 + ∣x∣2)(m/2)−12∣xn∣ ≤ m(1 + ∣x∣2)m/2 = mν1,m(β,x) for all x ∈ Rd and 1 ≤ n ≤ d, condition (iii) is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus V∞ is C1-controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Noting that for every m ∈ N and ε > 0 there is r > 0 such that (1 + ∣x∣2)m/2 (1 + ∣x∣2)m = (1 + ∣x∣2)−m/2 ≤ ε for all x ∉ Br(0), we obtain that ν1,m(β,x) ≤ εν1,2m(β,x) for all x ∉ Br(0) and β ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence V∞ fulfils condition (V∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Now, let us consider the multiplier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We already know that the generator for (OM,E) is strong and consistent by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10 because OM(R) is a Montel space, thus barrelled, by [83, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' II, §4, n○4, Théorème 16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 131] and its family of weights is continuous on Rd, thus locally bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ OM(R,E), g ∈ S(Rd), m ∈ N0 and β ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then (∂β)Ef ∈ OM(R,E) and hence ((∂β)Ef)g ∈ S(Rd,E), which implies that ((∂β)Ef)g ∈ C1 b (Rd,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' More- over, we choose h∶Rd → (0,∞), h(x) ∶= 1 + ∣x∣2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then ((∂β)Ef)gh is bounded on Rd and for ε > 0 there is r > 0 such that (1 + ∣x∣2)−1 ≤ ε for all x ∉ Br(0), yielding that Kβ,g ∶= acx(((∂β)Ef)g(Rd)) is absolutely convex and compact by Proposi- tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let j ⊂ S(Rd) be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since for each x ∈ Rd we have (∂β)Ef(x)maxg∈j ∣g(x)∣ = eiθ(∂β)Ef(x)̃g(x) for some ̃g ∈ j and θ ∈ [0,2π), we get Nj,m(f) = {(∂β)Ef(x)max g∈j ∣g(x)∣ ∣ x ∈ Rd, β ∈ Mm} ⊂ acx( ⋃ β∈Mm,g∈j Kβ,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES The set on the right-hand side is absolutely convex and compact by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Finally, our statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ For an alternative proof in the case of the Schwartz space we may also use Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 e) since S(Rd) is a Fréchet–Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 can also be used for an alternative proof of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8 if k = ∞ by observing that CW∞(Ω,E) = CV∞(Ω,E) for any lcHs E where V∞ ∶= {ν ∈ C∞ c (Ω) ∣ ν ≥ 0} and C∞ c (Ω) is the space of functions in C∞(Ω) with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Now, we improve Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 for the special case of spaces of ultradifferen- tiable functions E(Mp)(Ω,E) and E{Mp}(Ω,E) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 e) and f) where ωm ∶= Nd 0 × Ω for all m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For this purpose we recall the following conditions of Komatsu for the sequence (Mp)p∈N0 (see [99, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 26] and [101, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 653]): (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0) M0 = M1 = 1, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1) ∀ p ∈ N ∶ M 2 p ≤ Mp−1Mp+1, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2)’ ∃ A,C > 0 ∀ p ∈ N0 ∶ Mp+1 ≤ ACp+1Mp, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3)’ ∑∞ p=1 Mp−1 Mp < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, Ω ⊂ Rd open and (Mp)p∈N0 a sequence of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) E(Mp)(Ω,E) ≅ E(Mp)(Ω)εE if E is locally complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is complete or semi-Montel and in both cases (Mp)p∈N0 fulfils (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1) and (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' c) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is sequentially complete and (Mp)p∈N0 fulfils (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2)’ and (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator is strong and consistent by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10 since the family of weights given in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 e) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' f) is locally bounded away from zero on Ω and E(Mp)(Ω) is a Fréchet–Schwartz space in a) by [99, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 44] whereas E{Mp}(Ω) is a Montel space in b) and c) by [99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 65–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence the statements a) and b) follow from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let us turn to c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that E{Mp}(Ω,E) ⊂ E{Mp}(Ω,E)κ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 a) for any lcHs E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we claim that Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 c) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ′ ∈ E{Mp}(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to [101, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 677] there is a sequence (fn)n∈N in the space D{Mp}(Ω) of ultradifferentiable functions of class {Mp} of Roumieu-type with compact support which converges to f ′ in E{Mp}(Ω)′ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ E{Mp}(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We observe that for every e′ ∈ E′ ∣Rt f(fn)(e′)∣ = ∣∫ Ω fn(x)e′(f(x))dx∣ ≤ λ(supp(fn)) sup y∈Kn(f) ∣e′(y)∣ where λ is the Lebesgue measure, supp(fn) is the support of fn and Kn(f) ∶= {fn(x)f(x) ∣ x ∈ supp(fn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set Kn(f) is compact and metrisable by [34, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' IX, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, Proposition 17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 159] and thus the closure of its absolutely convex hull is compact in E as the sequentially complete space E has metric ccp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We conclude that Rt f(fn) ∈ (E′ κ)′ = J (E) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 c) is fulfilled, implying statement c) for sequentially complete E by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The results a) and b) in this example are new whereas c) is already proved in [101, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 678] in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, part a) improves [101, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 678] since Komatsu’s conditions (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='0), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2)’ and (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3)’ are not needed and the condition that E is sequentially complete is weakened to local completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We included c) to demonstrate an application of Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CHAPTER 4 Consistency 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The spaces AP(Ω,E) and consistency This section is dedicated to the properties of functions which are compatible with the ε-product in the sense that the space of functions having these properties can be chosen as the space AP(Ω,E) or ⋂m∈M domT E m in the Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7 b) of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This is done in a quite general way so that we are not tied to certain spaces and have to redo our argumentation, for example, if we consider the same generator (T E m,T K m)m∈M for two different spaces of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to the linearity and continuity of u ∈ FV(Ω)εE for a dom-space FV(Ω) and S(u) = u ○ δ with δ∶Ω → FV(Ω)′, x ↦ δx, these are properties which are purely pointwise or given by pointwise approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Among such properties of func- tions are continuity by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, Cauchy continuity by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3, uniform continuity by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, continuous extendability by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7, continuous differentiability by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, vanishing at infinity by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 and purely pointwise properties of a function like vanishing on a set by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We collect these properties in propositions and in follow-up lemmas we handle properties which can be described by compositions of defining operators T E m1 ○ T E m2 like continuous differentiability (of higher order) of Fourier transformations (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We fix the following notation for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For a dom-space FV(Ω) and linear T∶FV(Ω) → KΩ we set (δ ○ T)(x)(f) ∶= (δx ○ T)(f) ∶= T(f)(x) for all x ∈ Ω and f ∈ FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a topological Hausdorff space and FV(Ω) a dom-space such that FV(Ω) ⊂ C(Ω) as a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then S(u) ∈ C(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ C(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since S(u) = u○δ and δ ∈ C(Ω,FV(Ω)′ κ), we obtain that S(u) is in C(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Now, we tackle the problem of the continuity of δ∶Ω → FV(Ω)′ κ in the proposi- tion above and phrase our solution in a way such that it can be applied to show the continuity of the partial derivative (∂β)E(S(u)) as well (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We recall that a topological space Ω is called completely regular if for any non-empty closed subset A ⊂ Ω and x ∈ Ω ∖ A there is f ∈ C(Ω,[0,1]) such that f(x) = 0 and f(z) = 1 for all z ∈ A (see [88, Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 180]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Examples of completely regular spaces are uniformisable, particularly metrisable, spaces by [88, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 181] and locally convex Hausdorff spaces by [65, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A completely regular space Ω is a kR-space if for any completely regular space Y and any map f∶Ω → Y , whose restriction to each compact K ⊂ Ω is continuous, the map is already continuous on Ω (see [37, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7) Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Examples of kR-spaces are completely regular k-spaces by [63, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='21 Theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A topological space Ω is called k-space (compactly generated space) if it satisfies the following condition: A ⊂ Ω is closed if and only if A ∩ K is closed in K for every compact K ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Every locally compact Hausdorff space is a completely regular 37 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CONSISTENCY k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, every sequential Hausdorff space is a k-space by [63, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='20 The- orem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 152], in particular, every first-countable Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus metrisable spaces are completely regular Hausdorff k-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, the dual space (X′,τc) with the topology of compact convergence τc is an example of a completely regular Hausdorff k-space that is neither locally compact nor metrisable by [178, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 267] if X is an infinite-dimensional Fréchet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We denote by CW(Ω) the space of scalar-valued continuous functions on a topological Hausdorff space Ω with the topology τc of compact convergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the topology of uniform convergence on compact subsets, which itself is the weighted topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and by Cb(Ω) the space of scalar-valued bounded, continuous functions on Ω with the topology of uniform convergence on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a topological Hausdorff space, FV(Ω) a dom-space and T∶FV(Ω) → C(Ω) linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then δ ○T ∈ C(Ω,FV(Ω)′ γ) in each of the subsequent cases: (i) Ω is a kR-space and T∶FV(Ω) → CW(Ω) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (ii) T∶FV(Ω) → Cb(Ω) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we observe that (δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f) for every f ∈ FV(Ω) as T(f) is continuous on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (i) Verbatim as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (ii) There are j ∈ J, m ∈ M and C > 0 such that sup x∈Ω ∣(δx ○ T)(f)∣ = sup x∈Ω ∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m for every f ∈ FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This means that {δx ○T ∣ x ∈ Ω} is equicontinuous in FV(Ω)′, yielding the statement like before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The preceding lemma is just a modification of [16, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Lemma, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 198] where FV(Ω) = CV(Ω), the Nachbin-weighted space of continuous functions, and T = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Next, we turn to Cauchy continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' A function f∶Ω → E from a metric space Ω to an lcHs E is called Cauchy continuous if it maps Cauchy sequences to Cauchy sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We write CC(Ω,E) for the space of Cauchy continuous functions from Ω to E and set CC(Ω) ∶= CC(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (Cauchy continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a metric space and FV(Ω) a dom-space such that FV(Ω) ⊂ CC(Ω) as a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then S(u) ∈ CC(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ CC(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE and (xn) a Cauchy sequence in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then (δxn) is a Cauchy sequence in FV(Ω)′ κ since δ ∈ CC(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows that (S(u)(xn)) is a Cauchy sequence in E because u is uniformly continuous and u(δxn) = S(u)(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we conclude that S(u) ∈ CC(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ For the next lemma we equip the space CC(Ω) with the topology of uniform convergence on precompact subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),CC(Ω)) for a metric space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then δ ○ T ∈ CC(Ω,FV(Ω)′ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (xn) be a Cauchy sequence in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have (δxn○T)(f) = T(f)(xn) for every f ∈ FV(Ω), which implies that ((δxn ○ T)(f)) is a Cauchy sequence in K because T(f) ∈ CC(Ω) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since K is complete, it has a unique limit T∞(f) ∶= limn→∞(δxn ○ T)(f) defining a linear functional in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set N ∶= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' THE SPACES AP(Ω, E) AND CONSISTENCY 39 {xn ∣ n ∈ N} is precompact in Ω since Cauchy sequences are precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence there are j ∈ J, m ∈ M and C > 0 such that sup n∈N ∣(δxn ○ T)(f)∣ = sup x∈N ∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m for every f ∈ FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore the set {δxn○T ∣ n ∈ N} is equicontinuous in FV(Ω)′, which implies that T∞ ∈ FV(Ω)′ and the convergence of (δxn ○T) to T∞ in FV(Ω)′ γ due to the observation in the beginning and the fact that γ(FV(Ω)′,FV(Ω)) and σ(FV(Ω)′,FV(Ω)) coincide on equicontinuous sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In particular, (δxn ○ T) is a Cauchy sequence in FV(Ω)′ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Furthermore, for every x ∈ Ω we obtain from the choice xn = x for all n ∈ N that δx ○ T ∈ FV(Ω)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus the map δ ○ T∶Ω → FV(Ω)′ γ is well-defined and Cauchy continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The subsequent proposition and lemma handle the analogous statements for uniform continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For a metric space Ω we denote by Cu(Ω,E) the space of uniformly continuous functions from Ω to E and set Cu(Ω) ∶= Cu(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (uniform continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (Ω,d) be a metric space and FV(Ω) a dom-space such that FV(Ω) ⊂ Cu(Ω) as a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then S(u) ∈ Cu(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cu(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (zn), (xn) be sequences in Ω with limn→∞ d(zn,xn) = 0 and u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then (δzn −δxn) converges to 0 in FV(Ω)′ κ because δ ∈ Cu(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As a consequence (S(u)(zn) − S(u)(xn)) converges to 0 in E since u is uniformly continuous and u(δzn −δxn) = S(u)(zn)−S(u)(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we conclude that S(u) ∈ Cu(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ For the next lemma we mean by Cbu(Ω) the space of scalar-valued bounded, uniformly continuous functions equipped with the topology of uniform convergence on a metric space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),Cbu(Ω)) for a metric space (Ω,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then δ ○ T ∈ Cu(Ω,FV(Ω)′ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (zn) and (xn) be sequences in Ω such that limn→∞ d(zn,xn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have (δzn ○ T − δxn ○ T)(f) = T(f)(zn) − T(f)(xn) for every f ∈ FV(Ω), which implies that (δzn ○ T − δxn ○ T)(f) converges to 0 in K for every f ∈ FV(Ω) because T(f) ∈ Cu(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' There exist j ∈ J, m ∈ M and C > 0 such that sup n∈N ∣(δzn ○ T − δxn ○ T)(f)∣ ≤ 2sup x∈Ω ∣T(f)(x)∣ ≤ 2C∣f∣FV(Ω),j,m for every f ∈ FV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore the set {δzn ○ T − δxn ○ T ∣ n ∈ N} is equicontinuous in FV(Ω)′ and we conclude the statement like before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Let us turn to continuous extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let X be a metric space and Ω ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We write Cext(Ω,E) for the space of functions f ∈ C(Ω,E) which have a continuous extension to Ω and set Cext(Ω) ∶= Cext(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (continuous extendability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let X be a metric space, Ω ⊂ X and FV(Ω) a dom-space such that FV(Ω) ⊂ Cext(Ω) as a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then S(u) ∈ Cext(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cext(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 1Here, we use the symbol u for elements in FV(Ω)εE instead of the usual u to avoid confusion with the index u of Cu(Ω) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Cu(Ω, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CONSISTENCY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' There is δext ∈ C(Ω,FV(Ω)′ κ) such that δext = δ on Ω since δ ∈ Cext(Ω,FV(Ω)′ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, u○δext ∈ C(Ω,E) and equal to S(u) = u○δ on Ω, yielding S(u) ∈ Cext(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ For the next lemma we equip Cext(Ω) with the topology of uniform convergence on compact subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let X be a metric space, Ω ⊂ X, FV(Ω) a dom-space and T ∈ L(FV(Ω),Cext(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then δ ○ T ∈ Cext(Ω,FV(Ω)′ γ) if FV(Ω) is barrelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 (i) we derive that δ○T ∈ C(Ω,FV(Ω)′ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let x ∈ ∂Ω and (xn) be a sequence in Ω with xn → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then (δxn ○ T) is a sequence in FV(Ω)′ and lim n→∞(δxn ○ T)(f) = lim n→∞T(f)(xn) =∶ (δext x T)(f) in K for every f ∈ FV(Ω), which implies that (δxn○T) converges to δext x T pointwise on FV(Ω) because T(f) ∈ Cext(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As a consequence of the Banach–Steinhaus theorem we get (δext x T) ∈ FV(Ω)′ and the convergence in FV(Ω)′ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Let FV(Ω,E) be a dom-space, X a set, K a family of sets and π∶⋃m∈M ωm → X such that ⋃K∈K K ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We say that a function f ∈ ⋂m∈M domT E m vanishes at infinity in the weighted topology w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (π,K) if ∀ ε > 0, j ∈ J, m ∈ M, α ∈ A ∃ K ∈ K ∶ sup x∈ωm, π(x)∉K pα(T E m(f)(x))νj,m(x) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (16) Further, we set APπ,K(Ω,E) ∶= {f ∈ ⋂ m∈M domT E m ∣ f fulfils (16)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (vanishing at ∞ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' to (π,K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (T E m,T K m)m∈M be the generator for (FV,E), let FV(Ω,Y ) ⊂ APπ,K(Ω,Y ) as a linear subspace for Y ∈ {K,E} and K be closed under taking finite unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (i) If for all u ∈ FV(Ω)εE it holds that S(u) ∈ ⋂m∈M dom(T E m) and ∀ m ∈ M, x ∈ ωm ∶ (T E mS(u))(x) = u(T K m,x), (17) then S(u) ∈ APπ,K(Ω,E) for all u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (ii) If for all e′ ∈ E′ and f ∈ FV(Ω,E) it holds that e′ ○ f ∈ ⋂m∈M dom(T K m) and ∀ m ∈ M, x ∈ ωm ∶ T K m(e′ ○ f)(x) = (e′ ○ T E m(f))(x), (18) then e′ ○ f ∈ APπ,K(Ω) for all e′ ∈ E′ and f ∈ FV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (i) We set Bj,m ∶= {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} for j ∈ J and m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The topologies σ(FV(Ω)′,FV(Ω)) and κ(FV(Ω)′,FV(Ω)) coincide on the equicontinuous set B○ j,m and we deduce that the restriction of u to B○ j,m is σ(FV(Ω)′,FV(Ω))-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let ε > 0, j ∈ J, m ∈ M, α ∈ A and set Uα,ε ∶= {x ∈ E ∣ pα(x) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there are a finite set N ⊂ FV(Ω) and η > 0 such that u(f ′) ∈ Uα,ε for all f ′ ∈ VN,η where VN,η ∶= {f ′ ∈ FV(Ω)′ ∣ sup f∈N ∣f ′(f)∣ < η} ∩ B○ j,m because the restriction of u to B○ j,m is σ(FV(Ω)′,FV(Ω))-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since N ⊂ FV(Ω) is finite, FV(Ω) ⊂ APπ,K(Ω) and K is closed under taking finite unions, there is K ∈ K such that sup x∈ωm π(x)∉K ∣T K m(f)(x)∣νj,m(x) < η (19) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' FURTHER EXAMPLES OF ε-PRODUCTS 41 for every f ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows from (19) and (the proof of) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8 that Dπ⊄K,j,m ∶= {T K m,x(⋅)νj,m(x) ∣ x ∈ ωm, π(x) ∉ K} ⊂ VN,η and thus u(Dπ⊄K,j,m) ⊂ Uα,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore we have sup x∈ωm π(x)∉K pα(T E m(S(u))(x))νj,m(x) = (17) sup x∈ωm π(x)∉K pα(u(T K m,x))νj,m(x) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we conclude that S(u) ∈ APπ,K(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (ii) Let ε > 0, f ∈ FV(Ω,E) and e′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then there exist α ∈ A and C > 0 such that ∣e′(x)∣ ≤ Cpα(x) for every x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For j ∈ J and m ∈ M there is K ∈ K such that sup x∈ωm π(x)∉K pα(T E m(f)(x))νj,m(x) < ε C since FV(Ω,E) ⊂ APπ,K(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows that sup x∈ωm π(x)∉K ∣T K m(e′ ○ f)(x)∣νj,m(x) = (18) sup x∈ωm π(x)∉K ∣e′(T E m(f)(x))∣νj,m(x) < C ε C = ε, yielding e′ ○ f ∈ APπ,K(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The first part of the proof above adapts an idea in the proof of [16, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 Theo- rem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 199–200] where (T E m,T K m)m∈M = (idEΩ,idKΩ) which is a special case of our proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Our last proposition of this section is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For ω ⊂ Ω we set APω(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ ω ∶ f(x) = 0} and APω(Ω) ∶= APω(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proposition (vanishing on a subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let ω ⊂ Ω and FV(Ω) a dom- space such that FV(Ω) ⊂ APω(Ω) as a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then S(u) ∈ APω(Ω,E) for all u ∈ FV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further examples of ε-products In Chapter 3 we dealt with weighted spaces of continuously partially differen- tiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Now, we treat many examples of weighted spaces FV(Ω,E) of functions with less regularity on a set Ω with values in a locally convex Hausdorff space E over the field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Applying the results of the preceding sections, we give conditions on E such that FV(Ω) and FV(Ω,E) are ε-compatible, in particular, that FV(Ω,E) ≅ FV(Ω)εE holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We start with the simplest example of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a non-empty set and equip the space EΩ with the topology of pointwise convergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' the locally convex topology given by the seminorms ∣f∣K,α ∶= sup x∈K pα(f(x))χK(x), f ∈ EΩ, for finite K ⊂ Ω and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' To prove EN0 ≅ KN0εE for complete E is given as an exercise in [94, Aufgabe 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 259], which we generalise now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a non-empty set and E an lcHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then EΩ ≅ KΩεE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The strength and consistency of the generator (idEΩ,idKΩ) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ EΩ, K ⊂ Ω be finite and set NK(f) ∶= f(Ω)χK(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then we have NK(f) = f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus NK(f) is finite, hence compact, NK(f) ⊂ acx(f(K)) and acx(f(K)) is a subset of the finite dimensional subspace span(f(K)) of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows that acx(f(K)) is compact by [89, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 Proposition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 113], implying our statement by virtue of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CONSISTENCY The next example will give us the counterpart of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='9 a) on the level of sequence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a set, E an lcHs and V ∶= (νj)j∈J a directed family of weights νj∶Ω → [0,∞) on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We set ℓV(Ω,E) ∶= {f ∈ EΩ ∣ ∀ j ∈ J, α ∈ A ∶ ∣f∣j,α ∶= sup x∈Ω pα(f(x))νj(x) < ∞} and ℓV(Ω) ∶= ℓV(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, (Ω,d) a uniformly discrete metric space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' there is r > 0 such that d(x,y) ≥ r for all x,y ∈ Ω, x ≠ y, and V ∶= (νj)j∈J a directed family of weights on Ω such that ∀ j ∈ J ∃ i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ K ∶ νj(x) ≤ ενi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (20) If E is locally complete, then ℓV(Ω,E) ≅ ℓV(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ ℓV(Ω,E) and j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then fνj is bounded on Ω by definition of ℓV(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since (Ω,d) is uniformly discrete, there is r > 0 such that pα(f(x)νj(x) − f(y)νj(y)) d(x,y) ≤ 2 r ∣f∣j,α < ∞, x,y ∈ Ω, x ≠ y, for every α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore fνj ∈ C[1] b (Ω,E) where C[1] b (Ω,E) ∶= {g ∈ EΩ ∣ ∀α ∈ A ∶ sup x∈Ω pα(g(x)) < ∞ and sup x,y∈Ω x≠y pα(g(x) − g(y)) d(x,y) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to (20) there is i ∈ J such that for all ε > 0 there exists a compact set K ⊂ Ω such that νj(x) ≤ ενi(x) for all x ∈ Ω∖K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' As V is directed, we may assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' that νj(x) ≤ νi(x) for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This implies that the zeros of νi are zeros of νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We define h∶Ω → [0,∞) by h(x) ∶= νi(x)/νj(x) for x ∈ Ω with νj(x) ≠ 0 and h(x) ∶= 1 if νj(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We observe that h(x) > 0 for all x ∈ Ω as the zeros of νi are contained in the zeros of νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It follows that f(x)νj(x)h(x) = f(x)νi(x) for x ∈ Ω with νj(x) ≠ 0 and f(x)νj(x)h(x) = 0 for x ∈ Ω with νj(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence fνjh is bounded on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, εh(x) = ενi(x)/νj(x) ≥ 1, for x ∈ Ω ∖ K with νj(x) ≠ 0 because (20) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Moreover, the zeros of νj are contained in N ∶= {x ∈ Ω ∣ f(x)νj(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' This yields that acx(fνj(Ω)) is absolutely convex and compact by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' So our statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Let us apply the preceding result to some known sequence spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We recall that a matrix A ∶= (ak,j)k,j∈N of non-negative numbers is called Köthe matrix if it fulfils: (1) ∀ k ∈ N ∃ j ∈ N ∶ ak,j > 0, (2) ∀ k,j ∈ N ∶ ak,j ≤ ak,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that what we call k is usually called j and vice-versa (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' [131, Def- inition, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 326]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' But the notation we chose is more in line with the meaning of j in our Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 of a weight function and therefore we prefer to keep our notation consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' For an lcHs E we define the Köthe space λ∞(A,E) ∶= {x = (xk) ∈ EN ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup k∈N pα(xk)ak,j < ∞} 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' FURTHER EXAMPLES OF ε-PRODUCTS 43 and the spaces of E-valued rapidly decreasing sequences which we need for some theorems on Fourier expansions (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='13, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='14) by s(Ω,E) ∶= {x = (xk) ∈ EΩ ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup k∈Ω pα(xk)(1 + ∣k∣2)j/2 < ∞} with Ω = Nd, Nd 0, Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we set λ∞(A) ∶= λ∞(A,K) and s(Ω) ∶= s(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be a locally complete lcHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' a) If A ∶= (ak,j)k,j∈N is a Köthe matrix such that ∀ j ∈ N ∃ i ∈ N ∀ ε > 0 ∃ K ∈ N ∀ k ∈ N, k > K ∶ ak,j ≤ εak,i, (21) then λ∞(A,E) ≅ λ∞(A)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' b) s(Ω,E) ≅ s(Ω)εE for Ω = Nd, Nd 0, Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We observe that N and Ω are uniformly discrete metric spaces if they are equipped with the metric induced by the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, a set in a discrete space is compact if and only if it is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In case b) we set νj∶Ω → (0,∞), νj(k) ∶= (1 + ∣k∣2)j/2 for j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then for ε > 0 there is K ∈ N such that (1 + ∣k∣2)j/2 (1 + ∣k∣2)j = (1 + ∣k∣2)−j/2 ≤ ε for all k ∈ Ω with ∣k∣ > K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' In both cases the family of weights are directed, in case a) due to condition (2) of the definition of a Köthe matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we can apply Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Due to [131, Proposition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 330–331] condition (21) is equivalent to λ∞(A) being a Schwartz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since λ∞(A) is also a Fréchet space by [131, Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 326], another way to prove Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 a) (and b) as well) is given by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Our next examples are Favard-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs, 0 < γ ≤ 1, Ω a compact Hausdorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' ϕ(t + r,s) = ϕ(t,ϕ(r,s)) and ϕ(0,s) = s, t,r ∈ [0,∞), s ∈ Ω, and (̃T E t )t≥0 the induced semigroup given by ̃T E t ∶C(Ω,E) → C(Ω,E), ̃T E t (f) ∶= f(ϕ(t,⋅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The semigroup (̃T K t )t≥0 is (equi-)bounded and strongly continuous by [62, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' II, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='31 Exercises (1), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The vector-valued Favard space of order γ of the semigroup (̃T E t )t≥0 is defined by Fγ(Ω,E) ∶= {f ∈ C(Ω,E) ∣ ∀ α ∈ A ∶ sup x∈Ω,t>0 pα(̃T E t (f)(x) − f(x))t−γ < ∞} equipped with the system of seminorms given by ∣f∣α ∶= max(sup x∈Ω pα(f(x)), sup x∈Ω,t>0 pα(̃T E t (f)(x) − f(x))t−γ), f ∈ Fγ(Ω,E), for α ∈ A (see [39, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 160] and [39, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 160]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Further, we set Fγ(Ω) ∶= Fγ(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Fγ(Ω,E) is a dom-space, which follows from the setting ω ∶= [0,∞) × Ω, domT E ∶= C(Ω,E) and T E∶C(Ω,E) → Eω given by T E(f)(0,x) ∶= f(x) and T E(f)(t,x) ∶= ̃T E t (f)(x) − f(x), t > 0, x ∈ Ω, as well as AP(Ω,E) ∶= EΩ and the weight given by ν(0,x) ∶= 1 and ν(t,x) ∶= t−γ for t > 0 and x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be a semi-Montel space, 0 < γ ≤ 1, Ω a compact Haus- dorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then Fγ(Ω,E) ≅ Fγ(Ω)εE holds for the Favard space of order γ of the induced semigroup (̃T E t )t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' CONSISTENCY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator (T E,T K) for (Fγ,E) is consistent by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 b)(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Its strength is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus our statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ The space of càdlàg functions on a set Ω ⊂ R with values in an lcHs E is defined by D(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ Ω ∶ lim w→x+f(w) = f(x) and lim w→x−f(w) exists}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 Further, we set D(Ω) ∶= D(Ω,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Due to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 the maps given by ∣f∣K,α ∶= sup x∈Ω pα(f(x))χK(x), f ∈ D(Ω,E), for compact K ⊂ Ω and α ∈ A form a system of seminorms inducing a locally convex Hausdorff topology on D(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs and Ω ⊂ R locally compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is quasi- complete, then D(Ω)εE ≅ D(Ω,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' First, we show that the generator (idEΩ,idKΩ) for (D,E) is strong and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The strength is a consequence of a simple calculation, so we only prove the consistency explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have to show that S(u) ∈ D(Ω,E) for all u ∈ D(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let x ∈ Ω be an accumulation point of [x,∞) ∩ Ω resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' (−∞,x] ∩ Ω, (xn) be a sequence in Ω such that xn → x+ resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' xn → x−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We have δxn(f) = f(xn) → f(x) = δx(f), xn → x+, and δxn(f) = f(xn) → lim n→∞f(xn) =∶ T(f)(x), xn → x−, for every f ∈ D(Ω), which implies that (δxn) converges to δx if xn → x+, and to δx ○ T if xn → x− in D(Ω)′ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since Ω is locally compact, there are a compact neighbourhood U(x) ⊂ Ω of x and n0 ∈ N such that xn ∈ U(x) for all n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence we deduce sup n≥n0 ∣δxn(f)∣ ≤ ∣f∣U(x) for every f ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Therefore the set {δxn ∣ n ≥ n0} is equicontinuous in D(Ω)′, which implies that (δxn) converges to δx if xn → x+ and to δx ○ T if xn → x− in D(Ω)′ γ and thus in D(Ω)′ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' From S(u)(x) = u(δx) = lim n→∞u(δxn) = lim n→∞S(u)(xn), xn → x+, and u(δx ○ T) = lim n→∞u(δxn) = lim n→∞S(u)(xn), xn → x−, for every u ∈ D(Ω)εE follows the consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Second, let f ∈ D(Ω,E), K ⊂ Ω be compact and consider NK(f) = f(Ω)χK(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We observe that NK(f) = f(K)∪{0} if K ≠ Ω, and NK(f) = f(K) if K = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' We note that NK(f) ⊂ acx(f(K)) and acx(f(K)) is absolutely convex and compact by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1 because E is quasi-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus we derive our statement from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ We turn to Cauchy continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let Ω be a metric space, E an lcHs and the space CC(Ω,E) of Cauchy continuous functions from Ω to E be equipped with the system of seminorms given by ∣f∣K,α ∶= sup x∈K pα(f(x))χK(x), f ∈ CC(Ω,E), for K ⊂ Ω precompact and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 2We note that for x ∈ Ω we only demand limw→x+ f(w) = f(x) if x is an accumulation point of [x, ∞)∩Ω, and the existence of the limit limw→x− f(w) if x is an accumulation point of (−∞, x]∩Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' FURTHER EXAMPLES OF ε-PRODUCTS 45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs and Ω a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is a Fréchet space or a semi-Montel space, then CC(Ω,E) ≅ CC(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator (idEΩ,idKΩ) for (CC,E) is consistent by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Its strength follows from the uniform continuity of every e′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' First, we consider the case that E is a Fréchet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let f ∈ CC(Ω,E), K ⊂ Ω be precompact and consider NK(f) = f(Ω)χK(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Then NK(f) = f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The set f(K) is precompact in the metrisable space E by [13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 576].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Thus we obtain CC(Ω,E) ⊂ CC(Ω,E)κ by virtue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Since E is complete, the first part of the statement follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='4 with Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is a semi-Montel space, then it is a consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ Let (Ω,d) be a metric space, E an lcHs and the space Cbu(Ω,E) of bounded uniformly continuous functions from Ω to E be equipped with the system of semi- norms given by ∣f∣α ∶= sup x∈Ω pα(f(x)), f ∈ Cbu(Ω,E), for α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let E be an lcHs and (Ω,d) a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is a semi- Montel space, then Cbu(Ω,E) ≅ Cbu(Ω)εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The generator (idEΩ,idKΩ) for (Cbu,E) is consistent by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' It is also strong due to the uniform continuity of every e′ ∈ E′, yielding our statement by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If N is equipped with the metric induced by the absolut value, then Cbu(N,E) = ℓ∞(N,E) where ℓ∞(N,E) is the space of bounded E-valued sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' If E is a separable infinite-dimensional Hilbert space, then the map S∶Cbu(N)εE → Cbu(N,E) is not surjective by [17, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='8 Beispiel, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 140] and [94, Satz 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' 235–236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Hence one cannot drop the condition that E is a semi-Montel space in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' Let (Ω,d) be a metric space, z ∈ Ω, E an lcHs, 0 < γ ≤ 1 and define the space of E-valued γ-Hölder continuous functions on Ω that vanish at z by C[γ] z (Ω,E) ∶= {f ∈ EΩ ∣ f(z) = 0 and ∀ α ∈ A ∶ ∣f∣α < ∞} where ∣f∣α ∶= sup x,w∈Ω x≠w pα(f(x) − f(w)) d(x,w)γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFRT4oBgHgl3EQfpDe0/content/2301.13612v1.pdf'} +page_content=' The topological subspace C[γ] z,0(Ω,E) of γ-Hölder continuous functions that vanish at infinity consists of all f ∈ C[γ] z (Ω,E) such that for all ε > 0 there is δ > 0 with sup x,w∈Ω 0 10 10 . 5 M ⊙) tends to have larger molecular +g as and g as fraction than inactive g alaxies. And Franceschini et al. +⋆ E-mail: mohamedabdulazez9@gmail.com +( 1999 ) found that the histories of the BH accretion rate and stellar +formation in host galaxies are similar. Also Heckman et al. ( 2004 ) +studied the accretion-driven growth of BH at low redshift using type +2 AGN and found that the BH of masses less than 10 8 M ⊙ that reside +in moderately massive galaxies have accretion rate time scale that is +comparable to the age of universe. +Since AGN is powered by a central BH which resides at a +host galaxy of certain physical properties such as stellar mass and +star formation rate, the evolution of AGN should be related to its +host galaxy. In the following we will present some previous works +on studying the relation between AGN and host galaxy either by +observ ational e vidence or hydrodynamical simulations. +The accretion process caused by the central BH, results in a nuclear +activity that appears as a feedback on the ambient medium. This +activity has different scenarios such as winds, radiation pressure, and +jets (Zubovas & King 2014a ; Fabian, Vasude v an & Gandhi 2008 ). +The model made by King & Pounds ( 2003 ) explained the energy +and momentum of large scale outflows. According to this model, +the AGN radiation pressure can launch a relativistic wind from very +close in, where outflow can emerge from a photosphere of a radius +a few tens of schwarzchild radius ( R s ) given by R s = 2 GM +c 2 , where +M is the BH mass. This wind shocks against the ambient medium +producing an outflow. When the BH reaches a critical mass given by +Zubovas & King ( 2014b ), +M σ ≈ 3 . 67 × 10 8 σ 4 +200 M ⊙, +where σ is the velocity dispersion in host galaxy spheroid, the AGN +outflows become energy driven and cannot be cooled any more which +can cause gas loss and consequently affects the star formation in the +© 2022 The Author(s) +Published by Oxford University Press on behalf of Royal Astronomical Society + +2 +M. A. Abdulrahman et al. +MNRAS 00, 1 (2022) +host galaxy. Also Wagner et al. ( 2016 ) showed that the feedback +can be positive such as triggering star formation by using its energy +in pressure triggered collapse or ne gativ e such as quenching star +formation by the loss of material from the host galaxy. +On the other hand, time evolution simulations of the impact of +the jet emerging from AGN on the ambient interstellar medium +(ISM) made by Wagner, Bicknell & Umemura ( 2012 ) showed that +the effect of the jet depends on the size of the ambient clouds. For +large clouds ( > 50 pc ), it increases the star formation rate, but for +small clouds ( ∼10 pc ), it causes explosion of cloud and quenching +of star formation. So ne gativ e and positive feedback can co-exist +depending on the density of the ambient ISM. In the simulations +made by Dubois et al. ( 2013 ), the jet power can be considered as +a reason of the material loss which can transform the disk galaxy +into red elliptical galaxies by the quenching of star formation. Also +Mukherjee et al. ( 2018 ) performed simulations to study the effect +of a relativistic jet on the ambient gaseous disk and found that, +depending on jet power, the ISM density, and jet orientation, the star +formation can be enhanced or quenched. These authors found that jet +can contribute in increasing the velocity dispersion of ambient ISM. +Accordingly, the effect of AGN on its host galaxy is rele v ant to the +host galaxy properties. +Observational studies made by (Ferrarese & Merritt 2000 ; Geb- +hardt et al. 2000 ) revealed a correlation between the BH mass and +the velocity dispersion of the host bulge. Reines & Volonteri ( 2015 ) +found a correlation between the BH mass and the stellar mass of +the host galaxy in which the BH mass increases with increasing the +stellar mass of its host galaxy. This is consistent with the observations +made by Bilata-Woldeyes et al. ( 2020 ) using data from BAT-SWIFT +to study the relation between morphology of the host galaxy and the +AGN properties such as the Eddington ratio and BH mass. In these +observations, the BH masses are larger in elliptical galaxies than in +spiral galaxies. +Recent studies made by Dittmann & Miller ( 2020 ) investigated +the growth of the central BH by assuming a merging scenario with +the compact objects formed in the accretion disk. The study by +Tartenas & Zubovas ( 2019 ) showed the feeding of the AGN by the +dynamical perturbations. It also showed that a collision between +circumnuclear ring and molecular cloud that can be an efficient +fueling mechanism, depending on the angle of collision. So the +fueling mechanism has a crucial role in driving the relationship +between the AGN and its host galaxy. As a way to further examine +how such a relation between the AGN and host galaxy exists, +Smethurst et al. ( 2016 ) studied the star formation history of type +2 AGN and inactive galaxies. In this study, the gas reservoir in the +host galaxy is the main source for BH fueling as first examined and +mentioned by Magorrian et al. ( 1998 ), where both of nuclear activity +and star formation are related to the host galaxy gas reservoir. +To further understand this mutual effect between the AGN and +the host galaxy, Valentini et al. ( 2020 ) performed a simulation of the +galaxy disk to study gas accretion models. This simulation showed +how the AGN feedback on the multiphase of the ISM of hot and cold +phases can affect the co-evolution of the BH and its host galaxy and +found that the accretion of cold gas is more ef fecti ve in BH growth +rate than hot gas, and the gas accretion contributes in the BH growth +more than mergers with other BHs. +So in this paper we study the evolution of isolated spiral galaxies +in terms of AGN contribution by investigating the effect of AGN +on the spiral arms of spiral galaxies and how the spiral arms can +control the AGN activity as being a source of fueling of the central +BH. Depending on the fueling of AGN by driving the gas from the +spiral arm under the gravitational force of the central BH to derive +Figure 1. Schematic chart of a simple spiral galaxy having two arms. +an Equation for the AGN luminosity that depends on parameters +such as the BH mass and the spiral arm radius which is a measure +of the amount of gas in it. We also investigate the BH mass-AGN +luminosity relation and how it can be affected by changing physical +parameters such as the spiral arm radius and the torus length. +We hope to construct a physical scenario for how the AGN-galaxy +co-evolution works. This paper is presented as follows. The model +approach is explained in Sections 2 and the data of the galaxy sample +is given in Section 3 . Results and the discussions are presented +in Sections 4 and 5 , respectively, while the conclusion is given in +Section 6 . In this work, the cosmological parameters were taken +with H 0 = 70.0 km s −1 Mpc −1 , �m = 0.30, �λ = 0.70. +2 MODEL APPROACH +For a spiral galaxy having a simple galactic disk with two arms, we +assume that most of the mass of the galactic matter is concentrated +within these dense arms while the rest of the galactic medium of the +host disk is very diffuse medium with very low density that can be +neglected. Based on the simple morphology of the unified model, +we assume that the torus is divided to internal and external sides +around the the AGN as shown in the schematic chart in Fig. 1 . In this +chart, the internal torus is facing the AGN while the external torus +represents the outer region of the galactic disk. +The internal torus region is assumed to partially contain these +arms. In which the feedback of the BH activity is more efficient on +the matter with these arms. In this context, Yu et al. ( 2022 ) found that +the spiral arm is efficient in transporting the gas to central region. +Accordingly, we assume that the material transfers in a conical shape +from the closest region within the internal torus into the accretion +disk through a spiral arm. This shape makes the spiral arm radius +varies from smaller at the contact point with the accretion disk to +larger radius within deeper internal region of the torus (neglecting +the environmental effects on the galactic outskirts). +For a particle of mass ( m ) in the accretion disk at a distance ( r ) +from the central BH, it mo v es with a velocity ( υ) where its angular +momentum ( �) is given by, +� = m AD υAD r +(1) +where m AD refers to the mass of accretion disk. +Due to the gravitational effect of the BH, the particles around +the BH experience a torque pushing them inward into the BH. As +the matter in the accretion disk loses angular momentum the matter + +External +torus +Internaltorus +RA +Accretiondisk +Black hole +rThe impact of the AGN and the torus properties on the evolution of spiral galaxies +3 +MNRAS 00, 1 (2022) +spirals into the BH and this gives the chance to the material in internal +torus to mo v e to accretion disk. +So we can assume that the change of angular momentum equals +the torque on the gas in the accretion disk. Accordingly, the change +of r with time is a consequence of the change of the gravitational +radius ( r g ) of the BH with time which indicates the growth of BH or +its evolution, hence, +dr +dt = dr g +dt = +� gas +m AD υAD +(2) +where � gas is the torque on the gas in the accretion disk. +Considering a gaseous disk where the gravitational force is due to +a central BH, the torque on gas at a radius ( r ), as given by Netzer +( 2013 ), is: +� gas ( r) = ˙m ( GMr) 1 / 2 f ( r) +(3) +where, ˙m is the radial mass inflow rate, M is the BH mass, and f ( r ) = +1 − ( r in / r ) 1/2 . The r in parameter represents the radius at which the +torque on gas is zero and it falls in a non-circular orbit into the BH +which is known as the inner-most stable circular orbit (ISCO). +Using r g = GM +c 2 where M , G and c are the BH mass, the gravita- +tional constant and the speed of light, respectively, we have +dr g +dt = G +c 2 +dM +dt +(4) +It is known that the BH grows due to the accretion of matter with +taking into account that not all matter is accreted. A fraction ( δ) of +this matter does not accrete but it escapes due to the feedback of +the AGN such as radiation pressure and wind from accretion disk +(Zubovas & King 2014a ) and the star formation taking place in the +vicinity of BH (Dittmann & Miller 2020 ). So we can write the change +of the BH mass with time as, +dM +dt = (1 − δ) ˙M +(5) +where ˙M is the accretion rate given by ˙M = L +ηc 2 while L is the AGN +bolometric luminosity hereafter it is called L AGN , and η is the radiative +efficiency. +Then equation ( 5 ) becomes, +dM +dt = (1 − δ) L +ηc 2 +(6) +From equations ( 2 ), (3), (4) and (6), the L AGN can be written as, +L AGN = ηc 4 ˙m ( GM r ) 1 / 2 f ( r ) +G (1 − δ) m AD υAD +(7) +where a fast rotating BH has a spin parameter of 0.998, and r in = +1.24 r g . It is possible to put r = nr g where n takes values of 20– +40 indicating the mean disk size according to available data on +variability of high redshift luminous AGN (Netzer 2013 ). +By considering the conical shape of the spiral arm facing the +accretion disk, m AD can be written as, +dm AD +dt += −ξm T +(8) +where m T is the mass of the internal torus, and ξ is the transfer +efficiency which could be thought as the fraction of medium +clumpness (clumpy or smooth). +m T = π +3 R 2 +A ρT l T +(9) +where R A is the spiral arm radius within the internal torus region +characterized by a very dense conical shape. The parameters ρT and +l T are the density within the torus and its length, respectively. By +assuming that the material is concentrated in the spiral arms at a +certain time (t), for which, ρT becomes mostly the arm density ( ρA ). +Substituting by (9) into (8), and integrating ρA w.r.t time, the mass +of accretion disk becomes +m AD = π +6 ξR 2 +A ρ2 +A l T +(10) +Then the luminosity Equation in the present model becomes +L AGN = 5 . 156 × 10 25 +ηn +1 +2 f ( r ) M ˙m +υAD ξ(1 − δ) l T R 2 +A ρ2 +A +( erg s −1 ) +(11) +By setting δ = 0.1, ξ = 0.1, η = 0.1, n = 25, f ( r ) ∼ 0.777, υAD = +10 8 m / s as standard values, equation ( 11 ) can be re-written as +L AGN = 2 . 792 × 10 18 M ˙m +l T R 2 +A ρ2 +A +( erg s −1 ) +(12) +Therefore, the ef fecti ve parameters on the AGN luminosity are the +mass inflow rate ( ˙m ), the internal torus length ( l T ), the spiral arm +radius ( R A ), the spiral arm density ( ρA ), and the BH mass ( M ). +3 DATA +To compare with the calculations of equation ( 12 ) and in an attempt +to check the validity of our BH mass-luminosity relation, a sample +of the observational data was provided by a catalog of type 1 AGNs +from SDSS-DR7 (Oh et al. 2015a ). This catalog contains 5553 type +1 AGN with a redshift of z ≤ 0.2. In which, galaxies were selected +after applying some classification criteria for data of OSSY catalog. +This catalog provides us with the logarithm of bolometric luminosity +of AGN ( L bol ) derived from a method developed by Heckman et al. +( 2004 ) using the luminosity of the [O III ] λ5007 emission line L O III +as a tracer of nuclear activity where L bol ≈ 3500 L O III ( erg s −1 ). It +also provides us with the logarithm of the BH mass as derived by +Greene & Ho ( 2005 ) depending on the luminosity and line width of +the broad H α line. The bolometric luminosity ranges from 42.09 to +46.77 erg s −1 in logarithmic scale. Also the BH masses ranges from +6.13 to 9.29 M ⊙ in logarithmic scale. +As the galaxy selection is flux-limited, we have considered the +Malmquist bias as a selecting criterion of our galaxy sample where +the luminosity varies as a function of the redshift. In Fig. 2 , we plot +the bolometric luminosity across the redshift of 5553 galaxies as +shown in the top panel , then we employ a flux-limit cut off with +a minimum flux limit of 4 × 10 −12 ergs −1 cm −2 to estimate the +theoretical luminosity (solid line) using the formula of L = 4 πd L F +where d L is the luminosity distance. This luminosity distance was +estimated according to the analytical approximation considering the +case of flat cosmologies (Adachi & Kasai 2012 ). Accordingly, the +number of galaxies in the sample has been reduced to 4954 galaxies as +shown in the bottom panel . And the current range for the bolometric +luminosity becomes from 42.88 to 46.77 erg s −1 in logarithmic scale. +4 RESULTS +In this Section, we present the results that show the impact of +the escaped fraction ( δ) and the clumpness fraction ( ξ) mentioned +abo v e in equation ( 11 ) on the AGN luminosity. Then we used also +equation ( 12 ) to show the effect of the spiral arm density ( ρA ) and +the radius ( R A ) on this luminosity. We showed the AGN luminosity +versus the BH mass using the observational data along with those of +equation ( 12 ). By using derived relations from previous observational +works of (Dom ´ınguez S ´anchez et al. 2012 ; Baron & M ´enard 2019 ; +Huang et al. 2012 ), we estimated the star formation rate, gas mass, + +4 +M. A. Abdulrahman et al. +MNRAS 00, 1 (2022) +Figure 2. The bolometric luminosity of data sample (red dots) across redshift +before ( top panel ) and after ( bottom panel ) applying the Malmquist bias where +the solid line represents the flux-limit cut off of the galaxy sample. +and stellar mass of our sample the galaxies comparing the results +from both of the observed data and equation ( 12 ). Finally, we used +the results that we obtained in an attempt to put a suitable evolution +scenario of the spiral galaxy in terms of AGN. +4.1 The AGN luminosity versus torus properties +Using equation ( 11 ) with assuming R A = 100 pc, ρA = 10 −10 kg m −3 , +l T = 100 pc, ˙m = 0 . 1 M ⊙ yr −1 , Fig. 3 sho ws ho w the AGN +luminosity is affected by; the clumpness fraction of the medium +( ξ) and the velocity of the material in the accretion disk ( υAD ) in +the top panels ( left and right, respectively ); the radiative efficiency +( η) and the mean disk size ( n ) in the middle panels ( left and right, +respectively ); and the escaped fraction ( δ) in the bottom panel. For +each ef fecti ve parameter, this luminosity was estimated with keeping +the others constant. From this figure, the luminosity exponentially +decreases with increasing both ξ and υAD , while it exponentially +increases with increasing both of η, n, and in a logarithmic way with +δ. +In order to see how the BH mass affect the relation between the +AGN luminosity and each physical parameter in equation ( 12 ), we +present its behaviour at three different BH masses ( LogM = 6, 7 and +8 M ⊙) as shown in Fig. 4 ( the assumed value for each parameter +was chosen according to the best fit for the data sample shown later +in Fig. 5 ). From Fig. 4 , the luminosity has a similar trends showing +an exponential decrease with both of the R A , ρA , and l T while it has +differently an exponential increase with ˙m . +The spiral arm radius and density of a galaxy can be considered as +a morphological parameters that demonstrate the host galaxy shape +where the spiral arm contains most of the amount of material within +torus. This matter is considered as the main feeding source for the +central BH which controls the AGN luminosity. For which, the effect +of both of spiral arm radius and density is shown in the top panels (left +and right, respectively) of Fig. 4 . Also Masoura et al. ( 2018 ) found +that the AGN luminosity, in a host galaxy, depends on the position +of this host galaxy from the main sequence line, depending on the +available gas of the host galaxy. Since the accreted matter from the +spiral arm travels through a path from spiral arm to the BH, the length +of this path should also control the produced AGN luminosity. This +path length is represented in our model by the internal torus length. +The left top panel of Fig. 4 shows its correlation with the luminosity +at three different BH masses, in which, the luminosity increases fast +at smaller lengths but it decreases slowly at longer lengths. +All of the abo v e physical parameters are related to the properties +of the galaxy disk but the mass inflow rate or the accretion rate of +the BH is related to the properties of the central BH. Its correlation +with luminosity, shown in the left bottom panel Fig. 4 , shows the +remarkable increase of the luminosity with the mass inflow rate. +This means that the luminosity doesn’t depend only on the amount +of the material flowing toward the BH but also the time taken by this +material to reach the accretion disk. +For the data sample, the top panel in Fig. 5 shows the result with +implying equation ( 12 ) and those of observed data fitted with a slope +of 1.005 ± 0.029, for which, the standard deviation was adapted to +x = 1, y = 0.86, giving a correlation coefficient of 0.49. To account +for the dispersion of the scattered points, we calculated the residual +which shows a normal Gaussian distribution as shown in the bottom +panel. +Using the data sample, we present some physical properties +concerning the host galaxy, such as the SFR, the stellar mass ( M stellar ) +and the gas mass ( M gas ), to explain how these parameters can change +across redshift. The observed data of our galaxy sample provides +us with the H α emission line luminosity, L ( H α), which we used +to measure the SFR based on the SFR- L ( H α) dependence using +equation ( 1 ) given by Dom ´ınguez S ´anchez et al. ( 2012 ) where: +SFR (M ⊙ yr −1 ) = 7 . 9 × 10 −42 L (H α) ( erg s −1 ) +To estimate the stellar mass as a function of BH mass, we re- +arrange equation ( 9 ) given by Baron & M ´enard ( 2019 ) to be as +follows: +Log +� M stel l ar +10 11 M ⊙ +� += +Log( M BH +M ⊙ ) − (7 . 88 ± 0 . 13 ) +(1 . 64 ± 0 . 18 ) +By assuming that stars form in molecular clouds, the gas mass can +be measured using the HI 21 cm line. According to equation ( 1 ) given +by Huang et al. ( 2012 ) using the stellar mass of M stel l ar > 10 9 M ⊙ +since our data range is 9.94–11.859 M ⊙ in log scale, the gas mass +can be calculated in terms of the stellar masss as follows: +Log ( M H I ) ≈ 0 . 276 Log( M stel l ar ) +For a deeper understanding of the relation shown in Fig. 5 , we +further investigated the SFR distribution along this relation. From +Fig. 6 , we can see that at a certain BH mass, the AGN luminosity +decreases with decreasing the SFR. From equation ( 12 ), ρA can +be used as an indicator to the SFR. It is expected to have a +clumpy medium with increasing the density in the arm. When star +formation takes place, the ultraviolet radiation and the stellar wind + +47. +46- +45 +Lbol +Log +44- +43- +42 +0.00 +0.05 +0.10 +0.15 +0.20 +Redshift +48 +46 +44 +42- +0.00 +0.05 +0.10 +0.15 +0.20 +RedshiftThe impact of the AGN and the torus properties on the evolution of spiral galaxies +5 +MNRAS 00, 1 (2022) +Figure 3. The AGN luminosity distribution as a function of the clumpness fraction and the velocity in the accretion disk (left and right top panels, respectively), +the radiative efficiency and the mean disk size (left and right middle panels, respectively), and the escaped fraction from the BH accretion (bottom panel). +injected energy, caused by massive stars. These could be destructive +mechanisms to the molecular clouds leading to a decrease in the +ISM density or cloud dispersion (Grudi ´c et al. 2018 ; Gonz ´alez- +Samaniego & Vazquez-Semadeni 2020 ). Accordingly, the gas can +then flow easily to the central region leading to high AGN luminosity. +To implement the relation between SFR and AGN evolution, we +present the SFR variation with BH mass as shown in Fig. 7 . This +figure shows a linear relation with a slope of 0.86 and a correlation +coefficient of 0.49. It is obvious that this correlation fits well at +smaller BH masses but SFR shows flattening at higher BH masses, +deviating from the linear trend. Also from Figs 7 and 8 (bottom +panel), we can see a variation in the SFR versus the BH mass +and redshift, respectively. On average, the SFR is decreasing at low +redshift values and low BH mass. This variation points to the relation +between the BH mass and the SFR. In our work, we assume that the +spiral arm is the reservoir of gas needed for the BH feeding and +it can be seen from Fig. 7 along with Fig. 8 where gas in spiral +arm is consumed by both of star formation and the BH feeding. + +43.4 +47 +43.2 - +(s/B1a) +(s/810) +46 +43.0- +Lbol +45 +42.8 +42.6- +42.4- +43- +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.0 +2.0x105 +4.0x105 +6.0x105 +8.0x105 +1.0x106 +UAD (km/s) +43.6 +43.15 +43.4- +43.10- +43.2- +(s/8.1a) +43.0- +(erg) +43.05 +42.8- +42.6- +43.00 +42.4- +42.2 - +42.95 +42.0- +41.8 +42.90 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +20 +25 +30 +35 +40 +n +n +43.7 +43.6 +43.5- +S +43.4- +43.3 +43.1 +43.0- +42.9- +0.0 +0.2 +0.4 +0.6 +0.86 +M. A. Abdulrahman et al. +MNRAS 00, 1 (2022) +Figure 4. The bolometric luminosity of the AGN versus the spiral arm radius and the torus length (left and right top panels, respectively), and the spiral arm +density and the accretion rate (left and right bottom panels, respectiv ely). F or these physical parameters, the results were estimated at 3 different values of BH +mass of 6, 7, and 8 M ⊙ in log scale, assuming R A = 300 pc, ρA = 10 −12 kg m −3 , l T = 500 pc, ˙m = 0 . 1 M ⊙ yr −1 . +This gas consumption should decrease the gas content of the host +galaxy (Scoville et al. 2016 ; Genzel et al. 2015 ). In Fig. 8 (top and +middle panels), we can see that both of the gas mass and stellar mass +are following the same trend with decreasing redshift. The decrease +in each of them is slow and this can be used as evidence for the +BH feedback such as jets and winds which could make the growth +rate of the BH becomes slow. This can be used as evidence for the +stellar cycle where the gas is converted into stars and then returns +back through, for instance, the supernovae. Also previous work by +Tacconi et al. ( 2008 ) showed that the SFR / M gas ratio is relatively +constant. +4.2 Evolution of a spiral galaxy +We have showed that AGN luminosity is affected by the integration +of all aforementioned physical parameters. Therefore, we tried to put +an evolution scenario for an isolated spiral galaxy depending on these +parameter and neglecting any merging and environmental effects. +For a spiral galaxy hosting a central BH, at first, the gas content +of spiral arms is condensed to form clumpy clouds of gas as possible +candidates for star formation to take place. If the gas gets condensed +the amount of material available for the BH accretion decreases, +hence the AGN luminosity decreases. This is why we get low AGN +luminosity for high values of spiral arm radius and density. But due + +M= 6 Mo +46 +M= 6 Mo +45 +M= 7 Mo +M= 7 Mo +M= 8 Mo +45 +M= 8 Mo +44 +S +(erg/ +(erg/s +44 +43 +43 +Log +42 +42 +41- +41- +40- +40 +0 +2000 +4000 +6000 +8000 +0 +2000 +4000 +6000 +8000 +10000 +12000 +R (pc) +l1 (pc) +52 +M=6 Mo +36 +M= 7 Mo +M= 8 Mo +50 +S +33 +Lbol +48 +30 +Log +46- +M= 6 Mo +27 +M= 7 Mo +- +M= 8 Mo +44 +24 +0.0 +2.0x10-12 +4.0x10-12 +6.0x10-12 +8.0x10-12 +0.0 +5.0x1020 +1.0x1021 +1.98x102 +m (kg/s) +Pa (kg/m3)The impact of the AGN and the torus properties on the evolution of spiral galaxies +7 +MNRAS 00, 1 (2022) +Figure 5. The relation between the BH mass and the bolometric luminosity +of AGN for the observed data of the galaxy sample (top panel). The black line +is our results according to equation ( 12 ) and the blue line is the fit of observed +data. The regular residual of equation ( 12 ) showing a normal distribution +(bottom panel). +Figure 6. The BH mass and AGN luminosity against star formation rate +distribution. +Figure 7. The star formation rate variation with BH mass. +to the star formation the luminosity emitted from the disk should be +high. +As time goes and due to galaxy rotation and star formation, the +gas in the spiral arms became dispersed which make it easy for the +central BH to pull it producing high AGN luminosity. This is why +we get high AGN luminosity for small values of spiral arm radius +and density. +The accretion process has a time which is determined by the +internal torus length and also by the accretion rate of the central +BH. During the gas journey from the spiral arm to the central BH, it +travels a certain path which is considered as the internal torus length. +In our approach, we assumed that material is transferred through a +conical path which can be representative of unbarred galaxy. This +path length is in rele v ant to the galaxy size and also the existence +of its bar which should alter the AGN luminosity. According to +Alonso et al. ( 2014 ) who found that among their sample which +includes barred and unbarred AGN, the barred galaxies exhibit a +higher nuclear activity than unbarred ones. Also the length of the +internal torus would affect the AGN luminosity and the activity time +of AGN. Using the magnetic-hydrodynamical simulations, Rosas- +Gue v ara et al. ( 2022 ) studied the evolution of barred massive disk +galaxies. These authors found that barred galaxies have lower star +formation rate and lower gas fraction compared to unbarred ones. +This indicates that the existence or absence of a bar may increase +or decrease the gas transport efficiency from the galactic disk to the +accretion disk. +Linking the correlation between the AGN luminosity, gas mass, +and SFR Shangguan et al. ( 2020 ), we can deduce the evolution +scenario based on these physical properties of our galaxy sample. +Using Figs 6 and 9 we can divide the evolution of a galaxy into three +phases . +Phase 1 is the period of time where the gas is still condensed at +spiral arms that we have a large gas mass galaxy at a certain SFR, +the gas is consumed to form stars and there is not enough gas to be +accreted by the central BH. For this period we should observe low +AGN luminosity, low BH mass, and low stellar mass. +Phase 2 is the period of time where star formation has taken place +that the stellar mass increases and the gas is dispersed whether by +the galaxy rotation or even by stellar wind or stellar cycle, so the +available gas mass for accretion increases, leading to an increase in +the BH mass and the AGN luminosity. This period may vary from +a galaxy to another depending on its size and morphology (there +is a bar or not), where Alonso et al. ( 2014 ), Kim & Choi ( 2020 ) +showed that the nuclear activity is higher in galaxies having a bar, + +47 +46- +45 +Lbol +Log +44 +43 +equation(12)result +Fit of observation +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +Log M (Mo) +1000 +800- +600 +Counts +400 +200 +0 +-2 +-1 +0 +1 +2 +Regular residual2 +46.5 +1.5 +46.0 +S +1 +45.5 +1007 +0.5 +Log +0 +SFR +44.5 +44.0 +-1 +43.5 +43.0 +-1.5 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +Log M +(Mo)3 +2 +SFR (Mo-yr +0 +-1 +-2 +Fitof observation +5 +7 +9 +10 +Log M (M)8 +M. A. Abdulrahman et al. +MNRAS 00, 1 (2022) +Figure 8. The gas mass (top panel), stellar mass (middle panel), and star +formation rate (bottom panel) variation with redshift. +referring to the vital role of the bar in transporting the gas to the +central region. Generally, in this phase and through a transition for +the total luminosity of the g alaxy; the g alactic disk luminosity may +decrease gradually or becomes constant depending on the value of +SFR and the AGN luminosity starts to increase. This phase can also +be interpreted in terms of a study done by Zewdie et al. ( 2020 ) using +SDSS MPA-JHU catalogue with the stellar mass range of LogM ∗ = +10 . 73 −11 . 03 M ⊙. Using BPT-diagram, these authors found that the +AGN, in this stellar mass range, have lower star formation rates than +Figure 9. The relation between SFR, BH mass, and gas mass. +Figure 10. The 3 evolutionary phases of the spiral galaxy. In case of effect +of AGN feedback (positive) on SFR (orange circles) and without the effect +of AGN on SFR (black circles) feedback. +star-forming galaxies, and galaxies in this range mo v e from the blue +cloud to the red sequence. +Phase 3 is the period of time where the accretion rate of +central BH increases due to the large amount of gas coming from +spiral arms. Accordingly, a small amount of gas remains in the +spiral arms for star formation. So what we will observe is high +stellar mass and low galactic disk luminosity but high AGN +luminosity. +In Fig. 10 , we summarize the 3 phases for the spiral galaxy +evolution. If the AGN has no effect on the SFR of the host galaxy +(black circles) therefore it has a nearly constant SFR, for example, +for Log SFR ∼ −0.6 M ⊙ yr −1 . The stellar mass increases, leading +to an increase in the galactic disk luminosity due to the stellar +luminosities superposition but the increase in the AGN luminosity +is very small, it changes slightly between Log L bol ∼ 44 and ∼ +44.5 erg s −1 . In contrast, if the AGN has a positive feedback on +the host galaxy, the SFR increases. A factor of 25 increase in SFR +is related to an increase of 2 order of magnitudes in the AGN +luminosity. + +3.4 - +3.2 - +3.0 +W +Log +2.8 +2.6 - +0.00 +0.05 +0.10 +0.15 +0.20 +Redshift +12 - +(w) +11 +Log Mster +10- +0.00 +0.05 +0.10 +0.15 +0.20 +Redshift +3 - +2 - +SFR +0 +Log +-1 +-2 - +-3 - +4 +0.00 +0.05 +0.10 +0.15 +0.20 +Redshift9.0 +3.2 +8.5 +3.1 +Log Mgas +M +Log +7.5 +(Mo) +2.9 +7.0 +6.5 +2.8 +1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Log SFR (Mo/yr)11.8 +46.5 +11.6 +46 +11.4 +45.5 +45 +Log Lbol +44.5 +(erg/s) +10.6 +44 +10.4 +43.5 +10.2 +10.0 +43 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Log SFR (Mo/yr)The impact of the AGN and the torus properties on the evolution of spiral galaxies +9 +MNRAS 00, 1 (2022) +5 DISCUSSION +According to the results presented in the section 4 , we find out that +the torus properties such as spiral arm radius, spiral arm density +and torus length can be used as indicators of the the evolution of +the AGN. Also the accretion rate by the BH doesn’t only depend +on the BH physical parameters such as its spin, but also on the +torus properties or the host galaxy properties. From our results we +conclude that the AGN can have an ef fecti ve role in the evolution +of its host galaxy and vice v ersa. F or each galaxy, the luminosity +increases as the spiral arm decreases, which means that the gas in +spiral arms is the main source for the AGN luminosity and changing +it leads to a change in the AGN luminosity. Also accretion of stars +onto supermassive BHs can occur. But to determine which is more +efficient in the BH growth, the gas accretion or the star accretion, +Pfister et al. ( 2021 ) treated the tidal disruption events caused by +star accretion in cosmological simulations. These authors found +that contribution from stars’ accretion is negligible compared to gas +accretion. +Ho we ver this decrease in spiral arm gas content is not only due to +the accretion by the BH but partially caused by the feeding process +of BH and partially by the star formation taking place in spiral arms. +The star formation rate is high when the gas mass is high in spiral +arms, but to ease the gas flow from the spiral arms to the central BH, +the gas should be dispersed and this is done by the consumption of +gas in the star formation process and the dispersion caused by the +stellar wind or any damping mechanism of the formed stars (Thomp- +son, Quataert & Murray 2005 ; Hayward & Hopkins 2017 ; Lupi +2019 ). +The dispersion of gas which causes the gas to flow easily from +the spiral arm toward the BH leads to a high AGN luminosity. This +process is not continuous but occurs periodically depending on the +star formation rate at spiral arms and the amount of gas available for +it and it can be measured as a variability of AGN activity across the +galaxy life time. This variability could make the galaxy normal for a +period of time, and active for another period of time but, for further +investigation of this variability, time-dependent SEDs are needed +to be studied. Another discontinuity of this feeding process can be +caused by the interaction between AGN feedback (outflows) and the +material flowing from the spiral arm. +Due to the accretion process and feeding mechanism the spiral arm +could disappear in a short period of time. But the accretion process +is slowed down by the feedback coming from the AGN as its BH +reaches a critical mass calculated by Ishibashi & Fabian ( 2012 ). And +it is mentioned by Zubovas & King ( 2014b ) that the BH reaches a +critical mass in which the AGN begins to produce outflows and this +may be one of the reasons for slowing down the accretion process. +This leads to a slow increase in the AGN luminosity at higher BH +masses and seen as a flattening in Fig. 5 (top panel). This slow gas +consumption is also what might cause the spiral arm to be long +lived. +All of the abo v e results and discussion concern about an isolated +galaxy without taking into account the environment effects such +as merging of galaxies or the location of this galaxy in its cluster. +Mergers and location of host galaxy with respect to cluster center +can affect the time at which the gas of host galaxy disk is being +consumed due to the AGN activity but not the physical process +occurring between host galaxy disk and its BH. For example, wet +merger can quick the process of accretion by triggering the gas into +the BH. Also the motion of the galaxy can affect the life time of the +process which can be slowed down due to the ram pressure causing +loss of matter. +6 CONCLUSIONS +In this work, we have seen how the AGN can affect the evolution of +the spiral galaxies and that our approach provides an evolutionary +track for the AGN or specifically for the spiral galaxies in terms +of their AGN evolution. This track begins with a BH of low mass, +feeding on the gas mass of the host galaxy. Through its evolution and +its consumption of gas mass, the luminosity increases then decreases +slightly. +If we focus on this evolutionary track for AGN, we can see that as +time evolves the luminosity decreases due to the decrease or the lack +of gas mass in spiral arm which is also consistent with the decrease +in spiral arm radius, and this results also were obtained by Masoura +et al. ( 2018 ) for types 1 and 2 where the X-ray luminosity was found +to decrease as the redshift decreases indicating a decrease in AGN +activity. +From the observed data, the gas mass decreases with decreasing +the redshift. Since the spiral arm density indicates the gas mass +within the spiral arm, hence the spiral arm density also decreases +with decreasing the redshift. So we can say that, during the evolution +of AGN in spiral galaxies, the gas in spiral arms is consumed in +feeding the central BH which indicates that the AGN is affecting +the morphology of spiral galaxies. Hence we can link the AGN +luminosity to the spiral arm radius or the gas mass in the spiral arm +and use equation ( 11 ) to get the morphology of distant active galaxies +through observing their luminosity and vice versa by assuming best +fit values for each parameter that give the observed AGN luminosity. +This evolution has some consequence in between such as the +variable appearance observed in AGN or the AGN variability. In +studies done by Oh et al. ( 2015b ) and Suh et al. ( 2015 ) for type 1 +and type 2, the BH mass-luminosity relation was controlled by the +Eddington ratio which indicates a change in the accretion rate of +the central BH. This change can be explained in terms of our model +approach by considering the spiral arm radius or the gas content +which represents the gas reservoir for BH accretion rate. This also +shows that the accretion process of the gas in spiral arms is not +continuous but happens in phases or episodes of time. As mentioned +by Zubovas & King ( 2014b ) when the BH reaches a critical mass in +which the AGN begins to produce outflows, this may be one of the +causes for slowing down the accretion process. This slowing down +is what causes the spiral arm to be long lived. +ACKNOWLEDGEMENTS +DATA AVAILABILITY +No new data were generated or analysed in support of this research +REFERENCES +Adachi M., Kasai M., 2012, Progress Theor. Phys. , 127, 145 +Alonso S., Coldwell G., Lambas D. G., 2014, A&A , 572, A86 +Antonucci R., 1993, ARA&A , 31, 473 +Baron D., M ´enard B., 2019, MNRAS , 487, 3404 +Bilata-Wolde yes B., Po vi ´c M., Be yoro-Amado Z., Getachew-Woreta T., +Terefe S., 2020, preprint ( arXiv:2003.12416 ) +Dittmann A. J., Miller M. C., 2020, MNRAS , 493, 3732 +Dom ´ınguez S ´anchez H. et al., 2012, MNRAS , 426, 330 +Dubois Y., Gavazzi R., Peirani S., Silk J., 2013, MNRAS , 433, 3297 + +10 +M. A. 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R., 2014b, MNRAS , 439, 400 +This paper has been typeset from a T E X/L +A T E X file prepared by the author. + diff --git a/5tAyT4oBgHgl3EQfQPbi/content/tmp_files/load_file.txt b/5tAyT4oBgHgl3EQfQPbi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5110974aba23b9bfac83eb6befe1a69d4175ea4 --- /dev/null +++ b/5tAyT4oBgHgl3EQfQPbi/content/tmp_files/load_file.txt @@ -0,0 +1,728 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf,len=727 +page_content='MNRAS 00, 1 (2022) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1093/mnras/stac3602 The impact of the AGN and the torus properties on the evolution of spiral galaxies M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Abdulrahman , 1 ‹ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Gadallah , 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Ahmed 1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Elna wa wy 1 1 Astronomy, Space science and Meteorology Department, Faculty of Science, Cairo University, Egypt 2 Astronomy and Meteorology Department, Faculty of Science, Al-Azhar University, Nasr city, 11884, Cairo, Egypt Accepted 2022 December 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Received 2022 December 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' in original form 2022 September 3 A B S T R A C T For spiral galaxies, the active galactic nucleus (AGN) and some physical parameters that concern the host galaxy such as spiral arm radius and density can play an important role in the morphological evolution of these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Considering the gravitational effect of the central black hole as a feeding mechanism, the gas flows from spiral arms to the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, we constructed our approach and derived an equation for the AGN luminosity that depends on parameters such as the black hole mass and the spiral arm density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The galaxy samples were taken from a catalog of type 1 AGN from SDSS-DR7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In our model, we present the relation between the AGN luminosity and the black hole mass depending on the abo v e physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' We also investigated the relation between the black hole mass and the star formation rate for the galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The physical properties of the torus, such as the spiral arm radius, density, the torus length, and the gas mass, and the star formation rate were explained in terms of the variation of the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' These properties are more effective in the evolutionary scenario of the spiral galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Relative to the variation of the AGN luminosity, the evolutionary track is dif ferent based quantitati vely on the star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In which the variation in the star formation rate is positively correlated with the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' K ey words: galaxies: e volution – galaxies: active – galaxies: spiral – black hole physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 1 INTRODUCTION The first active galactic nuclei (AGN) in nearby galaxies were observed and described by Seyfert ( 1943 ) who found spiral galaxies having stronger emission lines from their nucleus than usual and was named Seyfert galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Later observations have revealed many types of AGN which exhibit different features such as seyfert type-1 and seyfert type-2 (hereafter, type 1 and type 2, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Based on their optical emission lines, type 1 shows broad emission lines and type 2 shows only narrow emission lines (Khachikian & Weedman 1974 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The standard unified model introduced by Antonucci ( 1993 ) defines the AGN as a central black hole (BH) surrounded by an accretion disk and dusty torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The AGN luminosity should be controlled by its evolution which depends on the feeding mechanism and available matter for feeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Concerning the feeding mechanism, Alonso, Coldwell & Lambas ( 2014 ) found that barred active galaxies show an excess of nuclear activity more than unbarred ones which refers to the importance of bars in gas inflow to central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' And Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2015 ) showed in their simulation that supernova (SN) feedback could alter the evolution of central BH during galaxy formation, where in strong SN feedback the energy released can destruct the dense clumps in the galaxy core preventing the evolution of BH due to the lack of cold gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Concerning the available matter, Koss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2021 ) found that AGN in massive galaxies ( > 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 5 M ⊙) tends to have larger molecular g as and g as fraction than inactive g alaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' And Franceschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ⋆ E-mail: mohamedabdulazez9@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='com ( 1999 ) found that the histories of the BH accretion rate and stellar formation in host galaxies are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2004 ) studied the accretion-driven growth of BH at low redshift using type 2 AGN and found that the BH of masses less than 10 8 M ⊙ that reside in moderately massive galaxies have accretion rate time scale that is comparable to the age of universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Since AGN is powered by a central BH which resides at a host galaxy of certain physical properties such as stellar mass and star formation rate, the evolution of AGN should be related to its host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In the following we will present some previous works on studying the relation between AGN and host galaxy either by observ ational e vidence or hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The accretion process caused by the central BH, results in a nuclear activity that appears as a feedback on the ambient medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This activity has different scenarios such as winds, radiation pressure, and jets (Zubovas & King 2014a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Fabian, Vasude v an & Gandhi 2008 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The model made by King & Pounds ( 2003 ) explained the energy and momentum of large scale outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' According to this model, the AGN radiation pressure can launch a relativistic wind from very close in, where outflow can emerge from a photosphere of a radius a few tens of schwarzchild radius ( R s ) given by R s = 2 GM c 2 , where M is the BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This wind shocks against the ambient medium producing an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' When the BH reaches a critical mass given by Zubovas & King ( 2014b ), M σ ≈ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 67 × 10 8 σ 4 200 M ⊙, where σ is the velocity dispersion in host galaxy spheroid, the AGN outflows become energy driven and cannot be cooled any more which can cause gas loss and consequently affects the star formation in the © 2022 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Abdulrahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' MNRAS 00, 1 (2022) host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2016 ) showed that the feedback can be positive such as triggering star formation by using its energy in pressure triggered collapse or ne gativ e such as quenching star formation by the loss of material from the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' On the other hand, time evolution simulations of the impact of the jet emerging from AGN on the ambient interstellar medium (ISM) made by Wagner, Bicknell & Umemura ( 2012 ) showed that the effect of the jet depends on the size of the ambient clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For large clouds ( > 50 pc ), it increases the star formation rate, but for small clouds ( ∼10 pc ), it causes explosion of cloud and quenching of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So ne gativ e and positive feedback can co-exist depending on the density of the ambient ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In the simulations made by Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2013 ), the jet power can be considered as a reason of the material loss which can transform the disk galaxy into red elliptical galaxies by the quenching of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2018 ) performed simulations to study the effect of a relativistic jet on the ambient gaseous disk and found that, depending on jet power, the ISM density, and jet orientation, the star formation can be enhanced or quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' These authors found that jet can contribute in increasing the velocity dispersion of ambient ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, the effect of AGN on its host galaxy is rele v ant to the host galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Observational studies made by (Ferrarese & Merritt 2000 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Geb- hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2000 ) revealed a correlation between the BH mass and the velocity dispersion of the host bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Reines & Volonteri ( 2015 ) found a correlation between the BH mass and the stellar mass of the host galaxy in which the BH mass increases with increasing the stellar mass of its host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This is consistent with the observations made by Bilata-Woldeyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2020 ) using data from BAT-SWIFT to study the relation between morphology of the host galaxy and the AGN properties such as the Eddington ratio and BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In these observations, the BH masses are larger in elliptical galaxies than in spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Recent studies made by Dittmann & Miller ( 2020 ) investigated the growth of the central BH by assuming a merging scenario with the compact objects formed in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The study by Tartenas & Zubovas ( 2019 ) showed the feeding of the AGN by the dynamical perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' It also showed that a collision between circumnuclear ring and molecular cloud that can be an efficient fueling mechanism, depending on the angle of collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So the fueling mechanism has a crucial role in driving the relationship between the AGN and its host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' As a way to further examine how such a relation between the AGN and host galaxy exists, Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2016 ) studied the star formation history of type 2 AGN and inactive galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In this study, the gas reservoir in the host galaxy is the main source for BH fueling as first examined and mentioned by Magorrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 1998 ), where both of nuclear activity and star formation are related to the host galaxy gas reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' To further understand this mutual effect between the AGN and the host galaxy, Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2020 ) performed a simulation of the galaxy disk to study gas accretion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This simulation showed how the AGN feedback on the multiphase of the ISM of hot and cold phases can affect the co-evolution of the BH and its host galaxy and found that the accretion of cold gas is more ef fecti ve in BH growth rate than hot gas, and the gas accretion contributes in the BH growth more than mergers with other BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So in this paper we study the evolution of isolated spiral galaxies in terms of AGN contribution by investigating the effect of AGN on the spiral arms of spiral galaxies and how the spiral arms can control the AGN activity as being a source of fueling of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Depending on the fueling of AGN by driving the gas from the spiral arm under the gravitational force of the central BH to derive Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Schematic chart of a simple spiral galaxy having two arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' an Equation for the AGN luminosity that depends on parameters such as the BH mass and the spiral arm radius which is a measure of the amount of gas in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' We also investigate the BH mass-AGN luminosity relation and how it can be affected by changing physical parameters such as the spiral arm radius and the torus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' We hope to construct a physical scenario for how the AGN-galaxy co-evolution works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This paper is presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The model approach is explained in Sections 2 and the data of the galaxy sample is given in Section 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Results and the discussions are presented in Sections 4 and 5 , respectively, while the conclusion is given in Section 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In this work, the cosmological parameters were taken with H 0 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 km s −1 Mpc −1 , �m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='30, �λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2 MODEL APPROACH For a spiral galaxy having a simple galactic disk with two arms, we assume that most of the mass of the galactic matter is concentrated within these dense arms while the rest of the galactic medium of the host disk is very diffuse medium with very low density that can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Based on the simple morphology of the unified model, we assume that the torus is divided to internal and external sides around the the AGN as shown in the schematic chart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In this chart, the internal torus is facing the AGN while the external torus represents the outer region of the galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The internal torus region is assumed to partially contain these arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In which the feedback of the BH activity is more efficient on the matter with these arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In this context, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2022 ) found that the spiral arm is efficient in transporting the gas to central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, we assume that the material transfers in a conical shape from the closest region within the internal torus into the accretion disk through a spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This shape makes the spiral arm radius varies from smaller at the contact point with the accretion disk to larger radius within deeper internal region of the torus (neglecting the environmental effects on the galactic outskirts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For a particle of mass ( m ) in the accretion disk at a distance ( r ) from the central BH, it mo v es with a velocity ( υ) where its angular momentum ( �) is given by, � = m AD υAD r (1) where m AD refers to the mass of accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Due to the gravitational effect of the BH, the particles around the BH experience a torque pushing them inward into the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' As the matter in the accretion disk loses angular momentum the matter External torus Internaltorus RA Accretiondisk Black hole rThe impact of the AGN and the torus properties on the evolution of spiral galaxies 3 MNRAS 00, 1 (2022) spirals into the BH and this gives the chance to the material in internal torus to mo v e to accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So we can assume that the change of angular momentum equals the torque on the gas in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, the change of r with time is a consequence of the change of the gravitational radius ( r g ) of the BH with time which indicates the growth of BH or its evolution, hence, dr dt = dr g dt = � gas m AD υAD (2) where � gas is the torque on the gas in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Considering a gaseous disk where the gravitational force is due to a central BH, the torque on gas at a radius ( r ), as given by Netzer ( 2013 ), is: � gas ( r) = ˙m ( GMr) 1 / 2 f ( r) (3) where, ˙m is the radial mass inflow rate, M is the BH mass, and f ( r ) = 1 − ( r in / r ) 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The r in parameter represents the radius at which the torque on gas is zero and it falls in a non-circular orbit into the BH which is known as the inner-most stable circular orbit (ISCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Using r g = GM c 2 where M , G and c are the BH mass, the gravita- tional constant and the speed of light, respectively, we have dr g dt = G c 2 dM dt (4) It is known that the BH grows due to the accretion of matter with taking into account that not all matter is accreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A fraction ( δ) of this matter does not accrete but it escapes due to the feedback of the AGN such as radiation pressure and wind from accretion disk (Zubovas & King 2014a ) and the star formation taking place in the vicinity of BH (Dittmann & Miller 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So we can write the change of the BH mass with time as, dM dt = (1 − δ) ˙M (5) where ˙M is the accretion rate given by ˙M = L ηc 2 while L is the AGN bolometric luminosity hereafter it is called L AGN , and η is the radiative efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Then equation ( 5 ) becomes, dM dt = (1 − δ) L ηc 2 (6) From equations ( 2 ), (3), (4) and (6), the L AGN can be written as, L AGN = ηc 4 ˙m ( GM r ) 1 / 2 f ( r ) G (1 − δ) m AD υAD (7) where a fast rotating BH has a spin parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='998, and r in = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='24 r g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' It is possible to put r = nr g where n takes values of 20– 40 indicating the mean disk size according to available data on variability of high redshift luminous AGN (Netzer 2013 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' By considering the conical shape of the spiral arm facing the accretion disk, m AD can be written as, dm AD dt = −ξm T (8) where m T is the mass of the internal torus, and ξ is the transfer efficiency which could be thought as the fraction of medium clumpness (clumpy or smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' m T = π 3 R 2 A ρT l T (9) where R A is the spiral arm radius within the internal torus region characterized by a very dense conical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The parameters ρT and l T are the density within the torus and its length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' By assuming that the material is concentrated in the spiral arms at a certain time (t), for which, ρT becomes mostly the arm density ( ρA ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Substituting by (9) into (8), and integrating ρA w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='t time, the mass of accretion disk becomes m AD = π 6 ξR 2 A ρ2 A l T (10) Then the luminosity Equation in the present model becomes L AGN = 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 156 × 10 25 ηn 1 2 f ( r ) M ˙m υAD ξ(1 − δ) l T R 2 A ρ2 A ( erg s −1 ) (11) By setting δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1, n = 25, f ( r ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='777, υAD = 10 8 m / s as standard values, equation ( 11 ) can be re-written as L AGN = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 792 × 10 18 M ˙m l T R 2 A ρ2 A ( erg s −1 ) (12) Therefore, the ef fecti ve parameters on the AGN luminosity are the mass inflow rate ( ˙m ), the internal torus length ( l T ), the spiral arm radius ( R A ), the spiral arm density ( ρA ), and the BH mass ( M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 3 DATA To compare with the calculations of equation ( 12 ) and in an attempt to check the validity of our BH mass-luminosity relation, a sample of the observational data was provided by a catalog of type 1 AGNs from SDSS-DR7 (Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2015a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This catalog contains 5553 type 1 AGN with a redshift of z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In which, galaxies were selected after applying some classification criteria for data of OSSY catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This catalog provides us with the logarithm of bolometric luminosity of AGN ( L bol ) derived from a method developed by Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2004 ) using the luminosity of the [O III ] λ5007 emission line L O III as a tracer of nuclear activity where L bol ≈ 3500 L O III ( erg s −1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' It also provides us with the logarithm of the BH mass as derived by Greene & Ho ( 2005 ) depending on the luminosity and line width of the broad H α line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The bolometric luminosity ranges from 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='09 to 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='77 erg s −1 in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also the BH masses ranges from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='13 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='29 M ⊙ in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' As the galaxy selection is flux-limited, we have considered the Malmquist bias as a selecting criterion of our galaxy sample where the luminosity varies as a function of the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2 , we plot the bolometric luminosity across the redshift of 5553 galaxies as shown in the top panel , then we employ a flux-limit cut off with a minimum flux limit of 4 × 10 −12 ergs −1 cm −2 to estimate the theoretical luminosity (solid line) using the formula of L = 4 πd L F where d L is the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This luminosity distance was estimated according to the analytical approximation considering the case of flat cosmologies (Adachi & Kasai 2012 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, the number of galaxies in the sample has been reduced to 4954 galaxies as shown in the bottom panel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' And the current range for the bolometric luminosity becomes from 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='88 to 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='77 erg s −1 in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 RESULTS In this Section, we present the results that show the impact of the escaped fraction ( δ) and the clumpness fraction ( ξ) mentioned abo v e in equation ( 11 ) on the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Then we used also equation ( 12 ) to show the effect of the spiral arm density ( ρA ) and the radius ( R A ) on this luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' We showed the AGN luminosity versus the BH mass using the observational data along with those of equation ( 12 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' By using derived relations from previous observational works of (Dom ´ınguez S ´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2012 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Baron & M ´enard 2019 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2012 ), we estimated the star formation rate, gas mass, 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Abdulrahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' MNRAS 00, 1 (2022) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The bolometric luminosity of data sample (red dots) across redshift before ( top panel ) and after ( bottom panel ) applying the Malmquist bias where the solid line represents the flux-limit cut off of the galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' and stellar mass of our sample the galaxies comparing the results from both of the observed data and equation ( 12 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Finally, we used the results that we obtained in an attempt to put a suitable evolution scenario of the spiral galaxy in terms of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1 The AGN luminosity versus torus properties Using equation ( 11 ) with assuming R A = 100 pc, ρA = 10 −10 kg m −3 , l T = 100 pc, ˙m = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 1 M ⊙ yr −1 , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 3 sho ws ho w the AGN luminosity is affected by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' the clumpness fraction of the medium ( ξ) and the velocity of the material in the accretion disk ( υAD ) in the top panels ( left and right, respectively );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' the radiative efficiency ( η) and the mean disk size ( n ) in the middle panels ( left and right, respectively );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' and the escaped fraction ( δ) in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For each ef fecti ve parameter, this luminosity was estimated with keeping the others constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From this figure, the luminosity exponentially decreases with increasing both ξ and υAD , while it exponentially increases with increasing both of η, n, and in a logarithmic way with δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In order to see how the BH mass affect the relation between the AGN luminosity and each physical parameter in equation ( 12 ), we present its behaviour at three different BH masses ( LogM = 6, 7 and 8 M ⊙) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 ( the assumed value for each parameter was chosen according to the best fit for the data sample shown later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 , the luminosity has a similar trends showing an exponential decrease with both of the R A , ρA , and l T while it has differently an exponential increase with ˙m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The spiral arm radius and density of a galaxy can be considered as a morphological parameters that demonstrate the host galaxy shape where the spiral arm contains most of the amount of material within torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This matter is considered as the main feeding source for the central BH which controls the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For which, the effect of both of spiral arm radius and density is shown in the top panels (left and right, respectively) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also Masoura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2018 ) found that the AGN luminosity, in a host galaxy, depends on the position of this host galaxy from the main sequence line, depending on the available gas of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Since the accreted matter from the spiral arm travels through a path from spiral arm to the BH, the length of this path should also control the produced AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This path length is represented in our model by the internal torus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The left top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 shows its correlation with the luminosity at three different BH masses, in which, the luminosity increases fast at smaller lengths but it decreases slowly at longer lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' All of the abo v e physical parameters are related to the properties of the galaxy disk but the mass inflow rate or the accretion rate of the BH is related to the properties of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Its correlation with luminosity, shown in the left bottom panel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4 , shows the remarkable increase of the luminosity with the mass inflow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This means that the luminosity doesn’t depend only on the amount of the material flowing toward the BH but also the time taken by this material to reach the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For the data sample, the top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 5 shows the result with implying equation ( 12 ) and those of observed data fitted with a slope of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='029, for which, the standard deviation was adapted to x = 1, y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='86, giving a correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' To account for the dispersion of the scattered points, we calculated the residual which shows a normal Gaussian distribution as shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Using the data sample, we present some physical properties concerning the host galaxy, such as the SFR, the stellar mass ( M stellar ) and the gas mass ( M gas ), to explain how these parameters can change across redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The observed data of our galaxy sample provides us with the H α emission line luminosity, L ( H α), which we used to measure the SFR based on the SFR- L ( H α) dependence using equation ( 1 ) given by Dom ´ınguez S ´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2012 ) where: SFR (M ⊙ yr −1 ) = 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 9 × 10 −42 L (H α) ( erg s −1 ) To estimate the stellar mass as a function of BH mass, we re- arrange equation ( 9 ) given by Baron & M ´enard ( 2019 ) to be as follows: Log � M stel l ar 10 11 M ⊙ � = Log( M BH M ⊙ ) − (7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 88 ± 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 13 ) (1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 64 ± 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 18 ) By assuming that stars form in molecular clouds, the gas mass can be measured using the HI 21 cm line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' According to equation ( 1 ) given by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2012 ) using the stellar mass of M stel l ar > 10 9 M ⊙ since our data range is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='94–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='859 M ⊙ in log scale, the gas mass can be calculated in terms of the stellar masss as follows: Log ( M H I ) ≈ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 276 Log( M stel l ar ) For a deeper understanding of the relation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 5 , we further investigated the SFR distribution along this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 6 , we can see that at a certain BH mass, the AGN luminosity decreases with decreasing the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From equation ( 12 ), ρA can be used as an indicator to the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' It is expected to have a clumpy medium with increasing the density in the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' When star formation takes place, the ultraviolet radiation and the stellar wind 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 46- 45 Lbol Log 44- 43- 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='20 Redshift 48 46 44 42- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='20 RedshiftThe impact of the AGN and the torus properties on the evolution of spiral galaxies 5 MNRAS 00, 1 (2022) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The AGN luminosity distribution as a function of the clumpness fraction and the velocity in the accretion disk (left and right top panels, respectively), the radiative efficiency and the mean disk size (left and right middle panels, respectively), and the escaped fraction from the BH accretion (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' injected energy, caused by massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' These could be destructive mechanisms to the molecular clouds leading to a decrease in the ISM density or cloud dispersion (Grudi ´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2018 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Gonz ´alez- Samaniego & Vazquez-Semadeni 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, the gas can then flow easily to the central region leading to high AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' To implement the relation between SFR and AGN evolution, we present the SFR variation with BH mass as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This figure shows a linear relation with a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='86 and a correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' It is obvious that this correlation fits well at smaller BH masses but SFR shows flattening at higher BH masses, deviating from the linear trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also from Figs 7 and 8 (bottom panel), we can see a variation in the SFR versus the BH mass and redshift, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' On average, the SFR is decreasing at low redshift values and low BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This variation points to the relation between the BH mass and the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In our work, we assume that the spiral arm is the reservoir of gas needed for the BH feeding and it can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 7 along with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 8 where gas in spiral arm is consumed by both of star formation and the BH feeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='4 47 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2 - (s/B1a) (s/810) 46 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0- Lbol 45 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6- 42.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='86 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Abdulrahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' MNRAS 00, 1 (2022) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The bolometric luminosity of the AGN versus the spiral arm radius and the torus length (left and right top panels, respectively), and the spiral arm density and the accretion rate (left and right bottom panels, respectiv ely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' F or these physical parameters, the results were estimated at 3 different values of BH mass of 6, 7, and 8 M ⊙ in log scale, assuming R A = 300 pc, ρA = 10 −12 kg m −3 , l T = 500 pc, ˙m = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 1 M ⊙ yr −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This gas consumption should decrease the gas content of the host galaxy (Scoville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 2015 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 8 (top and middle panels), we can see that both of the gas mass and stellar mass are following the same trend with decreasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The decrease in each of them is slow and this can be used as evidence for the BH feedback such as jets and winds which could make the growth rate of the BH becomes slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This can be used as evidence for the stellar cycle where the gas is converted into stars and then returns back through, for instance, the supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also previous work by Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2008 ) showed that the SFR / M gas ratio is relatively constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2 Evolution of a spiral galaxy We have showed that AGN luminosity is affected by the integration of all aforementioned physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Therefore, we tried to put an evolution scenario for an isolated spiral galaxy depending on these parameter and neglecting any merging and environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For a spiral galaxy hosting a central BH, at first, the gas content of spiral arms is condensed to form clumpy clouds of gas as possible candidates for star formation to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' If the gas gets condensed the amount of material available for the BH accretion decreases, hence the AGN luminosity decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This is why we get low AGN luminosity for high values of spiral arm radius and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' But due M= 6 Mo 46 M= 6 Mo 45 M= 7 Mo M= 7 Mo M= 8 Mo 45 M= 8 Mo 44 S (erg/ (erg/s 44 43 43 Log 42 42 41- 41- 40- 40 0 2000 4000 6000 8000 0 2000 4000 6000 8000 10000 12000 R (pc) l1 (pc) 52 M=6 Mo 36 M= 7 Mo M= 8 Mo 50 S 33 Lbol 48 30 Log 46- M= 6 Mo 27 M= 7 Mo M= 8 Mo 44 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x10-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x10-12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x10-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x10-12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x1020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0x1021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='98x102 m (kg/s) Pa (kg/m3)The impact of the AGN and the torus properties on the evolution of spiral galaxies 7 MNRAS 00, 1 (2022) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The relation between the BH mass and the bolometric luminosity of AGN for the observed data of the galaxy sample (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The black line is our results according to equation ( 12 ) and the blue line is the fit of observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The regular residual of equation ( 12 ) showing a normal distribution (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The BH mass and AGN luminosity against star formation rate distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The star formation rate variation with BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' to the star formation the luminosity emitted from the disk should be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' As time goes and due to galaxy rotation and star formation, the gas in the spiral arms became dispersed which make it easy for the central BH to pull it producing high AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This is why we get high AGN luminosity for small values of spiral arm radius and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The accretion process has a time which is determined by the internal torus length and also by the accretion rate of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' During the gas journey from the spiral arm to the central BH, it travels a certain path which is considered as the internal torus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In our approach, we assumed that material is transferred through a conical path which can be representative of unbarred galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This path length is in rele v ant to the galaxy size and also the existence of its bar which should alter the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' According to Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2014 ) who found that among their sample which includes barred and unbarred AGN, the barred galaxies exhibit a higher nuclear activity than unbarred ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also the length of the internal torus would affect the AGN luminosity and the activity time of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Using the magnetic-hydrodynamical simulations, Rosas- Gue v ara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2022 ) studied the evolution of barred massive disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' These authors found that barred galaxies have lower star formation rate and lower gas fraction compared to unbarred ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This indicates that the existence or absence of a bar may increase or decrease the gas transport efficiency from the galactic disk to the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Linking the correlation between the AGN luminosity, gas mass, and SFR Shangguan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2020 ), we can deduce the evolution scenario based on these physical properties of our galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Using Figs 6 and 9 we can divide the evolution of a galaxy into three phases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Phase 1 is the period of time where the gas is still condensed at spiral arms that we have a large gas mass galaxy at a certain SFR, the gas is consumed to form stars and there is not enough gas to be accreted by the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For this period we should observe low AGN luminosity, low BH mass, and low stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Phase 2 is the period of time where star formation has taken place that the stellar mass increases and the gas is dispersed whether by the galaxy rotation or even by stellar wind or stellar cycle, so the available gas mass for accretion increases, leading to an increase in the BH mass and the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This period may vary from a galaxy to another depending on its size and morphology (there is a bar or not), where Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2014 ), Kim & Choi ( 2020 ) showed that the nuclear activity is higher in galaxies having a bar, 47 46- 45 Lbol Log 44 43 equation(12)result Fit of observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 Log M (Mo) 1000 800- 600 Counts 400 200 0 2 1 0 1 2 Regular residual2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 S 1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 Log 0 SFR 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 Log M (Mo)3 2 SFR (Mo-yr 0 1 2 Fitof observation 5 7 9 10 Log M (M)8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Abdulrahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' MNRAS 00, 1 (2022) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The gas mass (top panel), stellar mass (middle panel), and star formation rate (bottom panel) variation with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' referring to the vital role of the bar in transporting the gas to the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Generally, in this phase and through a transition for the total luminosity of the g alaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' the g alactic disk luminosity may decrease gradually or becomes constant depending on the value of SFR and the AGN luminosity starts to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This phase can also be interpreted in terms of a study done by Zewdie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2020 ) using SDSS MPA-JHU catalogue with the stellar mass range of LogM ∗ = 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 73 −11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 03 M ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Using BPT-diagram, these authors found that the AGN, in this stellar mass range, have lower star formation rates than Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The relation between SFR, BH mass, and gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The 3 evolutionary phases of the spiral galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In case of effect of AGN feedback (positive) on SFR (orange circles) and without the effect of AGN on SFR (black circles) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' star-forming galaxies, and galaxies in this range mo v e from the blue cloud to the red sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Phase 3 is the period of time where the accretion rate of central BH increases due to the large amount of gas coming from spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Accordingly, a small amount of gas remains in the spiral arms for star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So what we will observe is high stellar mass and low galactic disk luminosity but high AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 10 , we summarize the 3 phases for the spiral galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' If the AGN has no effect on the SFR of the host galaxy (black circles) therefore it has a nearly constant SFR, for example, for Log SFR ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6 M ⊙ yr −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The stellar mass increases, leading to an increase in the galactic disk luminosity due to the stellar luminosities superposition but the increase in the AGN luminosity is very small, it changes slightly between Log L bol ∼ 44 and ∼ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 erg s −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In contrast, if the AGN has a positive feedback on the host galaxy, the SFR increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' A factor of 25 increase in SFR is related to an increase of 2 order of magnitudes in the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='4 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 W Log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='20 Redshift 12 - (w) 11 Log Mster 10- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='20 Redshift 3 - 2 - SFR 0 Log 1 2 - 3 - 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='20 Redshift9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='1 Log Mgas M Log 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 (Mo) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 Log SFR (Mo/yr)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6 46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 45 Log Lbol 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 (erg/s) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='6 44 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content='0 Log SFR (Mo/yr)The impact of the AGN and the torus properties on the evolution of spiral galaxies 9 MNRAS 00, 1 (2022) 5 DISCUSSION According to the results presented in the section 4 , we find out that the torus properties such as spiral arm radius, spiral arm density and torus length can be used as indicators of the the evolution of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also the accretion rate by the BH doesn’t only depend on the BH physical parameters such as its spin, but also on the torus properties or the host galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From our results we conclude that the AGN can have an ef fecti ve role in the evolution of its host galaxy and vice v ersa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' F or each galaxy, the luminosity increases as the spiral arm decreases, which means that the gas in spiral arms is the main source for the AGN luminosity and changing it leads to a change in the AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also accretion of stars onto supermassive BHs can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' But to determine which is more efficient in the BH growth, the gas accretion or the star accretion, Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2021 ) treated the tidal disruption events caused by star accretion in cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' These authors found that contribution from stars’ accretion is negligible compared to gas accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Ho we ver this decrease in spiral arm gas content is not only due to the accretion by the BH but partially caused by the feeding process of BH and partially by the star formation taking place in spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The star formation rate is high when the gas mass is high in spiral arms, but to ease the gas flow from the spiral arms to the central BH, the gas should be dispersed and this is done by the consumption of gas in the star formation process and the dispersion caused by the stellar wind or any damping mechanism of the formed stars (Thomp- son, Quataert & Murray 2005 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Hayward & Hopkins 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Lupi 2019 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' The dispersion of gas which causes the gas to flow easily from the spiral arm toward the BH leads to a high AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This process is not continuous but occurs periodically depending on the star formation rate at spiral arms and the amount of gas available for it and it can be measured as a variability of AGN activity across the galaxy life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This variability could make the galaxy normal for a period of time, and active for another period of time but, for further investigation of this variability, time-dependent SEDs are needed to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Another discontinuity of this feeding process can be caused by the interaction between AGN feedback (outflows) and the material flowing from the spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Due to the accretion process and feeding mechanism the spiral arm could disappear in a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' But the accretion process is slowed down by the feedback coming from the AGN as its BH reaches a critical mass calculated by Ishibashi & Fabian ( 2012 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' And it is mentioned by Zubovas & King ( 2014b ) that the BH reaches a critical mass in which the AGN begins to produce outflows and this may be one of the reasons for slowing down the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This leads to a slow increase in the AGN luminosity at higher BH masses and seen as a flattening in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 5 (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This slow gas consumption is also what might cause the spiral arm to be long lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' All of the abo v e results and discussion concern about an isolated galaxy without taking into account the environment effects such as merging of galaxies or the location of this galaxy in its cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Mergers and location of host galaxy with respect to cluster center can affect the time at which the gas of host galaxy disk is being consumed due to the AGN activity but not the physical process occurring between host galaxy disk and its BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' For example, wet merger can quick the process of accretion by triggering the gas into the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Also the motion of the galaxy can affect the life time of the process which can be slowed down due to the ram pressure causing loss of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' 6 CONCLUSIONS In this work, we have seen how the AGN can affect the evolution of the spiral galaxies and that our approach provides an evolutionary track for the AGN or specifically for the spiral galaxies in terms of their AGN evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This track begins with a BH of low mass, feeding on the gas mass of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Through its evolution and its consumption of gas mass, the luminosity increases then decreases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' If we focus on this evolutionary track for AGN, we can see that as time evolves the luminosity decreases due to the decrease or the lack of gas mass in spiral arm which is also consistent with the decrease in spiral arm radius, and this results also were obtained by Masoura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2018 ) for types 1 and 2 where the X-ray luminosity was found to decrease as the redshift decreases indicating a decrease in AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' From the observed data, the gas mass decreases with decreasing the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Since the spiral arm density indicates the gas mass within the spiral arm, hence the spiral arm density also decreases with decreasing the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' So we can say that, during the evolution of AGN in spiral galaxies, the gas in spiral arms is consumed in feeding the central BH which indicates that the AGN is affecting the morphology of spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Hence we can link the AGN luminosity to the spiral arm radius or the gas mass in the spiral arm and use equation ( 11 ) to get the morphology of distant active galaxies through observing their luminosity and vice versa by assuming best fit values for each parameter that give the observed AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This evolution has some consequence in between such as the variable appearance observed in AGN or the AGN variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' In studies done by Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2015b ) and Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ( 2015 ) for type 1 and type 2, the BH mass-luminosity relation was controlled by the Eddington ratio which indicates a change in the accretion rate of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This change can be explained in terms of our model approach by considering the spiral arm radius or the gas content which represents the gas reservoir for BH accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This also shows that the accretion process of the gas in spiral arms is not continuous but happens in phases or episodes of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' As mentioned by Zubovas & King ( 2014b ) when the BH reaches a critical mass in which the AGN begins to produce outflows, this may be one of the causes for slowing down the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' This slowing down is what causes the spiral arm to be long lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' ACKNOWLEDGEMENTS DATA AVAILABILITY No new data were generated or analysed in support of this research REFERENCES Adachi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=', Kasai M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=', 2012, Progress Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfQPbi/content/2301.00043v1.pdf'} +page_content=' Phys.' metadata={'source': 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a/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/2301.00996v1.pdf.txt b/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/2301.00996v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..258f78b90c13653a9f31974b53713f3cfd8c90e4 --- /dev/null +++ b/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/2301.00996v1.pdf.txt @@ -0,0 +1,1472 @@ +Transactional Composition of Nonblocking Data +Structures +Wentao Cai, Haosen Wen, and Michael L. Scott +{wcai6,hwen5,scott}@ur.rochester.edu +University of Rochester +Augest 18, 2022 +Abstract +This paper introduces nonblocking transaction composition (NBTC), +a new methodology for atomic composition of nonblocking operations on +concurrent data structures. Unlike previous software transactional mem- +ory (STM) approaches, NBTC leverages the linearizability of existing non- +blocking structures, reducing the number of memory accesses that must +be executed together, atomically, to only one per operation in most cases +(these are typically the linearizing instructions of the constituent opera- +tions). +Our obstruction-free implementation of NBTC, which we call Medley, +makes it easy to transform most nonblocking data structures into transac- +tional counterparts while preserving their nonblocking liveness and high +concurrency. In our experiments, Medley outperforms Lock-Free Trans- +actional Transform (LFTT), the fastest prior competing methodology, by +40–170%. The marginal overhead of Medley’s transactional composition, +relative to separate operations performed in succession, is roughly 2.2×. +For persistent data structures, we observe that failure atomicity for +transactions can be achieved “almost for free” with epoch-based periodic +persistence. Toward that end, we integrate Medley with nbMontage, a +general system for periodically persistent data structures. The resulting +txMontage provides ACID transactions and achieves throughput up to +two orders of magnitude higher than that of the OneFile persistent STM +system. +1 +Introduction +Nonblocking concurrent data structures, first explored in the 1970s, remain an +active topic of research today. In such structures, there is no reachable state +of the system that can prevent an individual operation from making forward +progress. This liveness property is highly desirable in multi-threaded programs +that aim for high scalability and are sensitive to high tail latency caused by +inopportune preemption of resource-holding threads. +1 +arXiv:2301.00996v1 [cs.DC] 3 Jan 2023 + +Many multi-threaded systems, including those for finance, travel [30], ware- +house management [6], and databases in general [39], need to compose opera- +tions into transactions that occur in an all-or-nothing fashion (i.e., atomically). +Concurrent data structures, however, ensure atomicity only for individual oper- +ations; composing a transaction across operations requires nontrivial program- +ming effort and introduces high overhead. Preserving nonblocking liveness for +every transaction is even more difficult. +One potential solution can be found in software transactional memory (STM) +systems, which convert almost arbitrary sequential code into speculative trans- +actions. Several STM systems provide nonblocking progress [10, 19, 25, 26, 37]. +Most instrument each memory access and arrange to restart operations that +conflict at the level of individual loads and stores. +The resulting program- +ming model is attractive, but the instrumentation typically imposes 3–10× over- +head [34, Sec. 9.2.3]. +Inspired by STM, Spiegelman et al. [36] proposed transactional data struc- +ture libraries (TDSL), which introduce (blocking) transactions for certain hand- +modified concurrent data structures. By observing that reads need to be tracked +only on critical nodes whose updates may indicate semantic conflicts, TDSL re- +duces read set size and achieves better performance than general STMs. +Herlihy and Koskinen [18] proposed transactional boosting, a (blocking) +methodology that allows an STM system to incorporate operations on exist- +ing concurrent data structures. Using a system of semantic locks (e.g., with +one lock per key in a mapping), transactions arrange to execute concurrently +so long as their boosted operations are logically independent, regardless of +low-level conflicts. A transaction that restarts due to a semantic conflict (or +to a low-level conflict outside the boosted code) will roll back any already- +completed boosted operations by performing explicitly identified inverse opera- +tions. An insert(k,v) operation, for example, would be rolled back by performing +remove(k).Transactional boosting leverages the potential for high concurrency +in existing data structures, but is intrinsically lock-based, and is not fully gen- +eral: operations on a single-linked FIFO queue, for example, have no obvious +inverse. +In work concurrent to TDSL, Zhang et al. [43] proposed the Lock-Free Trans- +actional Transform (LFTT), a nonblocking methodology to compose nonblock- +ing data structures, based on the observation that only certain nodes—those +critical to transaction semantics—really matter in conflict management. Each +operation on an LFTT structure publishes, on every critical node, a descrip- +tion of the transaction of which it is a part, so that conflicting transactions +can see and help each other. A remove(7) operation, for example, would pub- +lish a description of its transaction on the node in its structure with key 7. +Initially, LFTT supported only static transactions, whose constituent opera- +tions were all known in advance. Subsequently, LaBorde et al. [23] proposed a +Dynamic Transactional Transform (DTT) that generalizes LFTT to dynamic +transactions (specified as lambda expressions). Concurrently, Elizarov et al. [8] +proposed LOFT which is similar to LFTT but avoids LFTT’s bug of duplicated +helping. +2 + +Unfortunately, as in transactional boosting, the need to identify critical +nodes tends to limit LFTT and DTT to data structures representing sets and +mappings. +DTT’s publishing and helping mechanisms also require that the +“glue” code between operations be fully reentrant (to admit concurrent execu- +tion by helping threads [23]) and may result in redundant work when conflicts +arise. +Worse, for read-heavy workloads, LFTT and DTT require readers to +be visible to writers, introducing metadata updates that significantly increase +contention in the cache coherence protocol. +In our work, we propose NonBlocking Transaction Composition (NBTC), +a new methodology that can create transactional versions of a wide variety of +concurrent data structures while preserving nonblocking progress and incurring +significantly lower overhead than traditional STM. The intuition behind NBTC +is that in already nonblocking structures, only critical memory accesses—for +the most part, the linearizing load and compare-and-swap (CAS) instructions— +need to occur atomically, while most pre-linearization memory accesses can +safely be executed as they are encountered, and post-linearization accesses can +be postponed until after the transaction commits. +In comparison to STM, NBTC significantly reduces the number of mem- +ory accesses that must be instrumented—typically to only one per constituent +operation. Unlike transactional boosting and transactional transforms, NBTC +brings the focus back from semantics to low-level memory accesses, thereby +enabling mechanical transformation of existing structures and accommodating +almost arbitrary abstractions—much more than sets and mappings. NBTC also +supports dynamic transactions, invisible readers, and non-reentrant “glue” code +between the operations of a transaction. The one requirement for compatibility +is that the linearization points of constituent operations must be immediately +identifiable: each operation must be able to tell when it has linearized at run +time, without performing any additional shared-memory accesses. Most non- +blocking structures in the literature appear to meet this requirement. +To assess the practicality of NBTC, we have built an obstruction-free imple- +mentation, Medley, that uses a variant of Harris et al.’s multi-word CAS [16] +to execute the critical memory accesses of each transaction atomically, eagerly +resolving conflicting transactions as they are discovered. Using Medley, we have +created NBTC versions of Michael and Scott’s queue [29], Fraser’s skiplist [10], +the rotating skiplist of Dick et al. [7], Michael’s chained hash table [28], and +Natarajan and Mittal’s binary search tree [31]. All of the transformations were +straightforward. +In the traditional language of database transactions [15], Medley provides +isolation and consistency. Building on recent work on persistent memory, we +have also integrated Medley with the nbMontage system of Cai et al. [2] to +create a system, txMontage, that provides failure atomicity and durability as +well—i.e., full ACID transactions. Specifically, we leverage the epoch system of +nbMontage, which divides time into coarse-grain temporal intervals and recov- +ers, on failure, to the state of a recent epoch boundary. By folding a check of +the epoch number into its multi-word CAS, txMontage ensures that operations +of the same transaction always linearize in the same epoch, thereby obtaining +3 + +failure atomicity and durability “almost for free.” +Summarizing contributions: +• (Section 2) We introduce nonblocking transaction composition (NBTC), a +new methodology with which to compose the operations of nonblocking data +structures. +• (Section 3) Deploying NBTC, we implement Medley, a general system for +transactional nonblocking structures. Medley’s easy-to-use API and mechan- +ical transform make it easy to convert compatible nonblocking structures to +transactional form. +• (Section 4) We integrate Medley with nbMontage to create txMontage, provid- +ing not only transactional isolation and consistency, but also failure atomicity +and durability. +• (Section 5) We argue that using NBTC and Medley, transactions composed +of nonblocking structures are nonblocking and strictly serializable. We also +argue that transactions with txMontage provide a persistent variant of strict +serializability analogous to the buffered durable linearizability of Izraelevitz +et al. [21]. +• (Section 6) We present performance results, confirming that Medley imposes +relatively modest overhead and scales to large numbers of threads. Specif- +ically, Medley outperforms LFTT by 1.4× to 2.7× and outperforms TDSL +and the OneFile nonblocking STM [33] system by an order of magnitude. On +persistent memory, txMontage outperforms nonblocking persistent STM by +two orders of magnitude. +2 +Nonblocking Transaction Composition +Nonblocking transaction composition (NBTC) is a new methodology that fully +leverages the linearizability of nonblocking data structure operations. NBTC +obtains strict serializability by atomically performing only the critical memory +accesses of composed operations. It supports a large subset of the nonblocking +data structures in the literature (characterized more precisely below), preserving +the high concurrency and nonblocking liveness of the transformed structures. +2.1 +NBTC Composability +The key to NBTC composability is the immediately identifiable linearization +point. Specifically: +Definition 1. A data structure operation has an immediately identifiable lin- +earization point if: +1. statically, we can identify every instruction that may potentially serve as +the operation’s linearization point. Such an instruction must be a load for a +read-only operation or a compare-and-swap (CAS) for an update operation; +4 + +2. dynamically, after executing a potentially linearizing instruction, we can de- +termine whether it was indeed the linearization point. A linearizing load has +to be determined before the operation returns; a linearizing CAS has to be +determined without performing any additional shared-memory accesses. +There can be more than one potential linearization point in the code of an +operation, but only one of them will constitute the linearization point in any +given invocation. +Definition 2. A nonblocking data structure is NBTC-composable if each of its +operations has an immediately identifiable linearization point. +While it may be possible to relax this definition, the current version accom- +modates a very large number of existing nonblocking structures. +2.2 +The Methodology +It is widely understood that most nonblocking operations comprise a “planning” +phase and a “cleanup” phase, separated by a linearizing instruction [12, 38]. +Executing the planning phase does not commit the operation to success; cleanup, +if needed, can be performed by any thread. The basic strategy in NBTC is to +perform the planning for all constituent operations of the current transaction, +then linearize all those operations together, atomically, and finally perform all +cleanup. Our survey of existing data structures and composition patterns reveals +two principle complications with this strategy. +The first complication involves the notion of a publication point, where an +operation may become visible to other threads but not yet linearize. Because +publication can alter the behavior of other threads, it must generally (like a +linearization point) remain speculative until the entire transaction is ready to +commit. An example can be seen in the binary search tree of Natarajan and +Mittal [31], where an update operation o may perform a CAS that publishes its +intent to linearize soon but not quite yet. After this publication point, either o +itself or any other update that encounters the publication notice may attempt +to linearize o (in the interest of performance, a read operation will ignore it). +Notably, CAS instructions that serve to help other (already linearized) opera- +tions, without revealing the nature of the current operation, need not count as +publication. +The second complication arises when a transaction, t, performs two or more +operations on the same data structure and one of the later operations (call it +o2) depends on the outcome of an earlier operation (call it o1). Here the thread +executing t must proceed as if o1 has completed, but other threads must ignore +it. If o1 requires cleanup (something that NBTC will normally delay until after +transaction commit), o2 may need to help o1 before it can proceed, while other +transactions should not even be aware of o1’s existence. +Both complicating cases can be handled by introducing the notion of a spec- +ulation interval in which CAS instructions must be completed together for an +5 + +operation to take effect as part of a transaction. This is similar to the CAS ex- +ecutor phase in a normalized nonblocking data structure [38], but not the same, +largely due to the second complication. For an operation that becomes visible +before its linearization point, it suffices to include in the speculation interval all +CAS operations between the publication and linearization points, inclusive. For +an operation o2 that needs to see an earlier operation o1 in the same transaction, +it suffices to track the transaction’s writes and to start o2’s speculation interval +no later than the first instruction that accesses a location written by o1. +Definition 3. A bit more precise, we say +• A CAS instruction in operation o of thread t in history H is benign if there +is no extension H′ of H such that t executes no more instructions in H′ and +yet o linearizes in H′ nonetheless. +• The first CAS instruction of o that is not benign is o’s publication point (this +will often be the same as its linearization point). +• The speculation interval of o begins either at the publication point or at the +first instruction that sees a value speculatively written by some earlier opera- +tion in the same transaction (whichever comes first) and extends through o’s +linearization point. +• A load in a read-only operation is critical if it is the immediately identifiable +linearization point of the operation. A CAS in an update operation is critical +if it lies in the speculation interval. +Without loss of generality, we assume that all updates to shared memory +(other than initialization of objects not yet visible to other threads) are effected +via CAS. +Given these definitions, the NBTC methodology is straightforward: To atom- +ically execute a set of operations on NBTC-composable data structures, we +transform every operation such that (1) instructions prior to the speculation +interval and non-critical instructions in the speculation interval are executed on +the fly as a transaction encounters them; (2) critical instructions are executed +in a speculative fashion, so they will take effect, atomically, only on transaction +commit; and (3) instructions after the speculation interval are postponed until +after the commit. +3 +The Medley System +To illustrate NBTC, we have written a system, Medley, that (1) instruments +critical instructions, executes them speculatively, and commits them atomically +using M-compare-N-swap, our variant of the multi-word CAS of Harris et al. +[16]; (2) identifies and eagerly resolves transaction conflicts; and (3) delays non- +critical cleanup until transaction commit. +6 + +1 template class CASObj { // Augmented atomic object +2 +T nbtcLoad(); +3 +bool nbtcCAS(T expected, T desired, bool linPt, bool pubPt); +4 +/* Regular atomic methods: */ +5 +T load(); void store(T desired); bool CAS(T expected, T desired); +6 }; +7 class Composable { // Base class of all transactional objects +8 +template void addToReadSet(CASObj*,T); // Register load +9 +void addToCleanups(function); // Register post-critical work +10 +template T* tNew(...); // Create a new block +11 +template void tDelete(T*); // Delete a block +12 +template void tRetire(T*); // Epoch-based safe retire +13 +TxManager* mgr; // Tx metadata shared among Composables +14 +struct OpStarter { OpStarter(TxManager*); } // RAII op starter +15 }; +16 class TxManager { // Manager shared among composable objects +17 +void txBegin(); // Start a transaction +18 +void txEnd(); // Try to commit the transaction +19 +void txAbort(); // Explicitly abort the transaction +20 +void validateReads(); // Optional validation for opacity +21 }; +22 struct TransactionAborted : public std::exception{ }; +Figure 1: C++ API of Medley for transaction composition. +3.1 +API +Figure 1 summarizes Medley’s API. Using this API, we transform an NBTC- +composable data structure into a transactional structure as follows: +1. Replace critical loads and CASes with nbtcLoad and nbtcCAS, respectively. +Fields to which such accesses are made should be declared using the CASObj +template. +2. Invoke addToReadSet for the critical load in a read operation, recording the +address and the loaded value. +3. Register each operation’s post-critical work via addToCleanups. +4. Replace every new and delete with tNew and tDelete. Replace every retire +(for safe memory reclamation—SMR) with tRetire. +5. Declare an OpStarter object at the beginning of each operation. +CASObj¡T¿ augments each CAS-able 64-bit word (e.g., atomic¡Node*¿) with +additional metadata bits for speculation tracking (details in Section 3.2). It +provides a specialized load and CAS as well as the usual methods of atomic¡T¿. +To dynamically identify the speculation interval, nbtcCAS takes two extra argu- +ments, linPt and pubPt, that indicate whether this call, should it succeed, will +constitute its operation’s linearization or/and publication point. In a similar +vein, addToReadSet can be called after an nbtcLoad to indicate (after inspecting +the return value) that this was (or is likely to have been) the linearizing load of +a read-only operation, and should be tracked for validation at commit time. +Composable is a base class for transactional objects. It provides a variety of +NBTC-related methods, including support for safe memory reclamation (SMR), +used to ensure that nodes are not reclaimed until one can be certain that no +references remain among the private variables of other threads. Our current +7 + +implementation of SMR uses epoch-based reclamation [10, 17, 27]. For the sake +of generality, Composable also provides an API for transactional boosting, which +can be used to incorporate lock-based operations into Medley transactions (at +the cost, of course, of nonblocking progress). We do not discuss this mechanism +further in this paper. +The TxManager class manages transaction metadata and provides methods +to initiate, abort, and complete a transaction. A TxManager instance is shared +among all Composable instances intended for use in the same transactions. In +each operation call, the manager distinguishes (via OpStarter()) whether exe- +cution is currently inside or outside a transaction. If outside, all transactional +instrumentation is elided; if inside, instrumentation proceeds as specified by the +NBTC methodology. +Given that nonblocking operations can execute safely in any reachable state +of the system, there is usually no need to stop the execution of a doomed-to- +abort transaction as soon as a conflict arises—i.e., to guarantee opacity [14]. In +exceptional cases (e.g., when later operations of a transaction cannot be called +with certain combinations of parameters, or when aborts are likely enough that +delaying them may compromise performance), the validateReads method can be +used to determine whether previous reads remain correct. +To illustrate the use of Medley, Figure 2 highlights lines of code in Michael’s +nonblocking hash table [28] that must be modified for NBTC; Figure 3 then +shows an example transaction that modifies two hash tables. In a real appli- +cation, the catch block for TransactionAborted would typically loop back to the +beginning of the transaction code to try again, possibly with additional code +to detect and recover from livelock (e.g., via backoff or hints to the underly- +ing scheduler). In contrast to STM systems, Medley does not instrument the +intra-transaction “glue” code between data structure operations. +This code +is always executed as regular code outside a transaction and should always be +data-race free; if it has side effects, the catch block (written by the programmer) +for aborted transactions should compensate for these before the programmer +chooses to retry or give up. +3.2 +M-Compare-N-Swap +To execute the critical memory accesses of each transaction atomically, we em- +ploy a software-emulated M-compare-N-swap (MCNS) that builds on the double- +compare-single-swap (RDCSS) and multi-word CAS (CASN) of Harris et al. [16]. +Each transaction maintains a descriptor that contains a read set, a write set, +and a 64-bit triple of thread ID, serial number, and status, as shown in Fig- +ure 4. Descriptors are pre-allocated on a per-thread basis within a TxManager +instance, and are reused across transactions. A status can be InPrep (initial +state), InProg (ready to commit), Committed (only when InProg and validation +succeeds), or Aborted (from InPrep or due to failed validation). +Each originally 64-bit word at which a critical memory access may occur is +augmented with a 64-bit counter, together comprising an 128-bit CASObj. Each +critical CAS installs a pointer to its descriptor in the CASObj and increments +8 + +1 class MHashTable : +public Composable { +2 struct Node { K key; V val; CASObj next; }; +3 // from p, find c >= k; nbtcLoad and tRetire may be used +4 bool find(CASObj* &p, Node* &c, Node* &n, K k); +5 optional get(K key) { +6 +OpStarter starter(mgr); CASObj* prev = nullptr; +7 +Node *curr, *next; optional res = {}; +8 +if (find(prev,curr,next,key)) res = curr->val; +9 +addToReadSet(prev,curr); +10 +return res; +11 } +12 optional put(K key, V val) { // insert or replace if key exists +13 +OpStarter starter(mgr); +14 +CASObj* prev = nullptr; optional res = {}; +15 +Node *newNode = tNew(key, val), *curr, *next; +16 +while(true) { +17 +if (find(prev,curr,next,key)) { // update +18 +newNode->next.store(curr); +19 +if (curr->next.nbtcCAS(next,mark(newNode),true,true)) { +20 +res = curr->val; +21 +auto cleanup = [](){ +22 +if (prev->CAS(curr,newNode)) tRetire(curr); +23 +else find(prev,curr,next,key); +24 +}; +25 +addToCleanups(cleanup); // execute right away if not in tx +26 +break; +27 +} +28 +} else { // key does not exist; insert +29 +newNode->next.store(curr); +30 +if (prev->nbtcCAS(curr,newNode,true,true)) break; +31 +} +32 +} +33 +return res; +34 }}; +Figure 2: Michael’s lock-free hash table example (Medley-related parts +highlighted). +1 void doTx(MHashTable* ht1, MHashTable* ht2, V v, K a1, K a2) { +2 +TxManager* mgr=ht1->mgr; assert(mgr==ht2->mgr); +3 +try { // transfer ‘v’ from account ‘a1’ in ‘ht1’ to ‘a2’ in ‘ht2’ +4 +mgr->txBegin(); +5 +V v1 = ht1->get(a1); V v2 = ht2->get(a2); +6 +if (!v1.hasValue() or v1.value() < v) mgr->txAbort(); +7 +ht1->put(a1, v1.value() - v); ht2->put(a2, v + v2.valueOr(0)); +8 +mgr->txEnd(); +9 +} catch (TransactionAborted) { /* transaction aborted */ } +10 } +Figure 3: Transaction example on Michael’s hash table. +9 + +1 struct Desc { +2 +map* readSet; +3 +map* writeSet; +4 +atomic status;//63..50 tid 49..2 serialNumber 1..0 status +5 +enum STATUS { InPrep=0, InProg=1, Committed=2, Aborted=3 }; +6 }; +7 struct CASObj { atomic val_cnt; }; +Figure 4: Descriptor and CASObj structures. +the counter; at commit or abort, the descriptor is uninstalled and the counter +incremented again. We leverage 128-bit CAS instructions on the x86 to change +the original word and the counter together, atomically. The counter is odd when +CASObj contains a pointer to a descriptor and even when it is a real value. +Each instance of MCNS proceeds through phases that install descriptors, +finalize status, and uninstall descriptors. The first two phases are on the critical +path of a data structure operation. A new transaction initializes metadata in its +descriptor (at txBegin): it clears the read and write sets, increments the serial +number, and resets the status to InPrep. The installing phase then occurs over +the course of the transaction: Each critical load records its address, counter, +and value in the read set. Each critical CAS records its address, old counter, +old value, and desired new value in the write set; it then installs a pointer to +the descriptor in the CASObj. Pseudocode for the installing phase appears in +Figure 5. +To spare the programmer the need to reason about counters, nbtcLoad makes +a record of its ⟨counter, object⟩ pair (line 15 in Fig. 5); addToReadSet then adds +this pair (and the specified CASObj) to the transaction’s read set (line 20). +When a thread encounters its own descriptor, nbtcLoad returns the specu- +lated value from the write set (line 11). Likewise, nbtcCAS updates the write +entry (line 34). Such encounters automatically initiate the speculation interval +(lines 10, 30, and 32), which then extends through the linearization point of the +current operation (line 38). +If an operation encounters the descriptor of some other thread, it gets that +descriptor out of the way by calling tryFinalize (Fig. 6). This method aborts the +associated transaction if the descriptor is InPrep, helps complete the commit if +InProg, and in all cases uninstalls the descriptor from the CASObj in which it +was found. Similar actions occur when a thread is forced to abort or reaches the +end of its transaction and attempts to commit (lines 39–58). Whether helping +or acting on its own behalf, a thread performing an MCNS must verify that the +descriptor is still responsible for the CASObj through which it was discovered +(line 9) and (if committing) that the values in the read set are still valid (line 25). +After CAS-ing the status to Committed or Aborted, the thread uninstalls the +descriptor from all associated CASObjs, replacing pointers to the descriptor +with the appropriate updated values (lines 31 and 34). Once uninstalling is +complete, the owner thread calls cleanup routines (line 55) for a commit or +deallocates tNew-ed blocks (line 43) for an abort. +Our design adopts invisible readers and eager contention management for +efficiency and simplicity. Eager contention management admits the possibility +10 + +1 void TxManager::txBegin() { +2 +desc->readSet->clear(); desc->writeSet->clear(); +3 +status.store((status.load() & ~3) + 4); +4 } +5 T CASObj::nbtcLoad() { +6 retry: +7 +{val,cnt} = val_cnt.load(); +8 +if (cnt % 2) { // is descriptor +9 +if (val == desc) { +10 +startSpeculativeInterval(); +11 +return desc->writeSet[this].newVal; +12 +} else val->tryFinalize(this, {val,cnt}); +13 +goto retry; // until object has real value +14 +} +15 +... /* Record ‘this’ and ‘cnt’ to be added to readSet */ +16 +return val; +17 } +18 void Composable::addToReadSet(CASObj* obj, T val) { +19 +... /* Retrieve ‘cnt’ by ‘obj‘ */ +20 +mgr->readSet[obj] = {val,cnt}; +21 } +22 bool CASObj::nbtcCAS(T expected,T desired,bool linPt,bool pubPt){ +23 retry: +24 +{val,cnt} = val_cnt.load(); +25 +if (cnt % 2) { // is descriptor +26 +if (val != desc) { // not own descriptor +27 +val->tryFinalize(this, {val,cnt}); +28 +goto retry; // until object has real value +29 +} +30 +startSpeculativeInterval(); +31 +} else if (val != expected) return false; +32 +if (pubPt) startSpeculativeInterval(); +33 +if (inSpeculativeInterval()) { // Is critical CAS +34 +desc->writeSet[this] = {val,cnt,desired}; +35 +bool ret = true; +36 +if (!(cnt % 2)) ret = this->CAS({val,cnt},{desc,cnt+1}); +37 +if (!ret) desc->writeSet.remove(this); +38 +if (linPt and ret) endSpeculativeInterval(); +39 +return ret; +40 +} else return CAS(expected, desired); +41 } +Figure 5: Pseudocode for installing phase. +11 + +1 bool Desc::stsCAS(uint64 d, STATUS expected, STATUS desired) { +2 +d = d & ~3; return status.CAS(d + expected, d + desired); +3 } +4 bool Desc::setReady(){return stsCAS(status.load(),InPrep,InProg);} +5 bool Desc::commit(uint64 d){return stsCAS(d,InProg,Committed);} +6 bool Desc::abort(uint64 d){return stsCAS(d,d & 1,Aborted);} +7 void Desc::tryFinalize(CASObj* obj, uint128 var) { +8 +uint64 d = status.load(); +9 +if (obj->val_cnt.load() != var) // ensure d indicates right tx +10 +return; +11 +if (d & 3 == InPrep) { +12 +abort(d); +13 +uint64 newd = status.load(); +14 +if (newd & ~3 != d & ~3) return; // serial number mismatch +15 +d = newd; +16 +} +17 +if (d & 3 == InProg) { +18 +if (validateReads(d)) commit(d); +19 +else abort(d); +20 +} +21 +uninstall(status.load()); +22 } +23 bool Desc::validateReads() { +24 +for (e:*readSet) +25 +if ({e.val,e.cnt} != e.addr->load()) return false; +26 +return true; +27 } +28 void Desc::uninstall(uint64 d) { +29 +if (d % 3 == Committed) +30 +for (e:*writeSet) +31 +e.addr->CAS({this,e.cnt+1}, {e.newVal,e.cnt+2}); +32 +else // Aborted +33 +for (e:*writeSet) +34 +e.addr->CAS({this,e.cnt+1}, {e.oldVal,e.cnt+2}); +35 } +36 struct TxManager { +37 +threadLocal vector cleanups, allocs; +38 +threadLocal Desc* desc; +39 +void txAbort() { +40 +uint64 d = desc->status.load(); +41 +desc->abort(d); +42 +desc->uninstall(d); +43 +for (f:allocs) f(); // undo tNew +44 +throw TransactionAborted(); +45 +} +46 +void txEnd() { +47 +if (!desc->setReady()) txAbort(); +48 +else { +49 +uint64 d = desc->status.load(); +50 +if (!desc->validateReads()) desc->abort(d); +51 +else if (d & 3 == InProg) desc->commit(d); +52 +d = desc->status.load(); +53 +if (d & 3 == Committed) { +54 +desc->uninstall(d); +55 +for (f:cleanups) f(); +56 +} else txAbort(); +57 +} +58 +} +59 }; +Figure 6: Pseudocode of methods that finalize transactions. +12 + +of livelock—transactions that repeatedly abort each other—and therefore guar- +antees only obstruction freedom. Lazy (commit-time) contention management +along with some total order of descriptor installment might allow us to pre- +serve lock freedom for structures that provide it [35], but would significantly +complicate the tracking and retrieving of uncommitted changes, and would not +address starvation, which may be a bigger problem than livelock in practice; +we consider these implementation choices orthogonal to the effectiveness of the +NBTC methodology, and decide to explore them in the future. +4 +Persistent Memory +Transactions developed, historically, in the database community; transactional +memory (TM) adapted them to in-memory structures in multithreaded pro- +grams. The advent of cheap, low-power, byte-addressable nonvolatile memory +(NVM) presents the opportunity to merge these two historical threads in a way +that ideally leverages NBTC. Specifically, where TM aims to convert sequential +code to thread-safe parallel code, NBTC assumes—as in the database world— +that we are already in possession of efficient thread-safe structures and we wish +to combine their operations. Given this assumption, it seems appropriate (as +described at the end of Sec. 3.1) to assume that the programmer is responsi- +ble for the “glue” code between operations, and to focus on the atomicity and +durability of the composed operations. +4.1 +Durable Linearizability +On machines with volatile caches, data structures in NVM will generally be +consistent after a crash only if programs take pains to issue carefully chosen +write-back and fence instructions. To characterize desired behavior, Izraelevitz +et al. [21] introduced durable linearizability as a correctness criterion for persis- +tent structures. A structure is durably linearizable if it is linearizable during +crash-free execution and its long-term history remains linearizable when crash +events are elided. Equivalently [11], each operation should persist between its +invocation and response, and the order of persists should match the linearization +order. +Many durably linearizable nonblocking data structures have been designed +in recent years [3, 9, 11, 44]. Several groups have also proposed methodologies +by which existing nonblocking structures can be made durably linearizable [12, +13, 21]. +Other groups have developed persistent STM systems, but most of +these have been lock-based [4, 5, 24, 40]. OneFile [33] and QSTM [1] are, to the +best of our knowledge, the only nonblocking persistent STM systems. OneFile +serializes transactions using a global sequence number, eliminating the need for +a read set and improving read efficiency, but introducing the need for invasive +data structure modifications and a 128-bit wide CAS. QSTM employs a global +persistent queue for active transactions, avoiding the need for wide CAS and +invasive structural changes, but its execution remains inherently serial. +13 + +4.2 +Lowering Persistence Overhead +Unfortunately, write-back and fence instructions tend to have high latency. +Given the need for operations to persist before returning, durable linearizability +appears to be intrinsically expensive. Immediate persistence for STM introduces +additional overhead, as metadata for transaction concurrency control must also +be eagerly written back and fenced. +To move high latency instructions off the application’s critical path, Izraele- +vitz et al. [21] introduced the notion of buffered durable linearizability (BDL). +By allowing a modest suffix of pre-crash execution to be lost during post-crash +recovery (so long as the overall history remains linearizable), BDL allows write- +back and fence instructions to execute in batches, off the application’s critical +path. Applications that need to ensure persistence before communicating with +the outside world can employ a sync operation, reminiscent of those in tradi- +tional file systems and databases. +First proposed in the context of the Dal´ı persistent hash table [32], periodic +persistence was subsequently adopted by nbMontage [2], a general-purpose sys- +tem to create BDL versions of existing nonblocking structures. The nbMontage +system divides wall-clock time into “epochs” and persists operations in a batch +at the end of each epoch. In the wake of a crash in epoch e, the system recovers +all structures to their state as of the end of epoch e − 2. To maximize through- +put in the absence of crashes, nbMontage also distinguishes between data that +are semantically significant (a.k.a. “payloads”) and data that are merely per- +formance enhancing (e.g., indices); the latter can be kept in DRAM and rebuilt +during recovery. As an example, the payloads of a mapping are simply a pile of +key-value pairs; the associated hash table, tree, or skiplist resides in transient +DRAM. The payloads of a queue are ⟨serial number, item⟩ pairs. +To ensure that post-crash recovery always reflects a consistent state of each +structure, every nbMontage operation is forced to linearize in the epoch with +which its payloads have been labeled. Operations that take “too long” to com- +plete may be forced to abort and start over. The nbMontage system as a whole +is lock free; sync is actually wait free. +4.3 +Durable Strict Serializability +Linearizability, of course, is not suitable for transactions, which must remain +speculative until all operations can be made visible together. +STM systems +typically provide strict serializability instead: transactions in a crash-free history +appear to occur in a sequential order that respects real time (if A commits before +B begins, then A must serialize before B) [34, Sec. 3.1.2]. For a persistent version +of NBTC, we need to accommodate crashes. +Like Izraelevitz et al. [21], we assume a full-system crash failure model: +data structures continue to exist after a crash, but are accessed only by new +threads—the old threads disappear. Under this model: +Definition 4. An execution history H displays durable strict serializability +(DSS) if it is strictly serializable when crash events are elided. +14 + +Like durable linearizability, this definition requires all work completed before a +crash to be visible after the crash. The buffered analogue is similar: +Definition 5. An execution history H displays buffered durable strict serializ- +ability (BDSS) if there exists a happens-before–consistent cut of each inter-crash +interval such that H is strictly serializable when crash events are elided along +with the post-cut suffix of each inter-crash interval. +4.4 +Merging Medley with nbMontage +The epoch system of nbMontage provides a natural mechanism with which to +provide failure atomicity and durability for Medley transactions: if operations +of the same transaction always occur in the same epoch, then they will be +recovered (or lost) together in the wake of a crash. Building on this observation, +we merge the two systems to create txMontage. Payloads of all operations in a +given transaction are labeled with the same epoch number. That number is then +validated along with the rest of the read set during MCNS commit, ensuring +that the transaction commits in the expected epoch. While nbMontage itself is +quite complex, this one small change is all that is required to graft it (and all +its converted persistent data structures) onto Medley: persistence comes into +transactions “almost for free.” +5 +Correctness +In this section, we argue that histories comprising well-formed Medley transac- +tions are strictly serializable, that Medley is obstruction free, and that txMon- +tage provides buffered durable strict serializability. +Definition 6. A Medley transaction is well-formed if +1. it starts with txBegin and ends with txEnd, optionally with txAbort in between; +2. it contains operations of NBTC-transformed data structures; and +3. all other intra-transaction code is nonblocking and free from any side effects +not managed by handlers for the TransactionAborted exception. +5.1 +Strict Serializability +Lemma 1. At the implementation level (operating on the array of words that +comprises system memory), nbtcLoad, nbtcCAS, tryFinalize, txAbort, and txEnd +(MCNS) are linearizable operations. +Proof (sketch). Follows directly from Harris et al. [16]. Their RDCSS compares +(without changing) only a single location, and their CASN supports the up- +date of all touched words, but the proofs adapt in a straightforward way. In +particular, as in RDCSS, an unsuccessful tryFinalize or txEnd can linearize on a +(failed) validating read or a failed CAS of its status word. A tryFinalize or txEnd +15 + +whose status CAS is successful linearizes “in the past,” on the first of its vali- +dating reads. (Ironically, this means that MCNS does not have an immediately +identifiable linearization point.) +Lemma 2. In any history in which transaction t performs an nbtcLoad or nbtc- +CAS operation x on CASObj o, and in which t’s txEnd operation y succeeds, no +tryFinalize or txEnd for a different transaction that modifies o succeeds between +x and y. +Proof (sketch). Suppose the contrary, and call the transaction with the conflict- +ing tryFinalize or txEnd u. If u’s nbtcCAS of o occurs between x and y, it will +abort and uninstall t’s descriptor, or cause read validation to fail in y, contra- +dicting the assumption that t’s txEnd succeeds. If u’s nbtcCAS of o occurs before +x, then x will abort and uninstall u’s descriptor, contradicting the assumption +that u’s tryFinalize or txEnd succeeds after x. +Theorem 3. Histories comprising well-formed Medley transactions are strictly +serializable. +Proof (sketch). In an NBTC-transformed data structure, all critical memory +accesses will be performed using nbtcLoad or nbtcCAS. These will be followed, +at some point, by a call to txEnd. If that call succeeds, no conflicting tryFinalize +or txEnd succeeds in the interim, by Lemma 2. This in turn implies that our +Medley history is equivalent to a sequential history in which each operation takes +effect at the nbtcLoad or nbtcCAS corresponding to the linearization point of the +original data structure operation, prior to NBTC transformation. Moreover, all +operations of the same transaction are contiguous in this sequential history— +that is, our Medley history is strictly serializable. +5.2 +Obstruction Freedom +Theorem 4. When used to build well-formed transactions that retry on abort, +Medley is obstruction free. +Proof (sketch). In any reachable system state, if one thread continues to execute +while others are paused, every nbtcLoad or nbtcCAS that encounters a conflict +will first finalize (commit or abort) the encountered descriptor, uninstall it, +and install its own descriptor. If the thread encounters its own descriptor, a +nbtcLoad will return the speculated value and a nbtcCAS will update the write +set if the argument matches the previous new value in the write set. In either +case, the MCNS will make progress. If it eventually aborts, it may repeat one +round of a brand new MCNS which, with no newly introduced contention, must +commit. +5.3 +Buffered Durable Strict Serializability +Theorem 5. Histories comprising well-formed txMontage transactions exhibit +buffered durable strict serializability. +16 + +Proof (sketch). Each transaction reads the current epoch, e, in txBegin. It then +validates this epoch number during MCNS commit. Per Lemma 1, this MCNS +must linearize inside e. With nbMontage-provided failure atomicity of all oper- +ations in the same epoch, the theorem trivially holds. +6 +Performance Results +As noted in Section 1, we have used Medley to create NBTC versions of Michael +and Scott’s queue [29], Fraser’s skiplist [10], the rotating skiplist of Dick et al. [7], +Michael’s chained hash table [28], and Natarajan and Mittal’s binary search +tree [31]. All of the transformations were straightforward. In this section we +report on the performance on Medley and txMontage hash tables and skiplists, +comparing them to various alternatives from the literature. +Specifically, we tested the following transient systems: +Medley – as previously described (hash table and skip list) +OneFile – transient version of the lock-free STM of Ramalhete et al. [33] (hash +table and skip list) +TDSL – transactional data structure library of Spiegelman et al. [36] (authors’ +skiplist only) +LFTT – lock-free transactional transform of Zhang et al. [43] (authors’ skiplist +only) +We also tested the following persistent systems: +txMontage – Medley + nbMontage (hash table and skiplist) +POneFile – persistent version of OneFile [33] (hash table and skiplist) +6.1 +Experimental Setup +We report throughput for hash table and skiplist microbenchmarks and for +skiplists used to run a subset of TPC-C [6]. We also measure latency for skiplists. +All code will be made publicly available prior to publication; we intend to par- +ticipate in the artifact evaluation process. +All tests were conducted on a Linux 5.3.7 (Fedora 30) server with two Intel +Xeon Gold 6230 processors. Each socket has 20 physical cores and 40 hyper- +threads, totaling 80 hyperthreads. Threads in all experiments were pinned first +one per core on socket 0, then on the extra hyperthreads of that socket, and +then on socket 1. Each socket has 6 channels of 32 GB DRAMs and 6 channels +of 128 GB Optane DIMMs. We mount NVM from each socket as an indepen- +dent ext4 file system. In all experiments, DRAM is allocated across the two +sockets according to Linux’s default policy; in persistent data structures, only +NVM on socket 0 is used, in direct access (DAX) mode. In all cases, we report +the average of three trials, each of which runs for 30 seconds. +Our throughput and latency microbenchmark begins by pre-loading the +structure with 0.5 M key-value pairs, drawn from a key space of 1 M keys. Both +17 + +104 +105 +106 +106 +107 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +(a) get:insert:remove 0:1:1 +104 +105 +106 +106 +107 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +(b) get:insert:remove 2:1:1 +104 +105 +106 +106 +107 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +(c) get:insert:remove 18:1:1 +Figure 7: Throughput of transactional hash tables (log Y axis). +104 +105 +106 +106 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +TDSL +LFTT +(a) get:insert:remove 0:1:1 +104 +105 +106 +106 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +TDSL +LFTT +(b) get:insert:remove 2:1:1 +104 +105 +106 +106 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +POneFile +TDSL +LFTT +(c) get:insert:remove 18:1:1 +Figure 8: Throughput of transactional skiplists (log Y axis). +keys and values are 8-byte integers. In the benchmarking phase, each thread +composes and executes transactions comprising 1 to 10 operations each. Oper- +ations (on uniformly random keys) are chosen among get, insert, and remove in +a ratio specified as a parameter (0:1:1, 2:1:1, or 18:1:1 in our experiments). +In OneFile, we use a sequential chained hash table parallelized using STM. +In Medley, we use an NBTC-transformed version of Michael’s lock-free hash +table [28]. Each table has 1 M buckets. In OneFile and TDSL, skiplists are +derived from Fraser’s STM-based skiplist [10]. In LFTT and Medley, they are +derived from Fraser’s CAS-based nonblocking skiplist [10]. Each skiplist has up +to 20 levels. +For TPC-C, we are limited by the fact that Fraser’s skiplists do not sup- +port range queries. Following the lead of Yu et al. in their experiments with +DBx1000[42], we limit our experiments to TPC-C’s newOrder and payment trans- +actions, which we perform in a 1:1 ratio. These are the dominant transactions +in the benchmark; neither performs a range query. +6.2 +Throughput (Transient) +Throughput results for the hash table and skiplist microbenchmarks appear +in Figures 7 and 8, respectively. Solid lines represent transactions on transient +data structures; dotted lines represent persistent transactions. Considering only +the transient case for now, Medley consistently outperforms the transient ver- +sion of OneFile by more than an order of magnitude, on both hash tables and +skiplists, for anything more than a trivial number of threads. The gap becomes +larger when the workload has a higher percentage of writes. Despite its lack +of scalability, OneFile performs well at small thread counts, especially with a +18 + +read-mostly workload. We attribute this fact to its serialized transaction design, +which eliminates the need for read sets. +As described in Section 1, TDSL provides (blocking) transactions over vari- +ous specially constructed data structures. While conflicts still occur on writes, +read sets are limited to only semantically critical nodes, and the authors report +significant improvements in throughput relative to general-purpose STM [36]. +As shown in Figure 8, however, TDSL, like OneFile, has limited scalability, and +is dramatically outperformed by Medley. Somewhat to our surprise, TDSL also +fails to outperform OneFile on this microbenchmark, presumably because of the +latter’s elimination of read sets. +Among the various skiplist competitors, LFTT comes closest to rivaling +Medley, but still trails by a factor of 1.4–2× in the best (write-only)case. Re- +executing entire transactions in LFTT introduces considerableredundant work— +planning in particular. On read-mostly workloads, where Medley benefits from +invisible readers, LFTT trails by a factor of 2–2.7×. +As a somewhat more realistic benchmark, we repeated our comparison of +Medley, OneFile, and TDSL on the newOrder and payment transactions of TPC- +C. We were unable to include LFTT in these tests because it supports only static +transactions, in which the set of data structure operations is known in advance— +nor could we integrate its dynamic variant (DTT [23]), as the available version +of the code does not allow arbitrary key and value types. LaBorde et al. [23] +report, however, that DTT’s performance is similar to that of LFTT on simple +transactions. Given that DTT has to publish the entire transaction as a lambda +expression on all its critical nodes, we would expect DTT’s performance to be, +if anything, somewhat worse on the large transactions of TPC-C, and LFTT +was already about 2× slower than Medley on the microbenchmark. +TPC-C throughput for Medley, (transient) OneFile, and TDSL appears +in Figure 9. Because transactions on TPC-C are large, OneFile is impacted +severely. By ensuring the atomicity of only critical accesses, Medley still scales +for large numbers of threads and outperforms the competition by as much as +45×. +6.3 +Latency (Transient) +In an attempt to assess the marginal cost of transaction composition, we re-ran +our microbenchmark on Fraser’s original skiplist (Original—no transactions), +the NBTC-transformed skiplist without transactions (TxOff—no calls to txBe- +gin or txEnd), and the NBTC-transformed skiplist with transactions (TxOn—as +in Figure 8). +Figure 10a reports latency for structures placed in DRAM. Without trans- +actions, the transformed skiplist is 1.8× slower than the original. With trans- +actions turned on, it’s about 2.2× slower. These results suggest that the more- +than-doubled cost of CASes (installing and uninstalling descriptors) accounts +for about 2/3 of Medley’s overhead. +19 + +104 +105 +106 +106 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Threads +Throughput (txn/s) +Medley +txMontage +OneFile +TDSL +Figure 9: TPC-C skiplist throughput (log Y axis). +234 +169 +202 +421 +313 +366 +513 +383 +453 +0 +250 +500 +750 +1000 +0:1:1 +2:1:1 +18:1:1 +Latency (ns/txn) +Original +TxOff +TxOn +(a) on DRAM +847 +380 +593 +623 +330 +419 +650 +404 +502 +0 +250 +500 +750 +1000 +0:1:1 +2:1:1 +18:1:1 +Latency (ns/txn) +Original +TxOff +TxOn +(b) transient on NVM +673 +334 +443 +678 +408 +525 +0 +250 +500 +750 +1000 +0:1:1 +2:1:1 +18:1:1 +Latency (ns/txn) +TxOff +TxOn +(c) persistent on NVM +Figure 10: Average latency on skiplists at 40 threads. +X labels are ratio of get:insert:remove. +20 + +6.4 +Persistence +To evaluate the impact of failure atomicity and durability on the throughput of +txMontage, we can return to the dotted lines of Figures 7, 8, and 9. +Throughput +In the microbenchmark tests, with strict persistence and eager cache-line write- +back, persistent OneFile is an order of magnitude slower than its transient +version. With periodic persistence, however, the txMontage hash table achieves +half the throughput of Medley at 40 threads on the write-only workload—almost +two orders of magnitude faster than POneFile. With a read-mostly workload +on the hash table, or with any of the workloads on the skiplist (with its lower +overall concurrency), txMontage is almost as fast as Medley. In the extreme +write-heavy case (80 threads on the 0:1:1 hash table workload), we attribute +the roughly 4× slowdown of txMontage to NVM’s write bottleneck [22]—in +particular, to the phenomenon of write amplification [20, 41]. +Results are similar in TPC-C (Fig. 9). Transactions here are both large and +heavy on writes; allocating payloads on NVM limits txMontage’s throughput +to roughly a fifth of Medley’s, but that is still about 4× faster than transient +OneFile. POneFile, for its part, spent so long on the warm-up phase of TPC-C +that we lost patience and killed the test. +Latency +Figure 10b shows the latency of skiplist transactions when txMontage payloads +are allocated on NVM (and indices on DRAM) but persistence is turned off +(no epochs or explicit cache line write-back). +For comparison, we have also +shown the latency of the original, non-transactional skiplist with all data placed +in NVM. +Figure 10c shows the corresponding latencies for fully operational +txMontage. +Comparing Figures 10a and 10b, we see lower marginal overhead for trans- +actions when running on NVM. This may suggest that the hardware write bot- +tleneck is reducing overall throughput and thus contention. +On the write-only workload (leftmost groups of bars), moving payloads to +NVM introduces an overhead of almost 50% (Fig. 10a versus Fig. 10b). On the +read-mostly workload (rightmost bars), this penalty drops to 5%. Again, we +attribute the effect to NVM’s write bottleneck. The high latency of the original +skiplist entirely allocated on NVM (green bars in Figure 10b) appears to confirm +this hypothesis. +Comparing Figures 10b and 10c, txMontage pays less than 5%, relative to +Medley on NVM, for failure atomicity and durability. +21 + +7 +Conclusion +We have presented nonblocking transaction composition (NBTC), a new method- +ology that leverages the linearizability of existing nonblocking data structures +when building dynamic transactions. As concrete realizations, we introduced the +Medley system for transient structures and the txMontage system for (buffered) +persistent structures. Medley transactions are isolated and consistent; txMon- +tage transactions are also failure atomic and durable. Both systems are quite +fast: where even the best STM has traditionally suffered slowdowns of 3–10×, +Medley incurs more like 2.2×; txMontage, for its part, adds only 5–20% to the +overhead of nbMontage, allowing it to outperform existing nonblocking persis- +tent STM systems by nearly two orders of magnitude. +Given their eager contention management, Medley and txMontage main- +tain obstruction freedom for transactions on nonblocking structures. In future +work, we plan to explore lazy contention management, postponing installment +of descriptors until transactions are ready to commit. By sorting and installing +descriptors in canonical order, the resulting systems would preserve lock free- +dom. 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Conf. on Architectural Sup- +port for Programming Languages and Operating Systems (ASPLOS), pages +91–104, Newport Beach, CA, March 2011. +[41] Zixuan Wang, Xiao Liu, Jian Yang, Theodore Michailidis, Steven Swanson, +and Jishen Zhao. Characterizing and modeling non-volatile memory sys- +tems. In 53rd Intl. Symp. on Microarchitecture (MICRO), pages 496–508, +virtual conference, October 2020. +[42] Xiangyao Yu, George Bezerra, Andrew Pavlo, Srinivas Devadas, and +Michael Stonebraker. +Staring into the abyss: An evaluation of concur- +rency control with one thousand cores. Proc. of the VLDB Endowment, 8 +(3):209–220, November 2014. +[43] Deli Zhang and Damian Dechev. Lock-free transactions without rollbacks +for linked data structures. In 28th ACM Symp. on Parallelism in Algorithms +and Architectures (SPAA), pages 325–336, Pacific Grove, CA, July 2016. +[44] Yoav Zuriel, Michal Friedman, Gali Sheffi, Nachshon Cohen, and Erez Pe- +trank. Efficient lock-free durable sets. Proc. of the ACM on Programming +Languages, 3(OOPSLA):128:1–128:26, October 2019. +26 + diff --git a/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/load_file.txt b/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4650e906753fee8dcd63deb891ece72bc488f8fc --- /dev/null +++ b/6dAzT4oBgHgl3EQfEfrh/content/tmp_files/load_file.txt @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf,len=970 +page_content='Transactional Composition of Nonblocking Data Structures Wentao Cai, Haosen Wen, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Scott {wcai6,hwen5,scott}@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='rochester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='edu University of Rochester Augest 18, 2022 Abstract This paper introduces nonblocking transaction composition (NBTC), a new methodology for atomic composition of nonblocking operations on concurrent data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Unlike previous software transactional mem- ory (STM) approaches, NBTC leverages the linearizability of existing non- blocking structures, reducing the number of memory accesses that must be executed together, atomically, to only one per operation in most cases (these are typically the linearizing instructions of the constituent opera- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Our obstruction-free implementation of NBTC, which we call Medley, makes it easy to transform most nonblocking data structures into transac- tional counterparts while preserving their nonblocking liveness and high concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In our experiments, Medley outperforms Lock-Free Trans- actional Transform (LFTT), the fastest prior competing methodology, by 40–170%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The marginal overhead of Medley’s transactional composition, relative to separate operations performed in succession, is roughly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For persistent data structures, we observe that failure atomicity for transactions can be achieved “almost for free” with epoch-based periodic persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Toward that end, we integrate Medley with nbMontage, a general system for periodically persistent data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The resulting txMontage provides ACID transactions and achieves throughput up to two orders of magnitude higher than that of the OneFile persistent STM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 1 Introduction Nonblocking concurrent data structures, first explored in the 1970s, remain an active topic of research today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In such structures, there is no reachable state of the system that can prevent an individual operation from making forward progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This liveness property is highly desirable in multi-threaded programs that aim for high scalability and are sensitive to high tail latency caused by inopportune preemption of resource-holding threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='00996v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='DC] 3 Jan 2023 Many multi-threaded systems, including those for finance, travel [30], ware- house management [6], and databases in general [39], need to compose opera- tions into transactions that occur in an all-or-nothing fashion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', atomically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Concurrent data structures, however, ensure atomicity only for individual oper- ations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' composing a transaction across operations requires nontrivial program- ming effort and introduces high overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Preserving nonblocking liveness for every transaction is even more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' One potential solution can be found in software transactional memory (STM) systems, which convert almost arbitrary sequential code into speculative trans- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Several STM systems provide nonblocking progress [10, 19, 25, 26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Most instrument each memory access and arrange to restart operations that conflict at the level of individual loads and stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The resulting program- ming model is attractive, but the instrumentation typically imposes 3–10× over- head [34, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Inspired by STM, Spiegelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [36] proposed transactional data struc- ture libraries (TDSL), which introduce (blocking) transactions for certain hand- modified concurrent data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' By observing that reads need to be tracked only on critical nodes whose updates may indicate semantic conflicts, TDSL re- duces read set size and achieves better performance than general STMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Herlihy and Koskinen [18] proposed transactional boosting, a (blocking) methodology that allows an STM system to incorporate operations on exist- ing concurrent data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Using a system of semantic locks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', with one lock per key in a mapping), transactions arrange to execute concurrently so long as their boosted operations are logically independent, regardless of low-level conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A transaction that restarts due to a semantic conflict (or to a low-level conflict outside the boosted code) will roll back any already- completed boosted operations by performing explicitly identified inverse opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' An insert(k,v) operation, for example, would be rolled back by performing remove(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Transactional boosting leverages the potential for high concurrency in existing data structures, but is intrinsically lock-based, and is not fully gen- eral: operations on a single-linked FIFO queue, for example, have no obvious inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In work concurrent to TDSL, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [43] proposed the Lock-Free Trans- actional Transform (LFTT), a nonblocking methodology to compose nonblock- ing data structures, based on the observation that only certain nodes—those critical to transaction semantics—really matter in conflict management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each operation on an LFTT structure publishes, on every critical node, a descrip- tion of the transaction of which it is a part, so that conflicting transactions can see and help each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A remove(7) operation, for example, would pub- lish a description of its transaction on the node in its structure with key 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Initially, LFTT supported only static transactions, whose constituent opera- tions were all known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Subsequently, LaBorde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [23] proposed a Dynamic Transactional Transform (DTT) that generalizes LFTT to dynamic transactions (specified as lambda expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Concurrently, Elizarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [8] proposed LOFT which is similar to LFTT but avoids LFTT’s bug of duplicated helping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2 Unfortunately, as in transactional boosting, the need to identify critical nodes tends to limit LFTT and DTT to data structures representing sets and mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' DTT’s publishing and helping mechanisms also require that the “glue” code between operations be fully reentrant (to admit concurrent execu- tion by helping threads [23]) and may result in redundant work when conflicts arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Worse, for read-heavy workloads, LFTT and DTT require readers to be visible to writers, introducing metadata updates that significantly increase contention in the cache coherence protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In our work, we propose NonBlocking Transaction Composition (NBTC), a new methodology that can create transactional versions of a wide variety of concurrent data structures while preserving nonblocking progress and incurring significantly lower overhead than traditional STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The intuition behind NBTC is that in already nonblocking structures, only critical memory accesses—for the most part, the linearizing load and compare-and-swap (CAS) instructions— need to occur atomically, while most pre-linearization memory accesses can safely be executed as they are encountered, and post-linearization accesses can be postponed until after the transaction commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In comparison to STM, NBTC significantly reduces the number of mem- ory accesses that must be instrumented—typically to only one per constituent operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Unlike transactional boosting and transactional transforms, NBTC brings the focus back from semantics to low-level memory accesses, thereby enabling mechanical transformation of existing structures and accommodating almost arbitrary abstractions—much more than sets and mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' NBTC also supports dynamic transactions, invisible readers, and non-reentrant “glue” code between the operations of a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The one requirement for compatibility is that the linearization points of constituent operations must be immediately identifiable: each operation must be able to tell when it has linearized at run time, without performing any additional shared-memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Most non- blocking structures in the literature appear to meet this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To assess the practicality of NBTC, we have built an obstruction-free imple- mentation, Medley, that uses a variant of Harris et al.’s multi-word CAS [16] to execute the critical memory accesses of each transaction atomically, eagerly resolving conflicting transactions as they are discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Using Medley, we have created NBTC versions of Michael and Scott’s queue [29], Fraser’s skiplist [10], the rotating skiplist of Dick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [7], Michael’s chained hash table [28], and Natarajan and Mittal’s binary search tree [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' All of the transformations were straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In the traditional language of database transactions [15], Medley provides isolation and consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Building on recent work on persistent memory, we have also integrated Medley with the nbMontage system of Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [2] to create a system, txMontage, that provides failure atomicity and durability as well—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', full ACID transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Specifically, we leverage the epoch system of nbMontage, which divides time into coarse-grain temporal intervals and recov- ers, on failure, to the state of a recent epoch boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' By folding a check of the epoch number into its multi-word CAS, txMontage ensures that operations of the same transaction always linearize in the same epoch, thereby obtaining 3 failure atomicity and durability “almost for free.” Summarizing contributions: (Section 2) We introduce nonblocking transaction composition (NBTC), a new methodology with which to compose the operations of nonblocking data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (Section 3) Deploying NBTC, we implement Medley, a general system for transactional nonblocking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Medley’s easy-to-use API and mechan- ical transform make it easy to convert compatible nonblocking structures to transactional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (Section 4) We integrate Medley with nbMontage to create txMontage, provid- ing not only transactional isolation and consistency, but also failure atomicity and durability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (Section 5) We argue that using NBTC and Medley, transactions composed of nonblocking structures are nonblocking and strictly serializable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We also argue that transactions with txMontage provide a persistent variant of strict serializability analogous to the buffered durable linearizability of Izraelevitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (Section 6) We present performance results, confirming that Medley imposes relatively modest overhead and scales to large numbers of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Specif- ically, Medley outperforms LFTT by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='4× to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='7× and outperforms TDSL and the OneFile nonblocking STM [33] system by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' On persistent memory, txMontage outperforms nonblocking persistent STM by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2 Nonblocking Transaction Composition Nonblocking transaction composition (NBTC) is a new methodology that fully leverages the linearizability of nonblocking data structure operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' NBTC obtains strict serializability by atomically performing only the critical memory accesses of composed operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' It supports a large subset of the nonblocking data structures in the literature (characterized more precisely below), preserving the high concurrency and nonblocking liveness of the transformed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1 NBTC Composability The key to NBTC composability is the immediately identifiable linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Specifically: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A data structure operation has an immediately identifiable lin- earization point if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' statically, we can identify every instruction that may potentially serve as the operation’s linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Such an instruction must be a load for a read-only operation or a compare-and-swap (CAS) for an update operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' dynamically, after executing a potentially linearizing instruction, we can de- termine whether it was indeed the linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A linearizing load has to be determined before the operation returns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' a linearizing CAS has to be determined without performing any additional shared-memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' There can be more than one potential linearization point in the code of an operation, but only one of them will constitute the linearization point in any given invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A nonblocking data structure is NBTC-composable if each of its operations has an immediately identifiable linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' While it may be possible to relax this definition, the current version accom- modates a very large number of existing nonblocking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2 The Methodology It is widely understood that most nonblocking operations comprise a “planning” phase and a “cleanup” phase, separated by a linearizing instruction [12, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Executing the planning phase does not commit the operation to success;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' cleanup, if needed, can be performed by any thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The basic strategy in NBTC is to perform the planning for all constituent operations of the current transaction, then linearize all those operations together, atomically, and finally perform all cleanup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Our survey of existing data structures and composition patterns reveals two principle complications with this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The first complication involves the notion of a publication point, where an operation may become visible to other threads but not yet linearize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Because publication can alter the behavior of other threads, it must generally (like a linearization point) remain speculative until the entire transaction is ready to commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' An example can be seen in the binary search tree of Natarajan and Mittal [31], where an update operation o may perform a CAS that publishes its intent to linearize soon but not quite yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' After this publication point, either o itself or any other update that encounters the publication notice may attempt to linearize o (in the interest of performance, a read operation will ignore it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Notably, CAS instructions that serve to help other (already linearized) opera- tions, without revealing the nature of the current operation, need not count as publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The second complication arises when a transaction, t, performs two or more operations on the same data structure and one of the later operations (call it o2) depends on the outcome of an earlier operation (call it o1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Here the thread executing t must proceed as if o1 has completed, but other threads must ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If o1 requires cleanup (something that NBTC will normally delay until after transaction commit), o2 may need to help o1 before it can proceed, while other transactions should not even be aware of o1’s existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Both complicating cases can be handled by introducing the notion of a spec- ulation interval in which CAS instructions must be completed together for an 5 operation to take effect as part of a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This is similar to the CAS ex- ecutor phase in a normalized nonblocking data structure [38], but not the same, largely due to the second complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For an operation that becomes visible before its linearization point, it suffices to include in the speculation interval all CAS operations between the publication and linearization points, inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For an operation o2 that needs to see an earlier operation o1 in the same transaction, it suffices to track the transaction’s writes and to start o2’s speculation interval no later than the first instruction that accesses a location written by o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A bit more precise, we say A CAS instruction in operation o of thread t in history H is benign if there is no extension H′ of H such that t executes no more instructions in H′ and yet o linearizes in H′ nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The first CAS instruction of o that is not benign is o’s publication point (this will often be the same as its linearization point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The speculation interval of o begins either at the publication point or at the first instruction that sees a value speculatively written by some earlier opera- tion in the same transaction (whichever comes first) and extends through o’s linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A load in a read-only operation is critical if it is the immediately identifiable linearization point of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A CAS in an update operation is critical if it lies in the speculation interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Without loss of generality, we assume that all updates to shared memory (other than initialization of objects not yet visible to other threads) are effected via CAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given these definitions, the NBTC methodology is straightforward: To atom- ically execute a set of operations on NBTC-composable data structures, we transform every operation such that (1) instructions prior to the speculation interval and non-critical instructions in the speculation interval are executed on the fly as a transaction encounters them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (2) critical instructions are executed in a speculative fashion, so they will take effect, atomically, only on transaction commit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' and (3) instructions after the speculation interval are postponed until after the commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 The Medley System To illustrate NBTC, we have written a system, Medley, that (1) instruments critical instructions, executes them speculatively, and commits them atomically using M-compare-N-swap, our variant of the multi-word CAS of Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (2) identifies and eagerly resolves transaction conflicts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' and (3) delays non- critical cleanup until transaction commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6 1 template class CASObj { // Augmented atomic object 2 T nbtcLoad();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 bool nbtcCAS(T expected, T desired, bool linPt, bool pubPt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4 /* Regular atomic methods: */ 5 T load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' void store(T desired);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' bool CAS(T expected, T desired);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 7 class Composable { // Base class of all transactional objects 8 template void addToReadSet(CASObj*,T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Register load 9 void addToCleanups(function);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Register post-critical work 10 template T* tNew(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Create a new block 11 template void tDelete(T*);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Delete a block 12 template void tRetire(T*);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Epoch-based safe retire 13 TxManager* mgr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Tx metadata shared among Composables 14 struct OpStarter { OpStarter(TxManager*);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' } // RAII op starter 15 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 16 class TxManager { // Manager shared among composable objects 17 void txBegin();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Start a transaction 18 void txEnd();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Try to commit the transaction 19 void txAbort();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Explicitly abort the transaction 20 void validateReads();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // Optional validation for opacity 21 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 22 struct TransactionAborted : public std::exception{ };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 1: C++ API of Medley for transaction composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1 API Figure 1 summarizes Medley’s API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Using this API, we transform an NBTC- composable data structure into a transactional structure as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Replace critical loads and CASes with nbtcLoad and nbtcCAS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Fields to which such accesses are made should be declared using the CASObj template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Invoke addToReadSet for the critical load in a read operation, recording the address and the loaded value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Register each operation’s post-critical work via addToCleanups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Replace every new and delete with tNew and tDelete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Replace every retire (for safe memory reclamation—SMR) with tRetire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Declare an OpStarter object at the beginning of each operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' CASObj¡T¿ augments each CAS-able 64-bit word (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', atomic¡Node*¿) with additional metadata bits for speculation tracking (details in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' It provides a specialized load and CAS as well as the usual methods of atomic¡T¿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To dynamically identify the speculation interval, nbtcCAS takes two extra argu- ments, linPt and pubPt, that indicate whether this call, should it succeed, will constitute its operation’s linearization or/and publication point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In a similar vein, addToReadSet can be called after an nbtcLoad to indicate (after inspecting the return value) that this was (or is likely to have been) the linearizing load of a read-only operation, and should be tracked for validation at commit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Composable is a base class for transactional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' It provides a variety of NBTC-related methods, including support for safe memory reclamation (SMR), used to ensure that nodes are not reclaimed until one can be certain that no references remain among the private variables of other threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Our current 7 implementation of SMR uses epoch-based reclamation [10, 17, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For the sake of generality, Composable also provides an API for transactional boosting, which can be used to incorporate lock-based operations into Medley transactions (at the cost, of course, of nonblocking progress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We do not discuss this mechanism further in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The TxManager class manages transaction metadata and provides methods to initiate, abort, and complete a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A TxManager instance is shared among all Composable instances intended for use in the same transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In each operation call, the manager distinguishes (via OpStarter()) whether exe- cution is currently inside or outside a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If outside, all transactional instrumentation is elided;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' if inside, instrumentation proceeds as specified by the NBTC methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given that nonblocking operations can execute safely in any reachable state of the system, there is usually no need to stop the execution of a doomed-to- abort transaction as soon as a conflict arises—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', to guarantee opacity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In exceptional cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', when later operations of a transaction cannot be called with certain combinations of parameters, or when aborts are likely enough that delaying them may compromise performance), the validateReads method can be used to determine whether previous reads remain correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To illustrate the use of Medley, Figure 2 highlights lines of code in Michael’s nonblocking hash table [28] that must be modified for NBTC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 3 then shows an example transaction that modifies two hash tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In a real appli- cation, the catch block for TransactionAborted would typically loop back to the beginning of the transaction code to try again, possibly with additional code to detect and recover from livelock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', via backoff or hints to the underly- ing scheduler).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In contrast to STM systems, Medley does not instrument the intra-transaction “glue” code between data structure operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This code is always executed as regular code outside a transaction and should always be data-race free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' if it has side effects, the catch block (written by the programmer) for aborted transactions should compensate for these before the programmer chooses to retry or give up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2 M-Compare-N-Swap To execute the critical memory accesses of each transaction atomically, we em- ploy a software-emulated M-compare-N-swap (MCNS) that builds on the double- compare-single-swap (RDCSS) and multi-word CAS (CASN) of Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each transaction maintains a descriptor that contains a read set, a write set, and a 64-bit triple of thread ID, serial number, and status, as shown in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Descriptors are pre-allocated on a per-thread basis within a TxManager instance, and are reused across transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A status can be InPrep (initial state), InProg (ready to commit), Committed (only when InProg and validation succeeds), or Aborted (from InPrep or due to failed validation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each originally 64-bit word at which a critical memory access may occur is augmented with a 64-bit counter, together comprising an 128-bit CASObj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each critical CAS installs a pointer to its descriptor in the CASObj and increments 8 1 class MHashTable : public Composable { 2 struct Node { K key;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' V val;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' CASObj next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 // from p, find c >= k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' nbtcLoad and tRetire may be used 4 bool find(CASObj* &p, Node* &c, Node* &n, K k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5 optional get(K key) { 6 OpStarter starter(mgr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' CASObj* prev = nullptr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 7 Node *curr, *next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' optional res = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 8 if (find(prev,curr,next,key)) res = curr->val;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9 addToReadSet(prev,curr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 10 return res;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 11 } 12 optional put(K key, V val) { // insert or replace if key exists 13 OpStarter starter(mgr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 14 CASObj* prev = nullptr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' optional res = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 15 Node *newNode = tNew(key, val), *curr, *next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 16 while(true) { 17 if (find(prev,curr,next,key)) { // update 18 newNode->next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='store(curr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 19 if (curr->next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='nbtcCAS(next,mark(newNode),true,true)) { 20 res = curr->val;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 21 auto cleanup = [](){ 22 if (prev->CAS(curr,newNode)) tRetire(curr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 23 else find(prev,curr,next,key);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 24 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 25 addToCleanups(cleanup);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // execute right away if not in tx 26 break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 27 } 28 } else { // key does not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' insert 29 newNode->next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='store(curr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 30 if (prev->nbtcCAS(curr,newNode,true,true)) break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 31 } 32 } 33 return res;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 34 }};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 2: Michael’s lock-free hash table example (Medley-related parts highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 1 void doTx(MHashTable* ht1, MHashTable* ht2, V v, K a1, K a2) { 2 TxManager* mgr=ht1->mgr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' assert(mgr==ht2->mgr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 try { // transfer ‘v’ from account ‘a1’ in ‘ht1’ to ‘a2’ in ‘ht2’ 4 mgr->txBegin();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5 V v1 = ht1->get(a1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' V v2 = ht2->get(a2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='hasValue() or v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='value() < v) mgr->txAbort();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 7 ht1->put(a1, v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='value() - v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' ht2->put(a2, v + v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='valueOr(0));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 8 mgr->txEnd();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9 } catch (TransactionAborted) { /* transaction aborted */ } 10 } Figure 3: Transaction example on Michael’s hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9 1 struct Desc { 2 map* readSet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 map* writeSet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4 atomic status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='//63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='.50 tid 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='.2 serialNumber 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='.0 status 5 enum STATUS { InPrep=0, InProg=1, Committed=2, Aborted=3 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 7 struct CASObj { atomic val_cnt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 4: Descriptor and CASObj structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' the counter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' at commit or abort, the descriptor is uninstalled and the counter incremented again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We leverage 128-bit CAS instructions on the x86 to change the original word and the counter together, atomically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The counter is odd when CASObj contains a pointer to a descriptor and even when it is a real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each instance of MCNS proceeds through phases that install descriptors, finalize status, and uninstall descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The first two phases are on the critical path of a data structure operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A new transaction initializes metadata in its descriptor (at txBegin): it clears the read and write sets, increments the serial number, and resets the status to InPrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The installing phase then occurs over the course of the transaction: Each critical load records its address, counter, and value in the read set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each critical CAS records its address, old counter, old value, and desired new value in the write set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' it then installs a pointer to the descriptor in the CASObj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Pseudocode for the installing phase appears in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To spare the programmer the need to reason about counters, nbtcLoad makes a record of its ⟨counter, object⟩ pair (line 15 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' addToReadSet then adds this pair (and the specified CASObj) to the transaction’s read set (line 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' When a thread encounters its own descriptor, nbtcLoad returns the specu- lated value from the write set (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Likewise, nbtcCAS updates the write entry (line 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Such encounters automatically initiate the speculation interval (lines 10, 30, and 32), which then extends through the linearization point of the current operation (line 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If an operation encounters the descriptor of some other thread, it gets that descriptor out of the way by calling tryFinalize (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This method aborts the associated transaction if the descriptor is InPrep, helps complete the commit if InProg, and in all cases uninstalls the descriptor from the CASObj in which it was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Similar actions occur when a thread is forced to abort or reaches the end of its transaction and attempts to commit (lines 39–58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Whether helping or acting on its own behalf, a thread performing an MCNS must verify that the descriptor is still responsible for the CASObj through which it was discovered (line 9) and (if committing) that the values in the read set are still valid (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' After CAS-ing the status to Committed or Aborted, the thread uninstalls the descriptor from all associated CASObjs, replacing pointers to the descriptor with the appropriate updated values (lines 31 and 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Once uninstalling is complete, the owner thread calls cleanup routines (line 55) for a commit or deallocates tNew-ed blocks (line 43) for an abort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Our design adopts invisible readers and eager contention management for efficiency and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Eager contention management admits the possibility 10 1 void TxManager::txBegin() { 2 desc->readSet->clear();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' desc->writeSet->clear();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='store((status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load() & ~3) + 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4 } 5 T CASObj::nbtcLoad() { 6 retry: 7 {val,cnt} = val_cnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 8 if (cnt % 2) { // is descriptor 9 if (val == desc) { 10 startSpeculativeInterval();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 11 return desc->writeSet[this].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='newVal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 12 } else val->tryFinalize(this, {val,cnt});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 13 goto retry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // until object has real value 14 } 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' /* Record ‘this’ and ‘cnt’ to be added to readSet */ 16 return val;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 17 } 18 void Composable::addToReadSet(CASObj* obj, T val) { 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' /* Retrieve ‘cnt’ by ‘obj‘ */ 20 mgr->readSet[obj] = {val,cnt};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 21 } 22 bool CASObj::nbtcCAS(T expected,T desired,bool linPt,bool pubPt){ 23 retry: 24 {val,cnt} = val_cnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 25 if (cnt % 2) { // is descriptor 26 if (val !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='= desc) { // not own descriptor 27 val->tryFinalize(this, {val,cnt});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 28 goto retry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // until object has real value 29 } 30 startSpeculativeInterval();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 31 } else if (val !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='= expected) return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 32 if (pubPt) startSpeculativeInterval();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 33 if (inSpeculativeInterval()) { // Is critical CAS 34 desc->writeSet[this] = {val,cnt,desired};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 35 bool ret = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 36 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (cnt % 2)) ret = this->CAS({val,cnt},{desc,cnt+1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 37 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='ret) desc->writeSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='remove(this);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 38 if (linPt and ret) endSpeculativeInterval();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 39 return ret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 40 } else return CAS(expected, desired);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 41 } Figure 5: Pseudocode for installing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 11 1 bool Desc::stsCAS(uint64 d, STATUS expected, STATUS desired) { 2 d = d & ~3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' return status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='CAS(d + expected, d + desired);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3 } 4 bool Desc::setReady(){return stsCAS(status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load(),InPrep,InProg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='} 5 bool Desc::commit(uint64 d){return stsCAS(d,InProg,Committed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='} 6 bool Desc::abort(uint64 d){return stsCAS(d,d & 1,Aborted);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='} 7 void Desc::tryFinalize(CASObj* obj, uint128 var) { 8 uint64 d = status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9 if (obj->val_cnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load() !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='= var) // ensure d indicates right tx 10 return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 11 if (d & 3 == InPrep) { 12 abort(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 13 uint64 newd = status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 14 if (newd & ~3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='= d & ~3) return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // serial number mismatch 15 d = newd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 16 } 17 if (d & 3 == InProg) { 18 if (validateReads(d)) commit(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 19 else abort(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 20 } 21 uninstall(status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 22 } 23 bool Desc::validateReads() { 24 for (e:*readSet) 25 if ({e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='val,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='cnt} !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='= e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='addr->load()) return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 26 return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 27 } 28 void Desc::uninstall(uint64 d) { 29 if (d % 3 == Committed) 30 for (e:*writeSet) 31 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='addr->CAS({this,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='cnt+1}, {e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='newVal,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='cnt+2});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 32 else // Aborted 33 for (e:*writeSet) 34 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='addr->CAS({this,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='cnt+1}, {e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='oldVal,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='cnt+2});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 35 } 36 struct TxManager { 37 threadLocal vector cleanups, allocs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 38 threadLocal Desc* desc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 39 void txAbort() { 40 uint64 d = desc->status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 41 desc->abort(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 42 desc->uninstall(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 43 for (f:allocs) f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' // undo tNew 44 throw TransactionAborted();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 45 } 46 void txEnd() { 47 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='desc->setReady()) txAbort();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 48 else { 49 uint64 d = desc->status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 50 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='desc->validateReads()) desc->abort(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 51 else if (d & 3 == InProg) desc->commit(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 52 d = desc->status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='load();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 53 if (d & 3 == Committed) { 54 desc->uninstall(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 55 for (f:cleanups) f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 56 } else txAbort();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 57 } 58 } 59 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 6: Pseudocode of methods that finalize transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 12 of livelock—transactions that repeatedly abort each other—and therefore guar- antees only obstruction freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Lazy (commit-time) contention management along with some total order of descriptor installment might allow us to pre- serve lock freedom for structures that provide it [35], but would significantly complicate the tracking and retrieving of uncommitted changes, and would not address starvation, which may be a bigger problem than livelock in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' we consider these implementation choices orthogonal to the effectiveness of the NBTC methodology, and decide to explore them in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4 Persistent Memory Transactions developed, historically, in the database community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' transactional memory (TM) adapted them to in-memory structures in multithreaded pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The advent of cheap, low-power, byte-addressable nonvolatile memory (NVM) presents the opportunity to merge these two historical threads in a way that ideally leverages NBTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Specifically, where TM aims to convert sequential code to thread-safe parallel code, NBTC assumes—as in the database world— that we are already in possession of efficient thread-safe structures and we wish to combine their operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given this assumption, it seems appropriate (as described at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1) to assume that the programmer is responsi- ble for the “glue” code between operations, and to focus on the atomicity and durability of the composed operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1 Durable Linearizability On machines with volatile caches, data structures in NVM will generally be consistent after a crash only if programs take pains to issue carefully chosen write-back and fence instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To characterize desired behavior, Izraelevitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [21] introduced durable linearizability as a correctness criterion for persis- tent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A structure is durably linearizable if it is linearizable during crash-free execution and its long-term history remains linearizable when crash events are elided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Equivalently [11], each operation should persist between its invocation and response, and the order of persists should match the linearization order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Many durably linearizable nonblocking data structures have been designed in recent years [3, 9, 11, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Several groups have also proposed methodologies by which existing nonblocking structures can be made durably linearizable [12, 13, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Other groups have developed persistent STM systems, but most of these have been lock-based [4, 5, 24, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' OneFile [33] and QSTM [1] are, to the best of our knowledge, the only nonblocking persistent STM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' OneFile serializes transactions using a global sequence number, eliminating the need for a read set and improving read efficiency, but introducing the need for invasive data structure modifications and a 128-bit wide CAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' QSTM employs a global persistent queue for active transactions, avoiding the need for wide CAS and invasive structural changes, but its execution remains inherently serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2 Lowering Persistence Overhead Unfortunately, write-back and fence instructions tend to have high latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given the need for operations to persist before returning, durable linearizability appears to be intrinsically expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Immediate persistence for STM introduces additional overhead, as metadata for transaction concurrency control must also be eagerly written back and fenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To move high latency instructions off the application’s critical path, Izraele- vitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [21] introduced the notion of buffered durable linearizability (BDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' By allowing a modest suffix of pre-crash execution to be lost during post-crash recovery (so long as the overall history remains linearizable), BDL allows write- back and fence instructions to execute in batches, off the application’s critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Applications that need to ensure persistence before communicating with the outside world can employ a sync operation, reminiscent of those in tradi- tional file systems and databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' First proposed in the context of the Dal´ı persistent hash table [32], periodic persistence was subsequently adopted by nbMontage [2], a general-purpose sys- tem to create BDL versions of existing nonblocking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The nbMontage system divides wall-clock time into “epochs” and persists operations in a batch at the end of each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In the wake of a crash in epoch e, the system recovers all structures to their state as of the end of epoch e − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To maximize through- put in the absence of crashes, nbMontage also distinguishes between data that are semantically significant (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' “payloads”) and data that are merely per- formance enhancing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', indices);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' the latter can be kept in DRAM and rebuilt during recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As an example, the payloads of a mapping are simply a pile of key-value pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' the associated hash table, tree, or skiplist resides in transient DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The payloads of a queue are ⟨serial number, item⟩ pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' To ensure that post-crash recovery always reflects a consistent state of each structure, every nbMontage operation is forced to linearize in the epoch with which its payloads have been labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Operations that take “too long” to com- plete may be forced to abort and start over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The nbMontage system as a whole is lock free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' sync is actually wait free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='3 Durable Strict Serializability Linearizability, of course, is not suitable for transactions, which must remain speculative until all operations can be made visible together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' STM systems typically provide strict serializability instead: transactions in a crash-free history appear to occur in a sequential order that respects real time (if A commits before B begins, then A must serialize before B) [34, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For a persistent version of NBTC, we need to accommodate crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Like Izraelevitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [21], we assume a full-system crash failure model: data structures continue to exist after a crash, but are accessed only by new threads—the old threads disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Under this model: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' An execution history H displays durable strict serializability (DSS) if it is strictly serializable when crash events are elided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 14 Like durable linearizability, this definition requires all work completed before a crash to be visible after the crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The buffered analogue is similar: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' An execution history H displays buffered durable strict serializ- ability (BDSS) if there exists a happens-before–consistent cut of each inter-crash interval such that H is strictly serializable when crash events are elided along with the post-cut suffix of each inter-crash interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='4 Merging Medley with nbMontage The epoch system of nbMontage provides a natural mechanism with which to provide failure atomicity and durability for Medley transactions: if operations of the same transaction always occur in the same epoch, then they will be recovered (or lost) together in the wake of a crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Building on this observation, we merge the two systems to create txMontage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Payloads of all operations in a given transaction are labeled with the same epoch number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' That number is then validated along with the rest of the read set during MCNS commit, ensuring that the transaction commits in the expected epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' While nbMontage itself is quite complex, this one small change is all that is required to graft it (and all its converted persistent data structures) onto Medley: persistence comes into transactions “almost for free.” 5 Correctness In this section, we argue that histories comprising well-formed Medley transac- tions are strictly serializable, that Medley is obstruction free, and that txMon- tage provides buffered durable strict serializability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A Medley transaction is well-formed if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' it starts with txBegin and ends with txEnd, optionally with txAbort in between;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' it contains operations of NBTC-transformed data structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' all other intra-transaction code is nonblocking and free from any side effects not managed by handlers for the TransactionAborted exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1 Strict Serializability Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' At the implementation level (operating on the array of words that comprises system memory), nbtcLoad, nbtcCAS, tryFinalize, txAbort, and txEnd (MCNS) are linearizable operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Follows directly from Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Their RDCSS compares (without changing) only a single location, and their CASN supports the up- date of all touched words, but the proofs adapt in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In particular, as in RDCSS, an unsuccessful tryFinalize or txEnd can linearize on a (failed) validating read or a failed CAS of its status word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' A tryFinalize or txEnd 15 whose status CAS is successful linearizes “in the past,” on the first of its vali- dating reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (Ironically, this means that MCNS does not have an immediately identifiable linearization point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=') Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In any history in which transaction t performs an nbtcLoad or nbtc- CAS operation x on CASObj o, and in which t’s txEnd operation y succeeds, no tryFinalize or txEnd for a different transaction that modifies o succeeds between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Suppose the contrary, and call the transaction with the conflict- ing tryFinalize or txEnd u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If u’s nbtcCAS of o occurs between x and y, it will abort and uninstall t’s descriptor, or cause read validation to fail in y, contra- dicting the assumption that t’s txEnd succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If u’s nbtcCAS of o occurs before x, then x will abort and uninstall u’s descriptor, contradicting the assumption that u’s tryFinalize or txEnd succeeds after x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Histories comprising well-formed Medley transactions are strictly serializable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In an NBTC-transformed data structure, all critical memory accesses will be performed using nbtcLoad or nbtcCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' These will be followed, at some point, by a call to txEnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If that call succeeds, no conflicting tryFinalize or txEnd succeeds in the interim, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This in turn implies that our Medley history is equivalent to a sequential history in which each operation takes effect at the nbtcLoad or nbtcCAS corresponding to the linearization point of the original data structure operation, prior to NBTC transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Moreover, all operations of the same transaction are contiguous in this sequential history— that is, our Medley history is strictly serializable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2 Obstruction Freedom Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' When used to build well-formed transactions that retry on abort, Medley is obstruction free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In any reachable system state, if one thread continues to execute while others are paused, every nbtcLoad or nbtcCAS that encounters a conflict will first finalize (commit or abort) the encountered descriptor, uninstall it, and install its own descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If the thread encounters its own descriptor, a nbtcLoad will return the speculated value and a nbtcCAS will update the write set if the argument matches the previous new value in the write set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In either case, the MCNS will make progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' If it eventually aborts, it may repeat one round of a brand new MCNS which, with no newly introduced contention, must commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='3 Buffered Durable Strict Serializability Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Histories comprising well-formed txMontage transactions exhibit buffered durable strict serializability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 16 Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each transaction reads the current epoch, e, in txBegin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' It then validates this epoch number during MCNS commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Per Lemma 1, this MCNS must linearize inside e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' With nbMontage-provided failure atomicity of all oper- ations in the same epoch, the theorem trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6 Performance Results As noted in Section 1, we have used Medley to create NBTC versions of Michael and Scott’s queue [29], Fraser’s skiplist [10], the rotating skiplist of Dick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [7], Michael’s chained hash table [28], and Natarajan and Mittal’s binary search tree [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' All of the transformations were straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In this section we report on the performance on Medley and txMontage hash tables and skiplists, comparing them to various alternatives from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Specifically, we tested the following transient systems: Medley – as previously described (hash table and skip list) OneFile – transient version of the lock-free STM of Ramalhete et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [33] (hash table and skip list) TDSL – transactional data structure library of Spiegelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [36] (authors’ skiplist only) LFTT – lock-free transactional transform of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [43] (authors’ skiplist only) We also tested the following persistent systems: txMontage – Medley + nbMontage (hash table and skiplist) POneFile – persistent version of OneFile [33] (hash table and skiplist) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='1 Experimental Setup We report throughput for hash table and skiplist microbenchmarks and for skiplists used to run a subset of TPC-C [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We also measure latency for skiplists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' All code will be made publicly available prior to publication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' we intend to par- ticipate in the artifact evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' All tests were conducted on a Linux 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='7 (Fedora 30) server with two Intel Xeon Gold 6230 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each socket has 20 physical cores and 40 hyper- threads, totaling 80 hyperthreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Threads in all experiments were pinned first one per core on socket 0, then on the extra hyperthreads of that socket, and then on socket 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each socket has 6 channels of 32 GB DRAMs and 6 channels of 128 GB Optane DIMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We mount NVM from each socket as an indepen- dent ext4 file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In all experiments, DRAM is allocated across the two sockets according to Linux’s default policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' in persistent data structures, only NVM on socket 0 is used, in direct access (DAX) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In all cases, we report the average of three trials, each of which runs for 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Our throughput and latency microbenchmark begins by pre-loading the structure with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='5 M key-value pairs, drawn from a key space of 1 M keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Both ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Threads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Throughput (txn/s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Medley ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='txMontage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='OneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='POneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='(a) get:insert:remove 0:1:1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='10 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Threads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Throughput (txn/s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Medley ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='txMontage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='OneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='POneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='(b) get:insert:remove 2:1:1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Threads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Throughput (txn/s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Medley ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='txMontage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='OneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='POneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='(c) get:insert:remove 18:1:1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Figure 7: Throughput of transactional hash tables (log Y axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='106 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='txMontage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='OneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='POneFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='TDSL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='LFTT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='(c) get:insert:remove 18:1:1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='Figure 8: Throughput of transactional skiplists (log Y axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' keys and values are 8-byte integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In the benchmarking phase, each thread composes and executes transactions comprising 1 to 10 operations each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Oper- ations (on uniformly random keys) are chosen among get, insert, and remove in a ratio specified as a parameter (0:1:1, 2:1:1, or 18:1:1 in our experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In OneFile, we use a sequential chained hash table parallelized using STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In Medley, we use an NBTC-transformed version of Michael’s lock-free hash table [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each table has 1 M buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In OneFile and TDSL, skiplists are derived from Fraser’s STM-based skiplist [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In LFTT and Medley, they are derived from Fraser’s CAS-based nonblocking skiplist [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Each skiplist has up to 20 levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For TPC-C, we are limited by the fact that Fraser’s skiplists do not sup- port range queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Following the lead of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' in their experiments with DBx1000[42], we limit our experiments to TPC-C’s newOrder and payment trans- actions, which we perform in a 1:1 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' These are the dominant transactions in the benchmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' neither performs a range query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2 Throughput (Transient) Throughput results for the hash table and skiplist microbenchmarks appear in Figures 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Solid lines represent transactions on transient data structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' dotted lines represent persistent transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Considering only the transient case for now, Medley consistently outperforms the transient ver- sion of OneFile by more than an order of magnitude, on both hash tables and skiplists, for anything more than a trivial number of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The gap becomes larger when the workload has a higher percentage of writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Despite its lack of scalability, OneFile performs well at small thread counts, especially with a 18 read-mostly workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We attribute this fact to its serialized transaction design, which eliminates the need for read sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As described in Section 1, TDSL provides (blocking) transactions over vari- ous specially constructed data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' While conflicts still occur on writes, read sets are limited to only semantically critical nodes, and the authors report significant improvements in throughput relative to general-purpose STM [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As shown in Figure 8, however, TDSL, like OneFile, has limited scalability, and is dramatically outperformed by Medley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Somewhat to our surprise, TDSL also fails to outperform OneFile on this microbenchmark, presumably because of the latter’s elimination of read sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Among the various skiplist competitors, LFTT comes closest to rivaling Medley, but still trails by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='4–2× in the best (write-only)case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Re- executing entire transactions in LFTT introduces considerableredundant work— planning in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' On read-mostly workloads, where Medley benefits from invisible readers, LFTT trails by a factor of 2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='7×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As a somewhat more realistic benchmark, we repeated our comparison of Medley, OneFile, and TDSL on the newOrder and payment transactions of TPC- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We were unable to include LFTT in these tests because it supports only static transactions, in which the set of data structure operations is known in advance— nor could we integrate its dynamic variant (DTT [23]), as the available version of the code does not allow arbitrary key and value types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' LaBorde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [23] report, however, that DTT’s performance is similar to that of LFTT on simple transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given that DTT has to publish the entire transaction as a lambda expression on all its critical nodes, we would expect DTT’s performance to be, if anything, somewhat worse on the large transactions of TPC-C, and LFTT was already about 2× slower than Medley on the microbenchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' TPC-C throughput for Medley, (transient) OneFile, and TDSL appears in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Because transactions on TPC-C are large, OneFile is impacted severely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' By ensuring the atomicity of only critical accesses, Medley still scales for large numbers of threads and outperforms the competition by as much as 45×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='3 Latency (Transient) In an attempt to assess the marginal cost of transaction composition, we re-ran our microbenchmark on Fraser’s original skiplist (Original—no transactions), the NBTC-transformed skiplist without transactions (TxOff—no calls to txBe- gin or txEnd), and the NBTC-transformed skiplist with transactions (TxOn—as in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 10a reports latency for structures placed in DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Without trans- actions, the transformed skiplist is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='8× slower than the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' With trans- actions turned on, it’s about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2× slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' These results suggest that the more- than-doubled cost of CASes (installing and uninstalling descriptors) accounts for about 2/3 of Medley’s overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 19 104 105 106 106 0 10 20 30 40 50 60 70 80 Threads Throughput (txn/s) Medley txMontage OneFile TDSL Figure 9: TPC-C skiplist throughput (log Y axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 234 169 202 421 313 366 513 383 453 0 250 500 750 1000 0:1:1 2:1:1 18:1:1 Latency (ns/txn) Original TxOff TxOn (a) on DRAM 847 380 593 623 330 419 650 404 502 0 250 500 750 1000 0:1:1 2:1:1 18:1:1 Latency (ns/txn) Original TxOff TxOn (b) transient on NVM 673 334 443 678 408 525 0 250 500 750 1000 0:1:1 2:1:1 18:1:1 Latency (ns/txn) TxOff TxOn (c) persistent on NVM Figure 10: Average latency on skiplists at 40 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' X labels are ratio of get:insert:remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='4 Persistence To evaluate the impact of failure atomicity and durability on the throughput of txMontage, we can return to the dotted lines of Figures 7, 8, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Throughput In the microbenchmark tests, with strict persistence and eager cache-line write- back, persistent OneFile is an order of magnitude slower than its transient version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' With periodic persistence, however, the txMontage hash table achieves half the throughput of Medley at 40 threads on the write-only workload—almost two orders of magnitude faster than POneFile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' With a read-mostly workload on the hash table, or with any of the workloads on the skiplist (with its lower overall concurrency), txMontage is almost as fast as Medley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In the extreme write-heavy case (80 threads on the 0:1:1 hash table workload), we attribute the roughly 4× slowdown of txMontage to NVM’s write bottleneck [22]—in particular, to the phenomenon of write amplification [20, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Results are similar in TPC-C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Transactions here are both large and heavy on writes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' allocating payloads on NVM limits txMontage’s throughput to roughly a fifth of Medley’s, but that is still about 4× faster than transient OneFile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' POneFile, for its part, spent so long on the warm-up phase of TPC-C that we lost patience and killed the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Latency Figure 10b shows the latency of skiplist transactions when txMontage payloads are allocated on NVM (and indices on DRAM) but persistence is turned off (no epochs or explicit cache line write-back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' For comparison, we have also shown the latency of the original, non-transactional skiplist with all data placed in NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Figure 10c shows the corresponding latencies for fully operational txMontage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Comparing Figures 10a and 10b, we see lower marginal overhead for trans- actions when running on NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' This may suggest that the hardware write bot- tleneck is reducing overall throughput and thus contention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' On the write-only workload (leftmost groups of bars), moving payloads to NVM introduces an overhead of almost 50% (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 10a versus Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' On the read-mostly workload (rightmost bars), this penalty drops to 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Again, we attribute the effect to NVM’s write bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' The high latency of the original skiplist entirely allocated on NVM (green bars in Figure 10b) appears to confirm this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Comparing Figures 10b and 10c, txMontage pays less than 5%, relative to Medley on NVM, for failure atomicity and durability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 21 7 Conclusion We have presented nonblocking transaction composition (NBTC), a new method- ology that leverages the linearizability of existing nonblocking data structures when building dynamic transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As concrete realizations, we introduced the Medley system for transient structures and the txMontage system for (buffered) persistent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Medley transactions are isolated and consistent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' txMon- tage transactions are also failure atomic and durable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Both systems are quite fast: where even the best STM has traditionally suffered slowdowns of 3–10×, Medley incurs more like 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content='2×;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' txMontage, for its part, adds only 5–20% to the overhead of nbMontage, allowing it to outperform existing nonblocking persis- tent STM systems by nearly two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Given their eager contention management, Medley and txMontage main- tain obstruction freedom for transactions on nonblocking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In future work, we plan to explore lazy contention management, postponing installment of descriptors until transactions are ready to commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' By sorting and installing descriptors in canonical order, the resulting systems would preserve lock free- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Lazy contention management would also facilitate helping, as any installed descriptor would have status == InProg, and any other thread could push it to completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' As currently defined in NBTC, speculation intervals are easy to identify, but may unnecessarily instrument certain harmless helping instructions between publication and linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' We are currently working to develop a more pre- cise but still tractable definition of helping in order to reduce the number of “critical” memory accesses that must be performed atomically in each transac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Alan Beadle, Wentao Cai, Haosen Wen, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Non- blocking persistent software transactional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 27th Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on High Performance Computing, Data, and Analytics (HiPC), pages 283–293, virtual conference, December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [2] Wentao Cai, Haosen Wen, Vladimir Maksimovski, Mingzhe Du, Rafaello Sanna, Shreif Abdallah, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Fast Nonblocking Persis- tence for Concurrent Data Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 35th Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Distributed Computing (DISC), pages 14:1–14:20, Freiburg, Germany, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [3] Zhangyu Chen, Yu Huang, Bo Ding, and Pengfei Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Lock-free concurrent level hashing for persistent memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In Usenix Annual Technical Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' (ATC), pages 799–812, virtual conference, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [4] Joel Coburn, Adrian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Caulfield, Ameen Akel, Laura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Grupp, Ra- jesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Gupta, Ranjit Jhala, and Steven Swanson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' NV-Heaps: Making 22 persistent objects fast and safe with next-generation, non-volatile memo- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 16th Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Architectural Support for Programming Lan- guages and Operating Systems (ASPLOS), pages 105–118, Newport Beach, CA, March 2011.' metadata={'source': 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recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In Intl Conf on Management of Data (SIGMOD), pages 1049–1063, Philadelphia, PA, June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [21] Joseph Izraelevitz, Hammurabi Mendes, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Linearizabil- ity of persistent memory objects under a full-system-crash failure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 30th Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Distributed Computing (DISC), pages 313–327, Paris, France, September 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [22] Joseph Izraelevitz, Jian Yang, Lu Zhang, Juno Kim, Xiao Liu, Amirsaman Memaripour, Yun Joon Soh, Zixuan Wang, Yi Xu, Subramanya R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Dulloor, Jishen Zhao, and Steven Swanson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Basic performance measurements of the Intel Optane DC persistent memory module, August 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Principles and Practice of Parallel Programming (PPoPP), pages 227–236, Salt Lake City, UT, February 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [27] Paul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' McKenney, Dipankar Sarma, Andrea Arcangeli, Andi Kleen, Orran Krieger, and Rusty Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Read copy update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In Ottawa Linux Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=', pages 338–367, Ottawa, ON, Canada, July 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [28] Maged M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' High performance dynamic lock-free hash tables and list-based sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 14th ACM Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Parallelism in Algorithms and Architectures (SPAA), pages 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Morrey III, Dhruva R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Chakrabarti, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Dal´ı: A periodically per- sistent hash map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 31st Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Symp.' metadata={'source': 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Mnemosyne: Lightweight persistent memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 16th Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Architectural Sup- port for Programming Languages and Operating Systems (ASPLOS), pages 91–104, Newport Beach, CA, March 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [41] Zixuan Wang, Xiao Liu, Jian Yang, Theodore Michailidis, Steven Swanson, and Jishen Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Characterizing and modeling non-volatile memory sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 53rd Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Microarchitecture (MICRO), pages 496–508, virtual conference, October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [42] Xiangyao Yu, George Bezerra, Andrew Pavlo, Srinivas Devadas, and Michael Stonebraker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Staring into the abyss: An evaluation of concur- rency control with one thousand cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' of the VLDB Endowment, 8 (3):209–220, November 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [43] Deli Zhang and Damian Dechev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Lock-free transactions without rollbacks for linked data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' In 28th ACM Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' on Parallelism in Algorithms and Architectures (SPAA), pages 325–336, Pacific Grove, CA, July 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' [44] Yoav Zuriel, Michal Friedman, Gali Sheffi, Nachshon Cohen, and Erez Pe- trank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Efficient lock-free durable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' of the ACM on Programming Languages, 3(OOPSLA):128:1–128:26, October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfEfrh/content/2301.00996v1.pdf'} +page_content=' 26' metadata={'source': 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XX, NO. XX, XXX 2022 +1 +Joint Optimization of Video-based AI Inference +Tasks in MEC-assisted Augmented Reality Systems +Guangjin Pan, Heng Zhang, Shugong Xu, Fellow, IEEE, +Shunqing Zhang, Senior Member, IEEE, and Xiaojing Chen +Abstract—The high computational complexity and energy +consumption of artificial intelligence (AI) algorithms hinder their +application in augmented reality (AR) systems. However, mobile +edge computing (MEC) makes it possible to solve this problem. +This paper considers the scene of completing video-based AI +inference tasks in the MEC system. We formulate a mixed-integer +nonlinear programming problem (MINLP) to reduce inference +delays, energy consumption and to improve recognition accuracy. +We give a simplified expression of the inference complexity model +and accuracy model through derivation and experimentation. +The problem is then solved iteratively by using alternating opti- +mization. Specifically, by assuming that the offloading decision is +given, the problem is decoupled into two sub-problems, i.e., the +resource allocation problem for the devices set that completes the +inference tasks locally, and that for the devices set that offloads +tasks. For the problem of offloading decision optimization, we +propose a Channel-Aware heuristic algorithm. To further reduce +the complexity, we propose an alternating direction method of +multipliers (ADMM) based distributed algorithm. The ADMM- +based algorithm has a low computational complexity that grows +linearly with the number of devices. Numerical experiments +show the effectiveness of proposed algorithms. The trade-off +relationship between delay, energy consumption, and accuracy +is also analyzed. +Index Terms—Mobile augmented reality, edge intelligence, +mobile edge computing, resource allocation. +I. INTRODUCTION +R +ECENTLY, the development of networks, cloud com- +puting, edge computing, artificial intelligence, and other +technologies has triggered people’s infinite imagination of +the Metaverse [1]. To enable users to interact between the +real world and the virtual world, augmented reality (AR) +technology plays a vital role. At the same time, artificial +intelligence (AI), due to its learning and inference capabilities, +has demonstrated a powerful ability in many fields such as +automatic speech recognition (ASR) [2], natural language +G. Pan, H. Zhang, S. Xu, S. Zhang and X. Chen are with Shanghai Institute +for Advanced Communication and Data Science, Shanghai University, Shang- +hai 200444, China. Emails: {guangjin_pan, hengzhang, shugong, Shunqing, +jodiechen}@shu.edu.cn. +Part of this work has been accepted by Globecom-2022. This work was +supported in part by the National Natural Science Foundation of China +(NSFC) under Grant 61871262, 62071284, and 61901251, the National Key +R&D Program of China grants 2017YFE0121400, 2019YFE0196600 and +2022YFB2902000, the Innovation Program of Shanghai Municipal Science +and Technology Commission grants 20JC1416400 and 21ZR1422400, Pudong +New Area Science & Technology Development Fund, Key-Area Research +and Development Program of Guangdong Province grant 2020B0101130012, +Foshan Science and Technology Innovation Team Project grant FS0AA- +KJ919-4402-0060, and research funds from Shanghai Institute for Advanced +Communication and Data Science (SICS). The corresponding author is +Shugong Xu. +processing (NLP) [3], computer vision (CV) [4], and so on. +With the assistance of AI technology, AR can carry out deeper +scene understanding and more immersive interactions. +However, the computational complexity of AI algorithms, +especially deep neural networks (DNN), is usually very high. +It is challenging to complete DNN inference timely and reli- +ably on mobile devices with limited computation and energy +capacity. In [5], experiments show that a typical single-frame +image processing AI inference task takes about 600 ms even +with speedup from the mobile GPU. In addition, continuously +executing the above inference tasks can only last up to 2.5 +hours on commodity devices. The above issues result in only a +few AR applications currently using deep learning [6]. In order +to reduce the inference time of DNNs, one way is to perform +network pruning on the neural network [7], [8]. However, +it could be destructive to the model if pruning too many +channels, and it may not be possible to recover a satisfactory +accuracy by fine-tuning [7]. +Edge AI [9]–[11] is another approach to solving these +problems. Integrating mobile edge computing (MEC) and AI +technology has recently become a promising paradigm for +supporting computationally intensive tasks. Edge AI transfers +the inference and training process of AI models to the edge of +the network close to the data source. Therefore, it can alleviate +network traffic load, delay, and privacy problems. +A. Related Works +Many existing studies use MEC’s powerful computing capa- +bilities to reduce delay [12], energy consumption [13], or both +delay and energy consumption [14]–[16] through offloading. +For example, [12] formulated an optimization problem aimed +at minimizing the processing delay of eMBB and mMTC +users by optimizing the users’ transmit power in UAV-Assisted +MEC systems. [13] develops a smart pricing mechanism to +coordinate the computation offloading of multi-layer devices +and reduces energy consumption. [14] uses the Stackelberg +game method to optimize the task allocation coefficient, calcu- +lation resource allocation coefficient, and transmission power +to minimize the energy consumption and delay of the NOMA- +based MEC system. +For edge AI inference, existing research has made some +progress. The authors in [17] propose a framework for jointly +optimizing inference task selection and downlink coordinated +beamforming to minimize communication power consumption +in wireless networks. Similarly, [18] proposes an IRS-assisted +edge inference system and designs a task selection strategy +arXiv:2301.01010v1 [cs.NI] 3 Jan 2023 + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +2 +to minimize the energy consumption of uplink and downlink +transmission and calculation. The work in [19] analyzes and +models the transmission error probability, inference accu- +racy, and timeout probability of the AI-powered time-critical +services. The work in [20] uses a tandem queueing model +to analyze queueing and processing delays of DL tasks in +multiple DNN partitions. [21] joint optimizes the service +placement, computational and radio resource allocation to +minimize the users’ total delay and energy consumption. [8] +combines model pruning and DNN partitioning to achieve a +4.81x reduction on end-to-end delay. [22] designs the Edgent +framework that can jointly optimize DNN partitioning and +DNN right-sizing to maximize the inference accuracy while +promising application delay requirements. These studies mea- +sure the inference time by experiments [8], [22] or assume that +the inference task’s computational complexity is proportional +to the input data size but without derivation and proof [20], +[21]. However, these models of computational complexity are +not rigorous enough or can not be generalized to different +neural network models. +As for the accuracy model, the authors in [23] designs +an edge network orchestration algorithm named FACT, which +boosts the performance of an edge-based AR system by opti- +mizing the edge server assignment and video frame resolution +selection for AR users. However, [23] builds an accuracy +model by fitting an accuracy curve for specific tasks, which +is not general. The work in [24] compresses image resolution +locally and performs inference tasks on edge servers, aiming +to maximize learning accuracy under constraints of delay +and energy. [24] proposes using an abstract non-decreasing +function to describe the relationship between accuracy and +input image size, which cannot be used to analyze various AI +inference tasks discriminately. Joint optimization is required +when different tasks and models are jointly deployed. An +insufficiently generalized accuracy model or an overly abstract +model can adversely affect joint optimization. A general +accuracy model is needed to measure various AI tasks. +Among the above studies, most studies consider optimizing +one or two performance metrics among the delay, energy con- +sumption, and accuracy. The authors in [24] jointly considers +delay, energy consumption and accuracy in image recognition +scenarios. However, it aims at maximizing computational +capacity under constraints of delay, energy consumption and +accuracy, and the DNN model is only deployed in edge servers. +In [6], [23], [25], video analytics scenarios are considered, but +they do not jointly consider delay, energy and accuracy. +B. Contributions and Organizations +In this paper, we consider a multi-user MEC system and +assume that each device executes the video-based DNN in- +ference task. Each device can be AR glasses, mobile robots, +and so on. In order to deepen AR’s ability to understand the +scene, we need to use time dimension information to improve +perception. Therefore, we consider video-based application +scenarios.for video-based AI inference tasks, there are two +modes, e.g., frame-by-frame recognition mode (the input for +each recognition is one frame) and multi-frame recognition +� + Uplink for real-time captured content + Downlink for inference results +Fig. 1. Multi-user MEC System model. The inference task can be executed on +the local or the edge server. When the task is offloaded to the edge server, the +uplink transmits the content captured in real-time, and the downlink transmits +the inference result. +mode (the input for each recognition is multiple frames). The +frame-by-frame inference mode is used to deal with tasks with +weak temporal correlation, such as face recognition and target +tracking., and has been studied in [6], [23], [25]. In this paper, +we focus on multi-frame recognition tasks, such as gesture +recognition and action recognition tasks. Since sampling in +the spatial dimension brings extra computation [24], we only +sample in the temporal domain. At each inference, the device +selects the most recent several frames from the history frames +for transmission or inference. +As shown in Fig. 1, mobile devices can transmit captured +video to the edge server via wireless networks. The edge +servers execute inference tasks and send results back to mo- +bile devices. However, when communication and computing +resources of the edge server are insufficient, devices can +execute the inference task locally. We model the problem as a +multi-objective optimization problem to optimize delay, energy +consumption, and inference accuracy. The main contributions +of this paper are summarized as follows, +• Multi-dimensional target optimization. High accuracy, +low delay, and low energy consumption are indispensable +for AR applications and must be optimized jointly. To +explore the trade-off relationship between delay, energy, +and accuracy, we formulate the video-based offloading +problem as a mixed-integer nonlinear programming prob- +lem (MINLP), aiming to reduce service delays, energy +consumption and improve recognition accuracy. +• General computational complexity and accuracy models. +To measure the computational complexity of neural net- +work models with different architectures and different +input sizes, we introduce the number of multiply-and- +accumulate operations (MACs). We illustrate the main +factors affecting DNN inference delay through experi- +ments and show that MAC can be used as a good measure +of the computational complexity of DNN inference tasks. +We also propose a general model to represent the relation- +ship between the inference accuracy and the number of +input frames. This model is suitable for different video- + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +3 +based recognition tasks and different DNN architectures. +We give simple expressions of the inference complexity +and accuracy to simplify the optimization problem. +• Channel-Aware scheduling scheme. To solve the opti- +mization problem, we decompose the original problem. +First, assuming that the offloading decision is given, we +solve the resource allocation problems for the device set +that completes the inference locally and the device set +that offloads the tasks to the edge server, respectively. For +edge DNN inference, we propose two algorithms based +on search and geometric programming (GP) to solve the +problem. Then, to obtain the optimal offloading policy, +we propose a Channel-Aware heuristic algorithm. The +original problem is solved iteratively through alternating +optimization. +• ADMM-based distributed resource allocation scheme. To +avoid the high complexity of the heuristic algorithm, we +propose an algorithm based on the Alternating direction +method of multipliers (ADMM). The ADMM-based algo- +rithm decomposes the original problem into parallel and +tractable subproblems. Therefore, the total computational +complexity of ADMM-based algorithms is more scalable +than the heuristic algorithm, especially when the number +of devices is large. +The rest of this paper is organized as follows. In Section +II, we introduce system models, including delay, energy, +and accuracy models. In Section III, we formulate the joint +optimization problem and convert the original problem to a +more tractable problem. Section IV proposes a Channel-Aware +heuristic algorithm to solve the proposed problem. In Section +V, we propose another ADMM-based distributed resource +allocation algorithm for the proposed problem, and analyze the +computational complexity of the solution algorithm. Numeri- +cal results and analysis are presented in Section VI. Finally, +the paper is concluded in Section VII. +II. SYSTEM MODEL +In this section, we introduce a single-cell MEC system and +establish delay, energy consumption, and accuracy models. As +shown in Fig. 1, we consider a multi-user MEC system with +one base station (BS) and N mobile devices, denoted by the +set N = {1, 2, . . . N}. Each device has a camera and needs +to accomplish DNN inference tasks. Due to the limitation of +device computational resources, DNN inference tasks can be +placed on local or edge servers. The limited computational +resource will lead to longer computing delay and greater power +consumption when the inference task is executed locally. +However, when the inference task is executed on the edge +server, it will bring additional wireless transmission delay. In +addition, accuracy is also a very important optimization target +in DNN inference tasks. +A. Offloading Framework +In this paper, we only consider the binary offloading +method. Binary offloading requires the DNN inference task to +be fully executed either at the device or the MEC server. The +overview of the DNN computing offloading system is depicted +Sampling +management module +Video +offloading +Result +feedback +Complete +inference task +Execute locally +Complete +inference task +Complete +inference task +Fig. 2. +The overview of the video sampling and computing offloading +system. The video sampling management module can control the sampling +rate of the captured video and determine the number of video frames used +for AI inference. Devices can transmit the video to the edge server or +perform inference tasks locally based on the wireless channel information +and computing capabilities. +in Fig. 2. First, devices sample the video captured in real- +time in the temporal dimension to obtain a short video with +a certain number of frames. Second, the DNN inference tasks +are executed. These inference tasks can be executed locally +on devices or the edge server. Therefore, each device’s video +sampling management module needs to select an appropriate +video sampling rate (how many frames need to be input) and +choose whether to offload the task to the MEC server. Denote +Dn, En and φn to be the total delay, energy consumption and +recognition accuracy of the device n, respectively. The total +delay and energy consumption of the device n can be given +by, +Dn = (1 − xn)Dmd +n ++ xn(Dt +n + De +n), +(1) +En = (1 − xn)Emd +n ++ xnEt +n, +(2) +where xn indicates whether the inference task is executed on +local or edge servers. Dt +n is the transmission delay for uplink, +Dmd +n +is the local inference delay, and De +n is the delay for +completing inference at the edge server. Et +n and Emd +n +are the +transmission and computational energy consumption, respec- +tively. The delay and energy consumption for downloading +computation results can be reasonably neglected because of +the results’ small data sizes. +B. Delay and Energy Models for Inference +The inference delay depends on the DNN model’s architec- +ture, the device’s or server’s computing power, and the input +to the model. In this section, we first give a measure of the +computational complexity of the DNN model and then give an +expression for the inference delay and energy consumption. +Different AI recognition tasks may require different AI +model architectures, including classic AI models such as +Resnet-18, Resnet-34, Resnet-50, VGG-16, etc. [26], [27]. In +order to optimize AI inference tasks more reasonably, different +AI models need a common method to evaluate computational + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +4 +complexity. In this paper, we use the number of MACs [28] to +measure the computational complexity of AI inference tasks. +MACs calculation methods of layers (such as fully connected +(FC) layers, convolutional layers and so on) can be obtained +in [28]. Taking 3D Convolutional Neural Network (3DCNN) +as an example, the computational complexity (measured by +MACs) of the lth layer of the nth device can be expressed as, +cn,l = +olol+1 +�2 +j=0 Kj +l , �2 +j=0 M j +n,l+1, +(3) +where ol is the number of input channels, ol+1 is the number +of output channels, �2 +j=0 Kj +l is the size of the convolution +kernel, and �2 +j=0 M j +n,l+1 is the size of the output feature +map. j = 0 represents the temporal dimension (the number of +frames), j = 1, 2 represent spatial dimensions (pixels of one +frame). Note that ol, ol+1, and �2 +j=0 Kj +l are all determined +by the neural network architecture and �2 +j=0 M j +n,l+1 depends +on the input size. The relation between the output feature size +and the input size can be expressed as, +M j +n,l+1 = +M j +n,l − Kj +l + 2dl +rl ++ 1, +(4) +where rl is the stride and dl is the padding size. +As mentioned above, the computational complexity of a +DNN model is determined by the number of layers, the DNN +model’s architecture, and the input and output size. In this +paper, we mainly focus on the impact of the number of input +video frames Mn on recognition accuracy and the allocation +of communication and computing resources. The inference +result will be more accurate with more frames Mn input, but +the communication and calculation overhead will be greater. +The computational complexity of the nth device’s task can be +expressed as C(Mn). +Then we give the expression for the inference delay and +energy consumption. Denote f max and f max +n +(in CPU cycle/s) +to be the total computation resource of the edge server and +mobile device n, respectively. Let f e +n and f md +n +(in CPU +cycle/s) denote the computation resource to device n allocated +by the edge server and the device, respectively. Therefore, +the computing resources satisfy � +n∈N f e +n +≤ f max and +f md +n +≤ f max +n +. The computation delay of the device n and +MEC can be respectively expressed as, +Dmd +n += ρC(Mn) +f md +n +, +(5) +De +n = ρC(Mn) +f en +, +(6) +where ρ (cycle/MAC) represents the number of CPU cycles +required to complete a multiplication and addition, which +depends on the CPU model. +As for energy consumption, denote κ to be a coefficient +determined by the corresponding device [24], and the com- +putational energy consumption of device n can be expressed +as, +Emd +n += κρC(Mn)f md +n +2. +(7) +C. Delay and Energy Models for Transmission +We consider a time-division multiple access (TDMA) +method for channel access. Specifically, each radio frame is +divided into N time slots for transmission, and each device can +only transmit in its own time slot. We assume that the length +of each radio frame is ∆T, which is short enough (e.g., 10 +ms in LTE or NR system [24]), and the length of a time slot +is ∆Ttn. +Denote hn and pn to be the channel gain and transmission +power of the device n, respectively. According to [21], the +achievable data rate of device n can be expressed as, +Rn = Bwlog2 +� +1 + pnhn +BwN0 +� +, +(8) +where Bw and N0 are the bandwidth and the variance of +additive white Gaussian noise (AWGN), respectively. +Let d denote the data size of one video frame. Since we only +want to analyze the impact of time dimension information (the +number of input frames Mn) on recognition accuracy, d is a +constant value. In each radio frame, the data size that can +be transmitted is ∆TRntn. Therefore, for each transmission, +⌈ +Mnd +∆T Rntn ⌉ radio frames are required, where ⌈·⌉ means the +ceil function. Considering that the length of the radio frame +is much shorter than the transmission delay, the transmission +delay for offloading to MEC can be written as, +Dt +n = ⌈ +Mnd +∆TRntn +⌉∆T ≈ Mnd +Rntn +, +(9) +where tn is the proportion of time that device n transmits. In +addition, according to [24], the energy consumption of each +device to transmit its video can be expressed as, +Et +n = Mnd +Rn +pn. +(10) +D. Inference Tasks Accuracy Model +As mentioned above, we mainly focus on the impact of the +number of input video frames Mn on recognition accuracy. +We assume that the quality of the input video is the same +for different devices. For a certain task and DNN model, the +accuracy is only determined by the number of input frames. +Therefore, the accuracy of device n can be expressed as +φn = Φ(Mn). According to [29], more frames will lead to +better inference accuracy, and as the input frames continue to +increase, the performance gain will gradually decrease. Some +prior studies also show that the relationship between frame +rate and accuracy can be expressed as concave functions [23]. +Therefore, we define Φ(Mn) as a monotone non-decreasing +function to describe the relationship between the accuracy and +the number of input frames. +III. PROBLEM FORMULATION +In this section, we formulate the optimization problem to +reduce the system’s delay and devices’ energy consumption +and improve accuracy. We analyze the difficulty of solving +the problem. To simplify the problem, we make a reasonable +conversion of the problem. + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +5 +A. Original Problem Formulation +Based on the above analysis, combining (1), (2), (5)-(7), +(9), and(10), the nth device’s delay and energy consumption +can be expressed as, +Dn += +(1 − xn)ρC(Mn) +f md +n ++ xn(ρC(Mn) +f en ++ Mnd +Rntn +), (11) +En += +(1 − xn)κρC(Mn)f md +n +2 + xn(Mnd +Rn +pn). +(12) +Given the system model described previously, our goal +is to reduce end-to-end delay and energy consumption and +improve recognition accuracy. Each device follows the binary +offloading policy. The mathematical optimization problem of +the total cost (delay, energy consumption, and accuracy) can +be expressed as, +Problem P1 (Original Problem): +minimize +{Mn,tn,f md +n +,f e +n,xn} +� +n∈N +� +β1Dn + β2En − β3Φ(Mn) +� +, (13) +subject to +Φ(Mn) ≥ αn, ∀n ∈ N, +(13a) +Mn ≤ M max +n +, Mn ∈ Z, +(13b) +� +n∈N +xntn ≤ 1, +(13c) +� +n∈N +xnf e +n ≤ f max, +(13d) +tn, f e +n ≥ 0, ∀n ∈ N, +(13e) +0 ≤ f md +n +≤ f max +n +, ∀n ∈ N, +(13f) +xn ∈ {0, 1} , ∀n ∈ N, +(13g) +where αn represents the recognition accuracy requirement, β1, +β2, β3 are the weight factors. (13a) represents the recognition +accuracy requirement of each device. (13b) indicates the frame +limit for the input video, Z is the set of integers, and M max +n +is +the maximum number of frames of the input video. (13c) and +(13d) represent the communication and computation resource +limitation, respectively. (13f) limits the computation resource +of each device. +The optimization variables in original problem P1 are +the number of input video frames Mn, the proportion of +transmission time tn, the local computation resource f md +n , the +edge computation resource allocation f e +n, and the offloading +decision xn. In addition, the first item in (13) is to reduce the +total delay of computation and transmission, the second item +is to reduce the device’s energy consumption, and the last item +is to improve the number of input video frames as well as the +recognition accuracy because of the monotone non-decreasing +function Φ(Mn). +Problem P1 is a non-convex MINLP problem and is difficult +to be solved. First, the complexity function C(Mn) is discrete +and depends on the architecture of the DNN and the size +of the input video. As the number of input frames Mn +increases, the computational complexity also increases. This +kind of increase is irregular because it is affected by the +structure of DNN layers, such as the stride and padding size +of 3DCNN according to (4). Therefore, C(Mn) cannot be +used for optimization directly. Second, as mentioned above, +the accuracy function Φ(Mn) is non-decreasing. However, we +cannot give a deterministic expression for Φ(Mn), so we can +not optimize it. In addition, both Mn and xn are integers, +making the problem difficult to be solved. +B. Problem Conversion +To make the problem P1 more tractable, we convert the +problem. First, we give an approximate expression of the +computational complexity function C(Mn). According to (3) +and (4), the computational complexity of 3DCNN layers is +proportional to the size of the input data. We can also obtain +a similar conclusion in other types of layers, such as the FC +layer [28]. Based on the above conclusion and combined with +the experiments in Sec. VI-A, in order to simply express the +computational complexity model, C(Mn) can be written as, +C(Mn) = mc,0Mn + mc,1, +(14) +where mc,0 ≥ 0 and mc,1 are constants and depend on the +network model. +Second, we propose a general model to express the re- +lationship between the accuracy and the number of input +video frames. Considering that the function Φ(Mn) is mono- +tonically non-decreasing and that as the number of input +frames increases, the accuracy gain decreases, combining our +experiments in Sec. VI-A, we model function Φ(Mn) as, +Φ(Mn) = − +ma,0 +Mn + ma,1 ++ ma,2, +(15) +where ma,0 ≥ 0, ma,2 ≥ 0 and ma,1 > −1 are constants and +depend on the target of inference tasks and the architecture of +DNN models. +Finally, we relax the range of the variable Mn. Consid- +ering that Φ(Mn) is a monotone non-decreasing function +and depends on the recognition task and network archi- +tecture, in order not to lose generality, define M min +n += +arg minMn Φ(Mn), Φ(Mn) ≥ αn, Mn ∈ Z. We can also +relax Mn into a closed connected subset of the real axis, and +(13a), (13b) can be written as Mn ∈ +� +M min +n +, M max +n +� +. Then +[Mn] can be regarded as the number of input video frames, +where [·] indicates rounding. We define two sets of devices, i.e. +N0 = {n | xn = 0, n ∈ N} and N1 = {n | xn = 1, n ∈ N}. +F0,n and F1,n are the cost function of the device n in sets N0 +and N1, respectively. The problem P1 can be rewritten as, +Problem P2 (Converted Problem): +minimize +{Mn,tn,f md +n +,f e +n,xn} +� +n∈N0 +(1 − xn)F0,n(Mn, f md +n ) ++ +� +n∈N1 +xnF1,n(Mn, f e +n, tn), +(16) +subject to Mn ∈ +� +M min +n +, M max +n +� +, +(16a) +(13c) − (13g), +where +F0,n(Mn, f md +n ) = β1 +ρC(Mn) +f md +n ++ β2κρC(Mn)f md2 +n +− β3Φ(Mn), +(17) +F1,n(Mn, f e +n, tn) = β1 +ρC(Mn) +f en ++ β1 +Mnd +Rntn + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +6 ++ β2 +Mndpn +Rn +− β3Φ(Mn). +(18) +IV. OPTIMIZATION PROBLEM SOLVING +In this section, we decompose the problem P2 and propose +a Channel-Aware heuristic algorithm to solve it. First, sup- +posing that the offloading decision (i.e., {xn}) is given, we +solve optimization problems for sets N0 and N1, respectively. +Second, we propose a Channel-Aware heuristic algorithm to +optimize the offloading decision {xn}. +A. Optimization Problem Solving for N0 +For set N0, i.e., when the device executes inference tasks +locally, the optimization problem becomes, +Problem PN0 (Problem for N0): +minimize +{Mn,f md +n +} FPN0 ≜ +� +n∈N0 +F0,n(Mn, f md +n ), +(19) +subject to +(13f), (16a). +The optimization variables in PN0 are the number of input +video frames Mn and the local computation resource f md +n . +Let {M ∗ +n, f md∗ +n +} denote the optimal solution to PN0. We can +derive the optimal solution to PN0 in a closed-form expression. +Theorem 1: The optimal solution to PN0 is given by, +f md∗ +n += min{ +3 +� +( β1 +2β2κ), f max +n +}, +(20) +M ∗ +n = min{max{ +� +β3ma,0 +β1ρmc,0 +f md +n ++ β2κρmc,0f md2 +n +− ma,1, M min +n +}, M max +n +}. +(21) +Proof: Please refer to Appendix A. +From Theorem 1, we can see that the optimal local CPU- +cycle frequency f md +n +is determined by the weight factors β1, +β2, the coefficient of CPU energy consumption κ, and is +limited by its corresponding upper bound f max +n +. More specifi- +cally, f md +n +is proportional to β +1 +3 +1 and inversely proportional to +β +1 +3 +2 and κ +1 +3 . As for the number of input video frames, when +3� +( β1 +2β2κ) ≤ f max +n +, combining (20) and (21), we have, +M ∗ +n = min{max{3− 1 +2 2 +1 +3 ρ− 1 +2 κ− 1 +6 m +− 1 +2 +c,0 β +− 1 +3 +1 +β +− 1 +6 +2 +β +1 +2 +3 m +1 +2 +a,0 +− ma,1, M min +n +}, M max +n +}. +(22) +The optimization results corresponding to each device are +only related to the parameters of the device itself and are not +associated with the parameters of other devices. +B. Optimization Problem Solving for N1 +Then we solve the optimization problem of N1. The prob- +lem P2 can be written as, +Problem PN1 (Problem for N1): +minimize +{Mn,f e +n,tn} +� +n∈N1 +F1,n(Mn, f e +n, tn), +(23) +subject to +(13c), (13d), (13e), (16a). +Algorithm 1: Algorithm 1: Search-Based Algorithm +for solving PN1 +Input: The offloading policy N1, the channel gain +{hn}, and other system parameters. +Output: {M ⋆ +n, f e⋆ +n , t⋆ +n} +Initialize the result of cost function F⋆ +� +PN1 to a +sufficiently large value; +Calculate the achievable data rate {Rn} using (8); +foreach {Mn} ∈ M do +Compute F � +PN1 using (27); +if F � +PN1 < F⋆ +� +PN1 then +{M ⋆ +n} = {Mn}; F⋆ +� +PN1 = F � +PN1 ; +Calculate {f e⋆ +n } and {t⋆ +n} using (25) and (26); +return {M ⋆ +n}, {f e⋆ +n }, and {t⋆ +n}. +The optimization variables in the the problem PN1 are the +number of input video frames Mn, the edge computation +resource f e +n, and the proportion of transmission time tn. +Let {M ∗ +n, f e∗ +n , t∗ +n} denote the optimal solution to PN1. We +can obtain the optimal solution to PN1 using the method of +Lagrange multiplier. The partial Lagrangian function can be +written as, +LPN1= +� +n∈N1 +�β1ρC(Mn) +f en ++ β1Mnd +Rntn ++ β2Mndpn +Rn +− β3Φ(Mn) +� ++ µ0( +� +n∈N1 +tn − 1) + µ1( +� +n∈N1 +f e +n − f max), +(24) +First of all, according to (24), supposing that M ∗ +n is given, +we can solve the problem PN1 based on the Karush-Kuhn- +Tucker (KKT) condition. We can obtain the function expres- +sions of f e∗ +n and t∗ +n relative to Mn, as shown in the following +theorem. +Theorem 2: The function expressions of f e∗ +n and t∗ +n relative +to M ∗ +n are given by, +f e∗ +n = f max� +C(M ∗n) +� +i∈N1 +� +C(M ∗ +i ) +, +(25) +t∗ +n = +� +M ∗ +n +Rn +� +i∈N1 +� +M ∗ +i +Ri +. +(26) +Proof: Please refer to Appendix B. +Combining (23), (25) and (26), the problem PN1 can be +written as an optimized function containing only the variable +Mn as follows, +Problem � +PN1 (Mn Optimization Problem for N1 ): +minimize +{Mn} +F � +PN1 ≜ β1ρ +f max ( +� +n∈N1 +� +C(Mn))2 ++ β1d( +� +n∈N1 +� +Mn +Rn +)2 + β2dpn( +� +n∈N1 +Mn +Rn +) +− +� +n∈N1 +β3Φ(Mn), +(27) + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +7 +Algorithm 2: Algorithm 2: GP-Based Algorithm for +solving PN1 +Input: The offloading policy N1, the channel gain +{hn}, and other system parameters. +Output: {M ⋆ +n, f e⋆ +n , t⋆ +n} +Calculate the achievable data rate {Rn} using (8); +Use the CVX tool to solve (29) and get { ˆ +M ⋆n}; +{M ⋆ +n} = {[e ˆ +M ⋆ +n]}; +Calculate {f e⋆ +n } and {t⋆ +n} using (25) and (26); +return {M ⋆ +n}, {f e⋆ +n }, and {t⋆ +n}. +subject to +(16a). +Denote Mopt +n += {Mn | M min +n +≤ Mn ≤ M max +n +, Mn ∈ +Z} to be the optional video frame number of device n. The +optimal solution can be obtained by searching for {Mn} ∈ M, +where M = {{Mi} | Mi ∈ Mopt +i +, i ∈ N1}. The detail of the +search based algorithm is shown in Algorithm 1. +Considering that the problem PN1 is convex when Mn is +given, Algorithm 1 is global optimal. However, When the +number of devices grows large, the computational complexity +of the Search-based algorithm will become very high or even +unacceptable. In this paper, we also propose a GP-based sub- +optimal algorithm to solve the problem PN1. First, we relax +the objective function of the problem PN 1. We introduce the +function, �Φ(Mn) = − ma,0 +Mn + ma,2, and PN1 can be rewritten +as, +Problem PGPN1 (GP-based Problem for N1): +minimize +{Mn,f e +n,tn} +� +n∈N1 +� +β1 +ρC(Mn) +f en ++ β1 +Mnd +Rntn ++ β2 +Mndpn +Rn +− β3�Φ(Mn) +� +, +(28) +subject to +(13c), (13d), (13e), (16a). +It is a non-convex GP problem. Inspired by [30], the GP +problem can be transformed into a convex problem by +changing variables and transforming the objective and con- +straints. Therefore, introducing variables, +ˆ +Mn = ln Mn, ˆf en = +ln f e +n, ˆtn = ln tn, and the problem can be written as, +Problem � +PGPN1 (Converted GP-based Problem for N1): +minimize +{ ˆ +Mn, ˆ +tn, ˆ +f e +n} +� +n∈N1 +� +β1ρmc,0e +ˆ +Mn− ˆ +f e +n + β1ρmc,1e− ˆ +f e +n ++ β1de ˆ +Mn− ˆ +tn +Rn ++ β2dpne ˆ +Mn +Rn ++β3ma,0e− ˆ +Mn +� +, (29) +subject to +ˆ +Mn ∈ +� +ln M min +n +, ln M max +n +� +, ∀n ∈ N1, +(29a) +� +n∈N1 +xne ˆ +tn ≤ 1, +(29b) +� +n∈N1 +xne +ˆ +f e +n ≤ f max, +(29c) +which is strictly convex problem that can be solved using the +CVX tool [31]. Considering that Mn is an integer, the result +of CVX optimization needs to be post-processed. Details of +the GP-based algorithm are shown in Algorithm 2. +Algorithm 3: Algorithm 3: Channel-Aware heuristic +algorithm for Optimizing Offloading Policy {xn} +Input: Parameters corresponding to the problem P1. +Output: Offloading policy N0 and N1. +Calculate the cost function {F0,n} for the set N using +(20) and (21) ; +Set N0 = ∅, N1 = N; +Calculate the cost function {F1,n} corresponding to +the set N1 using Algorithm 1 or Algorithm 2; +Set Flag = 1; +while Flag == 1 do +k = argminnhn, n ∈ N1; +N ∗ +0 = N0 ∪ {k}, N ∗ +1 = N1 − {k}; +Calculate the cost function {F∗ +1,n} corresponding +to the set N ∗ +1 using Algorithm 1 or Algorithm 2; +if � +n∈N0 F0,n + � +n∈N1 F1,n > +� +n∈N ∗ +0 F0,n + � +n∈N ∗ +1 F∗ +1,n then +F1,n = F∗ +1,n, ∀n ∈ N ∗ +1 ; +N0 = N ∗ +0 ; N1 = N ∗ +1 ; +else +Flag = 0; +return N0 and N1. +C. Optimization of Offloading Policy {xn} +Considering the complexity of Search-based offloading pol- +icy algorithm becomes high when the number of devices N +grows large. In this section, we propose a Channel-Aware +heuristic algorithm to optimize the offloading decision {xn}. +Inspired by the Theorem 1 and Theorem 2, when executing +inference locally, the cost function F0,n and optimization vari- +ables f md +n , Mn only depend on the device’s own parameters. +However, for edge set N1, the cost function is related to +the number and parameters of devices in the set N1. The +Channel-Aware heuristic algorithm is shown in Algorithm +3. First, calculate the cost function {F0,n} of set N0 when +each device’s task is executed locally. Second, assuming that +all devices are offloaded to the edge server for inference +and |N1| = N. In each iteration, the cost function {F1,n} +corresponding to each device of N1 is obtained. We select the +device k with smallest channel gain in set N1. Try to put the +device k from the set N1 into the set N0 and compute the cost +of new sets. If the total cost of new sets is reduced, continue +the next iteration. Otherwise, put the device k back to the set +N1. +V. JOINT OPTIMIZATION USING ADMM-BASED +METHOD +The complexity of the Channel-Aware heuristic algorithm +becomes high when the number of UE grows. In this section, +We propose an ADMM-based algorithm. The ADMM-based +algorithm can decompose P2 into N parallel sub-problems. +Each user only needs to solve one sub-problem, and the +average complexity of each device will be reduced. + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +8 +A. ADMM-based Problem Conversion +To make the original problem tractable, we jointly consider +the problem P2 and problem � +PGPN1 , and we converted the +problem into a GP-based problem, +Problem P3 (Converted GP-based Problem): +minimize +� +ˆ +Mn, ˆ +tn, ˆ +f md +n +, ˆ +f e +n,xn +� +� +n∈N +� +(1 − xn) ˆ +F0,n( ˆ +Mn, +ˆ +f md +n ) ++ xn ˆ +F1,n( ˆ +Mn, ˆf en, ˆtn) +� +, +(30) +subject to +ˆ +f md +n +≤ ln f max +n +, ∀n ∈ N, +(30a) +(13g), (29a) − (29c), +where +ˆ +Mn = ln Mn, +ˆ +f md +n += ln f md +n , ˆf en = ln f e +n, and ˆtn = +ln tn. +ˆ +F0,n( ˆ +Mn, +ˆ +f md +n ) and +ˆ +F1,n( ˆ +Mn, ˆf en, ˆtn) are given by, +ˆ +F0,n( ˆ +Mn, +ˆ +f md +n ) = β1ρmc,0e +ˆ +Mn− ˆ +f md +n ++ β1ρmc,1e− ˆ +f md +n ++ β2κmc,0e +ˆ +Mn+2 ˆ +f md +n ++ β2κmc,1e2 ˆ +f md +n ++ β3ma,0e− ˆ +Mn, (31) +ˆ +F1,n( ˆ +Mn, ˆf en, ˆtn) = β1ρmc,0e +ˆ +Mn− ˆ +f e +n + β1ρmc,1e− ˆ +f e +n ++ β1de ˆ +Mn− ˆ +tn +Rn ++ β2dpne ˆ +Mn +Rn ++ β3ma,0e− ˆ +Mn, +(32) +The optimization variables { ˆtn, ˆf en} are coupled among the +devices in the constraints (29b) and (29c). To decompose the +problem P3, we introduce local variables {yn} and {zn}. +Then, the ADMM-based problem can be written as, +Problem P4 (ADMM-based Problem): +minimize +� +ˆ +Mn, ˆ +tn, ˆ +f md +n +, ˆ +f e +n,xn,yn,zn +� +� +n∈N +ˆ +Fn(xn, ˆ +Mn, +ˆ +f md +n , yn, zn) ++ g( ˆf en, ˆtn), +(33) +subject to yn = ˆf en, zn = ˆtn, +(33a) +(13g), (29a), (30a), +where, +ˆ +Fn(xn, ˆ +Mn, +ˆ +f md +n , yn, zn) = (1 − xn) ˆ +F0,n( ˆ +Mn, +ˆ +f md +n ) ++ xn ˆ +F1,n( ˆ +Mn, xn, yn), +(34) +g( ˆf en, ˆtn) = +� +0, +if( ˆf en, ˆtn) ∈ G, ++∞ +, otherwise, +(35) +and, +G = +� +( ˆf en, ˆtn)| +� +n∈N1 +xne ˆ +tn ≤ 1, +� +n∈N1 +xne +ˆ +f e +n ≤ f max +� +. (36) +B. ADMM-based Problem Solving +The problem P4 can be effectively solved using the ADMM +algorithm. We can write a partial augmented Lagrangian of the +problem P4 as, +L4(u, v, θ) = +� +n∈N +ˆ +Fn(xn, ˆ +Mn, +ˆ +f md +n , yn, zn) + g( ˆf en, ˆtn) ++ +� +n∈N +θf +n(yn − ˆf en) + +� +n∈N +θt +n(zn − ˆtn) ++ +� +n∈N +s +2(yn − ˆf en)2 + +� +n∈N +s +2(zn − ˆtn)2, +(37) +where u = {xn, ˆ +Mn, +ˆ +f md +n , yn, zn}, v = { ˆf en, ˆtn}, θ = +{θf +n, θt +n}, and s is a fixed step size. Therefore, the dual function +is, +p(θ) = minimize +u,v +L4(u, v, θ) +(38) +subject to (13g), (29a), (30a), +and the dual problem can be given by, +maximize +θ +p(θ), +(39) +The problem (38) can be solved by iteratively updating +u, v, and θ [32]. Let {ui, vi, θi} denote the values in the +ith iteration. In the ith iteration, the update strategies of the +variables are as follows, +1) Step 1: Local variables update. In this step, we first +update the local variables u. Given variable vi and θi, we +minimize L4(u, v, θ) by, +ui+1 = argminimize +u +L4(u, vi, θi). +(40) +The problem (39) can be decomposed into N parallel sub- +problems. For each subproblem, we consider two cases where +xn = 0 and xn = 1, and express the problem as, +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +minimize +{ ˆ +Mn, ˆ +f md +n +,yn,zn} +ˆ +F0,n( ˆ +Mn, +ˆ +f md +n ) =θf +nyn + +� +n∈N +s +2(yn − ˆf en)2 ++θt +nzn + +� +n∈N +s +2(zn − ˆtn)2, +if xn = 0, +minimize +{ ˆ +Mn,yn,zn} +ˆ +F1,n( ˆ +Mn, yn, zn)=θf +nyn + +� +n∈N +s +2(yn − ˆf en)2 ++θt +nzn + +� +n∈N +s +2(zn − ˆtn)2, +if xn = 1. +(41) +These problems are both strictly convex problems that can +be solved using the CVX tool [31]. Therefore, we can cal- +culate the objective value for xn = 0 and xn = 1 and +choose the smaller one as the final result. After solving N +parallel subproblems, the optimal solution to (40) is given by +ui+1 = {(xn)i+1, ( ˆ +Mn)i+1, ( ˆ +f md +n )i+1, (yn)i+1, (zn)i+1}. +2) Step 2: Global variables update. In the second step, we +update the global variables v. By the definition of g(v) in +(35), vi+1 ∈ G must hold at the optimum. Therefore, the +subproblem can be equivalently written as, +vi+1 = argminimize +{ ˆ +f e +n, ˆ +tn} +� +n∈N +(θf +n)i(− ˆf en) + +� +n∈N +(θt +n)i(− ˆtn) ++ +� +n∈N +s +2(yi+1 +n +− ˆf en)2 + +� +n∈N +s +2(zi+1 +n +− ˆtn)2, +(42) +subject to, +(29b), (29c). +The problem can also be solved by the CVX tool [31]. We +propose a low-complexity scheme to solve this subproblem. + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +9 +Algorithm 4: Algorithm 4: ADMM-Based Algorithm +Input: Parameters corresponding to the problem P1. +Output: {xn, Mn, f md +n , f e +n, tn} +Initialize i = 0, {ui, vi, θi} = 0, s = 0.5, +µ⋆ +f = µ⋆ +t = 106, δ = 10−4; +repeat +foreach n ∈ N do +Update ui+1 by solving (41) and choose +smaller results; +foreach n ∈ N do +Update global variables vi+1 using (43) and +(44); +foreach n ∈ N do +Update multipliers θi+1 using (45) and (46); +i = i + 1; +until |Fi − Fi+1| < δ; +Mn = e ˆ +Mn, f md +n += e +ˆ +f md +n , f e +n = e ˆ +f e +n, tn = e ˆ +tn; +return {xn, Mn, f md +n , f e +n, tn}. +Considering the constraints (29b) and (29c), let µf and µt +denote the Lagrangian multipliers. The closed-form optimal +solution of this subproblem can be expressed as, +( ˆf en)i+1 =yn +i+1 + (θf +n)i − µf +s +, +(43) +( ˆtn)i+1 =zn +i+1 + (θt +n)i − µt +s +, +(44) +where µf can be obtained by the bisection search method +over (0, µ⋆ +f), until � +n∈N1 xne ˆ +f e +n ≤ f max satisfies. µ⋆ +f is a +sufficiently large value. It is because when µf ≥ 0, ( ˆf en)i+1 is +non-increasing. Similarly, µt can be obtained by the bisection +search method over (0, µ⋆ +t ), where µ⋆ +t is a sufficiently large +value, until � +n∈N1 xne ˆ +tn ≤ 1 satisfies. +3) Step 3: Multipliers update. In this step, we update the +multipliers θ using the obtained global variables v and local +variables u. The updated method is, +(θf +n)i+1 =(θf +n)i + s(yi+1 +n +− ( ˆf en)i+1), +(45) +(θt +n)i+1 =zn +i+1 + s(zi+1 +n +− ( ˆtn)i+1), +(46) +Repeat the above three steps until the cost function no +longer decreases. The cost function is Fi = � +n∈N [(1 − +xi +n) ˆ +F0,n(( ˆ +Mn)i, ( +ˆ +f md +n )i) + xi +n ˆ +F1,n(( ˆ +Mn)i, ( ˆf en)i, ( ˆtn)i)]. We +summarize solving steps of the ADMM algorithm as Algo- +rithm 4. +As a distributed iterative algorithm, the ADMM-based +scheme performs iterations between devices and BS rather than +locally, enabling online optimization during the recognition +process. In each iteration, ui is calculated locally and sent +to the MEC. After receiving ui from all devices, the MEC +updates vi and θi, and sends them to the device to complete +an iteration. Therefore, the iteration of the ADMM algorithm is +an online convergence process that can adapt to slight changes +in the channel. +0 +4 +8 +12 +16 +The number of input video frames +0 +100 +200 +300 +400 +500 +600 +Latency / ms +Theory, Resnet-18, 2.8G +Experiment, Resnet-18, 2.8G +Fitting, Resnet-18, 2.8G +Theory, Resnet-34, 2.8G +Experiment, Resnet-34, 2.8G +Fitting, Resnet-34, 2.8G +Theory, Resnet-18, 2.2G +Experiment, Resnet-18, 2.2G +Fitting, Resnet-18, 2.2G +m +m +c,0=16.6 +c,1=40.0 +c,0=20.0 +c,1=48.1 +m +m +m +m +c,0=26.1 +c,1=79.7 +Fig. 3. +The theoretical delay curve, the experimental delay curve and the +fitted curve corresponding to the experimentalal delay. Resnet-18 and Resnet- +34 are two classic neural network architectures. The frequency of the CPU is +2.8G and 2.2G. +C. Algorithm Computational Complexity Analysis +In this part, we analyze the computational complexity of +proposed algorithms. First, the complexity of solving problem +PN0 is O(|N0|). Second, as mentioned above, the complexity +of Algorithm 1 is O(� +n∈N1 |Mopt +n |), and the complexity of +Algorithm 2 is O((3|N1|)3.5) by the interior-point method +according to [33]. When we use Algorithm 1 for solving +PN1 and use Search-based algorithm for optimizing offloading +policy, the computational complexity is O(2N � +n∈N |Mopt +n |). +When we use Algorithm 1 for solving PN1 and use Algorithm +3 for optimizing offloading policy, the computational com- +plexity is O(N � +n∈N |Mopt +n |). In addition, the computational +complexity of Algorithm 2 for solving PN1 and Algorithm 3 +for optimizing offloading policy is O(N 4.5). For the ADMM- +based algorithm, as the complexity of each steps is O(N), the +overall complexity of one iteration is O(N). +VI. NUMERICAL RESULTS +In this section, we evaluate the performance of the proposed +algorithms via simulations. For all the simulation results, +unless specified otherwise, we set the downlink bandwidth +as Bw = 5 MHz and the power spectral as N0 = −174 +dBm/Hz [24]. According to [17], the path loss is modelled +as PL = 128.1 + 37.6 log10(D) dB, where D is the dis- +tance between the device and the BS in kilometres. Devices +randomly distributed in the area within [500m 500m]. The +computational resource of the MEC server and devices are +set to be 1.8 GHz and 22 GHz, respectively. The recognition +accuracy requirement and the maximum number of input video +frames are set to αn = 0.86 and M max +n += 16, respectively. +The coefficient κ is determined by the corresponding device +and is set to be 10−28 in this paper according to [24]. The size +of the input video is 112∗112∗Mn. In addition, the coefficient +of computational complexity ρ is set to be 0.12 cycle/MAC, +which is obtained through several experiments in Sec.VI-A. +Weights β1, β2, β3 are set to be 0.2, 0.2, 0.6, respectively. +A. Model Verification +First, we obtain the complexity coefficient through experi- +mental measurement. The calculation method of the compu- +tational complexity coefficient is as follows. First, calculate + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +10 +0 +4 +8 +12 +16 +The number of input video frames +0.4 +0.55 +0.7 +0.85 +1 +Accuracy +Fitted curve, gesture, resnet-101 +Fitted curve, gesture, resnet-18 +Fitted curve, action, resnet-101 +Experiment, gesture, resnet-101 +Experiment, gesture, resnet-18 +Experiment, action, resnet-101 + a,0=0.910 +a,1=1.400 +a,2=0.993 + a,0=0.910 +a,1=1.400 +a,2=0.993 +m +m +m + a,0=0.707 +a,1=0.939 +a,2=0.990 +m +m +m +m +m +m +Fig. 4. +The experimental and fitted curves of gesture recognition task and +action recognition task. +the MACs of the DNN model when the number of input +video frames is different, recorded as {C}. We use the Flops +Counter tool [34] for MACs calculation. Second, execute 100 +times of inference tasks with a different number of input +video frames, and record the average inference delay as {t}. +Finally, calculate the coefficients between the inference delay +and MACs by ρ = sum({C}) +sum({t}) . We use Intel(R) Xeon(R) E5- +2630 CPU for testing. We use the Resnet-18 and the Resnet-34 +for testing and limit the maximum frequency of the CPU to +2.8G and 2.2G. Fig. 3 shows the theoretical (MAC-based) and +experimental delay curves and the fitted curve corresponding +to the experimental delay. We can observe from Fig. 3 that the +theoretical delay is similar to the experimental delay, proving +that MACs can be modelled as computational complexity. +We also find that the linear fitted curve can approximately +represent the computational complexity with 9 ms root mean +square error (RMSE) for Resnet-18 and 2.8G, 17 ms RMSE +for Resnet-34 and 2.8G, and 11 ms RMSE for Resnet-18 +and 2.2G. The inference delay is associated with the number +of input frames, DNN model’s architecture and the device’s +capabilities. In addition, the computational complexity co- +efficients under the three conditions are 0.128, 0.122, and +0.123, respectively. Therefore, in following experiments, we +set ρ = 0.12 cycle/MAC. +We select the gesture and action recognition tasks to verify +the accuracy model. We use the Jester datasets [35], the +largest publicly available hand gesture dataset, to test the +gesture recognition task. For the action recognition task, we +use Kinetics-400 datasets [36]. We choose Resnet-18 and +Resnet-101 for testing. As shown in Fig. 4, Under different +tasks and different network models, the accuracy curve all +conforms to the characteristics of a non-decreasing function. +What’s more, as the number of input frames increases, the +performance gain of accuracy will gradually decrease. This +is because the information gain introduced in the temporal +domain decreases when the number of input frames increases. +The fitted curve can approximately represent the relationship +between the accuracy and the number of input frames. In +the gesture recognition task with the Resnet-101 model, the +gesture recognition task with the Resnet-18 model, and the +action recognition task with the Resnet-101 model, the RMSE +are 0.0054, 0.0048 and 0.0095, respectively. We take the +4 +12 +20 +28 +36 +The number of devices +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +Average cost +Local +Edge +Random +CCCP +ADMM +GP+Heuristic +Performance bounds +Performance loss +CCCP: 2.1% +ADMM: 0.24% +GP+Heuristic: 0.03% +Fig. 5. The average cost of proposed schemes and baseline schemes under a +different number of devices. +Resnet-18 and the gesture recognition task as examples for +the following experiments. +B. Simulation Results of Average Cost +In this section, we compare proposed schemes and some +baseline schemes. We run 100 tests and can calculate the +average cost of each device and the average running time of +each test. We compare the following schemes. +1) Search+Search: We use the Search-based algorithm +to solve PN1 and use the heuristic algorithm to optimize +offloading policy. +2) Search+Heuristic: We use the Search-based algorithm +to solve PN1 and use the Search-based algorithm to optimize +offloading policy. +3) GP+Heuristic: We use the GP-based algorithm to solve +PN1 and use the Channel-Aware heuristic algorithm to opti- +mize offloading policy. +4) ADMM: We use the ADMM-based algorithm to solve +the original problem. +5) CCCP [37]: We use the concave-convex procedure +(CCCP) algorithm to decide whether to offload inference tasks +to edge servers. Then we use Theorem 1 and the GP-based +algorithm for resource allocation. +6) Random: All inference tasks are randomly executed on +local or the edge server. We use Theorem 1 and the GP-based +algorithm for resource allocation. +7) Local: All inference tasks are executed locally. We use +Theorem 1 for local resource allocation. +8) Edge: All inference tasks are executed on the edge +server. We use the GP-based algorithm for resource allocation +In Fig. 5, we plot the average cost of different schemes +under different devices. The Search+Heuristic scheme and +Search+Search scheme have the same performance, represent- +ing the performance bounds. When the number of devices +exceeds 16, the performance bounds are not shown due to +their unacceptable computational complexity. It can be seen +from Fig. 5 that the proposed schemes are better than the +baseline schemes. Compared with the performance bounds, the +performance of the GP+Heuristic scheme has a slight decrease +due to the relaxation of the accuracy function Φ(Mn). The +performance of the ADMM scheme is worse than that of the +GP+Heuristic scheme, and is better than that of the CCCP + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +11 +2 +6 +10 +14 +18 +22 +The number of devices +0 +50 +100 +150 +200 +250 +Running time / s +ADMM +GP+Heuristic +CCCP +Search+Heuristic +Search+Search +Fig. 6. The average running time of proposed algorithms under a different +number of devices. +scheme. For example, when the number of devices is 16, the +CCCP, ADMM, and GP+Heuristic schemes have performance +losses of 2.1%, 0.24%, and 0.03%, respectively, compared +with performance bounds. Moreover, when the number of +devices is less than 8, the cost of the scheme that executes +tasks only at the edge is almost equal to the cost of the +proposed GP+Heuristic scheme. It is because all devices can +benefit from performing inference on the edge server when +the number of devices is small. If the inference task is only +executed locally, the average cost of the device will not change +because the local resources among the equipment do not affect +each other. +In Fig. 6, we plot the average running time of different +schemes under different devices. When the number of de- +vices exceeds 6, the running time of the Search+Heuristic +and Search+Search scenarios becomes unacceptable. The +GP+Heuristic scheme improves the solution efficiency. The +running time of GP+Heuristic is shorter than that of CCCP +scheme. However, the complexity of the solution remains +unsatisfactory as the number of devices increases. As for the +ADMM-based scheme, since the ADMM-based algorithm is +a distributed algorithm and the complexity of updating global +variables is much smaller than that of updating local variables, +we only consider the average running time for each device. +The average running time of the ADMM-based scheme does +not improve as the number of devices increases. It is worth +noting that in the ADMM-based scheme, the iteration stops +when |Fi − Fi+1| < δ, where δ = 10−5. Threshold-based +stopping conditions result in a different number of iterations +in different cases. When the number of devices is different, +the average number of iterations is also different, resulting in +different running times. Therefore, the average running time +of 18 devices is shorter than that of 14 and 22 devices. +Assuming that the ADMM-based scheme iterates once every +time an inference task is performed, we plot the curve corre- +sponding to the cost function and the number of iterations. +As shown in Fig.7, the ADMM-based scheme can converge +to acceptable performance after completing 3-5 iterations. As +the number of iterations increases, the performance will be +closer to the optimal performance. It shows that the ADMM +algorithm can converge through online iterations. We also test +the running time per iteration on each device, and it takes an +average of about 278ms. +0 +5 +10 +15 +20 +The number of iterations +-0.5 +0 +0.5 +1 +1.5 +2 +Cost +ADMM +Performance bounds +Fig. 7. +The curve corresponding to the cost function and the number of +iterations. +TABLE I +DELAY, ENERGY CONSUMPTION, AND ACCURACY OF LOCAL DEVICES +AND EDGE DEVICES +Local devices +Edge devices +Number of devices +12.3 +12.7 +Average delay +0.24 s +0.52 s +Average energy +1.00 J +0.025 J +Average accuracy +0.886 +0.866 +C. Simulation Results of Delay, Energy, and Accuracy +This section compares the average delay, energy consump- +tion, accuracy, and the offloading rate (the proportion of de- +vices that perform inference on the edge server). We consider +the different number of devices, bandwidths, edge computing +resources, and weights β1, β2, β3. We use the GP+Heuristic +scheme for testing. Table. I shows a comparison of devices +that finish inference locally and devices that finish inference +at the edge under default experimental settings. On average, +12.7 devices choose to offload to the edge server to perform +inference. Compared with edge devices, local devices have a +lower delay and higher accuracy but have greater inference +energy consumption. +Fig. 8 shows the average delay, energy, accuracy, and +offloading rate under different numbers of devices, different +bandwidths, and different edge computing resources. In Fig. +8(a), we plot results with different numbers of devices. As +shown in Fig. 8(a), when the number of devices is small (less +than 10), all devices offload the task to the edge server (the +offloading rate is equal to 1). For edge devices, all delay comes +from transmission delay and the edge inference delay, and all +energy consumption comes from transmission energy. With +the number of devices increasing, communication resources +and the edge server’s computation resources are shared by +more devices, decreasing the number of input frames Mn. A +decrease in the number of input frames results in a decrease +in accuracy. Then as Mn decreases, the transmission data size +decreases, and the transmission energy decreases. Meanwhile, +Competition from more devices leads to increased delays. +Therefore, when the number of devices is small (less than +10), with the number of devices increasing, the average delay +increases, the average accuracy and the average energy con- + +IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +12 +5 +15 +25 +35 +45 +The number of devices +0 +0.2 +0.4 +0.6 +0.8 +1 +Delay /s, Energy / J, Offloading rate +0.8 +0.85 +0.9 +0.95 +1 +Accuracy +Delay +Energy +Offloading rate +Accuracy +(a) Different number of devices +2 +6 +10 +14 +18 +Bandwidth / MHz +0 +0.2 +0.4 +0.6 +0.8 +1 +Delay /s, Energy / J, Offloading rate +0.8 +0.85 +0.9 +0.95 +1 +Accuracy +Delay +Energy +Offloading rate +Accuracy +(b) Different bandwidth +6 +14 +22 +30 +38 +Edge computing resource / GHz +0 +0.2 +0.4 +0.6 +0.8 +1 +Delay /s, Energy / J, Offloading rate +0.8 +0.85 +0.9 +0.95 +1 +Accuracy +Delay +Energy +Offloading rate +Accuracy +(c) Different edge computing resource +Fig. 8. The average delay, energy, offloading rate, and accuracy under different +numbers of devices, different bandwidths, and different edge computing +resources. +sumption decrease. When the number of devices exceeds 10, +the average energy consumption and accuracy increase, and the +average delay and offload rate gradually decrease. Considering +different bandwidths and different edge computing resources, +we plot Fig. 8(b) and Fig. 8(c). In Fig. 8(b) and Fig. 8(c), as +the bandwidth and edge computing resource increase, devices +will be more inclined to offload computing to the edge, which +increases the offloading rate. According to Table. I, when +β1, β2 and β3 are fixed, edge devices have lower energy +consumption, lower accuracy and higher delay. More edge +devices mean a greater delay and lower power consumption. +Meanwhile, when the bandwidth increases, since the edge +computing resources are fixed, the number of video frames +will decrease to reduce edge computing overhead, resulting +in a decrease in accuracy. The same conclusion can also be +obtained when edge computing resources increase. Therefore, +Fig. 9. The relationship between the delay, energy consumption, and accuracy. +with the increase of bandwidth and edge computing resources, +more edge devices lead to increased delay and decreased +energy and accuracy. +We set the minimum number of input frames M min +n += 1. +We use different weights, β1, β2, β3 to study the trade-off +relationship between the average delay, energy consumption, +and accuracy. The constraint is β1 + β2 + β3 = 1. The perfor- +mance of the trade-off surface is obtained by the GP+Heuristic +scheme. Fig. 9 shows the delay, energy consumption, and +accuracy are mutually limited. Higher energy consumption +leads to higher accuracy when the delay is constant. From +another perspective, in order to improve the accuracy, it is +necessary to sacrifice the performance of delay and energy +consumption. In addition, with the same accuracy, according +to Table. I, higher energy consumption will make the device +more inclined to execute inference tasks locally, and the delay +decreases. +VII. CONCLUSION +This paper considers optimizing video-based AI inference +tasks in a multi-user MEC system. An MINLP is formulated +to minimize the total delay and energy consumption, and +improve the total accuracy, with the constraint of computation +and communication resources. A MAC-based computational +complexity model is introduced to model the calculation delay, +and a simple approximate expression is proposed to simplify +the problem. We also propose a general accuracy model to +characterize the relation between the recognition accuracy and +the number of input frames. After that, we first assume that the +offloading decision is given and decouple the original problem +into two sub-problems. The first sub-problem is to optimize the +resources of the devices that complete the DNN inference tasks +locally. We derive the closed-form solution to this problem. +The second sub-problem is optimizing the devices’ resources +that offload the DNN inference tasks to the edge server. We +propose the Search-based and GP-based algorithm to solve +the second sub-problem. For the problem of offloading de- +cision optimization, we propose the Channel-Aware heuristic +algorithm. We also propose a distributed algorithm based on +ADMM. The ADMM-based algorithm reduce computational +complexity at the cost of an acceptable performance loss. +Numerical simulation and experimental results demonstrate + +1 +0.9Accurac +0.8 +0.7 +0.6 +2 +1.5 +1 +Energy / J +0.5 +0.5 +0 +02 +1.5 +1 +Delay / sIEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. XX, NO. XX, XXX 2022 +13 +the effectiveness of the proposed algorithm. We also provide +a detailed analysis of the delay, energy consumption, and +accuracy for different device numbers, bandwidths and edge +computing resources. +APPENDIX A +PROOF OF THEOREM 1 +The partial derivative of FPN0 with respect to f md +n +is, +∂FPN0 +∂f md +n += −β1 +ρC(Mn) +f md2 +n ++ 2β2κρC(Mn)f md +n , +(47) +By setting +∂FPN0 +∂f md +n += 0, we have, +f md +n += +3 +� +( β1 +2β2κ), +(48) +Therefore, f md +n +decreases monotonically in the interval +(−∞, 3� +( β1 +2β2κ)) and increases monotonically in the interval +( 3� +( β1 +2β2κ), +∞). Considering the value range of f md +n , the +optimal solution can be given by, +f md∗ +n += min{ +3 +� +( β1 +2β2κ), f max +n +} +(49) +Then we analyze Mn. The partial derivative of FPN0 with +respect to Mn is, +∂FPN0 +∂Mn += β1ρmc,0 +f md +n ++ β2κρmc,0f md2 +n +− +β3ma,0 +(Mn + ma,1)2 , +(50) +By setting +∂FPN0 +∂Mn += 0, we have, +Mn = +� +β3ma,0 +β1ρmc,0 +f md +n ++ β2κρmc,0f md2 +n +− ma,1, +(51) +Considering the value range of Mn, the optimal solution can +be given by, +M ∗ +n = min{max{ +� +β3ma,0 +β1ρmc,0 +f md +n ++ β2κρmc,0f md2 +n +− ma,1, M min +n +}, M max +n +} +(52) +which completes the proof. +APPENDIX B +PROOF OF THEOREM 2 +According to the KKT conditions, we can obtain the fol- +lowing necessary and sufficient conditions, +∂LPN1 +∂f e∗ +n += −β1ρC(M ∗ +n) +f e∗2 +n ++ u∗ +1 = 0, f e∗ +n > 0, +(53) +∂LPN1 +∂t∗n += −β1M ∗ +nd +Rnt∗2 +n ++ u∗ +0 = 0, t∗ +n > 0, +(54) +µ∗ +0( +� +n∈N ∗ +t∗ +n − 1) = 0, +(55) +µ∗ +1( +� +n∈N ∗ +f e∗ +n − f max) = 0, +(56) +µ∗ +0, µ∗ +1 ≥ 0. +(57) +Because β1ρC(M ∗ +n) +f e∗2 +n +and β1M ∗ +nd +Rnt∗2 +n +are positive, µ∗ +0 and µ∗ +1 are +also positive. We can obtain, +� +n∈N +f e∗ +n − f max = 0, +(58) +� +n∈N +t∗ +n − 1 = 0, +(59) +f e∗ +n = +� +β1ρC(M ∗n) +Rnµ∗ +1 +, +(60) +t∗ +n = +� +β1M ∗nd +Rnµ∗ +0 +. +(61) +Combining (58) and (60), we can get the expression of f e∗ +n +corresponding to M ∗ +n, +f e∗ +n = f max� +C(M ∗n) +� +i∈N1 +� +C(M ∗ +i ) +. +(62) +Similarly, combining (59) and (61), we can get the expression +of t∗ +n corresponding to M ∗ +n, +t∗ +n = +� +M ∗ +n +Rn +� +i∈N1 +� +M ∗ +i +Ri +, +(63) +which completes the proof. +REFERENCES +[1] H. Ning, H. Wang, Y. Lin, W. Wang, S. Dhelim, F. Farha, J. Ding, and +M. Daneshmand, “A survey on metaverse: the state-of-the-art, technolo- +gies, applications, and challenges,” arXiv preprint arXiv:2111.09673, +2021. +[2] J. Li, L. Deng, Y. Gong, and R. Haeb-Umbach, “An overview of noise- +robust automatic speech recognition,” IEEE/ACM Trans. Audio, Speech, +Lang. Process., vol. 22, no. 4, pp. 745–777, 2014. +[3] D. W. Otter, J. R. Medina, and J. K. 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Technol., vol. 69, no. 2, pp. 2246–2262, +2020. + diff --git a/99AzT4oBgHgl3EQfFfoM/content/tmp_files/load_file.txt b/99AzT4oBgHgl3EQfFfoM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8562f42eaa6357d910b0cdec4a1704575a77d818 --- /dev/null +++ b/99AzT4oBgHgl3EQfFfoM/content/tmp_files/load_file.txt @@ -0,0 +1,1195 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf,len=1194 +page_content='IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 1 Joint Optimization of Video-based AI Inference Tasks in MEC-assisted Augmented Reality Systems Guangjin Pan, Heng Zhang, Shugong Xu, Fellow, IEEE, Shunqing Zhang, Senior Member, IEEE, and Xiaojing Chen Abstract—The high computational complexity and energy consumption of artificial intelligence (AI) algorithms hinder their application in augmented reality (AR) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, mobile edge computing (MEC) makes it possible to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' This paper considers the scene of completing video-based AI inference tasks in the MEC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We formulate a mixed-integer nonlinear programming problem (MINLP) to reduce inference delays, energy consumption and to improve recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We give a simplified expression of the inference complexity model and accuracy model through derivation and experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The problem is then solved iteratively by using alternating opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Specifically, by assuming that the offloading decision is given, the problem is decoupled into two sub-problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', the resource allocation problem for the devices set that completes the inference tasks locally, and that for the devices set that offloads tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For the problem of offloading decision optimization, we propose a Channel-Aware heuristic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To further reduce the complexity, we propose an alternating direction method of multipliers (ADMM) based distributed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The ADMM- based algorithm has a low computational complexity that grows linearly with the number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Numerical experiments show the effectiveness of proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The trade-off relationship between delay, energy consumption, and accuracy is also analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Index Terms—Mobile augmented reality, edge intelligence, mobile edge computing, resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' INTRODUCTION R ECENTLY, the development of networks, cloud com- puting, edge computing, artificial intelligence, and other technologies has triggered people’s infinite imagination of the Metaverse [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To enable users to interact between the real world and the virtual world, augmented reality (AR) technology plays a vital role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' At the same time, artificial intelligence (AI), due to its learning and inference capabilities, has demonstrated a powerful ability in many fields such as automatic speech recognition (ASR) [2], natural language G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Pan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Zhang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Chen are with Shanghai Institute for Advanced Communication and Data Science, Shanghai University, Shang- hai 200444, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Emails: {guangjin_pan, hengzhang, shugong, Shunqing, jodiechen}@shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Part of this work has been accepted by Globecom-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61871262,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 62071284,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and 61901251,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' the National Key R&D Program of China grants 2017YFE0121400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 2019YFE0196600 and 2022YFB2902000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' the Innovation Program of Shanghai Municipal Science and Technology Commission grants 20JC1416400 and 21ZR1422400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Pudong New Area Science & Technology Development Fund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Key-Area Research and Development Program of Guangdong Province grant 2020B0101130012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Foshan Science and Technology Innovation Team Project grant FS0AA- KJ919-4402-0060,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and research funds from Shanghai Institute for Advanced Communication and Data Science (SICS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The corresponding author is Shugong Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' processing (NLP) [3], computer vision (CV) [4], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' With the assistance of AI technology, AR can carry out deeper scene understanding and more immersive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, the computational complexity of AI algorithms, especially deep neural networks (DNN), is usually very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It is challenging to complete DNN inference timely and reli- ably on mobile devices with limited computation and energy capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In [5], experiments show that a typical single-frame image processing AI inference task takes about 600 ms even with speedup from the mobile GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, continuously executing the above inference tasks can only last up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 hours on commodity devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The above issues result in only a few AR applications currently using deep learning [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In order to reduce the inference time of DNNs, one way is to perform network pruning on the neural network [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, it could be destructive to the model if pruning too many channels, and it may not be possible to recover a satisfactory accuracy by fine-tuning [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Edge AI [9]–[11] is another approach to solving these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Integrating mobile edge computing (MEC) and AI technology has recently become a promising paradigm for supporting computationally intensive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Edge AI transfers the inference and training process of AI models to the edge of the network close to the data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, it can alleviate network traffic load, delay, and privacy problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Related Works Many existing studies use MEC’s powerful computing capa- bilities to reduce delay [12], energy consumption [13], or both delay and energy consumption [14]–[16] through offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For example, [12] formulated an optimization problem aimed at minimizing the processing delay of eMBB and mMTC users by optimizing the users’ transmit power in UAV-Assisted MEC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [13] develops a smart pricing mechanism to coordinate the computation offloading of multi-layer devices and reduces energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [14] uses the Stackelberg game method to optimize the task allocation coefficient, calcu- lation resource allocation coefficient, and transmission power to minimize the energy consumption and delay of the NOMA- based MEC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For edge AI inference, existing research has made some progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The authors in [17] propose a framework for jointly optimizing inference task selection and downlink coordinated beamforming to minimize communication power consumption in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Similarly, [18] proposes an IRS-assisted edge inference system and designs a task selection strategy arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='01010v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='NI] 3 Jan 2023 IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 2 to minimize the energy consumption of uplink and downlink transmission and calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The work in [19] analyzes and models the transmission error probability, inference accu- racy, and timeout probability of the AI-powered time-critical services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The work in [20] uses a tandem queueing model to analyze queueing and processing delays of DL tasks in multiple DNN partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [21] joint optimizes the service placement, computational and radio resource allocation to minimize the users’ total delay and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [8] combines model pruning and DNN partitioning to achieve a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='81x reduction on end-to-end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [22] designs the Edgent framework that can jointly optimize DNN partitioning and DNN right-sizing to maximize the inference accuracy while promising application delay requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' These studies mea- sure the inference time by experiments [8], [22] or assume that the inference task’s computational complexity is proportional to the input data size but without derivation and proof [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, these models of computational complexity are not rigorous enough or can not be generalized to different neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As for the accuracy model, the authors in [23] designs an edge network orchestration algorithm named FACT, which boosts the performance of an edge-based AR system by opti- mizing the edge server assignment and video frame resolution selection for AR users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, [23] builds an accuracy model by fitting an accuracy curve for specific tasks, which is not general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The work in [24] compresses image resolution locally and performs inference tasks on edge servers, aiming to maximize learning accuracy under constraints of delay and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [24] proposes using an abstract non-decreasing function to describe the relationship between accuracy and input image size, which cannot be used to analyze various AI inference tasks discriminately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Joint optimization is required when different tasks and models are jointly deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' An insufficiently generalized accuracy model or an overly abstract model can adversely affect joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A general accuracy model is needed to measure various AI tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Among the above studies, most studies consider optimizing one or two performance metrics among the delay, energy con- sumption, and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The authors in [24] jointly considers delay, energy consumption and accuracy in image recognition scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, it aims at maximizing computational capacity under constraints of delay, energy consumption and accuracy, and the DNN model is only deployed in edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In [6], [23], [25], video analytics scenarios are considered, but they do not jointly consider delay, energy and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Contributions and Organizations In this paper, we consider a multi-user MEC system and assume that each device executes the video-based DNN in- ference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Each device can be AR glasses, mobile robots, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In order to deepen AR’s ability to understand the scene, we need to use time dimension information to improve perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, we consider video-based application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='for video-based AI inference tasks, there are two modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', frame-by-frame recognition mode (the input for each recognition is one frame) and multi-frame recognition � Uplink for real-time captured content Downlink for inference results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Multi-user MEC System model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The inference task can be executed on the local or the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When the task is offloaded to the edge server, the uplink transmits the content captured in real-time, and the downlink transmits the inference result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' mode (the input for each recognition is multiple frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The frame-by-frame inference mode is used to deal with tasks with weak temporal correlation, such as face recognition and target tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', and has been studied in [6], [23], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this paper, we focus on multi-frame recognition tasks, such as gesture recognition and action recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Since sampling in the spatial dimension brings extra computation [24], we only sample in the temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' At each inference, the device selects the most recent several frames from the history frames for transmission or inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 1, mobile devices can transmit captured video to the edge server via wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The edge servers execute inference tasks and send results back to mo- bile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, when communication and computing resources of the edge server are insufficient, devices can execute the inference task locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We model the problem as a multi-objective optimization problem to optimize delay, energy consumption, and inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows, Multi-dimensional target optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' High accuracy, low delay, and low energy consumption are indispensable for AR applications and must be optimized jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To explore the trade-off relationship between delay, energy, and accuracy, we formulate the video-based offloading problem as a mixed-integer nonlinear programming prob- lem (MINLP), aiming to reduce service delays, energy consumption and improve recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' General computational complexity and accuracy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To measure the computational complexity of neural net- work models with different architectures and different input sizes, we introduce the number of multiply-and- accumulate operations (MACs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We illustrate the main factors affecting DNN inference delay through experi- ments and show that MAC can be used as a good measure of the computational complexity of DNN inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also propose a general model to represent the relation- ship between the inference accuracy and the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' This model is suitable for different video- IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 3 based recognition tasks and different DNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We give simple expressions of the inference complexity and accuracy to simplify the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Channel-Aware scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To solve the opti- mization problem, we decompose the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, assuming that the offloading decision is given, we solve the resource allocation problems for the device set that completes the inference locally and the device set that offloads the tasks to the edge server, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For edge DNN inference, we propose two algorithms based on search and geometric programming (GP) to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then, to obtain the optimal offloading policy, we propose a Channel-Aware heuristic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The original problem is solved iteratively through alternating optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ADMM-based distributed resource allocation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To avoid the high complexity of the heuristic algorithm, we propose an algorithm based on the Alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The ADMM-based algo- rithm decomposes the original problem into parallel and tractable subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the total computational complexity of ADMM-based algorithms is more scalable than the heuristic algorithm, especially when the number of devices is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Section II, we introduce system models, including delay, energy, and accuracy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Section III, we formulate the joint optimization problem and convert the original problem to a more tractable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Section IV proposes a Channel-Aware heuristic algorithm to solve the proposed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Section V, we propose another ADMM-based distributed resource allocation algorithm for the proposed problem, and analyze the computational complexity of the solution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Numeri- cal results and analysis are presented in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Finally, the paper is concluded in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' SYSTEM MODEL In this section, we introduce a single-cell MEC system and establish delay, energy consumption, and accuracy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 1, we consider a multi-user MEC system with one base station (BS) and N mobile devices, denoted by the set N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Each device has a camera and needs to accomplish DNN inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Due to the limitation of device computational resources, DNN inference tasks can be placed on local or edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The limited computational resource will lead to longer computing delay and greater power consumption when the inference task is executed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, when the inference task is executed on the edge server, it will bring additional wireless transmission delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, accuracy is also a very important optimization target in DNN inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Offloading Framework In this paper, we only consider the binary offloading method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Binary offloading requires the DNN inference task to be fully executed either at the device or the MEC server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The overview of the DNN computing offloading system is depicted Sampling management module Video offloading Result feedback Complete inference task Execute locally Complete inference task Complete inference task Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The overview of the video sampling and computing offloading system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The video sampling management module can control the sampling rate of the captured video and determine the number of video frames used for AI inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Devices can transmit the video to the edge server or perform inference tasks locally based on the wireless channel information and computing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, devices sample the video captured in real- time in the temporal dimension to obtain a short video with a certain number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, the DNN inference tasks are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' These inference tasks can be executed locally on devices or the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, each device’s video sampling management module needs to select an appropriate video sampling rate (how many frames need to be input) and choose whether to offload the task to the MEC server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Denote Dn, En and φn to be the total delay, energy consumption and recognition accuracy of the device n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The total delay and energy consumption of the device n can be given by, Dn = (1 − xn)Dmd n + xn(Dt n + De n), (1) En = (1 − xn)Emd n + xnEt n, (2) where xn indicates whether the inference task is executed on local or edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Dt n is the transmission delay for uplink, Dmd n is the local inference delay, and De n is the delay for completing inference at the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Et n and Emd n are the transmission and computational energy consumption, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The delay and energy consumption for downloading computation results can be reasonably neglected because of the results’ small data sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Delay and Energy Models for Inference The inference delay depends on the DNN model’s architec- ture, the device’s or server’s computing power, and the input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this section, we first give a measure of the computational complexity of the DNN model and then give an expression for the inference delay and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Different AI recognition tasks may require different AI model architectures, including classic AI models such as Resnet-18, Resnet-34, Resnet-50, VGG-16, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In order to optimize AI inference tasks more reasonably, different AI models need a common method to evaluate computational IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 4 complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this paper, we use the number of MACs [28] to measure the computational complexity of AI inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' MACs calculation methods of layers (such as fully connected (FC) layers, convolutional layers and so on) can be obtained in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Taking 3D Convolutional Neural Network (3DCNN) as an example, the computational complexity (measured by MACs) of the lth layer of the nth device can be expressed as, cn,l = olol+1 �2 j=0 Kj l , �2 j=0 M j n,l+1, (3) where ol is the number of input channels, ol+1 is the number of output channels, �2 j=0 Kj l is the size of the convolution kernel, and �2 j=0 M j n,l+1 is the size of the output feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' j = 0 represents the temporal dimension (the number of frames), j = 1, 2 represent spatial dimensions (pixels of one frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Note that ol, ol+1, and �2 j=0 Kj l are all determined by the neural network architecture and �2 j=0 M j n,l+1 depends on the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The relation between the output feature size and the input size can be expressed as, M j n,l+1 = M j n,l − Kj l + 2dl rl + 1, (4) where rl is the stride and dl is the padding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As mentioned above, the computational complexity of a DNN model is determined by the number of layers, the DNN model’s architecture, and the input and output size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this paper, we mainly focus on the impact of the number of input video frames Mn on recognition accuracy and the allocation of communication and computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The inference result will be more accurate with more frames Mn input, but the communication and calculation overhead will be greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The computational complexity of the nth device’s task can be expressed as C(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then we give the expression for the inference delay and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Denote f max and f max n (in CPU cycle/s) to be the total computation resource of the edge server and mobile device n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Let f e n and f md n (in CPU cycle/s) denote the computation resource to device n allocated by the edge server and the device, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the computing resources satisfy � n∈N f e n ≤ f max and f md n ≤ f max n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The computation delay of the device n and MEC can be respectively expressed as, Dmd n = ρC(Mn) f md n , (5) De n = ρC(Mn) f en , (6) where ρ (cycle/MAC) represents the number of CPU cycles required to complete a multiplication and addition, which depends on the CPU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As for energy consumption, denote κ to be a coefficient determined by the corresponding device [24], and the com- putational energy consumption of device n can be expressed as, Emd n = κρC(Mn)f md n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (7) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Delay and Energy Models for Transmission We consider a time-division multiple access (TDMA) method for channel access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Specifically, each radio frame is divided into N time slots for transmission, and each device can only transmit in its own time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We assume that the length of each radio frame is ∆T, which is short enough (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', 10 ms in LTE or NR system [24]), and the length of a time slot is ∆Ttn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Denote hn and pn to be the channel gain and transmission power of the device n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' According to [21], the achievable data rate of device n can be expressed as, Rn = Bwlog2 � 1 + pnhn BwN0 � , (8) where Bw and N0 are the bandwidth and the variance of additive white Gaussian noise (AWGN), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Let d denote the data size of one video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Since we only want to analyze the impact of time dimension information (the number of input frames Mn) on recognition accuracy, d is a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In each radio frame, the data size that can be transmitted is ∆TRntn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, for each transmission, ⌈ Mnd ∆T Rntn ⌉ radio frames are required, where ⌈·⌉ means the ceil function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering that the length of the radio frame is much shorter than the transmission delay, the transmission delay for offloading to MEC can be written as, Dt n = ⌈ Mnd ∆TRntn ⌉∆T ≈ Mnd Rntn , (9) where tn is the proportion of time that device n transmits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, according to [24], the energy consumption of each device to transmit its video can be expressed as, Et n = Mnd Rn pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (10) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Inference Tasks Accuracy Model As mentioned above, we mainly focus on the impact of the number of input video frames Mn on recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We assume that the quality of the input video is the same for different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For a certain task and DNN model, the accuracy is only determined by the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the accuracy of device n can be expressed as φn = Φ(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' According to [29], more frames will lead to better inference accuracy, and as the input frames continue to increase, the performance gain will gradually decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Some prior studies also show that the relationship between frame rate and accuracy can be expressed as concave functions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, we define Φ(Mn) as a monotone non-decreasing function to describe the relationship between the accuracy and the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we formulate the optimization problem to reduce the system’s delay and devices’ energy consumption and improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We analyze the difficulty of solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To simplify the problem, we make a reasonable conversion of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Original Problem Formulation Based on the above analysis, combining (1), (2), (5)-(7), (9), and(10), the nth device’s delay and energy consumption can be expressed as, Dn = (1 − xn)ρC(Mn) f md n + xn(ρC(Mn) f en + Mnd Rntn ), (11) En = (1 − xn)κρC(Mn)f md n 2 + xn(Mnd Rn pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (12) Given the system model described previously, our goal is to reduce end-to-end delay and energy consumption and improve recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Each device follows the binary offloading policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The mathematical optimization problem of the total cost (delay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' energy consumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and accuracy) can be expressed as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem P1 (Original Problem): minimize {Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='f e n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='xn} � n∈N � β1Dn + β2En − β3Φ(Mn) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13) subject to Φ(Mn) ≥ αn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13a) Mn ≤ M max n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Mn ∈ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13b) � n∈N xntn ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13c) � n∈N xnf e n ≤ f max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13d) tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' f e n ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13e) 0 ≤ f md n ≤ f max n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13f) xn ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 1} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13g) where αn represents the recognition accuracy requirement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' β3 are the weight factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13a) represents the recognition accuracy requirement of each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13b) indicates the frame limit for the input video, Z is the set of integers, and M max n is the maximum number of frames of the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13c) and (13d) represent the communication and computation resource limitation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (13f) limits the computation resource of each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The optimization variables in original problem P1 are the number of input video frames Mn, the proportion of transmission time tn, the local computation resource f md n , the edge computation resource allocation f e n, and the offloading decision xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, the first item in (13) is to reduce the total delay of computation and transmission, the second item is to reduce the device’s energy consumption, and the last item is to improve the number of input video frames as well as the recognition accuracy because of the monotone non-decreasing function Φ(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem P1 is a non-convex MINLP problem and is difficult to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, the complexity function C(Mn) is discrete and depends on the architecture of the DNN and the size of the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As the number of input frames Mn increases, the computational complexity also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' This kind of increase is irregular because it is affected by the structure of DNN layers, such as the stride and padding size of 3DCNN according to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, C(Mn) cannot be used for optimization directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, as mentioned above, the accuracy function Φ(Mn) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, we cannot give a deterministic expression for Φ(Mn), so we can not optimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, both Mn and xn are integers, making the problem difficult to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem Conversion To make the problem P1 more tractable, we convert the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, we give an approximate expression of the computational complexity function C(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' According to (3) and (4), the computational complexity of 3DCNN layers is proportional to the size of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can also obtain a similar conclusion in other types of layers, such as the FC layer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Based on the above conclusion and combined with the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' VI-A, in order to simply express the computational complexity model, C(Mn) can be written as, C(Mn) = mc,0Mn + mc,1, (14) where mc,0 ≥ 0 and mc,1 are constants and depend on the network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, we propose a general model to express the re- lationship between the accuracy and the number of input video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering that the function Φ(Mn) is mono- tonically non-decreasing and that as the number of input frames increases, the accuracy gain decreases, combining our experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' VI-A, we model function Φ(Mn) as, Φ(Mn) = − ma,0 Mn + ma,1 + ma,2, (15) where ma,0 ≥ 0, ma,2 ≥ 0 and ma,1 > −1 are constants and depend on the target of inference tasks and the architecture of DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Finally, we relax the range of the variable Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Consid- ering that Φ(Mn) is a monotone non-decreasing function and depends on the recognition task and network archi- tecture, in order not to lose generality, define M min n = arg minMn Φ(Mn), Φ(Mn) ≥ αn, Mn ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can also relax Mn into a closed connected subset of the real axis, and (13a), (13b) can be written as Mn ∈ � M min n , M max n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then [Mn] can be regarded as the number of input video frames, where [·] indicates rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We define two sets of devices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' N0 = {n | xn = 0, n ∈ N} and N1 = {n | xn = 1, n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' F0,n and F1,n are the cost function of the device n in sets N0 and N1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The problem P1 can be rewritten as, Problem P2 (Converted Problem): minimize {Mn,tn,f md n ,f e n,xn} � n∈N0 (1 − xn)F0,n(Mn, f md n ) + � n∈N1 xnF1,n(Mn, f e n, tn), (16) subject to Mn ∈ � M min n , M max n � , (16a) (13c) − (13g), where F0,n(Mn, f md n ) = β1 ρC(Mn) f md n + β2κρC(Mn)f md2 n − β3Φ(Mn), (17) F1,n(Mn, f e n, tn) = β1 ρC(Mn) f en + β1 Mnd Rntn IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 6 + β2 Mndpn Rn − β3Φ(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (18) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' OPTIMIZATION PROBLEM SOLVING In this section, we decompose the problem P2 and propose a Channel-Aware heuristic algorithm to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, sup- posing that the offloading decision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', {xn}) is given, we solve optimization problems for sets N0 and N1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, we propose a Channel-Aware heuristic algorithm to optimize the offloading decision {xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Optimization Problem Solving for N0 For set N0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=', when the device executes inference tasks locally, the optimization problem becomes, Problem PN0 (Problem for N0): minimize {Mn,f md n } FPN0 ≜ � n∈N0 F0,n(Mn, f md n ), (19) subject to (13f), (16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The optimization variables in PN0 are the number of input video frames Mn and the local computation resource f md n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Let {M ∗ n, f md∗ n } denote the optimal solution to PN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can derive the optimal solution to PN0 in a closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Theorem 1: The optimal solution to PN0 is given by, f md∗ n = min{ 3 � ( β1 2β2κ), f max n }, (20) M ∗ n = min{max{ � β3ma,0 β1ρmc,0 f md n + β2κρmc,0f md2 n − ma,1, M min n }, M max n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (21) Proof: Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' From Theorem 1, we can see that the optimal local CPU- cycle frequency f md n is determined by the weight factors β1, β2, the coefficient of CPU energy consumption κ, and is limited by its corresponding upper bound f max n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' More specifi- cally, f md n is proportional to β 1 3 1 and inversely proportional to β 1 3 2 and κ 1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As for the number of input video frames, when 3� ( β1 2β2κ) ≤ f max n , combining (20) and (21), we have, M ∗ n = min{max{3− 1 2 2 1 3 ρ− 1 2 κ− 1 6 m − 1 2 c,0 β − 1 3 1 β − 1 6 2 β 1 2 3 m 1 2 a,0 − ma,1, M min n }, M max n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (22) The optimization results corresponding to each device are only related to the parameters of the device itself and are not associated with the parameters of other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Optimization Problem Solving for N1 Then we solve the optimization problem of N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The prob- lem P2 can be written as, Problem PN1 (Problem for N1): minimize {Mn,f e n,tn} � n∈N1 F1,n(Mn, f e n, tn), (23) subject to (13c), (13d), (13e), (16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Algorithm 1: Algorithm 1: Search-Based Algorithm for solving PN1 Input: The offloading policy N1, the channel gain {hn}, and other system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Output: {M ⋆ n, f e⋆ n , t⋆ n} Initialize the result of cost function F⋆ � PN1 to a sufficiently large value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate the achievable data rate {Rn} using (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' foreach {Mn} ∈ M do Compute F � PN1 using (27);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' if F � PN1 < F⋆ � PN1 then {M ⋆ n} = {Mn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' F⋆ � PN1 = F � PN1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate {f e⋆ n } and {t⋆ n} using (25) and (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' return {M ⋆ n}, {f e⋆ n }, and {t⋆ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The optimization variables in the the problem PN1 are the number of input video frames Mn, the edge computation resource f e n, and the proportion of transmission time tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Let {M ∗ n, f e∗ n , t∗ n} denote the optimal solution to PN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can obtain the optimal solution to PN1 using the method of Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The partial Lagrangian function can be written as, LPN1= � n∈N1 �β1ρC(Mn) f en + β1Mnd Rntn + β2Mndpn Rn − β3Φ(Mn) � + µ0( � n∈N1 tn − 1) + µ1( � n∈N1 f e n − f max), (24) First of all, according to (24), supposing that M ∗ n is given, we can solve the problem PN1 based on the Karush-Kuhn- Tucker (KKT) condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can obtain the function expres- sions of f e∗ n and t∗ n relative to Mn, as shown in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Theorem 2: The function expressions of f e∗ n and t∗ n relative to M ∗ n are given by, f e∗ n = f max� C(M ∗n) � i∈N1 � C(M ∗ i ) , (25) t∗ n = � M ∗ n Rn � i∈N1 � M ∗ i Ri .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (26) Proof: Please refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Combining (23), (25) and (26), the problem PN1 can be written as an optimized function containing only the variable Mn as follows, Problem � PN1 (Mn Optimization Problem for N1 ): minimize {Mn} F � PN1 ≜ β1ρ f max ( � n∈N1 � C(Mn))2 + β1d( � n∈N1 � Mn Rn )2 + β2dpn( � n∈N1 Mn Rn ) − � n∈N1 β3Φ(Mn), (27) IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 7 Algorithm 2: Algorithm 2: GP-Based Algorithm for solving PN1 Input: The offloading policy N1, the channel gain {hn}, and other system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Output: {M ⋆ n, f e⋆ n , t⋆ n} Calculate the achievable data rate {Rn} using (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Use the CVX tool to solve (29) and get { ˆ M ⋆n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' {M ⋆ n} = {[e ˆ M ⋆ n]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate {f e⋆ n } and {t⋆ n} using (25) and (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' return {M ⋆ n}, {f e⋆ n }, and {t⋆ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' subject to (16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Denote Mopt n = {Mn | M min n ≤ Mn ≤ M max n , Mn ∈ Z} to be the optional video frame number of device n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The optimal solution can be obtained by searching for {Mn} ∈ M, where M = {{Mi} | Mi ∈ Mopt i , i ∈ N1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The detail of the search based algorithm is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering that the problem PN1 is convex when Mn is given, Algorithm 1 is global optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, When the number of devices grows large, the computational complexity of the Search-based algorithm will become very high or even unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this paper, we also propose a GP-based sub- optimal algorithm to solve the problem PN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, we relax the objective function of the problem PN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We introduce the function, �Φ(Mn) = − ma,0 Mn + ma,2, and PN1 can be rewritten as, Problem PGPN1 (GP-based Problem for N1): minimize {Mn,f e n,tn} � n∈N1 � β1 ρC(Mn) f en + β1 Mnd Rntn + β2 Mndpn Rn − β3�Φ(Mn) � , (28) subject to (13c), (13d), (13e), (16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It is a non-convex GP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Inspired by [30], the GP problem can be transformed into a convex problem by changing variables and transforming the objective and con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' introducing variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ Mn = ln Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en = ln f e n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn = ln tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and the problem can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem � PGPN1 (Converted GP-based Problem for N1): minimize { ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f e n} � n∈N1 � β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e ˆ Mn− ˆ f e n + β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1e− ˆ f e n + β1de ˆ Mn− ˆ tn Rn + β2dpne ˆ Mn Rn +β3ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e− ˆ Mn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29) subject to ˆ Mn ∈ � ln M min n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ln M max n � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29a) � n∈N1 xne ˆ tn ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29b) � n∈N1 xne ˆ f e n ≤ f max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29c) which is strictly convex problem that can be solved using the CVX tool [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering that Mn is an integer, the result of CVX optimization needs to be post-processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Details of the GP-based algorithm are shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Algorithm 3: Algorithm 3: Channel-Aware heuristic algorithm for Optimizing Offloading Policy {xn} Input: Parameters corresponding to the problem P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Output: Offloading policy N0 and N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate the cost function {F0,n} for the set N using (20) and (21) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Set N0 = ∅, N1 = N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate the cost function {F1,n} corresponding to the set N1 using Algorithm 1 or Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Set Flag = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' while Flag == 1 do k = argminnhn, n ∈ N1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' N ∗ 0 = N0 ∪ {k}, N ∗ 1 = N1 − {k};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Calculate the cost function {F∗ 1,n} corresponding to the set N ∗ 1 using Algorithm 1 or Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' if � n∈N0 F0,n + � n∈N1 F1,n > � n∈N ∗ 0 F0,n + � n∈N ∗ 1 F∗ 1,n then F1,n = F∗ 1,n, ∀n ∈ N ∗ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' N0 = N ∗ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' N1 = N ∗ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' else Flag = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' return N0 and N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Optimization of Offloading Policy {xn} Considering the complexity of Search-based offloading pol- icy algorithm becomes high when the number of devices N grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this section, we propose a Channel-Aware heuristic algorithm to optimize the offloading decision {xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Inspired by the Theorem 1 and Theorem 2, when executing inference locally, the cost function F0,n and optimization vari- ables f md n , Mn only depend on the device’s own parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, for edge set N1, the cost function is related to the number and parameters of devices in the set N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The Channel-Aware heuristic algorithm is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, calculate the cost function {F0,n} of set N0 when each device’s task is executed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, assuming that all devices are offloaded to the edge server for inference and |N1| = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In each iteration, the cost function {F1,n} corresponding to each device of N1 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We select the device k with smallest channel gain in set N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Try to put the device k from the set N1 into the set N0 and compute the cost of new sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' If the total cost of new sets is reduced, continue the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Otherwise, put the device k back to the set N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' JOINT OPTIMIZATION USING ADMM-BASED METHOD The complexity of the Channel-Aware heuristic algorithm becomes high when the number of UE grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this section, We propose an ADMM-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The ADMM-based algorithm can decompose P2 into N parallel sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Each user only needs to solve one sub-problem, and the average complexity of each device will be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ADMM-based Problem Conversion To make the original problem tractable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' we jointly consider the problem P2 and problem � PGPN1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and we converted the problem into a GP-based problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem P3 (Converted GP-based Problem): minimize � ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f e n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='xn � � n∈N � (1 − xn) ˆ F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ) + xn ˆ F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (30) subject to ˆ f md n ≤ ln f max n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ∀n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (30a) (13g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29a) − (29c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' where ˆ Mn = ln Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n = ln f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en = ln f e n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' and ˆtn = ln tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ) and ˆ F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn) are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ) = β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e ˆ Mn− ˆ f md n + β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1e− ˆ f md n + β2κmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e ˆ Mn+2 ˆ f md n + β2κmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1e2 ˆ f md n + β3ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e− ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (31) ˆ F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn) = β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e ˆ Mn− ˆ f e n + β1ρmc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1e− ˆ f e n + β1de ˆ Mn− ˆ tn Rn + β2dpne ˆ Mn Rn + β3ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0e− ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (32) The optimization variables { ˆtn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆf en} are coupled among the devices in the constraints (29b) and (29c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' To decompose the problem P3, we introduce local variables {yn} and {zn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' the ADMM-based problem can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Problem P4 (ADMM-based Problem): minimize � ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f e n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='zn � � n∈N ˆ Fn(xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' zn) + g( ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (33) subject to yn = ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' zn = ˆtn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (33a) (13g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (29a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (30a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ Fn(xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' zn) = (1 − xn) ˆ F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆ f md n ) + xn ˆ F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='n( ˆ Mn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' yn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (34) g( ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn) = � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' if( ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn) ∈ G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' +∞ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (35) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' G = � ( ˆf en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ˆtn)| � n∈N1 xne ˆ tn ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' � n∈N1 xne ˆ f e n ≤ f max � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (36) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' ADMM-based Problem Solving The problem P4 can be effectively solved using the ADMM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can write a partial augmented Lagrangian of the problem P4 as, L4(u, v, θ) = � n∈N ˆ Fn(xn, ˆ Mn, ˆ f md n , yn, zn) + g( ˆf en, ˆtn) + � n∈N θf n(yn − ˆf en) + � n∈N θt n(zn − ˆtn) + � n∈N s 2(yn − ˆf en)2 + � n∈N s 2(zn − ˆtn)2, (37) where u = {xn, ˆ Mn, ˆ f md n , yn, zn}, v = { ˆf en, ˆtn}, θ = {θf n, θt n}, and s is a fixed step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the dual function is, p(θ) = minimize u,v L4(u, v, θ) (38) subject to (13g), (29a), (30a), and the dual problem can be given by, maximize θ p(θ), (39) The problem (38) can be solved by iteratively updating u, v, and θ [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Let {ui, vi, θi} denote the values in the ith iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In the ith iteration, the update strategies of the variables are as follows, 1) Step 1: Local variables update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this step, we first update the local variables u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Given variable vi and θi, we minimize L4(u, v, θ) by, ui+1 = argminimize u L4(u, vi, θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (40) The problem (39) can be decomposed into N parallel sub- problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For each subproblem, we consider two cases where xn = 0 and xn = 1, and express the problem as, � � � � � � � � � � � � � � � � � � � � � � � � � � � minimize { ˆ Mn, ˆ f md n ,yn,zn} ˆ F0,n( ˆ Mn, ˆ f md n ) =θf nyn + � n∈N s 2(yn − ˆf en)2 +θt nzn + � n∈N s 2(zn − ˆtn)2, if xn = 0, minimize { ˆ Mn,yn,zn} ˆ F1,n( ˆ Mn, yn, zn)=θf nyn + � n∈N s 2(yn − ˆf en)2 +θt nzn + � n∈N s 2(zn − ˆtn)2, if xn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (41) These problems are both strictly convex problems that can be solved using the CVX tool [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, we can cal- culate the objective value for xn = 0 and xn = 1 and choose the smaller one as the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' After solving N parallel subproblems, the optimal solution to (40) is given by ui+1 = {(xn)i+1, ( ˆ Mn)i+1, ( ˆ f md n )i+1, (yn)i+1, (zn)i+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 2) Step 2: Global variables update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In the second step, we update the global variables v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' By the definition of g(v) in (35), vi+1 ∈ G must hold at the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the subproblem can be equivalently written as, vi+1 = argminimize { ˆ f e n, ˆ tn} � n∈N (θf n)i(− ˆf en) + � n∈N (θt n)i(− ˆtn) + � n∈N s 2(yi+1 n − ˆf en)2 + � n∈N s 2(zi+1 n − ˆtn)2, (42) subject to, (29b), (29c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The problem can also be solved by the CVX tool [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We propose a low-complexity scheme to solve this subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 9 Algorithm 4: Algorithm 4: ADMM-Based Algorithm Input: Parameters corresponding to the problem P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Output: {xn, Mn, f md n , f e n, tn} Initialize i = 0, {ui, vi, θi} = 0, s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5, µ⋆ f = µ⋆ t = 106, δ = 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' repeat foreach n ∈ N do Update ui+1 by solving (41) and choose smaller results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' foreach n ∈ N do Update global variables vi+1 using (43) and (44);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' foreach n ∈ N do Update multipliers θi+1 using (45) and (46);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' until |Fi − Fi+1| < δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Mn = e ˆ Mn, f md n = e ˆ f md n , f e n = e ˆ f e n, tn = e ˆ tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' return {xn, Mn, f md n , f e n, tn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering the constraints (29b) and (29c), let µf and µt denote the Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The closed-form optimal solution of this subproblem can be expressed as, ( ˆf en)i+1 =yn i+1 + (θf n)i − µf s , (43) ( ˆtn)i+1 =zn i+1 + (θt n)i − µt s , (44) where µf can be obtained by the bisection search method over (0, µ⋆ f), until � n∈N1 xne ˆ f e n ≤ f max satisfies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' µ⋆ f is a sufficiently large value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It is because when µf ≥ 0, ( ˆf en)i+1 is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Similarly, µt can be obtained by the bisection search method over (0, µ⋆ t ), where µ⋆ t is a sufficiently large value, until � n∈N1 xne ˆ tn ≤ 1 satisfies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 3) Step 3: Multipliers update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In this step, we update the multipliers θ using the obtained global variables v and local variables u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The updated method is, (θf n)i+1 =(θf n)i + s(yi+1 n − ( ˆf en)i+1), (45) (θt n)i+1 =zn i+1 + s(zi+1 n − ( ˆtn)i+1), (46) Repeat the above three steps until the cost function no longer decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The cost function is Fi = � n∈N [(1 − xi n) ˆ F0,n(( ˆ Mn)i, ( ˆ f md n )i) + xi n ˆ F1,n(( ˆ Mn)i, ( ˆf en)i, ( ˆtn)i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We summarize solving steps of the ADMM algorithm as Algo- rithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As a distributed iterative algorithm, the ADMM-based scheme performs iterations between devices and BS rather than locally, enabling online optimization during the recognition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In each iteration, ui is calculated locally and sent to the MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' After receiving ui from all devices, the MEC updates vi and θi, and sends them to the device to complete an iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the iteration of the ADMM algorithm is an online convergence process that can adapt to slight changes in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 0 4 8 12 16 The number of input video frames 0 100 200 300 400 500 600 Latency / ms Theory, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Experiment, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Fitting, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Theory, Resnet-34, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Experiment, Resnet-34, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Fitting, Resnet-34, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G Theory, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G Experiment, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G Fitting, Resnet-18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G m m c,0=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 c,1=40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0 c,0=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0 c,1=48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1 m m m m c,0=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1 c,1=79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The theoretical delay curve, the experimental delay curve and the fitted curve corresponding to the experimentalal delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Resnet-18 and Resnet- 34 are two classic neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The frequency of the CPU is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Algorithm Computational Complexity Analysis In this part, we analyze the computational complexity of proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, the complexity of solving problem PN0 is O(|N0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, as mentioned above, the complexity of Algorithm 1 is O(� n∈N1 |Mopt n |), and the complexity of Algorithm 2 is O((3|N1|)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5) by the interior-point method according to [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When we use Algorithm 1 for solving PN1 and use Search-based algorithm for optimizing offloading policy, the computational complexity is O(2N � n∈N |Mopt n |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When we use Algorithm 1 for solving PN1 and use Algorithm 3 for optimizing offloading policy, the computational com- plexity is O(N � n∈N |Mopt n |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, the computational complexity of Algorithm 2 for solving PN1 and Algorithm 3 for optimizing offloading policy is O(N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For the ADMM- based algorithm, as the complexity of each steps is O(N), the overall complexity of one iteration is O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we evaluate the performance of the proposed algorithms via simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For all the simulation results, unless specified otherwise, we set the downlink bandwidth as Bw = 5 MHz and the power spectral as N0 = −174 dBm/Hz [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' According to [17], the path loss is modelled as PL = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1 + 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 log10(D) dB, where D is the dis- tance between the device and the BS in kilometres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Devices randomly distributed in the area within [500m 500m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The computational resource of the MEC server and devices are set to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 GHz and 22 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The recognition accuracy requirement and the maximum number of input video frames are set to αn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='86 and M max n = 16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The coefficient κ is determined by the corresponding device and is set to be 10−28 in this paper according to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The size of the input video is 112∗112∗Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, the coefficient of computational complexity ρ is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='12 cycle/MAC, which is obtained through several experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Weights β1, β2, β3 are set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Model Verification First, we obtain the complexity coefficient through experi- mental measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The calculation method of the compu- tational complexity coefficient is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' First, calculate IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 10 0 4 8 12 16 The number of input video frames 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='85 1 Accuracy Fitted curve, gesture, resnet-101 Fitted curve, gesture, resnet-18 Fitted curve, action, resnet-101 Experiment, gesture, resnet-101 Experiment, gesture, resnet-18 Experiment, action, resnet-101 a,0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='910 a,1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='400 a,2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='993 a,0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='910 a,1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='400 a,2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='993 m m m a,0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='707 a,1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='939 a,2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='990 m m m m m m Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The experimental and fitted curves of gesture recognition task and action recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' the MACs of the DNN model when the number of input video frames is different, recorded as {C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use the Flops Counter tool [34] for MACs calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Second, execute 100 times of inference tasks with a different number of input video frames, and record the average inference delay as {t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Finally, calculate the coefficients between the inference delay and MACs by ρ = sum({C}) sum({t}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use Intel(R) Xeon(R) E5- 2630 CPU for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use the Resnet-18 and the Resnet-34 for testing and limit the maximum frequency of the CPU to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 3 shows the theoretical (MAC-based) and experimental delay curves and the fitted curve corresponding to the experimental delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 3 that the theoretical delay is similar to the experimental delay, proving that MACs can be modelled as computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also find that the linear fitted curve can approximately represent the computational complexity with 9 ms root mean square error (RMSE) for Resnet-18 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G, 17 ms RMSE for Resnet-34 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8G, and 11 ms RMSE for Resnet-18 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The inference delay is associated with the number of input frames, DNN model’s architecture and the device’s capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, the computational complexity co- efficients under the three conditions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='128, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='122, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='123, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, in following experiments, we set ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='12 cycle/MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We select the gesture and action recognition tasks to verify the accuracy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use the Jester datasets [35], the largest publicly available hand gesture dataset, to test the gesture recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For the action recognition task, we use Kinetics-400 datasets [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We choose Resnet-18 and Resnet-101 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 4, Under different tasks and different network models, the accuracy curve all conforms to the characteristics of a non-decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' What’s more, as the number of input frames increases, the performance gain of accuracy will gradually decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' This is because the information gain introduced in the temporal domain decreases when the number of input frames increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The fitted curve can approximately represent the relationship between the accuracy and the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In the gesture recognition task with the Resnet-101 model, the gesture recognition task with the Resnet-18 model, and the action recognition task with the Resnet-101 model, the RMSE are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0054, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0048 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='0095, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We take the 4 12 20 28 36 The number of devices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1 Average cost Local Edge Random CCCP ADMM GP+Heuristic Performance bounds Performance loss CCCP: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1% ADMM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='24% GP+Heuristic: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='03% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The average cost of proposed schemes and baseline schemes under a different number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Resnet-18 and the gesture recognition task as examples for the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Simulation Results of Average Cost In this section, we compare proposed schemes and some baseline schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We run 100 tests and can calculate the average cost of each device and the average running time of each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We compare the following schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 1) Search+Search: We use the Search-based algorithm to solve PN1 and use the heuristic algorithm to optimize offloading policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 2) Search+Heuristic: We use the Search-based algorithm to solve PN1 and use the Search-based algorithm to optimize offloading policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 3) GP+Heuristic: We use the GP-based algorithm to solve PN1 and use the Channel-Aware heuristic algorithm to opti- mize offloading policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 4) ADMM: We use the ADMM-based algorithm to solve the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 5) CCCP [37]: We use the concave-convex procedure (CCCP) algorithm to decide whether to offload inference tasks to edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then we use Theorem 1 and the GP-based algorithm for resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 6) Random: All inference tasks are randomly executed on local or the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use Theorem 1 and the GP-based algorithm for resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 7) Local: All inference tasks are executed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use Theorem 1 for local resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8) Edge: All inference tasks are executed on the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use the GP-based algorithm for resource allocation In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 5, we plot the average cost of different schemes under different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The Search+Heuristic scheme and Search+Search scheme have the same performance, represent- ing the performance bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When the number of devices exceeds 16, the performance bounds are not shown due to their unacceptable computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 5 that the proposed schemes are better than the baseline schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Compared with the performance bounds, the performance of the GP+Heuristic scheme has a slight decrease due to the relaxation of the accuracy function Φ(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The performance of the ADMM scheme is worse than that of the GP+Heuristic scheme, and is better than that of the CCCP IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 11 2 6 10 14 18 22 The number of devices 0 50 100 150 200 250 Running time / s ADMM GP+Heuristic CCCP Search+Heuristic Search+Search Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The average running time of proposed algorithms under a different number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For example, when the number of devices is 16, the CCCP, ADMM, and GP+Heuristic schemes have performance losses of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='1%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='24%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='03%, respectively, compared with performance bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Moreover, when the number of devices is less than 8, the cost of the scheme that executes tasks only at the edge is almost equal to the cost of the proposed GP+Heuristic scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It is because all devices can benefit from performing inference on the edge server when the number of devices is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' If the inference task is only executed locally, the average cost of the device will not change because the local resources among the equipment do not affect each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 6, we plot the average running time of different schemes under different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When the number of de- vices exceeds 6, the running time of the Search+Heuristic and Search+Search scenarios becomes unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The GP+Heuristic scheme improves the solution efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The running time of GP+Heuristic is shorter than that of CCCP scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' However, the complexity of the solution remains unsatisfactory as the number of devices increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As for the ADMM-based scheme, since the ADMM-based algorithm is a distributed algorithm and the complexity of updating global variables is much smaller than that of updating local variables, we only consider the average running time for each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The average running time of the ADMM-based scheme does not improve as the number of devices increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It is worth noting that in the ADMM-based scheme, the iteration stops when |Fi − Fi+1| < δ, where δ = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Threshold-based stopping conditions result in a different number of iterations in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When the number of devices is different, the average number of iterations is also different, resulting in different running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, the average running time of 18 devices is shorter than that of 14 and 22 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Assuming that the ADMM-based scheme iterates once every time an inference task is performed, we plot the curve corre- sponding to the cost function and the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7, the ADMM-based scheme can converge to acceptable performance after completing 3-5 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As the number of iterations increases, the performance will be closer to the optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' It shows that the ADMM algorithm can converge through online iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also test the running time per iteration on each device, and it takes an average of about 278ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 0 5 10 15 20 The number of iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 2 Cost ADMM Performance bounds Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The curve corresponding to the cost function and the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' TABLE I DELAY, ENERGY CONSUMPTION, AND ACCURACY OF LOCAL DEVICES AND EDGE DEVICES Local devices Edge devices Number of devices 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7 Average delay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='24 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='52 s Average energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='00 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='025 J Average accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='866 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Simulation Results of Delay, Energy, and Accuracy This section compares the average delay, energy consump- tion, accuracy, and the offloading rate (the proportion of de- vices that perform inference on the edge server).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We consider the different number of devices, bandwidths, edge computing resources, and weights β1, β2, β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use the GP+Heuristic scheme for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' I shows a comparison of devices that finish inference locally and devices that finish inference at the edge under default experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' On average, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7 devices choose to offload to the edge server to perform inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Compared with edge devices, local devices have a lower delay and higher accuracy but have greater inference energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8 shows the average delay, energy, accuracy, and offloading rate under different numbers of devices, different bandwidths, and different edge computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(a), we plot results with different numbers of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(a), when the number of devices is small (less than 10), all devices offload the task to the edge server (the offloading rate is equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For edge devices, all delay comes from transmission delay and the edge inference delay, and all energy consumption comes from transmission energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' With the number of devices increasing, communication resources and the edge server’s computation resources are shared by more devices, decreasing the number of input frames Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A decrease in the number of input frames results in a decrease in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Then as Mn decreases, the transmission data size decreases, and the transmission energy decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Meanwhile, Competition from more devices leads to increased delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, when the number of devices is small (less than 10), with the number of devices increasing, the average delay increases, the average accuracy and the average energy con- IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 12 5 15 25 35 45 The number of devices 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 1 Delay /s, Energy / J, Offloading rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='95 1 Accuracy Delay Energy Offloading rate Accuracy (a) Different number of devices 2 6 10 14 18 Bandwidth / MHz 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 1 Delay /s, Energy / J, Offloading rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='95 1 Accuracy Delay Energy Offloading rate Accuracy (b) Different bandwidth 6 14 22 30 38 Edge computing resource / GHz 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 1 Delay /s, Energy / J, Offloading rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='95 1 Accuracy Delay Energy Offloading rate Accuracy (c) Different edge computing resource Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The average delay, energy, offloading rate, and accuracy under different numbers of devices, different bandwidths, and different edge computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' sumption decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' When the number of devices exceeds 10, the average energy consumption and accuracy increase, and the average delay and offload rate gradually decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering different bandwidths and different edge computing resources, we plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 8(c), as the bandwidth and edge computing resource increase, devices will be more inclined to offload computing to the edge, which increases the offloading rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' According to Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' I, when β1, β2 and β3 are fixed, edge devices have lower energy consumption, lower accuracy and higher delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' More edge devices mean a greater delay and lower power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Meanwhile, when the bandwidth increases, since the edge computing resources are fixed, the number of video frames will decrease to reduce edge computing overhead, resulting in a decrease in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The same conclusion can also be obtained when edge computing resources increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Therefore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The relationship between the delay, energy consumption, and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' with the increase of bandwidth and edge computing resources, more edge devices lead to increased delay and decreased energy and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We set the minimum number of input frames M min n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We use different weights, β1, β2, β3 to study the trade-off relationship between the average delay, energy consumption, and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The constraint is β1 + β2 + β3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The perfor- mance of the trade-off surface is obtained by the GP+Heuristic scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' 9 shows the delay, energy consumption, and accuracy are mutually limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Higher energy consumption leads to higher accuracy when the delay is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' From another perspective, in order to improve the accuracy, it is necessary to sacrifice the performance of delay and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' In addition, with the same accuracy, according to Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' I, higher energy consumption will make the device more inclined to execute inference tasks locally, and the delay decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' CONCLUSION This paper considers optimizing video-based AI inference tasks in a multi-user MEC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' An MINLP is formulated to minimize the total delay and energy consumption, and improve the total accuracy, with the constraint of computation and communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' A MAC-based computational complexity model is introduced to model the calculation delay, and a simple approximate expression is proposed to simplify the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also propose a general accuracy model to characterize the relation between the recognition accuracy and the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' After that, we first assume that the offloading decision is given and decouple the original problem into two sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The first sub-problem is to optimize the resources of the devices that complete the DNN inference tasks locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We derive the closed-form solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The second sub-problem is optimizing the devices’ resources that offload the DNN inference tasks to the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We propose the Search-based and GP-based algorithm to solve the second sub-problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' For the problem of offloading de- cision optimization, we propose the Channel-Aware heuristic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also propose a distributed algorithm based on ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The ADMM-based algorithm reduce computational complexity at the cost of an acceptable performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Numerical simulation and experimental results demonstrate 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='9Accurac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='6 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 1 Energy / J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 0 02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content='5 1 Delay / sIEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' XX, XXX 2022 13 the effectiveness of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We also provide a detailed analysis of the delay, energy consumption, and accuracy for different device numbers, bandwidths and edge computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' APPENDIX A PROOF OF THEOREM 1 The partial derivative of FPN0 with respect to f md n is, ∂FPN0 ∂f md n = −β1 ρC(Mn) f md2 n + 2β2κρC(Mn)f md n , (47) By setting ∂FPN0 ∂f md n = 0, we have, f md n = 3 � ( β1 2β2κ), (48) Therefore, f md n decreases monotonically in the interval (−∞, 3� ( β1 2β2κ)) and increases monotonically in the interval ( 3� ( β1 2β2κ), +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' Considering the value range of f md n , the optimal solution can be given by, f md∗ n = min{ 3 � ( β1 2β2κ), f max n } (49) Then we analyze Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' The partial derivative of FPN0 with respect to Mn is, ∂FPN0 ∂Mn = β1ρmc,0 f md n + β2κρmc,0f md2 n − β3ma,0 (Mn + ma,1)2 , (50) By setting ∂FPN0 ∂Mn = 0, we have, Mn = � β3ma,0 β1ρmc,0 f md n + β2κρmc,0f md2 n − ma,1, (51) Considering the value range of Mn, the optimal solution can be given by, M ∗ n = min{max{ � β3ma,0 β1ρmc,0 f md n + β2κρmc,0f md2 n − ma,1, M min n }, M max n } (52) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 2 According to the KKT conditions, we can obtain the fol- lowing necessary and sufficient conditions, ∂LPN1 ∂f e∗ n = −β1ρC(M ∗ n) f e∗2 n + u∗ 1 = 0, f e∗ n > 0, (53) ∂LPN1 ∂t∗n = −β1M ∗ nd Rnt∗2 n + u∗ 0 = 0, t∗ n > 0, (54) µ∗ 0( � n∈N ∗ t∗ n − 1) = 0, (55) µ∗ 1( � n∈N ∗ f e∗ n − f max) = 0, (56) µ∗ 0, µ∗ 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (57) Because β1ρC(M ∗ n) f e∗2 n and β1M ∗ nd Rnt∗2 n are positive, µ∗ 0 and µ∗ 1 are also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' We can obtain, � n∈N f e∗ n − f max = 0, (58) � n∈N t∗ n − 1 = 0, (59) f e∗ n = � β1ρC(M ∗n) Rnµ∗ 1 , (60) t∗ n = � β1M ∗nd Rnµ∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (61) Combining (58) and (60), we can get the expression of f e∗ n corresponding to M ∗ n, f e∗ n = f max� C(M ∗n) � i∈N1 � C(M ∗ i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' (62) Similarly, combining (59) and (61), we can get the expression of t∗ n corresponding to M ∗ n, t∗ n = � M ∗ n Rn � i∈N1 � M ∗ i Ri , (63) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfFfoM/content/2301.01010v1.pdf'} +page_content=' REFERENCES 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mode 100644 index 0000000000000000000000000000000000000000..2a5707a5b3eb805fdeb8a2f94476ce8984ed5e30 --- /dev/null +++ b/G9AyT4oBgHgl3EQfrfnZ/content/tmp_files/2301.00563v1.pdf.txt @@ -0,0 +1,1961 @@ +arXiv:2301.00563v1 [cond-mat.stat-mech] 2 Jan 2023 +Sample-to-sample fluctuations of transport coefficients in the totally asymmetric +simple exclusion process with quenched disorders +Issei Sakai1 and Takuma Akimoto1, ∗ +1Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan +(Dated: January 3, 2023) +We consider the totally asymmetric simple exclusion processes on quenched random energy land- +scapes. We show that the current and the diffusion coefficient differ from those for homogeneous +environments. Using the mean-field approximation, we analytically obtain the site density when the +particle density is low or high. As a result, the current and the diffusion coefficient are described +by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to +the many-body effect, the current and the diffusion coefficient differ from those for single-particle +dynamics. +The current is almost constant and becomes the maximal value in the intermediate +regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate +regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based +on the renewal theory. The deepest energy depth plays a central role in determining the maximal +current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient +depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find +that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized +by the Weibull distribution. We show that the disorder averages of the maximal current and the +diffusion coefficient converge to zero as the system size is increased and quantify the degree of the +non-self-averaging effect for the maximal current and the diffusion coefficient. +I. +INTRODUCTION +The one-dimensional asymmetric simple exclusion pro- +cess (ASEP) is a pedagogical model for non-equilibrium +systems [1]. +In particular, it describes various non- +equilibrium phenomena such as traffic flow [2] and pro- +tein synthesis by ribosomes [3–5]. The ASEP is a stochas- +tic process where particles with hard-core interactions +diffuse on a one-dimensional lattice. The ASEP can be +mapped to a model of interface growth in the Kardar- +Parisi-Zhang (KPZ) universality class [6]. +Hopping to +the right site in the ASEP corresponds to an increase +in the interface. +The distribution of interface height +was solved [7–9]. +Using the weak asymmetric limit of +the ASEP, the KPZ equation was rigorously solved an- +alytically [10, 11]. Moreover, the large deviation func- +tion of the time-averaged current was obtained [12, 13]. +The ASEP has been extended in various ways such as +Brownian ASEP [14], non-Poissonian hopping rates [15], +and disordered hopping rates [16–23]. +When particles +only hop to uni-direction, it is called the totally ASEP +(TASEP). For TASEPs, it is well known that the current- +density relation is given by [1] +J = 1 +τ ρ(1 − ρ), +(1) +where J is the particle current, ρ is particle density, and +τ is the inverse of the jump rate, i.e., the mean waiting +time. Moreover, in Refs. [24], the variance of the tagged +particle displacement, δxt, in time t is derived as a func- +∗ takuma@rs.tus.ac.jp +tion of ρ: +⟨δx2 +t ⟩ − ⟨δxt⟩2 +t +∼ +√π +2τ +(1 − ρ)3/2 +(Lρ)1/2 +(2) +for L → ∞ and t → ∞, where ⟨·⟩ is the ensemble average +and L is the system size. +Effects of disorder in the ASEP have been investi- +gated for decades [16–23]. +Due to the disorder in the +ASEP under the periodic boundary condition, a current- +density relation deviates from that in the ASEP with a +homogeneous jump rate, i.e., Eq. (1). More precisely, it +becomes flat and the current is maximized on the flat +regime [16, 18–23]. +Moreover, in the flat regime, the +low- and high-density phases coexist. In the ASEP on +networks, the flat regime also exists [25–27]. Under the +open boundary condition, the first-order phase transition +point between the low- and high-density phases depends +on the disorder [17]. +Random walks in heterogeneous environments show +anomalous diffusion. The heterogeneous environment is +characterized by a random energy landscape. There are +two types of random energy landscapes. One is an an- +nealed energy landscape, where the landscape randomly +changes with time. +The continuous-time random walk +is a diffusion model on the annealed energy landscape, +and its mean-squared displacement shows anomalous dif- +fusion when the mean waiting time diverges [28]. The +other is a quenched energy landscape, where the land- +scape is configured randomly and does not change with +time. +The quenched trap model (QTM) is a diffusion +model on the quenched energy landscape [29]. The mean- +squared displacement of the QTM on an infinite system +shows anomalous diffusion when the mean waiting time +diverges [29]. In the QTM on a finite system, the dif- +fusion coefficient exhibits sample-to-sample fluctuations + +2 +[30–32]. +The diffusivity of interacting many-body sys- +tems on the annealed energy landscape has been inves- +tigated [33, 34]. However, the diffusivity of interacting +many-body systems on the quenched energy landscape +has never been investigated. Such a heterogeneous envi- +ronment is realized experimentally. In protein synthesis +by ribosomes, the codon decoding times become hetero- +geneous due to the heterogeneity of transfer RNA concen- +tration [5]. In other words, the distribution of the waiting +time depends on the site, i.e., ribosomes diffuse on the +quenched random environment. There are other diffusion +phenomena in such heterogeneous environments, such as +train delays, proteins on DNA [35, 36], and water trans- +portation in aquaporin [37]. +In this paper, we investigate sample-to-sample fluctu- +ations of the diffusivity for the TASEP on a quenched +random energy landscape. +In our previous study, we +show sample-to-sample fluctuations of the current [38]. +When an observable does not depend on the disorder re- +alization, it is called self-averaging [29]. +In the QTM, +it is known that the diffusion coefficient [30–32], the +mobility [32], and the mean first passage time [39] are +non-self-averaging. Is such a non-self-averaging behavior +still observed when the N-body effect is introduced in +the quenched random energy landscape? This is a non- +trivial question in diffusion in a heterogeneous environ- +ment. In particular, it is non-trivial that the TASEP with +disordered waiting-time distributions exhibits sample-to- +sample fluctuations for the current and the diffusion co- +efficient. Therefore, it is important to provide an exact +result for the current and the diffusion coefficient in het- +erogeneous quenched environments. +Our paper is organized as follows. In Sec. II, we formu- +late the TASEP on a quenched random energy landscape +and define averaging procedures. In Sec. III, we show the +numerical results of the current-density relation and the +density profile. In Sec. IV, we present derivations of the +density profile. In Sec. V, we present derivations of the +current and the diffusion coefficient. In Sec. VI, we dis- +cuss the self-averaging properties of the current and the +diffusion coefficient. In Sec. VII, we conclude this paper. +In Appendix A, we derive the passage time distribution. +In Appendix B, we derive the Fr´echet distribution. +II. +MODEL +We consider the TASEP on a quenched random en- +ergy landscape on a one-dimensional lattice. +It com- +prises N particles on the lattice of L sites with periodic +boundary conditions. +Each site can hold at most one +particle. Quenched disorder means that when realizing +the random energy landscape, it does not change with +time. At each lattice point, the depth E > 0 of the en- +ergy trap is randomly assigned. The depths are indepen- +dent identically distributed (IID) random variables with +an exponential distribution, φ(E) = T −1 +g +exp (−E/Tg), +where Tg is called the glass temperature. +A particle +can escape from a trap. Escape times from a trap are +IID random variables following an exponential distribu- +tion and follow the Arrhenius law, i.e., the mean escape +time of the kth site is given by τk = τc exp (Ek/T ), +where Ek is the depth of the energy at site k, T the +temperature, and τc a typical time. +The probability +of the escape time τ that is smaller than x is given +by Pr(τ ≤ x) ∼= Pr(E ≤ T ln(x/τc)). +It follows that +the probability density function (PDF) ψα(τ) of waiting +times follows a power-law distribution: +� ∞ +τ +dτ ′ψα(τ ′) = +� τ +τc +�−α +(τ ≥ τc) +(3) +with α ≡ T/Tg [30]. +The dynamics of the particle are described by the +Markovian one in the sense that the waiting time is +memory-less. In particular, the waiting times at site k +are assigned IID random variables following an exponen- +tial distribution, ψk(ti) = τ −1 +k +exp (−ti/τk). +After the +waiting time elapses, the particle attempts to hop the +neighboring site on its right. The hop is accepted only +if the site is empty. When the attempt is a success or +failure, the particle is assigned a new waiting time from +ψk+1(ti) or ψk(ti), respectively. +Here, we consider three averaging procedures, i.e., en- +semble average, disorder average, and time average. The +ensemble average of observable O(t) is an average with +respect to a stationary ensemble for a single disorder re- +alization denoted by ⟨O(t)⟩. The disorder average of ob- +servable O(t) is an average with respect to different dis- +order realizations denoted by ⟨O(t)⟩dis. The time average +of observable O(t) is defined by +¯O(T ) = 1 +T +� T +0 +O(t)dt. +(4) +Furthermore, we consider a stationary initial condition. +For the ASEP on a finite system, the variance of the +displacement of the tagged particle depends on whether +the initial conditions are identical or not, especially for a +short time [40]. However, the asymptotic behavior does +not depend on the initial condition. In this paper, we are +interested in the asymptotic behavior of the current and +the diffusivity. Therefore, the initial conditions in this +paper are not fixed. In numerical simulations, particles +start from the stationary ensemble of configurations. The +stationary ensemble is given by the configuration after +particles arrange randomly and diffuse for a long time. +III. +NUMERICAL RESULTS OF +CURRENT-DENSITY RELATION AND +DENSITY PROFILE +We numerically show that the current-density relation +for a disordered TASEP (DTASEP) deviates from that +for a TASEP with a homogeneous jump rate, i.e., the +homogeneous TASEP. Figure 1 shows the steady-state + +3 +current J against particle density ρ = N/L, i.e., the +current-density relation, for a DTASEP. For low and high +densities, the current-density relation is the same as that +of the homogeneous TASEP (see Fig. 1). However, there +is a distinct difference between them in the intermedi- +ate regime. In particular, the current for the DTASEP +becomes almost flat and smaller than that for the ho- +mogeneous TASEP in the intermediate regime. On the +other hand, there is no flat regime for the homogeneous +TASEP. The flat regime in the DTASEP is observed in +other disordered systems [16, 18–20, 23]. +Thus, it is +a manifestation of the existence of a disorder. In this +regime, the current is independent of the particle density +and maximized. In the following, we classify the density +into three regimes: the low density (LD) (0 < ρ ≤ ρ∗), +the maximal current (MC) (ρ∗ < ρ < 1 − ρ∗), and the +high density (HD) (1 − ρ∗ ≤ ρ < 1) regimes (Fig. 1). +We explicitly derive the transition density ρ∗ later (see +Eq. (13)). +Here, we numerically show the density profiles. +For +the LD and HD regimes, the system is homogeneous +on a macroscopic scale (Figs. 2(a) and (b)). +For the +MC regime, there is a macroscopic density segregation +(Figs. 2(c) and (d)). +The segregation is classified into +high- and low-density phases by the deepest trap. Com- +paring Figs. 2(c) and (d), we observe that the high- +density regime becomes large when the particle density +is increased. This result is qualitatively similar to that +in a system with one defect bond, studied in Ref. [41]. +!" +#" +$% +0 +0.01 +0.02 +0.03 +0 +0.2 +0.4 +0.6 +0.8 +1 +J +ρ +FIG. 1. Current-density relations for homogeneous and dis- +ordered TASEPs. The circles are obtained by the numerical +simulation of dynamics of the DTASEP (L = 5000, α = 2.5, +and τc = 1). +The solid line represents the current-density +relation, Eq. (1), for the homogeneous TASEP with τ being +set to equal to the sample average of the waiting times of the +DTASEP. ρ∗ is given by Eq. (13). +IV. +DERIVATION OF THE DENSITY PROFILE +Here, we derive the density profile by the mean-field +approximation. This derivation is almost the same as our +previous study [38]. Let Jk be the mean current across +the bond between site k and k+1. In the DTASEP, a hop +occurs with a rate 1/τk whenever site k is occupied, and +site k + 1 is not. Thus, the mean current is represented +by +Jk = +� 1 +τk +nk(1 − nk+1) +� +, +(5) +where nk denotes the number of a particle, which is 1 +if the site k is occupied and 0 otherwise. +If the sys- +tem is in a steady state, the ensemble average is equal +to the time average in the long-time limit, i.e., the sys- +tem is ergodic. The ensemble average in Eq. (5) coin- +cides with the long-time average if the system is ergodic. +The periodic boundary condition implies nL+1 = n1 and +τL+1 = τ1. The probability of finding a particle at site k +is given by ρk = ⟨nk⟩. In the mean-field approximation, +one can ignore correlations between nk and nk+1, which +means +⟨nknk+1⟩ = ⟨nk⟩ ⟨nk+1⟩ . +(6) +In +the +steady +state, +the +site +densities +are +time- +independent. Moreover, from the continuity of the cur- +rent, the current is independent of k, i.e., Jk = J for all +k. Therefore, we have the current-density relation: +J = 1 +τk +ρk(1 − ρk+1). +(7) +We note that the right-hand side of Eq. (7) is independent +of k. +We derive a simpler form of the site density by approx- +imating Eq. (7) for the LD and HD regimes. For the LD +regime, we can assume ρkρk+1 ≪ 1 because the particle +density is small. Ignoring ρkρk+1 in Eq. (7), we obtain +J ∼= 1 +τk +ρk. +(8) +Using the conservation of particles, � +i ρi = N, the site +density has the form +ρk ∼= τk +µ ρ, +(9) +for the LD regime, where µ is the sample average of the +waiting times, µ = � +i τi/L. +This result is the same +as the steady-state density for the QTM [30]. For the +HD regime, the particle density is high. Using the hole +density, σk = 1 − ρk, instead of ρk, we can derive the site +density in the same way as in the LD regime. The result +becomes +ρk = 1 − σk ∼= 1 − τk−1 +µ (1 − ρ). +(10) + +4 +(a) +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +1000 +2000 +3000 +4000 +5000 +ρk +k +(b) +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0 +1000 +2000 +3000 +4000 +5000 +ρk +k +(c) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +1000 +2000 +3000 +4000 +5000 +ρk +k +(d) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +1000 +2000 +3000 +4000 +5000 +ρk +k +FIG. 2. Density profiles: (a) ρ = 0.01, (b) ρ = 0.99, (c) ρ = 0.5, and (d) ρ = 0.8 (L = 5000, α = 2.5, and τc = 1). The squares +are the results of the numerical simulation of the dynamics of the DTASEP. Triangles are Eqs. (9) and (10) for (a) and (b), +respectively. +Figures 2(a) and 2(b) show the density profiles for LD +and HD regimes, respectively. The densities are well de- +scribed by the set of site densities {ρk}. Therefore, Eqs +(9) and (10) are good approximated forms of the site +densities. +The results for the LD and HD regimes re- +produce the current-density relation for a homogeneous +TASEP. In other words, the system is homogeneous on a +macroscopic scale. +Next, we approximately obtain a density ρ∗ which is +the boundary density between LD and MC regimes in +the current-density relation (see Fig. 1). By Eq. (7), the +current between sites m and m + 1 is given by ρm(1 − +ρm+1)/τm. +The steady-state current at the boundary +density between the LD and MC regimes can be described +by Eq. (1). At the boundary density, the current between +sites m and m + 1 is equal to the steady-state current: +1 +τm +ρm(1 − ρm+1) ∼= 1 +µρ∗(1 − ρ∗). +(11) +We find numerically find that the site with the maximal +mean waiting time is always the boundary the HD and +the LD phases. When the mean waiting time is maxi- +mized at site m, sites m and m + 1 exist in high- and +low-density phases, respectively. The site densities are +given by Eq. (10) and the hole density is ρ∗ in the high- +density phase. On the other hand, the site densities are +given by Eq. (9) and the particle density is ρ∗ in the +low-density phase. Therefore, we substitute Eq. (10) and +Eq. (9) into ρm and ρm+1, respectively, +1 +τm +� +1 − τm−1 +µ +ρ∗ +� � +1 − τm+1 +µ +ρ∗ +� +∼= 1 +µρ∗(1 − ρ∗). (12) +Solving this equation for ρ∗, we have +ρ∗ ∼µ(τm−1 + τm + τm+1) +2(τm−1τm+1 + τmµ) +− µ +� +(τm−1 + τm + τm+1)2 − 4(τm−1τm+1 + τmµ) +2(τm−1τm+1 + τmµ) +. +(13) +This formula depends crucially on the disorder realiza- +tion. In the following, we assume that the mean waiting +time is maximized at site m. For L → ∞, τm is much +longer than τm−1 and τm+1. Therefore, Eq. (13) can be +approximated as +ρ∗ ∼ 1 +2 − +1 +2τm +� +τ 2m − 4τmµ ∼ µ +τm +. +(14) +By the extreme value theory [42], the scaling of τm follows +τm = O(L1/α) +(15) + +5 +for L → ∞. For α > 1, the first moment of the waiting +times exists; i.e., µ → ⟨τ⟩ ≡ � ∞ +0 +τψα(τ)dτ (L → ∞). +Hence, the scaling of ρ∗ becomes +ρ∗ ∝ L−1/α. +(16) +For α ≤ 1, the first moment of the waiting times diverges. +The scaling of the sum of τi follows +L +� +i=1 +τi = O(L1/α) +(17) +for L → ∞. It follows that the scaling of ρ∗ becomes +ρ∗ ∼ L−1 +� +i τi +τm +∝ L−1. +(18) +Therefore, ρ∗ → 0 for L → ∞. +V. +DERIVATION OF CURRENT AND +DIFFUSIVITY +A. +LD and HD regimes +Here, we derive the current in the LD and HD regimes. +For single-particle dynamics on the quenched random +energy landscape, i.e., the QTM, the mean number of +events that a particle passes a site until time t is given +by [32] +⟨Qt⟩ +t +∼ 1 +Lµ (t → ∞), +(19) +where Qt is the number of events that a particle passes +a site until time t. +For the DTASEP in the LD and +HD regimes, the current depends on the particle density, +which is identical for the homogeneous TASEP (Eq. (1)). +Hence, the current in the LD and HD regimes is given by +J ∼ aρ(1 − ρ) +(20) +for L → ∞. +When ρ = 1/L, the current is equal to +Eq. (19) for L → ∞, i.e., the constant a is given by +a = 1/µ. Therefore, we have the current in the LD and +HD regimes: +J ∼ 1 +µρ(1 − ρ) +(21) +for L → ∞. +Next, we derive the diffusion coefficient in the LD and +HD regimes. δxt denotes the displacement of the tagged +particle until time t. For the QTM, the variance of the +displacement is given by [32] +lim +t→∞ +⟨δx2 +t ⟩ − ⟨δxt⟩2 +t +∼ σ2 +µ3 +(22) +for L → ∞, where σ2 is the sample mean of the squared +waiting times, σ2 = � +i τ 2 +i /L. For the DTASEP in the +LD and HD regimes, the variance of the displacement +depends on the particle density, which is identical for +the homogeneous TASEP (Eq. (2)). Hence, the diffusion +coefficient, D ≡ limt→∞(⟨δx2 +t ⟩ − ⟨δxt⟩2)/t, is given by +D ∼ b +√π +2 +(1 − ρ)3/2 +ρ1/2 +L−1/2 +(23) +for L → ∞. When ρ = 1/L, the diffusion coefficient is +equal to Eq. (22) for L → ∞, i.e., the constant b is given +by b = 2σ2/µ3√π. The diffusion coefficient in the LD +and HD regimes is given by +D ∼ σ2 +µ3 +(1 − ρ)3/2 +ρ1/2 +L−1/2 +(24) +for L → ∞. +B. +MC regime +Here, we derive the maximal current and the diffusion +coefficient in the MC regime by the renewal theory. We +define the passage time as a time interval between consec- +utive events that particles pass a site. We note that the +passage time differs from the first passage time because +the particles which pass a site are different. When the +target site is m, the mean and the variance of the passage +time Tm are obtained in Ref. [38] (see also Appendix A): +⟨Tm⟩ = τm + τm−1 +ρm−1 ++ +ρm−1 +τm−1 +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +τm+1 +1 − ρm+2 +, +(25) +⟨T 2 +m⟩ − ⟨Tm⟩2 =τ 2 +m + +� τm−1 +ρm−1 +�2 ++ +� +τm+1 +1 − ρm+2 +�2 +− +3 +� +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +�2 . +(26) +We consider the number of events Qt that particles +pass site m until time t to obtain the maximal current and +the diffusion coefficient. For the LD and HD regimes, the +density profile is homogeneous on a macroscopic scale. +However, local densities around the target site are fluc- +tuating, i.e., dense or dilute, which affects the passage +time. Therefore, the passage times are not IID random +variables for the LD and HD regimes. For the MC regime, +macroscopic density segregation exists. When the target +locates site m, particles are constantly dense on the left of +the target and dilute on the right. This segregation does +not vary with time. Therefore, the passage times are con- +sidered to be IID random variables for MC regime and +the process of Qt can be described by a renewal process +[43]. +By renewal theory [43], the mean number of re- +newals becomes ⟨Qt⟩ ∼ t/ ⟨Tm⟩ for t → ∞. The current +is represented through the mean number of the passing +events until time t: J = limt→∞ ⟨Qt⟩ /t. Thus, we have +Jmax ∼ +1 +⟨Tm⟩ +(27) + +6 +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +J +ρ +×10−5 +(b) +0 +0.001 +0.002 +0.003 +0.004 +0 +0.2 +0.4 +0.6 +0.8 +1 +J +ρ +(c) +0 +0.01 +0.02 +0.03 +0 +0.2 +0.4 +0.6 +0.8 +1 +J +ρ +FIG. 3. Current-density relation for different α, i.e., (a) α = 0.5, (b) α = 1.5, and (c) α = 2.5, where the fixed quenched +disorders. The circles are obtained by the numerical simulation of the dynamics of the DTASEP (L = 1000 for (a) and 5000 +for other cases). The dashed and the solid lines represent Eqs. (21) and (27), respectively. +(a) +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0 +0.2 +0.4 +0.6 +0.8 +1 +D +ρ +(b) +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +0 +0.2 +0.4 +0.6 +0.8 +1 +D +ρ +(c) +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +0 +0.2 +0.4 +0.6 +0.8 +1 +D +ρ +FIG. 4. Diffusion coefficient-density relation for different α, i.e., (a) α = 0.5, (b) α = 1.5, and (c) α = 2.5, where the fixed +quenched disorders. The circles are obtained by the numerical simulation of the dynamics of the DTASEP (L = 100 for (a), +500 for (b), and 1000 for (c)). The dashed and the solid lines represent Eqs. (24) and (31), respectively. +for L → ∞. The current depends on the disorder real- +ization. Figure 3 shows a good agreement between nu- +merical simulations and the theory. +Using the number of the passing events, we can de- +rive the mean displacement and the variance of the dis- +placement of a tagged particle. While the tagged particle +starting from site m+1 returns to site m+1, all particles +pass between site m and site m + 1. Therefore, in the +large-t limit, the displacement, δxt, is represented by +δxt ∼ LQt +N += Qt +ρ . +(28) +By renewal theory [43], the mean displacement and the +variance of the displacement are represented by +⟨δxt⟩ ∼ ⟨Qt⟩ +ρ +∼ +t +ρ⟨Tm⟩, +(29) +⟨δx2 +t ⟩ − ⟨δxt⟩2 ∼ 1 +ρ2 (⟨Q2 +t⟩ − ⟨Qt⟩2) +∼ 1 +ρ2 +⟨T 2 +m⟩ − ⟨Tm⟩2 +⟨Tm⟩3 +t +(30) +for t → ∞. Therefore, the diffusion coefficient for the +MC regimes is given by +D ∼ 1 +ρ2 +⟨T 2 +m⟩ − ⟨Tm⟩2 +⟨Tm⟩3 +(31) +for L → ∞. Figure 4 shows a good agreement between +numerical simulations and the theory. +VI. +SAMPLE-TO-SAMPLE FLUCTUATIONS OF +CURRENT AND DIFFUSIVITY +A. +Current +Here, we consider sample-to-sample fluctuations of the +current. To quantify the self-averaging (SA) property of +the current, we consider the SA parameter defined as [30] +SA(L; J) ≡ ⟨J(L)2⟩dis − ⟨J(L)⟩2 +dis +⟨J(L)⟩2 +dis +, +(32) + +7 +where J(L) is the current. If the SA parameter becomes +0, there is no sample-to-sample fluctuation, which means +SA. +1. +LD and HD regimes +Using Eq. (21), the SA parameter becomes +SA(L; J) = ⟨1/µ2⟩dis − ⟨1/µ⟩2 +dis +⟨1/µ⟩2 +dis +, +(33) +which is the same as the SA parameter for the diffusion +coefficient in the QTM [30]. When the mean waiting time +⟨τ⟩ ≡ +� ∞ +0 +τψα(τ)dτ is finite (α > 1), we have µ → ⟨τ⟩ +(L → ∞) by the law of large numbers. Therefore, in the +large-L limit, the current does not depend on the disorder +realization. Hence, the current is SA for α > 1. When +the mean waiting time diverges (α ≤ 1), the law of the +large numbers breaks down. +However, the generalized +central limit theorem is still valid. The PDF of the nor- +malized sum of the waiting times follows the one-sided +L´evy distribution [44], +�L +i=1 τi +L1/α +⇒ Xα (L → ∞), +(34) +where Xα is a random variable following the one-sided +L´evy distribution of index α. The PDF of Xα denoted +by lα(x) with x > 0 is given by [44] +lα(x) = − 1 +πx +∞ +� +k=1 +Γ(kα + 1) +k! +(−cx−α)k sin (kπα), +(35) +(a) +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +102 +103 +104 +105 +α = 0.7 +α = 0.5 +α = 0.3 +⟨J⟩dis(ρ(1 − ρ))−1 +L +(b) +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +102 +103 +104 +105 +106 +α = 2.5 +α = 1.5 +α = 0.5 +α = 0.3 +⟨Jmax⟩dis +L +FIG. 5. Disorder average of the current as a function of L +for several α: (a) LD and HD regimes and (b) MC regimes. +Solid lines show the asymptotic results, i.e., Eqs. (38) and +(47). Squares are the results of numerical simulations, where +we calculated the maximal currents (Eq. (27)) for different +disorder realizations by Monte Carlo simulations. We used +104 disorder realizations. Triangles are the results of the nu- +merical simulation of dynamics of the DTASEP (N = 1 for +(a) and ρ = 0.5 for (b)). We used 103 for L = 104 in the MC +regime and 104 disorder realizations for others. +where c = Γ(1 − α)τ α +c is the scale parameter. The first +and the second moment of X−1 +α +are given by [30] +⟨X−1 +α ⟩ = Γ(1/α) +αc1/α , ⟨X−2 +α ⟩ = Γ(2/α) +αc2/α . +(36) +The current can be represented by +J(L) ∼ ρ(1 − ρ) +L +L1/α +L1/α +�L +k=1 τk +∼ ρ(1 − ρ)L1−1/αX−1 +α +(37) +for L → ∞. Thus, the PDF of J is described by the in- +verse L´evy distribution. Using the first moment of the in- +verse L´evy distribution [30], we obtain the exact asymp- +totic behavior of the disorder average of the current, +⟨J(L)⟩dis ∼ ρ(1 − ρ)Γ(α−1) +ατcΓ(1 − α)1/α L1−1/α. +(38) +Hence, the current becomes 0 (see Fig. 5(a)). We note +that since the scaling of ρ∗ follows Eq. (18), we do not +simulate at the same density. +Using the first and the second moments of 1/µ, we have +the SA parameter +lim +L→∞ SA(L; J) = + + + + + +0 +(α > 1) +αΓ(2/α) +Γ(1/α)2 − 1 +(α ≤ 1). +(39) +For α ≤ 1, the SA parameter is a nonzero constant, and +thus J becomes non-SA. Therefore, there is a transition +of SA property in the LD and HD regimes. +2. +MC regime +When the system size is increased, we find a deeper and +deeper energy trap, that is, τm gets longer and longer. +Hence, Eq. (25) can be approximated as ⟨Tm⟩ ∼ τm, i.e., +we can approximate the maximal current: +Jmax ∼ 1 +τm +. +(40) +Since the PDF of the waiting times follow a power-law +distribution Eq. (3), the PDF of the normalized τm fol- +lows the Fr´echet distribution [42]: +τm +τcL1/α ⇒ Yα (L → ∞), +(41) +where Yα is a random variable following the Fr´echet dis- +tribution of index α. As derived in Appendix B, the PDF +of Yα, denoted fα(y) with y > 0, can be expressed as +fα(y) = αy−α−1 exp (−y−α). +(42) +Using Eq. (41), the maximal current can be represented +by +Jmax(L) ∼ +1 +τcL1/α +τcL1/α +τm +∼ +1 +τcL1/α Y −1 +α +(43) + +8 +for L → ∞. Thus, the PDF of Jmax is described by the +inverse Fr´echet distribution. +The PDF of Y −1 +α +can be explicitly represented by the +Fr´echet distribution: +Pr(Y −1 +α +≤ z) = Pr(Yα ≥ z−1) = +� ∞ +z−1 fα(y)dy. +(44) +The distribution is the Weibull distribution. We obtain +the PDF of Y −1 +α , denoted by wα(z): +wα(z) = αzα−1 exp (−zα). +(45) +The first and second moments of the Weibull distribution +are given by +⟨Y −1 +α ⟩ = Γ +� +1 + 1 +α +� +, +⟨Y −2 +α ⟩ = Γ +� +1 + 2 +α +� +. +(46) +From Eq. (46), we obtain the exact asymptotic behavior +of the disorder average of the maximal current, +⟨Jmax(L)⟩dis ∼ +1 +τcL1/α Γ +� +1 + 1 +α +� +. +(47) +Therefore, the maximal current decreases with the sys- +tem size L (see Fig. 5(b)). +Let us consider the SA property for the maximal cur- +rent. The SA parameter is defined as +SA(L; Jmax) ≡ ⟨Jmax(L)2⟩dis − ⟨Jmax(L)⟩2 +dis +⟨Jmax(L)⟩2 +dis +. +(48) +Using Eq. (43), we have +lim +L→∞ SA(L; Jmax) = ⟨Y −2 +α ⟩ − ⟨Y −1 +α ⟩ +2 +⟨Y −1 +α ⟩ +2 += Γ (1 + 2/α) +Γ (1 + 1/α)2 − 1. +(49) +The SA parameter becomes a nonzero constant, i.e., the +maximal current becomes non-SA (see Fig. 6(a)). This +result differs from LD and HD, and there is no transition +from SA to non-SA behavior for all α. +B. +Diffusivity +Here, we consider sample-to-sample fluctuations of the +diffusion coefficient. +In the homogeneous TASEP, the +diffusion coefficient becomes 0 for L → ∞ (Eq. (2)) be- +cause of the many-body effect. D = 0 in the homoge- +neous TASEP on a finite system implies the subdiffusion +in that on an infinite system [45]. +1. +LD and HD regimes +For the LD regime, ρ = N/L and 1 − ρ ∼ 1 for +L → ∞ and N ≪ L. We define the number of holes +(a) +10−2 +10−1 +100 +101 +102 +103 +104 +0 +0.5 +1 +1.5 +2 +2.5 +3 +SA +α +(b) +α +ρ +ρ∗ +1 − ρ∗ +1 +0 +non-SA +non-SA +SA +SA +non-SA +1/2 +2 +⟨D⟩dis = 0 +⟨D⟩dis = ∞ +⟨D⟩dis = 0 +⟨D⟩dis = 0 +⟨D⟩dis > 0 +⟨D⟩dis = 0 +LD +MC +HD +FIG. 6. (a) Self-averaging parameter as a function of α. The +squares and circles are the results of numerical simulations, +where we calculated the maximal currents (Eq. (27)) and the +diffusion coefficient (Eq. (31)) for different disorder realiza- +tions by Monte Carlo simulations (L = 105), respectively. +The triangles show the self-averaging parameter of the maxi- +mal current obtained by the numerical simulation of the dy- +namics of the DTASEP (L = 1000 and N = 500). We used +104 disorder realizations. The solid line represents Eq. (49). +(b) Phase diagram based on diffusivity in the LD, MC, and +HD regimes. +as M ≡ L − N, i.e., 1−ρ = M/L. Therefore, for the HD +regime, ρ = (L − M)/L ∼ 1 for L → ∞ and M ≪ L. +Using Eq. (24), the disorder average of the diffusion co- +efficient is given by +⟨D(L)⟩dis ∼ + + + + + + + + + +N −1/2 +�σ2 +µ3 +� +dis +(LD regime) +M 3/2L−2 +�σ2 +µ3 +� +dis +(HD regime) +(50) + +9 +(a) +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +103 +102 +103 +104 +105 +106 +α = 0.8 +α = 1.8 +α = 0.4 +⟨D⟩dis(Lρ)1/2(1 − ρ)−3/2 +L +(b) +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +102 +103 +104 +105 +106 +α = 2.5 +α = 1.5 +α = 0.8 +α = 0.4 +⟨D⟩dis +L +FIG. 7. Disorder average of the diffusion coefficient as a func- +tion of L for several α: (a) LD and HD regimes and (b) MC +regimes. +Squares are the results of numerical simulations, +where we calculated the diffusion coefficient (Eqs. (24) and +(31)) for different disorder realizations by Monte Carlo sim- +ulations (N = 1 for (a) and ρ = 0.5 for (b)). We used 104 +disorder realizations. Solid lines show the asymptotic results, +i.e., Eqs. (53) and (61). +for L → ∞. When the second moment of the waiting +time ⟨τ 2⟩ ≡ +� ∞ +0 +τ 2φα(τ)dτ is finite (α > 2), we have +σ2 → ⟨τ 2⟩ (L → ∞) by the law of large numbers. It +follows that the disorder average of D(L) is finite and +given by +⟨D(L)⟩dis ∼ + + + + + + + + + +N −1/2 ⟨τ 2⟩ +⟨τ⟩3 +(LD regime) +M 3/2L−2 ⟨τ 2⟩ +⟨τ⟩3 +(HD regime) +(51) +for L → ∞ and α > 2. Hence, the diffusion coefficient +become non-zero constant for the LD regime, whereas it +becomes 0 for the HD regime. +For α < 2, the second moment of the waiting time di- +verges. The disorder average of σ2/µ3, which was derived +in Ref. [32], is obtained as +�σ2 +µ3 +� +dis +∝ +� +L2−α +(1 < α < 2) +L2−1/α +(α < 1). +(52) +Therefore, the disorder average of the diffusion coefficient +is given by +⟨D(L)⟩dis ∝ +� +L2−α +(1 < α < 2) +L2−1/α +(α < 1) +(53) +for the LD regime and +⟨D(L)⟩dis ∝ +� +L−α +(1 < α < 2) +L−1/α +(α < 1) +(54) +for the HD regime, respectively. +Hence, the diffusion +coefficient for the LD regime diverges for 1 < α < 2 and +1/2 < α < 1, whereas it becomes 0 for α < 1/2 (see +Fig. 7(a)). The diffusion coefficient for the HD regime +becomes 0 for all α. The zero diffusion coefficient is a +signature of many-body effect. +Let us consider the SA property for the diffusion co- +efficient in LD and HD regimes. The SA parameter is +defined as +SA(L; D) ≡ ⟨D(L)2⟩dis − ⟨D(L)⟩2 +dis +⟨D(L)⟩2 +dis +. +(55) +The SA parameter goes to 0 in the large-L limit when +the diffusion coefficient is SA. +For α > 2, the second moment of waiting times exists; +i.e., ⟨τ 2⟩ = � ∞ +0 +τ 2ψα(τ)dτ. +Thus, σ2/µ3 converges to +⟨τ 2⟩ / ⟨τ⟩2 for L → ∞. Therefore, ⟨D(L)2⟩dis −⟨D(L)⟩2 +dis +converges to 0 for L → ∞, so that the diffusion coefficient +is SA for α > 2. +For 1 < α < 2, the second moment of σ2/µ3 was cal- +culated in Ref. [32]. The SA parameter diverges as +SA(L; D) ∝ ⟨D(L)2⟩dis +⟨D(L)⟩2 +dis +∝ Lα−1 +(56) +for L → ∞. Therefore, the diffusion coefficient is non-SA +for 1 < α < 2. +For α < 1, both the first and the second moments of +the waiting times diverge. σ2/µ3 can be represented as +σ2 +µ3 = L2−1/αC(L), +(57) +where C(L) = L1/α �L +i=1 τ 2 +i /(�L +i=1 τi)3 is a random vari- +able depending on the disorder realization. Hence, the +SA parameter becomes +SA(L; D) = ⟨D(L)2⟩dis +⟨D(L)⟩2 +dis +− 1 = ⟨C(L)2⟩dis +⟨C(L)⟩2 +dis +− 1. +(58) +Because �L +i=1 τ 2 +i < (�L +i=1 τ)3, 1/(�L +i=1 τi)3 < C(L) < +1, i.e., 0 < ⟨C(L)⟩dis < 1 and 0 < ⟨C(L)2⟩dis < 1, the SA +parameter is a finite value, i.e., the diffusion coefficient +is non-SA for α < 1. These results are the same as those +for the QTM. +2. +MC regime +When the system size is increased, we find a deeper and +deeper energy trap, that is, τm gets longer and longer. +Hence, Eq. (26) can be approximated as ⟨T 2 +m⟩ − ⟨Tm⟩2 ∼ +τ 2 +m, i.e., we can approximate the diffusion coefficient: +D ∼ ρ−2 +τm +. +(59) +By Eq. (41), the diffusion coefficient can be represented +by +D(L) ∼ +ρ−2 +τcL1/α +τcL1/α +τm +∼ +ρ−2 +τcL1/α Y −1 +α +(60) + +10 +for L → ∞. Therefore, the PDF of the diffusion coeffi- +cient is also described by the Weibull distribution. Using +the first moment of the Weibull distribution, we obtain +the exact asymptotic behavior of the disorder average of +the diffusion coefficient, +⟨D(L)⟩dis ∼ +ρ−2 +τcL1/α Γ(1 + 1/α). +(61) +Therefore, the diffusion coefficient also decreases with the +system size L (see Fig. 7(b)). +Next, we consider the SA parameter of the diffusion +coefficient in the MC regime. Using Eq. (60), we have +lim +L→∞ SA(L; D) = ⟨Y −2 +α ⟩ − ⟨Y −1 +α ⟩ +2 +⟨Y −1 +α ⟩ +2 += Γ (1 + 2/α) +Γ (1 + 1/α)2 − 1, +(62) +which is the same as the SA parameter for the maximal +current (see Fig. 6(a)). The transition point from SA to +non-SA, which exists for the LD and HD regimes, dis- +appears, and the diffusion coefficient is non-SA for all α +(see Fig. 6(b)). +VII. +CONCLUSION +In this paper, we have studied the TASEP on a +quenched random energy landscape. In the LD and HD +regimes, i.e., the dilute limit, the dynamics of the dis- +ordered TASEP can be approximately described by the +single-particle dynamics. +On the other hand, the dy- +namics in the MC regime become completely different +from that in the dilute limit due to the many-body ef- +fect. In particular, the LD and HD phases coexist in the +MC regimes. By renewal theory, we provided exact re- +sults for the current and diffusion coefficient. In the LD +regime, the disorder average of the diffusion coefficient +becomes 0 for α < 1/2, diverges for 1/2 < α < 2, and is +non-zero constant for α > 2, which is the same as in the +single-particle dynamics (Fig. 6(b)). On the other hand, +in the HD and MC regimes, it becomes 0 in the large-L +limit for all α (Fig. 6(b)) due to the many-body effect. +Moreover, we introduced the SA parameter to quantify +the SA property. We obtained a self-averaging and non- +self-averaging transition for the current and the diffusion +coefficient in the LD and HD regimes, which is the same +as in the single-particle dynamics. However, in the MC +regime, the current and diffusion coefficient are non-SA +for all α, which is different from the LD and HD regimes. +Therefore, many-body effects in quenched random energy +landscapes decrease the diffusion coefficient and lead to +a strong non-self-averaging feature. +ACKNOWLEDGMENTS +We thank K. Saito for fruitful discussions. T.A. was +supported by JSPS Grant-in-Aid for Scientific Research +(No. C JP21K033920). +Appendix A: Passage time distribution +In this Appendix, we derive the distribution of the pas- +sage time Tm site m in the MC regime, where m is the +site with the maximal mean waiting time. The passage +time can be divided into the hole escape time xm and +the particle escape time ym. At time t1, a particle es- +capes from site m. +At time t2, the subsequent parti- +cle arrives at site m. +The hole escape time is defined +as xm = t2 − t1 (Fig. 8). At time t3, the particle es- +capes from site m. The particle escape time is defined as +ym = t3 − t2 (Fig. 8). To obtain the hole escape time at +site m, we consider the hole dynamics. At site m, when +the hole jump succeeds by the ith attempt, the PDF of +the hole escape time xm follows the distribution of the +sum of i IID variables following the exponential distri- +bution, ψm−1(t) = τ −1 +m−1 exp (−t/τm−1), i.e., the Erlang +distribution +Er(xm; i, τm−1) = +xi−1 +m +(i − 1)!τ i +m−1 +exp +� +− xm +τm−1 +� +, +(A1) +and the success probability is given by ρm−1(1 − +ρm−1)i−1. Therefore, the PDF f(xm) of xm follows the +exponential distribution +f(xm) =ρm−1 +∞ +� +i=1 +(1 − ρm−1)i−1Er(xm; i, τm−1) +=ρm−1 +τm−1 +exp +� +− xm +τm−1 +� +× +∞ +� +i=1 +1 +(i − 1)! +�(1 − ρm−1)xm +τm−1 +�i−1 +=Ex +� +xm; τm−1 +ρm−1 +� +, +(A2) +where Ex(x; τ) ≡ exp (−x/τ)/τ is the exponential distri- +bution. +Because a particle can not escape from site m until the +neighbor site becomes empty, we must consider the effect +k +? +? +m +? +particle jump +k +? +m +? +? +hole jump +k +? +? +m +? +particle jump +t +t1 +t2 +t3 +xm +ym +Tm +FIG. 8. +Particle dynamics during the passage time. +The +filled and dashed-line circles denote particles and holes, re- +spectively. The question mark is either a particle or a hole. + +11 +of site m + 1. Using the same way of the derivation of +Eq. (A2), the PDF g(ym+1) of the particle escape time +ym+1 at site m + 1 is given by +g(ym+1) = Ex +� +ym+1; +τm+1 +1 − ρm+2 +� +. +(A3) +Using Eq. (A3), we derive the joint PDF of the hole es- +cape time xm and the particle escape time ym. When the +sum of the hole escape time xm and the particle escape +time ym is larger than the particle escape time ym+1, a +particle at site m can jump to site m+1. When a prticle +succeeds to jump to site m+1 once, i.e., xm+ym > ym+1, +the weighted joint PDF h1(xm, ym) of xm and ym is given +by +h1(xm, ym) = f(xm)Ex(ym; τm) +� xm+ym +0 +dym+1 g(ym+1). +(A4) +When a particle jump succeeds on the ith attempts (i > +1), xm + y′ +m < ym+1 < xm + ym, where y′ +m follows the +Erlang distribution Er(y′ +m; i − 1, τm) and ym is sum of +y′ +m and the IID random variable y with the exponential +distribution Ex(y; τm). +Then, the weighted joint PDf +hi(xm, ym) of xm and ym is given by +hi(xm, ym) = f(xm) +� ym +0 +dy′ +m Ex(ym − y′ +m; τm)Er(y′ +m; i − 1, τm) +� xm+ym +xm+y′m +dym+1 g(ym+1). +(A5) +Therefore, the joint PDF h(xm, ym) of xm and ym is given by +h(xm, ym) = +∞ +� +i=1 +hi(xm, ym) += f(xm)Ex(ym; τm) + +τm+1 +1−ρm+2 +τm − +τm+1 +1−ρm+2 +exp +� +−1 − ρm+2 +τm+1 +xm +� +f(xm)(Ex(ym; τm) − g(ym)). +(A6) +By the convolutional intergration of h(xm, ym), we have the PDF Φ(Tm) of the passage time Tm +Φ(Tm) = +� Tm +0 +dx h(x, Tm − x) += τm +ρm−1 +τm−1 +(ζ1 + ζ2ζ3)Ex(Tm; τm) − ζ1f(Tm) − ζ2g(Tm) + ζ3E +� +Tm; +1 +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +� +, +(A7) +where +ζ1 ≡ +1 +τm +ρm−1 +τm−1 − 1, ζ2 ≡ +1 +τm +1−ρm+2 +τm+1 +− 1 +, ζ3 ≡ +1 +τm +� +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +� +− 1 +. +Next, we derive the mean and variance of the passage time. The Laplace transform of Φ(Tm) with respect to s is +given by +ˆΦ(s) ≡ +� ∞ +0 +dTm e−sTmΦ(Tm) += τm +ρm−1 +τm−1 +(ζ1 + ζ2ζ3) +1 +τms + 1 − +ζ1 +τm−1 +ρm−1 s + 1 − +ζ2 +τm+1 +1−ρm+2 s + 1 + +ζ3 +s +ρm−1 +τm−1 + +1−ρm+2 +τm+1 ++ 1. +(A8) +It follows that the mean and variance of the passage time are given by +⟨Tm⟩ = τm + τm−1 +ρm−1 ++ +ρm−1 +τm−1 +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +τm+1 +1 − ρm+2 +, +(A9) +⟨T 2 +m⟩ − ⟨Tm⟩2 = τ2 +m + +� τm−1 +ρm−1 +�2 ++ +� +τm+1 +1 − ρm+2 +�2 +− +3 +� +ρm−1 +τm−1 + 1−ρm+2 +τm+1 +�2 . +(A10) +Appendix B: Fr´echet distribution +Here, we derive that when random variables follow a +power-law distribution (Eq. (3)), the maximum of those +follows the Fr´echet distribution using the extreme value + +12 +theory [42]. 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Phys. 63, 47 (1991). + diff --git a/G9AyT4oBgHgl3EQfrfnZ/content/tmp_files/load_file.txt b/G9AyT4oBgHgl3EQfrfnZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b071d65049896b0106490e2a9c3e88a0d95b094d --- /dev/null +++ b/G9AyT4oBgHgl3EQfrfnZ/content/tmp_files/load_file.txt @@ -0,0 +1,951 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf,len=950 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='00563v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='stat-mech] 2 Jan 2023 Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorders Issei Sakai1 and Takuma Akimoto1, ∗ 1Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan (Dated: January 3, 2023) We consider the totally asymmetric simple exclusion processes on quenched random energy land- scapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We show that the current and the diffusion coefficient differ from those for homogeneous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The current is almost constant and becomes the maximal value in the intermediate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', non-self-averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' INTRODUCTION The one-dimensional asymmetric simple exclusion pro- cess (ASEP) is a pedagogical model for non-equilibrium systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In particular, it describes various non- equilibrium phenomena such as traffic flow [2] and pro- tein synthesis by ribosomes [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The ASEP is a stochas- tic process where particles with hard-core interactions diffuse on a one-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The ASEP can be mapped to a model of interface growth in the Kardar- Parisi-Zhang (KPZ) universality class [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hopping to the right site in the ASEP corresponds to an increase in the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The distribution of interface height was solved [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the weak asymmetric limit of the ASEP, the KPZ equation was rigorously solved an- alytically [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, the large deviation func- tion of the time-averaged current was obtained [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The ASEP has been extended in various ways such as Brownian ASEP [14], non-Poissonian hopping rates [15], and disordered hopping rates [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When particles only hop to uni-direction, it is called the totally ASEP (TASEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For TASEPs, it is well known that the current- density relation is given by [1] J = 1 τ ρ(1 − ρ), (1) where J is the particle current, ρ is particle density, and τ is the inverse of the jump rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the mean waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [24], the variance of the tagged particle displacement, δxt, in time t is derived as a func- ∗ takuma@rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='tus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='jp tion of ρ: ⟨δx2 t ⟩ − ⟨δxt⟩2 t ∼ √π 2τ (1 − ρ)3/2 (Lρ)1/2 (2) for L → ∞ and t → ∞, where ⟨·⟩ is the ensemble average and L is the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Effects of disorder in the ASEP have been investi- gated for decades [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Due to the disorder in the ASEP under the periodic boundary condition, a current- density relation deviates from that in the ASEP with a homogeneous jump rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' More precisely, it becomes flat and the current is maximized on the flat regime [16, 18–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, in the flat regime, the low- and high-density phases coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the ASEP on networks, the flat regime also exists [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Under the open boundary condition, the first-order phase transition point between the low- and high-density phases depends on the disorder [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Random walks in heterogeneous environments show anomalous diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The heterogeneous environment is characterized by a random energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' There are two types of random energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' One is an an- nealed energy landscape, where the landscape randomly changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The continuous-time random walk is a diffusion model on the annealed energy landscape, and its mean-squared displacement shows anomalous dif- fusion when the mean waiting time diverges [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The other is a quenched energy landscape, where the land- scape is configured randomly and does not change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The quenched trap model (QTM) is a diffusion model on the quenched energy landscape [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The mean- squared displacement of the QTM on an infinite system shows anomalous diffusion when the mean waiting time diverges [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the QTM on a finite system, the dif- fusion coefficient exhibits sample-to-sample fluctuations 2 [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The diffusivity of interacting many-body sys- tems on the annealed energy landscape has been inves- tigated [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, the diffusivity of interacting many-body systems on the quenched energy landscape has never been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Such a heterogeneous envi- ronment is realized experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In protein synthesis by ribosomes, the codon decoding times become hetero- geneous due to the heterogeneity of transfer RNA concen- tration [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In other words, the distribution of the waiting time depends on the site, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', ribosomes diffuse on the quenched random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' There are other diffusion phenomena in such heterogeneous environments, such as train delays, proteins on DNA [35, 36], and water trans- portation in aquaporin [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In this paper, we investigate sample-to-sample fluctu- ations of the diffusivity for the TASEP on a quenched random energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In our previous study, we show sample-to-sample fluctuations of the current [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When an observable does not depend on the disorder re- alization, it is called self-averaging [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the QTM, it is known that the diffusion coefficient [30–32], the mobility [32], and the mean first passage time [39] are non-self-averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Is such a non-self-averaging behavior still observed when the N-body effect is introduced in the quenched random energy landscape?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This is a non- trivial question in diffusion in a heterogeneous environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In particular, it is non-trivial that the TASEP with disordered waiting-time distributions exhibits sample-to- sample fluctuations for the current and the diffusion co- efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, it is important to provide an exact result for the current and the diffusion coefficient in het- erogeneous quenched environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' II, we formu- late the TASEP on a quenched random energy landscape and define averaging procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' III, we show the numerical results of the current-density relation and the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' IV, we present derivations of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' V, we present derivations of the current and the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' VI, we dis- cuss the self-averaging properties of the current and the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' VII, we conclude this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Appendix A, we derive the passage time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In Appendix B, we derive the Fr´echet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' MODEL We consider the TASEP on a quenched random en- ergy landscape on a one-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' It com- prises N particles on the lattice of L sites with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Each site can hold at most one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Quenched disorder means that when realizing the random energy landscape, it does not change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At each lattice point, the depth E > 0 of the en- ergy trap is randomly assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The depths are indepen- dent identically distributed (IID) random variables with an exponential distribution, φ(E) = T −1 g exp (−E/Tg), where Tg is called the glass temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' A particle can escape from a trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Escape times from a trap are IID random variables following an exponential distribu- tion and follow the Arrhenius law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the mean escape time of the kth site is given by τk = τc exp (Ek/T ), where Ek is the depth of the energy at site k, T the temperature, and τc a typical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The probability of the escape time τ that is smaller than x is given by Pr(τ ≤ x) ∼= Pr(E ≤ T ln(x/τc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' It follows that the probability density function (PDF) ψα(τ) of waiting times follows a power-law distribution: � ∞ τ dτ ′ψα(τ ′) = � τ τc �−α (τ ≥ τc) (3) with α ≡ T/Tg [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The dynamics of the particle are described by the Markovian one in the sense that the waiting time is memory-less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In particular, the waiting times at site k are assigned IID random variables following an exponen- tial distribution, ψk(ti) = τ −1 k exp (−ti/τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' After the waiting time elapses, the particle attempts to hop the neighboring site on its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The hop is accepted only if the site is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the attempt is a success or failure, the particle is assigned a new waiting time from ψk+1(ti) or ψk(ti), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Here, we consider three averaging procedures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', en- semble average, disorder average, and time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The ensemble average of observable O(t) is an average with respect to a stationary ensemble for a single disorder re- alization denoted by ⟨O(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The disorder average of ob- servable O(t) is an average with respect to different dis- order realizations denoted by ⟨O(t)⟩dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The time average of observable O(t) is defined by ¯O(T ) = 1 T � T 0 O(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (4) Furthermore, we consider a stationary initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the ASEP on a finite system, the variance of the displacement of the tagged particle depends on whether the initial conditions are identical or not, especially for a short time [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, the asymptotic behavior does not depend on the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In this paper, we are interested in the asymptotic behavior of the current and the diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the initial conditions in this paper are not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In numerical simulations, particles start from the stationary ensemble of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The stationary ensemble is given by the configuration after particles arrange randomly and diffuse for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' NUMERICAL RESULTS OF CURRENT-DENSITY RELATION AND DENSITY PROFILE We numerically show that the current-density relation for a disordered TASEP (DTASEP) deviates from that for a TASEP with a homogeneous jump rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the homogeneous TASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Figure 1 shows the steady-state 3 current J against particle density ρ = N/L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the current-density relation, for a DTASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For low and high densities, the current-density relation is the same as that of the homogeneous TASEP (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, there is a distinct difference between them in the intermedi- ate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In particular, the current for the DTASEP becomes almost flat and smaller than that for the ho- mogeneous TASEP in the intermediate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' On the other hand, there is no flat regime for the homogeneous TASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The flat regime in the DTASEP is observed in other disordered systems [16, 18–20, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, it is a manifestation of the existence of a disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In this regime, the current is independent of the particle density and maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the following, we classify the density into three regimes: the low density (LD) (0 < ρ ≤ ρ∗), the maximal current (MC) (ρ∗ < ρ < 1 − ρ∗), and the high density (HD) (1 − ρ∗ ≤ ρ < 1) regimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We explicitly derive the transition density ρ∗ later (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Here, we numerically show the density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the LD and HD regimes, the system is homogeneous on a macroscopic scale (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2(a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the MC regime, there is a macroscopic density segregation (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The segregation is classified into high- and low-density phases by the deepest trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Com- paring Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2(c) and (d), we observe that the high- density regime becomes large when the particle density is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This result is qualitatively similar to that in a system with one defect bond, studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='" #" $% 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 J ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Current-density relations for homogeneous and dis- ordered TASEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The circles are obtained by the numerical simulation of dynamics of the DTASEP (L = 5000, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, and τc = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The solid line represents the current-density relation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (1), for the homogeneous TASEP with τ being set to equal to the sample average of the waiting times of the DTASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' ρ∗ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' DERIVATION OF THE DENSITY PROFILE Here, we derive the density profile by the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This derivation is almost the same as our previous study [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Let Jk be the mean current across the bond between site k and k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the DTASEP, a hop occurs with a rate 1/τk whenever site k is occupied, and site k + 1 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, the mean current is represented by Jk = � 1 τk nk(1 − nk+1) � , (5) where nk denotes the number of a particle, which is 1 if the site k is occupied and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' If the sys- tem is in a steady state, the ensemble average is equal to the time average in the long-time limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the sys- tem is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The ensemble average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (5) coin- cides with the long-time average if the system is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The periodic boundary condition implies nL+1 = n1 and τL+1 = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The probability of finding a particle at site k is given by ρk = ⟨nk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the mean-field approximation, one can ignore correlations between nk and nk+1, which means ⟨nknk+1⟩ = ⟨nk⟩ ⟨nk+1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (6) In the steady state, the site densities are time- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, from the continuity of the cur- rent, the current is independent of k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', Jk = J for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, we have the current-density relation: J = 1 τk ρk(1 − ρk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (7) We note that the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (7) is independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We derive a simpler form of the site density by approx- imating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (7) for the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the LD regime, we can assume ρkρk+1 ≪ 1 because the particle density is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Ignoring ρkρk+1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (7), we obtain J ∼= 1 τk ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (8) Using the conservation of particles, � i ρi = N, the site density has the form ρk ∼= τk µ ρ, (9) for the LD regime, where µ is the sample average of the waiting times, µ = � i τi/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This result is the same as the steady-state density for the QTM [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the HD regime, the particle density is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the hole density, σk = 1 − ρk, instead of ρk, we can derive the site density in the same way as in the LD regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The result becomes ρk = 1 − σk ∼= 1 − τk−1 µ (1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (10) 4 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='25 0 1000 2000 3000 4000 5000 ρk k (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='95 1 0 1000 2000 3000 4000 5000 ρk k (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 0 1000 2000 3000 4000 5000 ρk k (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 0 1000 2000 3000 4000 5000 ρk k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Density profiles: (a) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='01, (b) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='99, (c) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, and (d) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 (L = 5000, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, and τc = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The squares are the results of the numerical simulation of the dynamics of the DTASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Triangles are Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (9) and (10) for (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Figures 2(a) and 2(b) show the density profiles for LD and HD regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The densities are well de- scribed by the set of site densities {ρk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, Eqs (9) and (10) are good approximated forms of the site densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The results for the LD and HD regimes re- produce the current-density relation for a homogeneous TASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In other words, the system is homogeneous on a macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Next, we approximately obtain a density ρ∗ which is the boundary density between LD and MC regimes in the current-density relation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (7), the current between sites m and m + 1 is given by ρm(1 − ρm+1)/τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The steady-state current at the boundary density between the LD and MC regimes can be described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At the boundary density, the current between sites m and m + 1 is equal to the steady-state current: 1 τm ρm(1 − ρm+1) ∼= 1 µρ∗(1 − ρ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (11) We find numerically find that the site with the maximal mean waiting time is always the boundary the HD and the LD phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the mean waiting time is maxi- mized at site m, sites m and m + 1 exist in high- and low-density phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The site densities are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (10) and the hole density is ρ∗ in the high- density phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' On the other hand, the site densities are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (9) and the particle density is ρ∗ in the low-density phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (9) into ρm and ρm+1, respectively, 1 τm � 1 − τm−1 µ ρ∗ � � 1 − τm+1 µ ρ∗ � ∼= 1 µρ∗(1 − ρ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (12) Solving this equation for ρ∗, we have ρ∗ ∼µ(τm−1 + τm + τm+1) 2(τm−1τm+1 + τmµ) − µ � (τm−1 + τm + τm+1)2 − 4(τm−1τm+1 + τmµ) 2(τm−1τm+1 + τmµ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (13) This formula depends crucially on the disorder realiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the following, we assume that the mean waiting time is maximized at site m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For L → ∞, τm is much longer than τm−1 and τm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (13) can be approximated as ρ∗ ∼ 1 2 − 1 2τm � τ 2m − 4τmµ ∼ µ τm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (14) By the extreme value theory [42], the scaling of τm follows τm = O(L1/α) (15) 5 for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For α > 1, the first moment of the waiting times exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', µ → ⟨τ⟩ ≡ � ∞ 0 τψα(τ)dτ (L → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the scaling of ρ∗ becomes ρ∗ ∝ L−1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (16) For α ≤ 1, the first moment of the waiting times diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The scaling of the sum of τi follows L � i=1 τi = O(L1/α) (17) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' It follows that the scaling of ρ∗ becomes ρ∗ ∼ L−1 � i τi τm ∝ L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (18) Therefore, ρ∗ → 0 for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' DERIVATION OF CURRENT AND DIFFUSIVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' LD and HD regimes Here, we derive the current in the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For single-particle dynamics on the quenched random energy landscape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the QTM, the mean number of events that a particle passes a site until time t is given by [32] ⟨Qt⟩ t ∼ 1 Lµ (t → ∞), (19) where Qt is the number of events that a particle passes a site until time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the DTASEP in the LD and HD regimes, the current depends on the particle density, which is identical for the homogeneous TASEP (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the current in the LD and HD regimes is given by J ∼ aρ(1 − ρ) (20) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When ρ = 1/L, the current is equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (19) for L → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the constant a is given by a = 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, we have the current in the LD and HD regimes: J ∼ 1 µρ(1 − ρ) (21) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Next, we derive the diffusion coefficient in the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' δxt denotes the displacement of the tagged particle until time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the QTM, the variance of the displacement is given by [32] lim t→∞ ⟨δx2 t ⟩ − ⟨δxt⟩2 t ∼ σ2 µ3 (22) for L → ∞, where σ2 is the sample mean of the squared waiting times, σ2 = � i τ 2 i /L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the DTASEP in the LD and HD regimes, the variance of the displacement depends on the particle density, which is identical for the homogeneous TASEP (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the diffusion coefficient, D ≡ limt→∞(⟨δx2 t ⟩ − ⟨δxt⟩2)/t, is given by D ∼ b √π 2 (1 − ρ)3/2 ρ1/2 L−1/2 (23) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When ρ = 1/L, the diffusion coefficient is equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (22) for L → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the constant b is given by b = 2σ2/µ3√π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The diffusion coefficient in the LD and HD regimes is given by D ∼ σ2 µ3 (1 − ρ)3/2 ρ1/2 L−1/2 (24) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' MC regime Here, we derive the maximal current and the diffusion coefficient in the MC regime by the renewal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We define the passage time as a time interval between consec- utive events that particles pass a site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We note that the passage time differs from the first passage time because the particles which pass a site are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the target site is m, the mean and the variance of the passage time Tm are obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [38] (see also Appendix A): ⟨Tm⟩ = τm + τm−1 ρm−1 + ρm−1 τm−1 ρm−1 τm−1 + 1−ρm+2 τm+1 τm+1 1 − ρm+2 , (25) ⟨T 2 m⟩ − ⟨Tm⟩2 =τ 2 m + � τm−1 ρm−1 �2 + � τm+1 1 − ρm+2 �2 − 3 � ρm−1 τm−1 + 1−ρm+2 τm+1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (26) We consider the number of events Qt that particles pass site m until time t to obtain the maximal current and the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the LD and HD regimes, the density profile is homogeneous on a macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, local densities around the target site are fluc- tuating, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', dense or dilute, which affects the passage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the passage times are not IID random variables for the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For the MC regime, macroscopic density segregation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the target locates site m, particles are constantly dense on the left of the target and dilute on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This segregation does not vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the passage times are con- sidered to be IID random variables for MC regime and the process of Qt can be described by a renewal process [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' By renewal theory [43], the mean number of re- newals becomes ⟨Qt⟩ ∼ t/ ⟨Tm⟩ for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The current is represented through the mean number of the passing events until time t: J = limt→∞ ⟨Qt⟩ /t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, we have Jmax ∼ 1 ⟨Tm⟩ (27) 6 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 J ρ ×10−5 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='004 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 J ρ (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 J ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Current-density relation for different α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, (b) α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, and (c) α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, where the fixed quenched disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The circles are obtained by the numerical simulation of the dynamics of the DTASEP (L = 1000 for (a) and 5000 for other cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The dashed and the solid lines represent Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (21) and (27), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (a) 10−6 10−5 10−4 10−3 10−2 10−1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 D ρ (b) 10−6 10−5 10−4 10−3 10−2 10−1 100 101 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 D ρ (c) 10−6 10−5 10−4 10−3 10−2 10−1 100 101 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 1 D ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Diffusion coefficient-density relation for different α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, (b) α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, and (c) α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5, where the fixed quenched disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The circles are obtained by the numerical simulation of the dynamics of the DTASEP (L = 100 for (a), 500 for (b), and 1000 for (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The dashed and the solid lines represent Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (24) and (31), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The current depends on the disorder real- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Figure 3 shows a good agreement between nu- merical simulations and the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the number of the passing events, we can de- rive the mean displacement and the variance of the dis- placement of a tagged particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' While the tagged particle starting from site m+1 returns to site m+1, all particles pass between site m and site m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, in the large-t limit, the displacement, δxt, is represented by δxt ∼ LQt N = Qt ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (28) By renewal theory [43], the mean displacement and the variance of the displacement are represented by ⟨δxt⟩ ∼ ⟨Qt⟩ ρ ∼ t ρ⟨Tm⟩, (29) ⟨δx2 t ⟩ − ⟨δxt⟩2 ∼ 1 ρ2 (⟨Q2 t⟩ − ⟨Qt⟩2) ∼ 1 ρ2 ⟨T 2 m⟩ − ⟨Tm⟩2 ⟨Tm⟩3 t (30) for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the diffusion coefficient for the MC regimes is given by D ∼ 1 ρ2 ⟨T 2 m⟩ − ⟨Tm⟩2 ⟨Tm⟩3 (31) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Figure 4 shows a good agreement between numerical simulations and the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' SAMPLE-TO-SAMPLE FLUCTUATIONS OF CURRENT AND DIFFUSIVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Current Here, we consider sample-to-sample fluctuations of the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' To quantify the self-averaging (SA) property of the current, we consider the SA parameter defined as [30] SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' J) ≡ ⟨J(L)2⟩dis − ⟨J(L)⟩2 dis ⟨J(L)⟩2 dis , (32) 7 where J(L) is the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' If the SA parameter becomes 0, there is no sample-to-sample fluctuation, which means SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' LD and HD regimes Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (21), the SA parameter becomes SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' J) = ⟨1/µ2⟩dis − ⟨1/µ⟩2 dis ⟨1/µ⟩2 dis , (33) which is the same as the SA parameter for the diffusion coefficient in the QTM [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the mean waiting time ⟨τ⟩ ≡ � ∞ 0 τψα(τ)dτ is finite (α > 1), we have µ → ⟨τ⟩ (L → ∞) by the law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, in the large-L limit, the current does not depend on the disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the current is SA for α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the mean waiting time diverges (α ≤ 1), the law of the large numbers breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, the generalized central limit theorem is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The PDF of the nor- malized sum of the waiting times follows the one-sided L´evy distribution [44], �L i=1 τi L1/α ⇒ Xα (L → ∞), (34) where Xα is a random variable following the one-sided L´evy distribution of index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The PDF of Xα denoted by lα(x) with x > 0 is given by [44] lα(x) = − 1 πx ∞ � k=1 Γ(kα + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (−cx−α)k sin (kπα), (35) (a) 10−12 10−10 10−8 10−6 10−4 10−2 100 102 103 104 105 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='7 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='3 ⟨J⟩dis(ρ(1 − ρ))−1 L (b) 10−12 10−10 10−8 10−6 10−4 10−2 100 102 103 104 105 106 α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='3 ⟨Jmax⟩dis L FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Disorder average of the current as a function of L for several α: (a) LD and HD regimes and (b) MC regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Solid lines show the asymptotic results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (38) and (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Squares are the results of numerical simulations, where we calculated the maximal currents (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (27)) for different disorder realizations by Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We used 104 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Triangles are the results of the nu- merical simulation of dynamics of the DTASEP (N = 1 for (a) and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 for (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We used 103 for L = 104 in the MC regime and 104 disorder realizations for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' where c = Γ(1 − α)τ α c is the scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The first and the second moment of X−1 α are given by [30] ⟨X−1 α ⟩ = Γ(1/α) αc1/α , ⟨X−2 α ⟩ = Γ(2/α) αc2/α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (36) The current can be represented by J(L) ∼ ρ(1 − ρ) L L1/α L1/α �L k=1 τk ∼ ρ(1 − ρ)L1−1/αX−1 α (37) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, the PDF of J is described by the in- verse L´evy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the first moment of the in- verse L´evy distribution [30], we obtain the exact asymp- totic behavior of the disorder average of the current, ⟨J(L)⟩dis ∼ ρ(1 − ρ)Γ(α−1) ατcΓ(1 − α)1/α L1−1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (38) Hence, the current becomes 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We note that since the scaling of ρ∗ follows Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (18), we do not simulate at the same density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the first and the second moments of 1/µ, we have the SA parameter lim L→∞ SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' J) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 (α > 1) αΓ(2/α) Γ(1/α)2 − 1 (α ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (39) For α ≤ 1, the SA parameter is a nonzero constant, and thus J becomes non-SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, there is a transition of SA property in the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' MC regime When the system size is increased, we find a deeper and deeper energy trap, that is, τm gets longer and longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (25) can be approximated as ⟨Tm⟩ ∼ τm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', we can approximate the maximal current: Jmax ∼ 1 τm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (40) Since the PDF of the waiting times follow a power-law distribution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (3), the PDF of the normalized τm fol- lows the Fr´echet distribution [42]: τm τcL1/α ⇒ Yα (L → ∞), (41) where Yα is a random variable following the Fr´echet dis- tribution of index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' As derived in Appendix B, the PDF of Yα, denoted fα(y) with y > 0, can be expressed as fα(y) = αy−α−1 exp (−y−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (42) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (41), the maximal current can be represented by Jmax(L) ∼ 1 τcL1/α τcL1/α τm ∼ 1 τcL1/α Y −1 α (43) 8 for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, the PDF of Jmax is described by the inverse Fr´echet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The PDF of Y −1 α can be explicitly represented by the Fr´echet distribution: Pr(Y −1 α ≤ z) = Pr(Yα ≥ z−1) = � ∞ z−1 fα(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (44) The distribution is the Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We obtain the PDF of Y −1 α , denoted by wα(z): wα(z) = αzα−1 exp (−zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (45) The first and second moments of the Weibull distribution are given by ⟨Y −1 α ⟩ = Γ � 1 + 1 α � , ⟨Y −2 α ⟩ = Γ � 1 + 2 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (46) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (46), we obtain the exact asymptotic behavior of the disorder average of the maximal current, ⟨Jmax(L)⟩dis ∼ 1 τcL1/α Γ � 1 + 1 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (47) Therefore, the maximal current decreases with the sys- tem size L (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Let us consider the SA property for the maximal cur- rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The SA parameter is defined as SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Jmax) ≡ ⟨Jmax(L)2⟩dis − ⟨Jmax(L)⟩2 dis ⟨Jmax(L)⟩2 dis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (48) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (43), we have lim L→∞ SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Jmax) = ⟨Y −2 α ⟩ − ⟨Y −1 α ⟩ 2 ⟨Y −1 α ⟩ 2 = Γ (1 + 2/α) Γ (1 + 1/α)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (49) The SA parameter becomes a nonzero constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the maximal current becomes non-SA (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' This result differs from LD and HD, and there is no transition from SA to non-SA behavior for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Diffusivity Here, we consider sample-to-sample fluctuations of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the homogeneous TASEP, the diffusion coefficient becomes 0 for L → ∞ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (2)) be- cause of the many-body effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' D = 0 in the homoge- neous TASEP on a finite system implies the subdiffusion in that on an infinite system [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' LD and HD regimes For the LD regime, ρ = N/L and 1 − ρ ∼ 1 for L → ∞ and N ≪ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We define the number of holes (a) 10−2 10−1 100 101 102 103 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 3 SA α (b) α ρ ρ∗ 1 − ρ∗ 1 0 non-SA non-SA SA SA non-SA 1/2 2 ⟨D⟩dis = 0 ⟨D⟩dis = ∞ ⟨D⟩dis = 0 ⟨D⟩dis = 0 ⟨D⟩dis > 0 ⟨D⟩dis = 0 LD MC HD FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (a) Self-averaging parameter as a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The squares and circles are the results of numerical simulations, where we calculated the maximal currents (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (27)) and the diffusion coefficient (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (31)) for different disorder realiza- tions by Monte Carlo simulations (L = 105), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The triangles show the self-averaging parameter of the maxi- mal current obtained by the numerical simulation of the dy- namics of the DTASEP (L = 1000 and N = 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We used 104 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The solid line represents Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (b) Phase diagram based on diffusivity in the LD, MC, and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' as M ≡ L − N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', 1−ρ = M/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, for the HD regime, ρ = (L − M)/L ∼ 1 for L → ∞ and M ≪ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (24), the disorder average of the diffusion co- efficient is given by ⟨D(L)⟩dis ∼ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 N −1/2 �σ2 µ3 � dis (LD regime) M 3/2L−2 �σ2 µ3 � dis (HD regime) (50) 9 (a) 10−4 10−3 10−2 10−1 100 101 102 103 102 103 104 105 106 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 ⟨D⟩dis(Lρ)1/2(1 − ρ)−3/2 L (b) 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 102 103 104 105 106 α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='8 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='4 ⟨D⟩dis L FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Disorder average of the diffusion coefficient as a func- tion of L for several α: (a) LD and HD regimes and (b) MC regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Squares are the results of numerical simulations, where we calculated the diffusion coefficient (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (24) and (31)) for different disorder realizations by Monte Carlo sim- ulations (N = 1 for (a) and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='5 for (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We used 104 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Solid lines show the asymptotic results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (53) and (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the second moment of the waiting time ⟨τ 2⟩ ≡ � ∞ 0 τ 2φα(τ)dτ is finite (α > 2), we have σ2 → ⟨τ 2⟩ (L → ∞) by the law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' It follows that the disorder average of D(L) is finite and given by ⟨D(L)⟩dis ∼ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 N −1/2 ⟨τ 2⟩ ⟨τ⟩3 (LD regime) M 3/2L−2 ⟨τ 2⟩ ⟨τ⟩3 (HD regime) (51) for L → ∞ and α > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the diffusion coefficient become non-zero constant for the LD regime, whereas it becomes 0 for the HD regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For α < 2, the second moment of the waiting time di- verges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The disorder average of σ2/µ3, which was derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [32], is obtained as �σ2 µ3 � dis ∝ � L2−α (1 < α < 2) L2−1/α (α < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (52) Therefore, the disorder average of the diffusion coefficient is given by ⟨D(L)⟩dis ∝ � L2−α (1 < α < 2) L2−1/α (α < 1) (53) for the LD regime and ⟨D(L)⟩dis ∝ � L−α (1 < α < 2) L−1/α (α < 1) (54) for the HD regime, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the diffusion coefficient for the LD regime diverges for 1 < α < 2 and 1/2 < α < 1, whereas it becomes 0 for α < 1/2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The diffusion coefficient for the HD regime becomes 0 for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The zero diffusion coefficient is a signature of many-body effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Let us consider the SA property for the diffusion co- efficient in LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The SA parameter is defined as SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' D) ≡ ⟨D(L)2⟩dis − ⟨D(L)⟩2 dis ⟨D(L)⟩2 dis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (55) The SA parameter goes to 0 in the large-L limit when the diffusion coefficient is SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For α > 2, the second moment of waiting times exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', ⟨τ 2⟩ = � ∞ 0 τ 2ψα(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Thus, σ2/µ3 converges to ⟨τ 2⟩ / ⟨τ⟩2 for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, ⟨D(L)2⟩dis −⟨D(L)⟩2 dis converges to 0 for L → ∞, so that the diffusion coefficient is SA for α > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For 1 < α < 2, the second moment of σ2/µ3 was cal- culated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The SA parameter diverges as SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' D) ∝ ⟨D(L)2⟩dis ⟨D(L)⟩2 dis ∝ Lα−1 (56) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the diffusion coefficient is non-SA for 1 < α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' For α < 1, both the first and the second moments of the waiting times diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' σ2/µ3 can be represented as σ2 µ3 = L2−1/αC(L), (57) where C(L) = L1/α �L i=1 τ 2 i /(�L i=1 τi)3 is a random vari- able depending on the disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, the SA parameter becomes SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' D) = ⟨D(L)2⟩dis ⟨D(L)⟩2 dis − 1 = ⟨C(L)2⟩dis ⟨C(L)⟩2 dis − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (58) Because �L i=1 τ 2 i < (�L i=1 τ)3, 1/(�L i=1 τi)3 < C(L) < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', 0 < ⟨C(L)⟩dis < 1 and 0 < ⟨C(L)2⟩dis < 1, the SA parameter is a finite value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the diffusion coefficient is non-SA for α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' These results are the same as those for the QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' MC regime When the system size is increased, we find a deeper and deeper energy trap, that is, τm gets longer and longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (26) can be approximated as ⟨T 2 m⟩ − ⟨Tm⟩2 ∼ τ 2 m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', we can approximate the diffusion coefficient: D ∼ ρ−2 τm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (59) By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (41), the diffusion coefficient can be represented by D(L) ∼ ρ−2 τcL1/α τcL1/α τm ∼ ρ−2 τcL1/α Y −1 α (60) 10 for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the PDF of the diffusion coeffi- cient is also described by the Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the first moment of the Weibull distribution, we obtain the exact asymptotic behavior of the disorder average of the diffusion coefficient, ⟨D(L)⟩dis ∼ ρ−2 τcL1/α Γ(1 + 1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (61) Therefore, the diffusion coefficient also decreases with the system size L (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Next, we consider the SA parameter of the diffusion coefficient in the MC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (60), we have lim L→∞ SA(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' D) = ⟨Y −2 α ⟩ − ⟨Y −1 α ⟩ 2 ⟨Y −1 α ⟩ 2 = Γ (1 + 2/α) Γ (1 + 1/α)2 − 1, (62) which is the same as the SA parameter for the maximal current (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The transition point from SA to non-SA, which exists for the LD and HD regimes, dis- appears, and the diffusion coefficient is non-SA for all α (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' CONCLUSION In this paper, we have studied the TASEP on a quenched random energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the LD and HD regimes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the dilute limit, the dynamics of the dis- ordered TASEP can be approximately described by the single-particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' On the other hand, the dy- namics in the MC regime become completely different from that in the dilute limit due to the many-body ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In particular, the LD and HD phases coexist in the MC regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' By renewal theory, we provided exact re- sults for the current and diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' In the LD regime, the disorder average of the diffusion coefficient becomes 0 for α < 1/2, diverges for 1/2 < α < 2, and is non-zero constant for α > 2, which is the same as in the single-particle dynamics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' On the other hand, in the HD and MC regimes, it becomes 0 in the large-L limit for all α (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 6(b)) due to the many-body effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Moreover, we introduced the SA parameter to quantify the SA property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We obtained a self-averaging and non- self-averaging transition for the current and the diffusion coefficient in the LD and HD regimes, which is the same as in the single-particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' However, in the MC regime, the current and diffusion coefficient are non-SA for all α, which is different from the LD and HD regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, many-body effects in quenched random energy landscapes decrease the diffusion coefficient and lead to a strong non-self-averaging feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Saito for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' was supported by JSPS Grant-in-Aid for Scientific Research (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' C JP21K033920).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Appendix A: Passage time distribution In this Appendix, we derive the distribution of the pas- sage time Tm site m in the MC regime, where m is the site with the maximal mean waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The passage time can be divided into the hole escape time xm and the particle escape time ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At time t1, a particle es- capes from site m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At time t2, the subsequent parti- cle arrives at site m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The hole escape time is defined as xm = t2 − t1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At time t3, the particle es- capes from site m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The particle escape time is defined as ym = t3 − t2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' To obtain the hole escape time at site m, we consider the hole dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' At site m, when the hole jump succeeds by the ith attempt, the PDF of the hole escape time xm follows the distribution of the sum of i IID variables following the exponential distri- bution, ψm−1(t) = τ −1 m−1 exp (−t/τm−1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', the Erlang distribution Er(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i, τm−1) = xi−1 m (i − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='τ i m−1 exp � − xm τm−1 � , (A1) and the success probability is given by ρm−1(1 − ρm−1)i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Therefore, the PDF f(xm) of xm follows the exponential distribution f(xm) =ρm−1 ∞ � i=1 (1 − ρm−1)i−1Er(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i, τm−1) =ρm−1 τm−1 exp � − xm τm−1 � × ∞ � i=1 1 (i − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' �(1 − ρm−1)xm τm−1 �i−1 =Ex � xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm−1 ρm−1 � , (A2) where Ex(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τ) ≡ exp (−x/τ)/τ is the exponential distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Because a particle can not escape from site m until the neighbor site becomes empty, we must consider the effect k ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' m ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' particle jump k ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' m ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' hole jump k ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' m ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' particle jump t t1 t2 t3 xm ym Tm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Particle dynamics during the passage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The filled and dashed-line circles denote particles and holes, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The question mark is either a particle or a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 11 of site m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Using the same way of the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A2), the PDF g(ym+1) of the particle escape time ym+1 at site m + 1 is given by g(ym+1) = Ex � ym+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm+1 1 − ρm+2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A3) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A3), we derive the joint PDF of the hole es- cape time xm and the particle escape time ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When the sum of the hole escape time xm and the particle escape time ym is larger than the particle escape time ym+1, a particle at site m can jump to site m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' When a prticle succeeds to jump to site m+1 once, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=', xm+ym > ym+1, the weighted joint PDF h1(xm, ym) of xm and ym is given by h1(xm, ym) = f(xm)Ex(ym;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm) � xm+ym 0 dym+1 g(ym+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A4) When a particle jump succeeds on the ith attempts (i > 1), xm + y′ m < ym+1 < xm + ym, where y′ m follows the Erlang distribution Er(y′ m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i − 1, τm) and ym is sum of y′ m and the IID random variable y with the exponential distribution Ex(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Then, the weighted joint PDf hi(xm, ym) of xm and ym is given by hi(xm, ym) = f(xm) � ym 0 dy′ m Ex(ym − y′ m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm)Er(y′ m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' i − 1, τm) � xm+ym xm+y′m dym+1 g(ym+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A5) Therefore, the joint PDF h(xm, ym) of xm and ym is given by h(xm, ym) = ∞ � i=1 hi(xm, ym) = f(xm)Ex(ym;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm) + τm+1 1−ρm+2 τm − τm+1 1−ρm+2 exp � −1 − ρm+2 τm+1 xm � f(xm)(Ex(ym;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm) − g(ym)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A6) By the convolutional intergration of h(xm, ym), we have the PDF Φ(Tm) of the passage time Tm Φ(Tm) = � Tm 0 dx h(x, Tm − x) = τm ρm−1 τm−1 (ζ1 + ζ2ζ3)Ex(Tm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' τm) − ζ1f(Tm) − ζ2g(Tm) + ζ3E � Tm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' 1 ρm−1 τm−1 + 1−ρm+2 τm+1 � , (A7) where ζ1 ≡ 1 τm ρm−1 τm−1 − 1, ζ2 ≡ 1 τm 1−ρm+2 τm+1 − 1 , ζ3 ≡ 1 τm � ρm−1 τm−1 + 1−ρm+2 τm+1 � − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' Next, we derive the mean and variance of the passage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The Laplace transform of Φ(Tm) with respect to s is given by ˆΦ(s) ≡ � ∞ 0 dTm e−sTmΦ(Tm) = τm ρm−1 τm−1 (ζ1 + ζ2ζ3) 1 τms + 1 − ζ1 τm−1 ρm−1 s + 1 − ζ2 τm+1 1−ρm+2 s + 1 + ζ3 s ρm−1 τm−1 + 1−ρm+2 τm+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A8) It follows that the mean and variance of the passage time are given by ⟨Tm⟩ = τm + τm−1 ρm−1 + ρm−1 τm−1 ρm−1 τm−1 + 1−ρm+2 τm+1 τm+1 1 − ρm+2 , (A9) ⟨T 2 m⟩ − ⟨Tm⟩2 = τ2 m + � τm−1 ρm−1 �2 + � τm+1 1 − ρm+2 �2 − 3 � ρm−1 τm−1 + 1−ρm+2 τm+1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (A10) Appendix B: Fr´echet distribution Here, we derive that when random variables follow a power-law distribution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (3)), the maximum of those follows the Fr´echet distribution using the extreme value 12 theory [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We define τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' , τL as the random variables which follow the power-law distribution with exponent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' The probability for τm = max{τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' , τL} ≤ s is given by Pr(τm ≤ s) = L � i=1 Pr(τi ≤ s) = G(s)L, (B1) where G(s) = Pr(τi ≤ s) = 1 − (s/τc)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' We normalize τm as Xα = τm τcL1/α (B2) for L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' It follows that Pr(Xα ≤ x) = Fα(x) is given by Fα(x) = lim L→∞ G(τcL1/αx)L = exp (−x−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' (B3) Therefore, the normalized τm follows the Fr´echet distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQfrfnZ/content/2301.00563v1.pdf'} +page_content=' [1] B.' metadata={'source': 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restrictive setting where +the underlying repeated game does not change over time. +Much less is known about the +convergence of no-regret learning algorithms in dynamic multiagent settings. In this paper, +we characterize the convergence of optimistic gradient descent (OGD) in time-varying games by +drawing a strong connection with dynamic regret. Our framework yields sharp convergence +bounds for the equilibrium gap of OGD in zero-sum games parameterized on the minimal first-order +variation of the Nash equilibria and the second-order variation of the payoff matrices, subsuming +known results for static games. Furthermore, we establish improved second-order variation +bounds under strong convexity-concavity, as long as each game is repeated multiple times. Our +results also apply to time-varying general-sum multi-player games via a bilinear formulation of +correlated equilibria, which has novel implications for meta-learning and for obtaining refined +variation-dependent regret bounds, addressing questions left open in prior papers. Finally, we +leverage our framework to also provide new insights on dynamic regret guarantees in static games. +arXiv:2301.11241v1 [cs.LG] 26 Jan 2023 + +1 +Introduction +Most of the classical results in the literate on learning in games—exemplified by, among others, +the work of Hart and Mas-Colell [HM00], Foster and Vohra [FV97], and Freund and Schapire +[FS99]—rest on the assumption that the underlying repeated game remains invariant throughout the +learning process. Yet, in many learning environments that is unrealistic [Duv+22; Zha+22; Car+19; +MS21]. One such class is settings where the underlying game is actually changing, such as routing +problems on the internet [MO11], online advertising auctions [LST16], and dynamic mechanism +design [Pap+22; DMZ21]. Another such class consists of settings in which many similar games need +to be solved [Har+22]. For example, one may want to solve variations of a game for the purpose +of sensitivity analysis with respect to the modeling assumptions used to construct the game model. +Another example is solving multiple versions of a game any one of which might be faced in the future. +Despite the considerable interest in such dynamic multiagent environments, much less is known +about the convergence of no-regret learning algorithms in time-varying games. No-regret dynamics +are natural learning algorithms that have desirable convergence properties in static settings. Also, +the state-of-the-art algorithms for finding minimax equilibria in two-player zero-sum games are based +on advanced forms of no-regret dynamics [FKS21; BS19a]. Indeed, all the superhuman milestones +in poker have used them in the equilibrium-finding module of their architectures [Bow+15; BS17; +BS19b]. +In this paper, we seek to fill this knowledge gap by understanding properties of no-regret dy- +namics in time-varying games. In particular, we primarily investigate the convergence of optimistic +gradient descent (OGD) [Chi+12; RS13] in time-varying games. Unlike traditional no-regret learning +algorithms, such as (online) gradient descent, OGD has been recently shown to exhibit last-iterate +convergence in static (two-player) zero-sum games [Das+18; GPD20; COZ22; GTG22]. For the more +challenging scenario where the underlying game can vary in every round, a fundamental question +arises: Under what conditions on the sequence of games does OGD (with high probability) approximate +the sequence of Nash equilibria? +1.1 +Our Results +In this paper, we build a new framework that enables us to characterize the convergence of OGD +in time-varying games. Specifically, our first contribution is to identify natural variation measures +on the sequence of games whose sublinear growth guarantees that almost all iterates of OGD are +(approximate) Nash equilibria in time-varying (two-player) zero-sum games (Corollary 3.6). More +precisely, in Theorem 3.5 we derive a sharp non-asymptotic characterization of the equilibrium gap of +OGD as a function of the variation measures we identify: the minimal first-order variation of the Nash +equilibria and the second-order variation of the payoff matrices. It is a compelling property, in light of +the multiplicity of Nash equilibria, that the variation of the Nash equilibria is measured in terms of the +most favorable—i.e., one that minimizes the variation—such sequence. Additionally, we show that +our convergence bounds can be further improved by considering a variation measure that depends on +the deviation of approximate Nash equilibria of the games, a measure that could be arbitrarily smaller +than the one based on (even the least varying) sequence of exact Nash equilibria (Proposition 3.3). +From a technical standpoint, our analysis revolves around a new connection we draw between the +convergence of OGD in time-varying games and dynamic regret. In particular, the first key observation +is that dynamic regret is always nonnegative under any sequence of Nash equilibria (Property 3.2). +By combining that property with a dynamic RVU bound—in the sense of Syrgkanis et al. [Syr+15]— +1 + +that we derive (Lemmas 3.1 and A.1), we obtain in Theorem 3.4 a variation-dependent bound for +the second-order path length of OGD in time-varying games. In turn, this leads to our main result, +Theorem 3.5, discussed above. As such, we extend the regret-based framework of Anagnostides et al. +[Ana+22b] from static to time-varying games. In the special case of static games, our result reduces +to a tight T −1/2 rate. It is worth stressing that Property 3.2 is in fact more general, being intricately +tied to the admission of a minimax theorem (Property A.3), and applies even under a sequence of +approximate Nash equilibria—with slackness that gently degrades with the approximation thereof. +Moreover, for strongly convex-concave time-varying games, we obtain a refined second-order +variation bound on the sequence of Nash equilibria, as long as each game is repeated multiple +times (Theorem 3.8); this is inspired by an improved second-order bound for dynamic regret under +analogous conditions due to Zhang et al. [Zha+17]. As a byproduct of our techniques, we point out +that any no-regret learners are approaching a Nash equilibrium under strong convexity-concavity +(Proposition 3.9). Those results apply even in non-strongly convex-concave settings by suitably +trading-off the magnitude of a regularizer that makes the game strongly convex-concave. This offers +significant gains in the meta-learning setting as well, wherein each game is repeated multiple times. +Next, we extend our results to time-varying general-sum multi-player games via a bilinear formu- +lation of correlated equilibria. As such, we recover similar convergence bounds parameterized on the +variation of the correlated equilibria (Theorem 3.12). To illustrate the power of our framework, we +immediately recover natural and algorithm-independent similarity measures for the meta-learning +setting (Proposition A.13) even in general games (Corollary A.22), thereby addressing an open +question of Harris et al. [Har+22]. Our techniques also imply new per-player regret bounds in +zero-sum and general-sum games (Corollaries 3.7 and A.23), the latter addressing a question left +open by Zhang et al. [Zha+22]. We further parameterize the convergence of (vanilla) gradient descent +in time-varying potential games in terms of the deviation of the potential functions (Theorem 3.10). +Finally, building on our techniques in time-varying games, we investigate the best dynamic-regret +guarantees possible in static games. Although this is a basic question, it has apparently eluded +prior research. We first show that instances of optimistic mirror descent guarantee O( +√ +T) dynamic +per-player regret (Proposition 3.13), matching the known rate of (online) gradient descent but for +the significantly weaker notion of external regret. We further point out that O(log T) dynamic +regret is attainable, but in a stronger two-point feedback model. In stark contrast, even obtaining +sublinear dynamic regret for each player is precluded in general-sum games (Proposition 3.15). This +motivates studying a relaxation of dynamic regret that constrains the number of switches in the +comparator, for which we derive accelerates rates in general games (Theorem 3.16) by leveraging the +techniques of Syrgkanis et al. [Syr+15] in conjunction with our dynamic RVU bound (Lemma 3.1). +1.2 +Further Related Work +Even in static (two-player) zero-sum games, the pointwise convergence of no-regret learning algo- +rithms is a tenuous affair. Indeed, traditional learning dynamics within the no-regret framework, such +as (online) mirror descent, may even diverge away from the equilibrium; e.g., see [SAF02; MPP18; +Vla+20; GVM21]. Notwithstanding, the empirical frequency of no-regret learners is well-known +to approach the set of Nash equilibria in zero-sum games [FS99], and the set of coarse correlated +equilibria in general-sum games [HM00]—a standard relaxation of the Nash equilibrium [MV78; +Aum74]. Unfortunately, those classical results are of little use beyond static games, thereby offering +a crucial impetus for investigating iterate-convergence in games with a time-varying component—a +ubiquitous theme in many practical scenarios of interest [DMZ21; Pap+22; Ven21; Gar17; Van10; +2 + +RG22; PKB22; YH15; RJW21]. +Indeed, there has been a considerable effort endeavoring to extend the scope of traditional +game-theoretic results to the time-varying setting, approached from a variety of different stand- +points [LST16; Zha+22; Car+19; MO11; MS21; Duv+22]. In particular, our techniques in Section 3.1 +share similarities with the ones used by Zhang et al. [Zha+22], but our primary focus is very different: +Zhang et al. [Zha+22] were mainly interested in obtaining variation-dependent regret bounds, while +our results revolve around iterate-convergence to Nash equilibria. We stress again that minimizing +regret and approaching Nash equilibria are two inherently distinct problems, although connections +have emerged [Ana+22b], and are further cultivated in this paper. +Another closely related direction is on meta-learning in games [Har+22], wherein each game +can be repeated for multiple iterations. Such considerations are motivated in part by a number of +use-cases in which many “similar” games—or multiple game variations—ought to be solved [BS16], +such as Poker with different stack-sizes. While the meta-learning problem is a special case of our +general setting, our results are strong enough to have new implications for meta-learning in games, +even though the algorithms considered herein are not tailored to operate in that setting. +Finally, although our focus is on the convergence of OGD in time-varying games, some of our +results—namely, the ones formalized in Appendix A.1.7—can be viewed as part of an ongoing effort +to characterize the class of variational inequalities (VIs) that are amenable to efficient algorithms; +see [DDJ21; CZ22; Azi+20; BMW21; CP04; DL15; MV21; MRS20; Nou+19; Son+20; YKH20; +Das22], and references therein. We also highlight that the techniques used to establish last-iterate +convergence even in monotone (time-invariant) settings are particularly involved [GPD20; COZ22; +GTG22]; the simplicity of our framework, therefore, in the more challenging time-varying regime is +a compelling aspect of this paper. +2 +Preliminaries +Notation +We let N := {1, 2, . . . , } be the set of natural numbers. For a number p ∈ N, we let +[[p]] := {1, . . . , p}. For a vector w ∈ Rd, we use ∥w∥2 to represent its Euclidean norm; we also +overload that notation so that ∥ · ∥2 denotes the spectral norm when the argument is a matrix. +For a two-player zero-sum game, we denote by X ⊆ Rdx and Y ⊆ Rdy the strategy sets of the two +players—namely, Player x and Player y, respectively—where dx, dy ∈ N represent the corresponding +dimensions. It is assumed that X and Y are nonempty convex and compact sets. For example, in +the special case where X := ∆dx and Y := ∆dy—each set corresponds to a probability simplex—the +game is said to be in normal form. Further, we denote by DX the ℓ2-diameter of X, and by ∥X∥2 +the maximum ℓ2-norm attained by a point in X. We will always assume that the strategy sets +remain invariant, while the payoff matrix can change in each round. For notational convenience, we +will denote by z := (x, y) the concatenation of x and y, and by Z := X × Y the Cartesian product +of X and Y. In general n−player games, we instead use subscripts indexed by i ∈ [[n]] to specify +quantities related to a player. Superscripts are typically reserved to identify the time index. Finally, +to simplify the exposition, we use the O(·) notation to suppress time-independent parameters of +the problem; precise statements are given in Appendix A. +Dynamic regret +We operate in the usual online learning setting under full-feedback. Namely, at +every time t ∈ N the learner decides on a strategy x(t) ∈ X, and then observes a utility x �→ ⟨x, u(t) +x ⟩, +for u(t) +x ∈ Rdx. Following Daskalakis, Deckelbaum, and Kim [DDK11], we will insist on allowing only +3 + +O(1) previous utilities to be stored; this will preclude trivial exploration protocols when learning in +games. +A strong performance benchmark in this online setting is dynamic regret, defined for a time +horizon T ∈ N as follows: +DReg(T) +x (s(T) +x ) := +T +� +t=1 +⟨x(t,⋆) − x(t), u(t) +x ⟩, +(1) +where s(T) +x +:= (x(1,⋆), . . . , x(T,⋆)) ∈ X T is the sequence of comparators; setting x(1,⋆) = x(2,⋆) = · · · = +x(T,⋆) in (1) we recover the standard notion of (external) regret (denoted simply by Reg(T) +x ), which is +commonly used to establish convergence of the time-average strategies in static two-player zero-sum +games [FS99]. On the other hand, the more general notion of dynamic regret, introduced in (1), +has been extensively used in more dynamic environments; e.g., [Zha+20; Zha+17; Jad+15; Ces+12; +HS09]. We also let DReg(T) +x +:= maxs(T ) +x +∈X T DReg(T) +x (s(T) +x ). While ensuring o(T) dynamic regret is +clearly hopeless in a truly adversarial environment, Section 3.4 reveals that non-trivial guarantees +are possible when learning in zero-sum games. +Optimistic gradient descent +Optimistic gradient descent (OGD) [Chi+12; RS13] is a no-regret +algorithm defined with the following update rule: +x(t) := ΠX +� +ˆx(t) + ηm(t) +x +� +, +ˆx(t+1) := ΠX +� +ˆx(t) + ηu(t) +x +� +. +(OGD) +Here, η > 0 is the learning rate; ˆx(1) := arg minˆx∈X ∥ˆx∥2 +2 represents the initialization of OGD; +m(t) +x +∈ Rdx is the prediction vector at time t, and it is set as m(t) +x +:= u(t−1) +x +when t ≥ 2, and +m(1) +x +:= 0dx; and finally, ΠX (·) represents the Euclidean projection to the set X, which is well- +defined, and can be further computed efficiently for structured sets, such as the probability simplex. +For our purposes, we will posit access to a projection oracle for the set X, in which case the update +rule (OGD) is efficiently implementable. +In a multi-player n-player game, each Player i ∈ [[n]] is associated with a utility function +ui :× +n +i=1 Xi → R. We recall the following fundamental definition [Nas50]. +Definition 2.1 (Approximate Nash equilibrium). A joint strategy profile (x⋆ +1, . . . , x⋆ +n) ∈× +n +i=1 Xi is +an ϵ-approximate Nash equilibrium (NE), for an ϵ ≥ 0, if for any Player i ∈ [[n]] and any possible +deviation x′ +i ∈ Xi, +ui(x⋆ +1, . . . , x⋆ +i , . . . , x⋆ +n) ≥ ui(x⋆ +1, . . . , x′ +i, . . . , x⋆ +n) − ϵ. +3 +Convergence in Time-Varying Games +In this section, we formalize our results regarding convergence in time-varying games. We organize +this section as follows: First, in Section 3.1, we build the foundations of our framework by studying +the convergence of OGD in time-varying bilinear saddle-point problems, culminating in the non- +asymptotic characterization of Theorem 3.5; Section 3.2 formalizes our improvements under strong +convexity-concavity; we then extend our results (in Section 3.3) to time-varying multi-player general- +sum and potential games; and finally, Section 3.4 concerns dynamic regret guarantees in static games. +4 + +3.1 +Bilinear Saddle-Point Problems +We first study an online learning setting wherein two players interact in a sequence of time- +varying bilinear saddle-point problems. More precisely, we assume that in every repetition t ∈ [[T]] +the players select a pair of strategies (x(t), y(t)) ∈ X × Y. Then, Player x receives the utility +u(t) +x +:= −A(t)y(t) ∈ Rdx, where A(t) ∈ Rdx×dy represents the payoff matrix at the t-th repetition; +similarly, Player y receives the utility u(t) +y +:= (A(t))⊤x(t) ∈ Rdy. The proofs of this subsection are +included in Appendix A.1. +Dynamic RVU bound +The first key ingredient that we need is the property of regret bounded by +variation in utilities (RVU), in the sense of Syrgkanis et al. [Syr+15], but with respect to dynamic +regret; such a bound is established below. +Lemma 3.1 (RVU bound for dynamic regret). Consider any sequence of utilities (u(1) +x , . . . , u(T) +x ) +up to time T ∈ N. The dynamic regret (1) of OGD with respect to any sequence of comparators +(x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded by +D2 +X +2η + DX +η +T−1 +� +t=1 +∥x(t+1,⋆) − x(t,⋆)∥2+η +T +� +t=1 +∥u(t) +x − m(t) +x ∥2 +2 +− 1 +2η +T +� +t=1 +� +∥x(t) − ˆx(t)∥2 +2 + ∥x(t) − ˆx(t+1)∥2 +2 +� +. +(2) +In the special case of external regret—x(1,⋆) = x(2,⋆) = · · · = x(T,⋆)—(2) recovers the bound for +OGD of Syrgkanis et al. [Syr+15]. The key takeaway from Lemma 3.1 is that the overhead of dynamic +regret in (2) grows with the first-order variation of the sequence of comparators. In Lemma A.1 +we also articulate an extension of Lemma 3.1 for the more general optimistic mirror descent (OMD) +algorithm under a certain class of Bregman divergences. +Having established Lemma 3.1, we next point out a crucial property: by selecting a sequence of +Nash equilibria (recall Definition 2.1) as the comparators, the sum of the players’ dynamic regrets +is always nonnegative: +Property 3.2. Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of +the t-th game. Then, for s(T) +x += (x(t,⋆))1≤t≤T and s(T) +y += (y(t,⋆))1≤t≤T , +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) ≥ −2 +T +� +t=1 +ϵ(t). +In particular, if ϵ(t) = 0 for all t ∈ [[T]], we have +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) ≥ 0. +(3) +In fact, as we show in Property A.3, Property 3.2 applies even in certain (time-varying) nonconvex- +nonconcave min-max optimization problems, and it is a consequence of the minimax theorem; +Property 3.2 also holds for time-varying variational inequalities (VIs) that satisfy the so-called MVI +property (see Remark A.4). For comparison, it is evident that under a static sequence of two-player +zero-sum games, it holds that Reg(T) +x ++ Reg(T) +y +≥ 0. +5 + +Next, let us introduce some natural measures of the games’ variation. First, the first-order +variation of the Nash equilibria is defined for T ≥ 2 as +V(T) +NE := +inf +z(t,⋆)∈Z(t,⋆),∀t∈[[T]] +T−1 +� +t=1 +∥z(t+1,⋆) − z(t,⋆)∥2, +(4) +where Z(t,⋆) is the (nonempty) set of Nash equilibria of the t-th game. We recall that there can be a +multiplicity of Nash equilibria [van91]; as such, a compelling feature of the variation measure (4) is +that it depends on the most favorable sequence of Nash equilibria—one that minimizes the first-order +variation. +It is also important to point out the well-known fact that Nash equilibria can change abruptly +even under a “small” perturbation in the payoff matrix (see Example A.5), which is a caveat of the +variation (4). To address this, and in accordance with Property 3.2, we consider a more favorable +variation measure, defined as +V(T) +ϵ−NE := inf +�T−1 +� +t=1 +∥z(t+1,⋆) − z(t,⋆)∥2 + C +T +� +t=1 +ϵ(t) +� +, +for a sufficiently large parameter C > 0; the infimum above is subject to ϵ(t) ∈ R≥0 and z(t,⋆) ∈ Z(t,⋆) +ϵ(t) +for all t ∈ [[T]], where we denote by Z(t,⋆) +ϵ(t) +the set of ϵ(t)-approximate NE. It is evident that +V(T) +ϵ−NE ≤ V(T) +NE since one can take ϵ(1) = · · · = ϵ(T) = 0; in fact, V(T) +ϵ−NE can be arbitrarily smaller: +Proposition 3.3. For any T ≥ 4, there is a sequence of T games such that V(T) +NE ≥ T +2 while +V(T) +ϵ−NE ≤ δ, for any δ > 0. +Moreover, we also introduce a quantity that captures the variation of the payoff matrices: +V(T) +A +:= +T−1 +� +t=1 +∥A(t+1) − A(t)∥2 +2, +(5) +where we recall that here ∥ · ∥2 denotes the spectral norm. Unlike (4), the variation measure (5) +depends on the second-order variation (of the payoff matrices), which could translate to a lower-order +impact compared to (4) (see, e.g., Corollary A.8). We stress that while our convergence bounds +will be parameterized based on (4) and (5), the underlying algorithm—namely OGD—will remain +oblivious to those variation measures. +We are ready now to establish a refined bound on the second-order path-length of OGD in +time-varying zero-sum games. +Theorem 3.4 (Detailed version in Theorem A.6). Suppose that both players employ OGD with learning +rate η ≤ +1 +4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2. Then, for +any T ∈ N, the second-order path length �T +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +can be bounded by +O +� +1 + V(T) +ϵ−NE + V(T) +A +� +. +(6) +6 + +It is worth noting that when the deviation of the payoff matrices is controlled by the deviation of +the players’ strategies, in the sense that �T−1 +t=1 ∥A(t+1)−A(t)∥2 +2 ≤ W 2 �T−1 +t=1 ∥z(t+1)−z(t)∥2 +2 for some +parameter W ∈ R>0, the variation measure V(T) +A +in (6)—and in the subsequent convergence bounds— +can be entirely eliminated; see Corollary A.8. The same, in fact, applies under an improved prediction +mechanism (Remark A.12), but that prediction is not implementable in our online learning setting. +Armed with Theorem 3.4, we are ready to establish Theorem 3.5. The key observation is that the +Nash equilibrium gap at the t-th game can be bounded in terms of the quantity ∥z(t)− ˆz(t)∥2+∥z(t)− +ˆz(t+1)∥2, which in turn allows us to use (6) to bound the cumulative (squared) Nash equilibrium +gaps across the sequence of games; the aforeclaimed property was established in [Ana+22b] for +static games (Claim A.9), but readily extends to our setting as well, and in fact applies to any +member of OMD under a smooth regularizer. Below, we use the notation EqGap(t)(z(t)) ∈ R≥0 to +represent the Nash equilibrium gap of the joint strategy profile z(t) ∈ Z at the t-th game. +Theorem 3.5 (Main result; detailed version in Theorem A.10). Suppose that both players em- +ploy OGD with learning rate η = +1 +4L in a time-varying bilinear saddle-point problem, where L := +maxt∈[[T]] ∥A(t)∥2. Then, +T +� +t=1 +� +EqGap(t)(z(t)) +�2 += O +� +1 + V(T) +ϵ−NE + V(T) +A +� +, +(7) +where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD. +We next state some immediate consequences of this result. (Item 2 below follows from (7) by +Jensen’s inequality.) +Corollary 3.6. In the setting of Theorem 3.5, +1. If at least a δ-fraction of the iterates of OGD have at least ϵ > 0 Nash equilibrium gap, then +ϵ2δ ≤ O +� +1 +T +� +V(T) +ϵ−NE + V(T) +A ++ 1 +�� +; +2. The average Nash equilibrium gap of OGD is bounded as O +�� +1 +T +� +V(T) +ϵ−NE + V(T) +A ++ 1 +�� +. +In particular, in terms of asymptotic implications, if limT→+∞ +V(T ) +ϵ−NE +T +, limT→+∞ +V(T ) +A +T += 0, then +(i) for any ϵ > 0 the fraction of iterates of OGD with at least an ϵ Nash equilibrium gap converges to +0; and (ii) the average Nash equilibrium gap of the iterates of OGD converges to 0. +In the special case where V(T) +ϵ−NE, V(T) +A += O(1), Theorem 3.5 recovers the T −1/2 rate of OGD in +static bilinear saddle-point problems. It is also worth pointing out that Theorem 3.5 readily extends +to more general time-varying variational inequality problems as well (Remark A.4). +We also state below another interesting consequence of Theorem 3.4, which bounds each player’s +individual regret parameterized based on the variation measures. +Corollary 3.7 (Detailed version in Corollary A.11). In the setup of Theorem 3.4, it holds that +Reg(T) +x , Reg(T) +y += O +� +1 +η + η(V(T) +NE + V(T) +A ) +� +. +7 + +The O(·) notation here is considered in the regime η ≪ 1. Hence, selecting optimally the learning +rate gives an O( +� +V(T) +NE + V(T) +A ) bound on the individual regret of each player; while that optimal +value depends on the variation measures, which are not known to the learners, there are techniques +that would allow bypassing this [Zha+22]. Corollary 3.7 can also be readily parameterized in +terms of the improved variation measure V(T) +ϵ−NE. Finally, in Appendix A.1.7 we highlight certain +implications of our framework on solving (static) general VIs. +Meta-Learning +Our results also have immediate applications in the meta-learning setting [Har+22]. +More precisely, meta-learning in games is a special case of time-varying games which consists of a +sequence of H ∈ N separate games, each of which is repeated for m ∈ N consecutive rounds, so that +T := m × H. The central goal in meta-learning is to obtain convergence bounds parameterized by +the similarity of the games; identifying suitable similarity metrics is a central question in that line +of work. +In this context, we highlight that Theorem 3.5 readily provides a meta-learning guarantee +parameterized by the following notion of similarity between the Nash equilibria: +inf +z(h,⋆)∈Z(h,⋆),∀h∈[[H]] +H−1 +� +h=1 +∥z(h+1,⋆) − z(h,⋆)∥2, +(8) +where Z(h,⋆) is the set of Nash equilibria of the h-th game in the meta-learning sequence,1 as well +as the similarity of the payoff matrices—corresponding to the term V(T) +A +in (7). In fact, under a +suitable prediction—the one used by Harris et al. [Har+22]—the dependence on V(T) +A +can be entirely +removed; see Proposition A.13 for our formal result. A compelling aspect of our meta-learning +guarantee is that the considered algorithm is oblivious to the boundaries of the meta-learning. We +further provide some novel results on meta-learning in general-sum games in Section 3.3. +3.2 +Strongly Convex-Concave Games +In this subsection, we show that under additional structure we can significantly improve the variation +measures established in Theorem 3.4. More precisely, we first assume that each objective function +f(x, y) is µ-strongly convex with respect to x and µ-strongly concave with respect to y. Our second +assumption is that each game is played for multiple rounds m ∈ N, instead of only a single round; +this is akin to the meta-learning setting. The key insight is that, as long as m is large enough, +m = Ω(1/µ), those two assumptions suffice to obtain a second-order variation bound in terms of +the sequence of Nash equilibria, +S(H) +NE := +H−1 +� +h=1 +∥z(h+1,⋆) − z(h,⋆)∥2 +2, +(9) +where z(h,⋆) is a Nash equilibrium of the h-th game. This significantly refines the result of Theo- +rem 3.4, and is inspired by the improved dynamic regret bounds obtained by Zhang et al. [Zha+17]. +Below we sketch the key ideas of the improvement; the proofs are included in Appendix A.2. +In this setting, it is assumed that Player x obtains the utility u(t) +x := −∇xf(t)(x(t), y(t)) at every +time t ∈ [[T]], while its regret will be denoted by Reg(T) +L,y; similar notation applies for Player y. The +1In accordance to Theorem 3.5, (8) can be refined using a sequence of approximate Nash equilibria. +8 + +first observation is that, focusing on a single (static) game, under strong convexity-concavity the +sum of the players’ regrets are strongly nonnegative (Lemma A.15): +Reg(m) +L,x(x⋆) + Reg(m) +L,y (y⋆) ≥ µ +2 +m +� +t=1 +∥z(t) − z⋆∥2 +2, +(10) +for any Nash equilibrium z⋆ ∈ Z of the game. In turn, this can be cast in terms of dynamic regret +over the sequence of the h games (Lemma A.16). Next, combining those dynamic-regret lower +bounds with a suitable RVU-type property leads to a refined second-order path length bound as +long as that m = Ω(1/µ), which in turn leads to our main result below. Before we present its +statement, let us introduce the following measure of variation of the gradients: +V(H) +∇f := +H−1 +� +h=1 +max +z∈Z ∥F (h+1)(z) − F (h)(z)∥2 +2, +(11) +where let F : z := (x, y) �→ (∇xf(x, y), −∇yf(x, y)). This variation measure is analogous to V(T) +A +we introduced in (5) for time-varying bilinear saddle-point problems. +Theorem 3.8 (Detailed version in Theorem A.18). Let f(h) : X × Y be a µ-strongly convex-concave +and L-smooth function, for h ∈ [[H]]. Suppose that both players employ OGD with learning rate +η = min +� +1 +8L, 1 +2µ +� +for T repetitions, where T = m × H and m ≥ +2 +ηµ. Then, +T +� +t=1 +� +EqGap(t)(z(t)) +�2 += O(1 + S(H) +NE + V(H) +∇f ), +where S(H) +NE and V(H) +∇f are defined in (9) and (11). +Our techniques also imply improved regret bounds in this setting, as we formalize in Corol- +lary A.19. +There is another immediate but important implication of (10): any no-regret algorithm in a +(static) strongly convex-concave setting ought to be approaching the Nash equilibrium; in contrast, +this property is spectacularly false in (general) monotone settings [MPP18]. +Proposition 3.9. Let f : X × Y → R be a µ-strongly convex-concave function. If players incur +regrets such that Reg(T) +L,x + Reg(T) +L,y ≤ CT 1−ω, for some parameters C > 0 and ω ∈ (0, 1], then for any +ϵ > 0 and T > +� +2C +µϵ2 +�1/ω +there is a pair of strategies z(t) ∈ Z such that ∥z(t) − z⋆∥2 ≤ ϵ, where z⋆ +is a Nash equilibrium. +The insights of this subsection are also of interest in general monotone settings by incorpo- +rating a strongly convex regularizer; tuning its magnitude allows us to trade-off between a better +approximation and the benefits of strong convexity-concavity revealed in this subsection. +3.3 +General-Sum Multi-player Games +Next, we turn our attention to general-sum multi-player games. For simplicity, in this subsection +we posit that the game is represented in normal form, so that each Player i ∈ [[n]] has a finite set of +available actions Ai, and Xi := ∆(Ai). The proofs of this subsection are included in Appendix A.3. +9 + +Potential Games +First, we study the convergence of (online) gradient descent (GD) in time- +varying potential games (see Definition A.20 for the formal description).2 In our time-varying +setup, it is assumed that each round t ∈ [[T]] corresponds to a different potential game described +with a potential function Φ(t). We further let d : (Φ, Φ′) �→ maxz∈×n +i=1 Xi (Φ(z) − Φ′(z)), so that +V(T) +Φ +:= �T−1 +t=1 d(Φ(t), Φ(t+1)); we emphasize the fact that d(·, ·) is not symmetric. Analogously +to Theorem 3.5, we use EqGap(t)(z(t)) ∈ R≥0 to represent the NE gap of the joint strategy profile +z(t) := (x(t) +1 , . . . , x(t) +n ) at the t-th game. +Theorem 3.10. Suppose that each player employs (online) GD with a sufficiently small learning +rate. Then, +T +� +t=1 +� +EqGap(t)(z(t)) +�2 += O(Φmax + V(T) +Φ ), +where Φmax is such that |Φ(t)(·)| ≤ Φmax for any t ∈ [[T]]. +We refer to Appendix B for some illustrative experiments. +General games +Unfortunately, unlike the settings considered thus far, computing Nash equilibria +in general games is computationally hard [DGP08; CDT09]. Instead, learning algorithms are known +to converge to relaxations of the Nash equilibrium, known as (coarse) correlated equilibria. For +our purposes, we will employ a bilinear formulation of (coarse) correlated equilibria, which dates +back to the seminal work of Hart and Schmeidler [HS89]. This will allow us to translate the results +of Section 3.1 to general multi-player games. +Specifically, correlated equilibria3 can be expressed via a game between the n players and a medi- +ator. Intuitively, the mediator is endeavoring to identify a correlated strategy µ ∈ Ξ := ∆ +�× +n +i=1 Ai +� +for which no player has an incentive to deviate from the recommendation. In contrast, the players +are trying to optimally deviate so as to maximize their own utility. More precisely, there exist +matrices A1, . . . , An, with each matrix Ai depending solely on the utility of Player i, for which the +bilinear problem can be expressed as +min +µ∈Ξ +max +(¯x1,...,¯xn)∈×n +i=1 ¯ +Xi +n +� +i=1 +µ⊤Ai ¯xi, +(12) +where ¯ +Xi := conv(Xi, 0); incorporating the 0 vector will be useful for our purposes. This zero-sum +game has the property that there exists a strategy µ⋆ ∈ Ξ such that max¯xi∈ ¯ +Xi(µ⋆)⊤Ai ¯xi ≤ 0, for +any Player i ∈ [[n]], which corresponds to a correlated equilibrium. +Before we proceed, it is important to note that the learning paradigm considered here deviates +from the traditional one in that there is an additional learning agent, resulting in a less decentralized +protocol. Yet, the dynamics induced by solving (12) via online algorithms remain uncoupled [HM00], +in the sense that each player obtains feedback—corresponding to the deviation benefit—that depends +solely on its own utility. +Now in the time-varying setting, the matrices A1, . . . , An that capture the players’ utilities can +change in each repetition. Crucially, we show that the structure of the induced bilinear problem (12) +2Unlike two-player zero-sum games, gradient descent is known to approach Nash equilibria in potential games. +3The following bilinear formulation applies to coarse correlated equilibria as well (with different payoff matrices), +but we will focus solely on the stronger variant (CE) for the sake of exposition. +10 + +is such that there is a sequence of correlated equilibria that guarantee nonnegative dynamic regret; +this refines Property 3.2 in that only one player’s strategies suffice to guarantee nonnegativity, even +if the strategies of the other player remain invariant. Below, we denote by DReg(T) +µ +the dynamic +regret of the min player in (12), and by Reg(T) +i +the regret of each Player i up to time T ∈ N, so +that the regret of the max player in (12) can be expressed as �n +i=1 Reg(T) +i +. +Property 3.11. Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]]. +Then, +DReg(T) +µ (µ(1,⋆), . . . , µ(T,⋆)) + +n +� +i=1 +Reg(T) +i +≥ 0. +As a result, this enables us to apply Theorem 3.5 parameterized on (i) the variation of the CE +V(T) +CE := +inf +µ(t,⋆)∈Ξ(t,⋆),∀t∈[[T]] +T−1 +� +t=1 +∥µ(t+1,⋆) − µ(t,⋆)∥2, +where Ξ(t,⋆) denotes the set of CE of the t-th game, and (ii) the variation in the players’ utilities +V(T) +A +≤ �n +i=1 +�T−1 +t=1 ∥A(t+1) +i +− A(t) +i ∥2 +2; below, we denote by CeGap(t)(µ(t)) the CE gap of µ(t) ∈ Ξ +at the t-th game. +Theorem 3.12. Suppose that each player employs OGD in (12) with a suitable learning rate. Then, +T +� +t=1 +� +CeGap(t)(µ(t)) +�2 += O(1 + V(T) +CE + V(T) +A ). +There are further interesting implications of our framework that are worth highlighting. First, +we obtain meta-learning guarantees for general games that depend on the (algorithm-independent) +similarity of the correlated equilibria (Corollary A.22); that was left as an open question by Harris +et al. [Har+22], where instead algorithm-dependent similarity metrics were derived. Further, by +applying Corollary 3.7, we derive natural variation-dependent per-player regret bounds in general +games (Corollary A.23), addressing a question left by Zhang et al. [Zha+22]; we suspect that +obtaining such results—parameterized on the variation of the CE—are not possible without the +presence of the additional player. +3.4 +Dynamic Regret Bounds in Static Games +Finally, in this subsection we switch gears by investigating dynamic regret guarantees when learning +in static games. The proofs of this subsection are included in Appendix A.4. +First, we point out that while traditional no-regret learning algorithms guarantee O( +√ +T) external +regret, instances of OMD—a generalization of OGD; see (13) in Appendix A—in fact guarantee O( +√ +T) +dynamic regret in two-player zero-sum games, which is a much stronger performance measure: +Proposition 3.13. Suppose that both players in a (static) two-player zero-sum game employ OMD +with a smooth regularizer. Then, DReg(T) +x , DReg(T) +y += O( +√ +T). +11 + +In proof, the dynamic regret for each player under OMD with a smooth regularizer can be +bounded by the first-order path length of that player’s strategies, which in turn can be bounded +by O( +√ +T) given that the second-order path length is O(1) (Theorem 3.4). +In fact, Theo- +rem 3.4 readily extends Proposition 3.13 to time-varying zero-sum games as well, implying that +DReg(T) +x , DReg(T) +y += O +�√ +T(1 + V(T) +ϵ−NE + V(T) +A ) +� +. +A question that arises from Proposition 3.13 is whether the O( +√ +T) guarantee for dynamic +regret of OMD can be improved in the online learning setting. Below, we point out a significant +improvement to O(log T), but under a stronger two-point feedback model; namely, we posit that in +every round each player can select an additional auxiliary strategy, and each player then gets to +additionally observe the utility corresponding to the auxiliary strategies. Notably, this is akin to +how the extra-gradient method works [Hsi+19] (also cf. [RS13, Section 4.2] for multi-point feedback +models in the bandit setting). +Observation 3.14. Under two-point feedback, there exist learning algorithms that guarantee +DReg(T) +x , DReg(T) +y += O(log T) in two-player zero-sum games. +In particular, it suffices for each player to employ OMD, but with the twist that the first strategy +in each round is the time-average of OMD; the auxiliary strategy is the standard output of OMD. +Then, the dynamic regret of each player will grow as O +��T +t=1 +1 +t +� += O(log T) since the duality +gap of the average strategies is decreasing with a rate of T −1 [RS13]. It is an interesting question +whether Observation 3.14 can be improved to O(1). +General-sum games +In contrast, no (efficient) sublinear dynamic-regret guarantees are possible +in general games: +Proposition 3.15. Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n +i=1 DReg(T) +i += Ω(T), +even if n = 2, where Ω(·) here hides polynomial factors. +Indeed, this follows since computing a Nash equilibrium to (1/poly) accuracy in two-player +games is PPAD-hard [CDT09]. In fact, Proposition 3.15 applies beyond the online learning setting. +This motivates considering a relaxation of dynamic regret, wherein the sequence of comparators is +subject to the constraint �T−1 +t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1, for some parameter K ∈ N; this will +be referred to as K-DReg(T) +x . Naturally, external regret coincides with K-DReg(T) +x +under K = 1. +In this context, we employ Lemma 3.1 to bound K-DReg(T) under OGD: +Theorem 3.16 (Detailed version in Theorem A.24). Suppose that all n players employ OGD in an +L-smooth game. Then, for any K ∈ N, +1. �n +i=1 K-DReg(T) +i += O(K√nL); +2. K-DReg(T) +i += O(K3/4T 1/4n1/4√ +L), for i ∈ [[n]]. +One question that arises here is whether the per-player bound of O(K3/4T 1/4) (Item 2) can +be improved to ˜O(K), where ˜O(·) hides logarithmic factors. The main challenge is that, even for +K = 1, all known methods that obtain ˜O(1) [DFG21; PSS21; Ana+22a; Far+22] rely on non- +smooth regularizers that violate the preconditions of Lemma A.1—our dynamic RVU bound that +generalizes Lemma 3.1 beyond (squared) Euclidean regularization. It would also be interesting to give +12 + +a natural game-theoretic interpretation to the limit point of no-regret learners with K-DReg = o(T), +even for a fixed K ∈ N; for K = 1, it corresponds to the fundamental coarse correlated equilibrium. +At a superficial level, it seems to be related to the variant considered by Harrow, Natarajan, and +Wu [HNW16]. +4 +Conclusions and Future Work +In this paper, we developed a new framework for characterizing iterate-convergence of no-regret +learning algorithms—primarily optimistic gradient descent (OGD)—in time-varying games. There +are many promising avenues for future research. Besides closing the obvious gaps we highlighted +in Section 3.4, it is important to characterize the behavior of no-regret learning algorithms in +other fundamental multiagent settings, such as Stackelberg (security) games [Bal+15]. Moreover, +our results operate in the full-feedback model where each player receives feedback on all possible +actions. 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Before we proceed, it will be useful to +express the update rule (OGD) in the following equivalent form: +x(t) := arg max +x∈X +� +Ψ(t) +x (x) := ⟨x, m(t) +x ⟩ − 1 +ηBφx(x ∥ ˆx(t)) +� +, +ˆx(t+1) := arg max +ˆx∈X +� +ˆΨ(t) +x (ˆx) := ⟨ˆx, u(t) +x ⟩ − 1 +ηBφx(ˆx ∥ ˆx(t)) +� +. +(13) +Here, Bφx(· ∥ ·) denotes the Bregman divergence induced by the (squared) Euclidean regularizer +φx : x �→ 1 +2∥x∥2 +2; namely, Bφx(x ∥ x′) := φ(x)−φ(x′)−⟨∇φ(x′), x−x′⟩ = 1 +2∥x−x′∥2 +2, for x, x′ ∈ X. +The update rule (13) for general Bregman divergences will be referred to as optimistic mirror descent +(OMD). +A.1.1 +Dynamic RVU Bounds +We now show Lemma 3.1, the statement of which is recalled below for the convenience of the reader. +Then, in Lemma A.1 we provide an extension of Lemma 3.1 to a broader class of regularizers. +Lemma 3.1 (RVU bound for dynamic regret). Consider any sequence of utilities (u(1) +x , . . . , u(T) +x ) +up to time T ∈ N. The dynamic regret (1) of OGD with respect to any sequence of comparators +(x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded by +D2 +X +2η + DX +η +T−1 +� +t=1 +∥x(t+1,⋆) − x(t,⋆)∥2 + η +T +� +t=1 +∥u(t) +x − m(t) +x ∥2 +2 − 1 +2η +T +� +t=1 +� +∥x(t) − ˆx(t)∥2 +2 + ∥x(t) − ˆx(t+1)∥2 +2 +� +. +Proof. First, by (1/η)-strong convexity of the function Ψ(t) +x +(defined in (13)) for any time t ∈ [[T]], +we have that +⟨x(t), m(t) +x ⟩ − 1 +2η∥x(t) − ˆx(t)∥2 +2 − ⟨ˆx(t+1), m(t) +x ⟩ + 1 +2η∥ˆx(t+1) − ˆx(t)∥2 +2 ≥ 1 +2η∥x(t) − ˆx(t+1)∥2 +2, +(14) +where we used [Sha12, Lemma 2.8, p. 135]. Similarly, by (1/η)-strong convexity of the function ˆΨ(t) +x +(defined in (13)) for any time t ∈ [[T]], we have that for any comparator x(t,⋆) ∈ X, +⟨ˆx(t+1), u(t) +x ⟩ − 1 +2η∥ˆx(t+1) − ˆx(t)∥2 +2 − ⟨x(t,⋆), u(t) +x ⟩ + 1 +2η∥x(t,⋆) − ˆx(t)∥2 +2 ≥ 1 +2η∥ˆx(t+1) − x(t,⋆)∥2 +2. (15) +Thus, adding (14) and (15), +⟨x(t,⋆) − ˆx(t+1), u(t) +x ⟩ + ⟨ˆx(t+1) − x(t), m(t) +x ⟩ ≤ 1 +2η +� +∥ˆx(t) − x(t,⋆)∥2 +2 − ∥ˆx(t+1) − x(t,⋆)∥2 +2 +� +− 1 +2η +� +∥x(t) − ˆx(t)∥2 +2 + ∥x(t) − ˆx(t+1)∥2 +2 +� +. +(16) +19 + +We further see that +⟨x(t,⋆) − x(t), u(t) +x ⟩ = ⟨x(t) − ˆx(t+1), m(t) +x − u(t) +x ⟩ + ⟨x(t,⋆) − ˆx(t+1), u(t) +x ⟩ + ⟨ˆx(t+1) − x(t), m(t) +x ⟩. (17) +Now the first term on the right-hand side can be upper bounded using the fact that, by (14) and (15), +⟨x(t) − ˆx(t+1), m(t) +x − u(t) +x ⟩ ≥ 1 +η∥ˆx(t+1) − x(t)∥2 +2 =⇒ ∥ˆx(t+1) − x(t)∥2 ≤ η∥m(t) +x − u(t) +x ∥2, +by Cauchy-Schwarz, in turn implying that ⟨x(t) − ˆx(t+1), m(t) +x − u(t) +x ⟩ ≤ η∥m(t) +x − u(t) +x ∥2 +2. Thus, the +proof follows by combining this bound with (16) and (17), along with the fact that +T +� +t=1 +� +∥ˆx(t) − x(t,⋆)∥2 +2 − ∥ˆx(t+1) − x(t,⋆)∥2 +2 +� +≤ ∥ˆx(1) − x(1,⋆)∥2 +2 ++ +T−1 +� +t=1 +� +∥ˆx(t+1) − x(t+1,⋆)∥2 +2 − ∥ˆx(t+1) − x(t,⋆)∥2 +2 +� +≤ D2 +X + 2DX +T−1 +� +t=1 +∥x(t+1,⋆) − x(t,⋆)∥2, +where the last bound follows since +∥ˆx(t+1) − x(t+1,⋆)∥2 +2 − ∥ˆx(t+1) − x(t,⋆)∥2 +2 ≤ 2DX +���∥ˆx(t+1) − x(t+1,⋆)∥2 − ∥ˆx(t+1) − x(t,⋆)∥2 +��� +≤ 2DX ∥x(t+1,⋆) − x(t,⋆)∥2, +where we recall that DX denotes the ℓ2-diameter of X. +As an aside, we remark that assuming that m(t) +x := 0 and ∥u(t) +x ∥2 ≤ 1 for any t ∈ [[T]], Lemma 3.1 +implies that dynamic regret can be upper bounded by O +�� +(1 + �T−1 +t=1 ∥x(t+1,⋆) − x(t,⋆)∥2)T +� +, +for any (bounded)—potentially adversarially selected—sequence of utilities (u(1) +x , . . . , u(T) +x ), for +η := +� +D2 +X +2T + DX +�T −1 +t=1 ∥x(t+1,⋆)−x(t,⋆)∥2 +T +, which is a well-known result in online optimization [Zin03]; +while that requires setting the learning rate based on the first-order variation of the (optimal) +comparators, there are standard techniques that would allow bypassing that assumption. +Next, we provide an extension of Lemma 3.1 to the more general OMD algorithm under a broad +class of regularizers. +Lemma A.1 (Extension of Lemma 3.1 beyond Euclidean regularization). Consider a 1-strongly +convex continuously differentiable regularizer φ with respect to a norm ∥·∥ such that (i) ∥∇φ(x)∥∗ ≤ G +for any x, and (ii) Bφx(x ∥ x′) ≤ L∥x − x′∥ for any x, x′. Then, for any sequence of utilities +(u(1) +x , . . . , u(T) +x ) up to time T ∈ N the dynamic regret (1) of OMD with respect to any sequence of +comparators (x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded as +Bφx(x(1,⋆) ∥ ˆx(1)) +η ++ L + 2G +η +T−1 +� +t=1 +∥x(t+1,⋆) − x(t,⋆)∥+η +T +� +t=1 +∥u(t) +x − m(t) +x ∥2 +∗ +− 1 +2η +T +� +t=1 +� +∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2� +. +20 + +The proof is analogous to that of Lemma 3.1, and relies on the well-known three-point identity +for the Bregman divergence: +Bφx(x ∥ x′) = Bφx(x ∥ x′′) + Bφx(x′′ ∥ x′) − ⟨x − x′′, ∇φ(x′) − ∇φ(x′′)⟩. +(18) +In particular, along with the assumptions of Lemma A.1 imposed on the regularizer φx, (18) implies +that the term �T−1 +t=1 +� +Bφx(x(t+1,⋆) ∥ ˆx(t+1)) − Bφx(x(t,⋆) ∥ ˆx(t+1)) +� +is equal to +T−1 +� +t=1 +� +Bφx(x(t+1,⋆) ∥ x(t,⋆)) − ⟨x(t+1,⋆) − x(t,⋆), ∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))⟩ +� +≤ (L + 2G) +T−1 +� +t=1 +∥x(t+1,⋆) − x(t,⋆)∥, +since Bφx(x(t+1,⋆) ∥ x(t,⋆)) ≤ L∥x(t+1,⋆) − x(t,⋆)∥ (by assumption) and +⟨x(t+1,⋆) − x(t,⋆), ∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))⟩≤ ∥x(t+1,⋆) − x(t,⋆)∥∥∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))∥∗ +(19) +≤ +� +∥∇φ(ˆx(t+1))∥∗+ ∥∇φ(x(t,⋆))∥∗ +� +∥x(t+1,⋆) − x(t,⋆)∥ +(20) +≤ 2G∥x(t+1,⋆) − x(t,⋆)∥, +(21) +where (19) follows from the Cauchy-Schwarz inequality; (20) uses the triangle inequality for the +dual norm ∥ · ∥∗; and (21) follows from the assumption of Lemma A.1 that ∥∇φ(·)∥∗ ≤ G. The rest +of the proof of Lemma A.1 is analogous to Lemma 3.1, and it is therefore omitted. An important +question is whether Lemma A.1 can be extended under any regularizer; as we explain in Section 3.4, +this is the main obstacle to improving Theorem 3.16. +A.1.2 +Nonnegativity of Dynamic Regret +We next proceed with the proof of Property 3.2. To provide additional intuition, we first prove the +following special case; the proof of Property 3.2 is then analogous. +Property A.2 (Special case of Property 3.2). Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆))) is a +Nash equilibrium of the t-th game, for any time t ∈ [[T]]. Then, for s(T) +x += (x(t,⋆))1≤t≤T and +s(T) +y += (y(t,⋆))1≤t≤T , +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) ≥ 0. +Proof. Let v(t) := ⟨x(t,⋆), A(t)y(t,⋆)⟩ be the value of the t-th game, for some t ∈ [[T]]. Then, we +have that v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x, A(t)y(t,⋆)⟩ for any x ∈ X, since x(t,⋆) is a best response to +y(t,⋆); similarly, v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y⟩ for any y ∈ Y. Hence, ⟨x(t), A(t)y(t,⋆)⟩ − +⟨x(t,⋆), A(t)y(t)⟩ ≥ 0, or equivalently, ⟨x(t,⋆), u(t) +x ⟩ + ⟨y(t,⋆), u(t) +y ⟩ ≥ 0. But given that the game is +zero-sum, it holds that ⟨x(t), u(t) +x ⟩ + ⟨y(t), u(t) +y ⟩ = 0, so the last inequality can be in turn cast as +⟨x(t,⋆), u(t) +x ⟩ − ⟨x(t), u(t) +x ⟩ + ⟨y(t,⋆), u(t) +y ⟩ − ⟨y(t), u(t) +y ⟩ ≥ 0, +21 + +for any t ∈ [[T]]. As a result, summing over all t ∈ [[T]] we have shown that +DReg(T) +x (x(1,⋆), . . . , x(T,⋆))+ DReg(T) +y +(y(1,⋆), . . . , y(T,⋆)) += +T +� +t=1 +⟨x(t,⋆), u(t) +x ⟩ − ⟨x(t), u(t) +x ⟩ + ⟨y(t,⋆), u(t) +y ⟩ − ⟨y(t), u(t) +y ⟩ ≥ 0. +Property 3.2. Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of +the t-th game. Then, for s(T) +x += (x(t,⋆))1≤t≤T and s(T) +y += (y(t,⋆))1≤t≤T , +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) ≥ −2 +T +� +t=1 +ϵ(t). +Proof. Given that (x(t,⋆), y(t,⋆)) ∈ Z is an ϵ(t)-approximate Nash equilibrium of the t-th game, it +follows that ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x(t), A(t)y(t,⋆)⟩+ϵ(t) +x and ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩−ϵ(t) +y , +for some ϵ(t) +x , ϵ(t) +y +≤ ϵ(t). Thus, we have that ⟨x(t), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩ − ϵ(t) +x − ϵ(t) +y , or +equivalently, ⟨x(t,⋆), u(t) +x ⟩ + ⟨y(t,⋆), u(t) +y ⟩ ≥ −ϵ(t) +x − ϵ(t) +y +≥ −2ϵ(t). As a result, +⟨x(t,⋆), u(t) +x ⟩ − ⟨x(t), u(t) +x ⟩ + ⟨y(t,⋆), u(t) +y ⟩ − ⟨y(t), u(t) +y ⟩ ≥ −2ϵ(t), +(22) +for any t ∈ [[T]], and the statement follows by summing (22) over all t ∈ [[T]]. +In fact, as we show below (in Property A.3), Property A.2 is a more general consequence of the +minimax theorem. In particular, for a nonlinear online learning problem, we define dynamic regret +with respect to a sequence of comparators (x(1,⋆), . . . , x(T,⋆)) ∈ X T as follows: +DReg(T) +x (x(1,⋆), . . . , x(T,⋆)) := +T +� +t=1 +� +u(t) +x (x(t,⋆)) − u(t) +x (x(t)) +� +, +(23) +where u(1) +x , . . . , u(T) +x +: x �→ R are the continuous utility functions observed by the learner, which +could be in general nonconcave, and (x(t))1≤t≤T is the sequence of strategies produced by the +learner; (23) generalizes the notion of dynamic regret (1) in online linear optimization, that is, when +u(t) +x : x �→ ⟨x, u(t) +x ⟩, where u(t) +x ∈ Rdx, for any time t ∈ [[T]]. +Property A.3. Suppose that f(t) : X × Y → R is a continuous function such that for any t ∈ [[T]], +min +x∈X max +y∈Y f(t)(x, y) = max +y∈Y min +x∈X f(t)(x, y). +Let also x(t,⋆) ∈ arg minx∈X maxy∈Y f(t)(x, y) and y(t,⋆) ∈ arg maxy∈Y minx∈X f(t)(x, y), for any +t ∈ [[T]]. Then, for s(T) +x += (x(t,⋆))1≤t≤T and s(T) +y += (y(t,⋆))1≤t≤T , +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) ≥ 0. +22 + +Proof. By definition of dynamic regret (23), it suffices to show that f(t)(x(t), y(t,⋆)) ≥ f(t)(x(t,⋆), y(t)), +for any time t ∈ [[T]]. Indeed, +f(t)(x(t), y(t,⋆)) ≥ min +x∈X f(t)(x, y(t,⋆)) +(24) += max +y∈Y min +x∈X f(t)(x, y) +(25) += min +x∈X max +y∈Y f(t)(x, y) +(26) += max +y∈Y f(t)(x(t,⋆), y) +(27) +≥ f(t)(x(t,⋆), y(t)), +(28) +where (24) and (28) are obvious; (25) and (27) follow from the definition of y(t,⋆) ∈ Y and x(t,⋆) ∈ X, +respectively; and (26) holds by assumption. This concludes the proof. +Remark A.4 (MVI property). Property (3) can also be generalized beyond time-varying bilinear +saddle-point problems to more general time-varying variational inequality (VI) problems as follows. +Let F (t) : Z → Z be the (single-valued) operator of the VI problem at time t. F (t) is said to +satisfy the MVI property if there exists a point z(t,⋆) ∈ Z such that ⟨z − z(t,⋆), F (t)(z)⟩ ≥ 0 for +any z ∈ Z. For example, in the special case of a bilinear saddle-point problem, we have that +F : z := (x, y) �→ (Ay, −A⊤x), and the MVI property is satisfied by virtue of Von Neumann’s +minimax theorem. It is direct to see that Property A.2 applies to any time-varying VI with respect +to the sequence (z(t,⋆))1≤t≤T as long as every operator in the sequence (F (1), . . . , F (T)) satisfies +the MVI property. (Even more broadly, it suffices if almost all operators in the sequence satisfy +the MVI property—in that their fraction converges to 1 as T → +∞.) This observation enables +extending Theorem 3.5 beyond time-varying bilinear saddle-point problems. +A.1.3 +Variation of the Nash Equilibria +In our next example, we point out that an arbitrarily small change in the entries of the payoff +matrix can lead to a substantial deviation in the Nash equilibrium. +Example A.5. Consider a 2 × 2 (two-player) zero-sum game, where X := ∆2, Y := ∆2, described by +the payoff matrix +A := +�2δ +0 +0 +δ +� +, +(29) +for some δ > 0. Then, it is easy to see that the unique Nash equilibrium of this game is such that +x⋆, y⋆ := ( 1 +3, 2 +3) ∈ ∆2. Suppose now that the original payoff matrix (29) is perturbed to a new +matrix +A′ := +�δ +0 +0 +2δ +� +. +(30) +The new (unique) Nash equilibrium now reads x⋆, y⋆ := ( 2 +3, 1 +3) ∈ ∆2. We conclude that an arbitrarily +small deviation in the entries of the payoff matrix can lead to a non-trivial change in the Nash +equilibrium. +Next, we leverage the simple observation of the example above to establish Proposition 3.3, the +statement of which is recalled below. +23 + +Proposition 3.3. For any T ≥ 4, there is a sequence of T games such that V(T) +NE ≥ T +2 while +V(T) +ϵ−NE ≤ δ, for any δ > 0. +Proof. We consider a sequence of T games such that X, Y := ∆2, and +A(t) = +� +A +if t +mod 2 = 1, +A′ +if t +mod 2 = 0, +where A, A′ are the payoff matrices defined in (29) and (30), and are parameterized by δ > 0 +(Example A.5). Then, the exact Nash equilibria read +x(t,⋆), y(t,⋆) = +� +( 1 +3, 2 +3) +if t +mod 2 = 1, +( 2 +3, 1 +3) +if t +mod 2 = 0. +As a result, it follows that V(T) +NE := �T−1 +t=1 ∥z(t+1,⋆) − z(t,⋆)∥2 = 2 +3(T − 1) ≥ T +2 , for T ≥ 4. In contrast, +it is clear that V(T) +ϵ−NE ≤ CδT, which follows by simply considering the sequence of strategies wherein +both players are always selecting actions uniformly at random; we recall that C > 0 here is the +value that parameterizes V(T) +ϵ−NE. Thus, taking δ := +δ′ +CT , for some arbitrarily small δ′ > 0, concludes +the proof. +A.1.4 +Main Result +Next, we proceed with the proof of our main result, Theorem 3.5. The key ingredient is Theorem 3.4, +which bounds the second-order path length of OGD in terms of the considered variation measures. +We first give the precise statement of Theorem 3.4, and we then proceed with its proof. +Theorem A.6 (Detailed version of Theorem 3.4). Suppose that both players employ OGD with +learning rate η ≤ +1 +4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2. +Then, for any time horizon T ∈ N, +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≤ 2D2 +Z + 4η2L2∥Z∥2 +2 + 4DZV(T) +ϵ−NE + 8η2∥Z∥2 +2V(T) +A . +Proof of Theorem 3.4. First, for any t ≥ 2 we have that ∥u(t) +x − m(t) +x ∥2 +2 is equal to +∥A(t)y(t) − A(t−1)y(t−1)∥2 +2 ≤ 2∥A(t)(y(t) − y(t−1))∥2 +2 + 2∥(A(t) − A(t−1))y(t−1)∥2 +2 +(31) +≤ 2∥A(t)∥2 +2∥y(t) − y(t−1)∥2 +2 + 2∥A(t) − A(t−1)∥2 +2∥y(t−1)∥2 +2 +(32) +≤ 2L2∥y(t) − y(t−1)∥2 +2 + 2∥Y∥2 +2∥A(t) − A(t−1)∥2 +2, +(33) +where (31) uses the triangle inequality for the norm ∥ · ∥2 along with the inequality 2ab ≤ a2 + b2 for +any a, b ∈ R; (32) follows from the definition of the operator norm; and (33) uses the assumption +that ∥A(t)∥2 ≤ L and ∥y∥2 ≤ ∥Y∥2 for any y ∈ Y. A similar derivaiton shows that for t ≥ 2, +∥u(t) +y − m(t) +y ∥2 +2 ≤ 2L2∥x(t) − x(t−1)∥2 +2 + 2∥X∥2 +2∥A(t) − A(t−1)∥2 +2. +(34) +Further, for t = 1 we have that ∥u(1) +x +− m(1) +x ∥2 = ∥u(1) +x ∥2 = ∥ − A(1)y(1)∥2 ≤ L∥Y∥2, and +∥u(1) +y +− m(1) +y ∥2 = ∥u(1) +y ∥2 = ∥(A(1))⊤x(1)∥2 ≤ L∥X∥2. Next, we will use the following simple +corollary, which follows similarly to Lemma 3.1. +24 + +Corollary A.7. For any sequence s(T) +z +:= (z(t,⋆))1≤t≤T , the dynamic regret DReg(T) +z +(s(T) +z +) := +DReg(T) +x (s(T) +x ) + DReg(T) +y +(s(T) +y +) can be bounded by +D2 +Z +2η + DZ +η +T−1 +� +t=1 +∥z(t+1,⋆) −z(t,⋆)∥2 +η +T +� +t=1 +∥u(t) +z −m(t) +z ∥2 +2 − 1 +2η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +where m(t) +z +:= (m(t) +x , m(t) +y ) and u(t) +z +:= (u(t) +x , u(t) +y ) for any t ∈ [[T]]. +As a result, combining (34) and (33) with Corollary A.7 applied for the dynamic regret +of both players with respect to the sequence of comparators ((x(t,⋆), y(t,⋆)))1≤t≤T yields that +DReg(T) +x (x(1,⋆), . . . , x(T,⋆)) + DReg(T) +y +(y(1,⋆), . . . , y(T,⋆)) is upper bounded by +D2 +Z +2η + ηL2∥Z∥2 +2 + DZ +η +T−1 +� +t=1 +∥z(t+1,⋆) − z(t,⋆)∥2+2η∥Z∥2 +2V(T) +A +− 1 +4η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +where we used the fact that +2ηL2 +T +� +t=2 +∥z(t) − z(t−1)∥2 +2 − 1 +4η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≤ +� +2ηL2 − 1 +8η +� +T +� +t=2 +∥z(t) − z(t−1)∥2 +2 ≤ 0, +for η ≤ +1 +4L. Finally, using the fact that DReg(T) +x (x(1,⋆), . . . , x(T,⋆)) + DReg(T) +y +(y(1,⋆), . . . , y(T,⋆)) ≥ +−2 �T +t=1 ϵ(t) for a suitable sequence of ϵ(t)-approximate Nash equilibria (Property 3.2)—one that +attains the variation measure V(T) +ϵ−NE—yields that +0 ≤ D2 +Z +2η + ηL2∥Z∥2 +2 + DZ +η V(T) +ϵ−NE + 2η∥Z∥2 +2V(T) +A +− 1 +4η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +where it suffices if the parameter C of V(T) +ϵ−NE is such that 2 ≤ DZ +η C. Thus, rearranging the last +displayed inequality concludes the proof. +Next, we refine this theorem in time-varying games in which the deviation of the payoff matrices +is bounded by the deviation of the players’ strategies, in the following formal sense. +Corollary A.8. Suppose that both players employ OGD with learning rate η ≤ min +� +1 +4L, +1 +8W∥Z∥ +� +in a +time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2 and V(T) +A +≤ W 2 �T−1 +t=1 ∥z(t+1)− +z(t)∥2 +2, for some parameter W ∈ R>0. Then, for any time horizon T ∈ N, +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥ˆz(t) − ˆz(t+1)∥2 +2 +� +≤ 4D2 +Z + 8η2L2∥Z∥2 +2 + 8DZV(T) +NE , +where V(T) +NE is defined in (4). +25 + +Proof. Following the proof of Theorem 3.4, we have that for any η ≤ +1 +4L, +0 ≤ D2 +Z +2η + ηL2∥Z∥2 +2 + DZ +η V(T) +NE + 2η∥Z∥2 +2V(T) +A +− 1 +4η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +. +Further, for η ≤ +1 +8W∥Z∥2 , +2η∥Z∥2 +2V(T) +A +− 1 +8η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≤ +� +2η∥Z∥2 +2W 2 − +1 +16η +� T−1 +� +t=1 +∥z(t+1) − z(t)∥2 +2 ≤ 0. +Thus, we have shown that +0 ≤ D2 +Z +2η + ηL2∥Z∥2 +2 + DZ +η V(T) +NE − 1 +8η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +and rearranging concludes the proof. +Thus, in such time-varying games it is the first-order variation term, V(T) +NE , that will drive our +convergence bounds. +Now before proving Theorem 3.5, we state the connection between the equilibrium gap and the +deviation of the players’ strategies +� +∥z(t) − ˆz(t)∥2 + ∥z(t) − ˆz(t+1)∥2 +� +. In particular, the following +claim can be extracted by [Ana+22b, Claim A.14]. (We caution that we use a slightly different +indexing for the secondary sequence (ˆx(t) +i ) in the definition of OMD (13) compared to [Ana+22b].) +Claim A.9. Suppose that the sequences (x(t) +i )1≤t≤T and (ˆx(t) +i )1≤t≤T+1 are produced by OMD under +a G-smooth regularizer 1-strongly convex with respect to a norm ∥ · ∥. Then, for any time t ∈ [[T]] +and any xi ∈ Xi, +⟨x(t) +i , u(t) +i ⟩ ≥ ⟨xi, u(t) +i ⟩ − G +η ∥ˆx(t+1) +i +− ˆx(t) +i ∥ − ∥u(t) +i ∥∗∥x(t) +i +− ˆx(t+1) +i +∥. +We are now ready to prove Theorem 3.5, the precise version of which is stated below. +Theorem A.10 (Detailed version of Theorem 3.5). Suppose that both players employ OGD with +learning rate η = +1 +4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2. +Then, +T +� +t=1 +� +EqGap(t)(z(t)) +�2 +≤ 2L2(4 + ∥Z∥2)2 � +2D2 +Z + 4η2L2∥Z∥2 +2 + 4DZV(T) +ϵ−NE + 8η2∥Z∥2 +2V(T) +A +� +, +where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD. +26 + +Proof. Let us first fix a time t ∈ [[T]]. For convenience, we denote by BR(t) +x (x(t)) := maxx∈X {⟨x, u(t) +x ⟩}− +⟨x(t), u(t) +x ⟩, the best response gap of Player’s x strategy x(t) ∈ X, and similarly for BR(t) +y (y)(t). By +definition, it holds that EqGap(t) := max{BR(t) +x (x(t)), BR(t) +y (y(t))}. By Claim A.9, we have that +BR(t) +x (x(t)) ≤ 1 +η∥ˆx(t+1) − ˆx(t)∥2 + ∥u(t) +x ∥2∥x(t) − ˆx(t+1)∥2 +(35) +≤ 4L∥ˆx(t+1) − ˆx(t)∥2 + L∥Y∥2∥x(t) − ˆx(t+1)∥2 +(36) +≤ L (4 + ∥Z∥2) +� +∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2 +� +, +(37) +where (35) follows from Claim A.9 for G = 1 (since the squared Euclidean regularizer φx : x �→ 1 +2∥x∥2 +2) +is 1-smooth; (36) uses the fact that η := +1 +4L and ∥u(t) +x ∥2 = ∥ − A(t)y(t)∥2 ≤ L∥Y∥; and (37) follows +from the triangle inequality. A similar derivation shows that +BR(t) +y (y(t)) ≤ L(4 + ∥Z∥) +� +∥y(t) − ˆy(t)∥2 + ∥y(t) − ˆy(t+1)∥2 +� +. +(38) +Thus, +T +� +t=1 +� +EqGap(t)(z(t)) +�2 += +T +� +t=1 +� +max{BR(t) +x (x(t)), BR(t) +y (y(t))} +�2 +≤ +T +� +t=1 +�� +BR(t) +x (x(t)) +�2 ++ +� +BR(t) +y (y(t)) +�2� +≤ 2L2(4 + ∥Z∥2)2 +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +(39) +where the last bound uses (37) and (38). Combining (39) with Theorem A.6 concludes the proof. +A.1.5 +Variation-Dependent Regret Bounds +Here we state an important implication of Theorem 3.5 for deriving variation-dependent regret +bounds in time-varying bilinear saddle-point problems; cf. [Zha+22]. +Corollary A.11 (Detailed version of Corollary 3.7). In the setup of Theorem 3.4, it holds that +Reg(T) +x +≤ D2 +X +η ++ 8ηL2D2 +Z + ηL2∥Y∥2 +2 + 16η3L4∥Z∥2 +2 + 16ηL2DZV(T) +NE + (2η∥Y∥2 +2 + 32η3L2∥Z∥2 +2)V(T) +A , +and +Reg(T) +y +≤ D2 +Y +η ++ 8ηL2D2 +Z + ηL2∥X∥2 +2 + 16η3L4∥Z∥2 +2 + 16ηL2DZV(T) +NE + (2η∥X∥2 +2 + 32η3L2∥Z∥2 +2)V(T) +A . +Proof. First, applying Lemma 3.1 under x(1,⋆) = · · · = x(T,⋆), we have +Reg(T) +x +≤ D2 +X +η ++ ηL2∥Y∥2 +2 + 2ηL2 +T +� +t=2 +∥y(t) − y(t−1)∥2 +2 + 2η∥Y∥2 +2 +T +� +t=2 +∥A(t) − A(t−1)∥2 +2, +(40) +27 + +and similarly, +Reg(T) +y +≤ D2 +Y +η ++ ηL2∥X∥2 +2 + 2ηL2 +T +� +t=2 +∥x(t) − x(t−1)∥2 +2 + 2η∥X∥2 +2 +T +� +t=2 +∥A(t) − A(t−1)∥2 +2. +(41) +Now, by Theorem 3.4 we have +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≤ 2D2 +Z + 4η2L2∥Z∥2 +2 + 4DZV(T) +NE + 8η2∥Z∥2 +2V(T) +A . +(42) +Further, +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≥ +T +� +t=1 +� +∥x(t) − ˆx(t)∥2 +2 + ∥x(t) − ˆx(t+1)∥2 +2 +� +≥ 1 +2 +T +� +t=2 +∥x(t) − x(t−1)∥2 +2. +Combining this bound with (42) and (41) gives the claimed regret bound on Reg(T) +y +, and a similar +derivation also gives the claimed bound on Reg(T) +x . +A.1.6 +Meta-Learning +We next provide the implication of Theorem 3.5 in the meta-learning setting. We first make a +remark regarding the effect of the prediction of OGD to Theorem 3.5, and how that relates to an +assumption present in [Har+22]. +Remark A.12 (Improved predictions). Throughout Section 3.1, we have considered the standard +prediction m(t) +x +:= u(t−1) +x += −A(t−1)y(t−1) for t ≥ 2, and similarly for Player y. It is easy to see +that using the predictions +m(t) +x := −A(t)y(t−1) and m(t) +y +:= (A(t))⊤x(t−1) +(43) +for t ≥ 1 (where z(0) := ˆz(1)) entirely removes the dependency on V(T) +A +on all our convergence +bounds. While such a prediction cannot be implemented in the standard online learning model, +there are settings in which we might know the sequence of matrices in advance; the meta-learning +setting offers such examples, and indeed, Harris et al. [Har+22] use the improved prediction of (43). +Proposition A.13 (Meta-learning). Suppose that both players employ OGD with learning rate η = +1 +4L, +where L := maxh∈[[H]] ∥A(h)∥2, and the prediction of (43) in a meta-learning bilinear saddle-point +problem with H ∈ N games, each repeated for m ∈ N consecutive iterations. Then, for an average +game, +� +P +Hϵ2 + P ′V(H) +NE +Hϵ2 +� +(44) +iterations suffice to reach an ϵ-approximate Nash equilibrium, where P := 4L2(4 + ∥Z∥2)2D2 +Z, +P ′ := 8L2(4 + ∥Z∥)2DZ, and +V(H) +NE := +inf +z(h,⋆)∈Z(h,⋆),∀h∈[[H]] +H−1 +� +h=1 +∥z(h+1,⋆) − z(h,⋆)∥2. +28 + +The proof is a direct application of Theorem A.10, where we remark that the term depending +on V(T) +A +and the term 4η2L2∥Z∥2 +2 from Theorem A.10 are eliminated because of the improved +prediction of Remark A.12. The first term in the iteration complexity bound (44) vanishes in +the meta-learning regime—as the number of games increases H ≫ 1—while the second term is +proportional to V(H) +NE +H , a natural similarity measure; (44) always recovers the m−1/2 rate, but offers +significant gains if the games as similar, in the sense that V(H) +NE +H +≪ 1. It is worth noting that, unlike +the similarity measure derived in [Har+22], V(H) +NE +H +depends on the order of the games. We further +remark that Proposition A.13 can be readily extended even if each game in the meta-learning +sequence is not repeated for the same number of iterations. +A.1.7 +General Variational Inequalities +Although our main focus in this paper is on the convergence of learning algorithms in time-varying +games, our techniques could also be of interest for solving (static) general variational inequality +(VI) problems. +In particular, let F : Z → Z be a single-valued operator. Solving general VIs is well-known to +be computationally intractable, and so instead focus has been on identifying broad subclasses that +elude those intractability barriers (see our overview in Section 1.2). Our framework in Section 3.1 +motivates introducing the following measure of complexity for a VI problem: +C(F) := +inf +z(1,⋆),...,z(T,⋆)∈Z +T−1 +� +t=1 +∥z(t+1,⋆) − z(t,⋆)∥2, +(45) +subject to +DReg(T) +z +(z(1,⋆), . . . , z(T,⋆)) ≥ 0 ⇐⇒ +T +� +t=1 +⟨z(t) − z(t,⋆), F(z(t))⟩ ≥ 0. +(46) +In words, (45) expresses the infimum first-order variation that a sequence of comparators must have +in order to guarantee nonnegative dynamic regret (46); it is evident that (46) always admits a feasible +sequence, namely s(T) +z +:= (z(t))1≤t≤T . We note that, in accordance to our results in Section 3.1, one +can also consider an approximate version of the complexity measure (45), which could behave much +more favorably (recall Proposition 3.3). +Now in a (static) bilinear saddle-point problem, it holds that C(F) = 0 given that there exists a +static comparator that guarantees nonnegativity of the dynamic regret. More broadly, our techniques +imply O(poly(1/ϵ)) iteration-complexity bounds for any VI problem such that C(F) ≤ CT 1−ω, for a +time-independent parameter C > 0 and ω ∈ (0, 1]: +Proposition A.14. Consider a variational inequality problem described with the operator F : Z → Z +such that F is L-Lipschitz continuous, in the sense that ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, and +C(F) ≤ CT 1−ω for C > 0 and ω ∈ (0, 1]. Then, OGD with learning rate η = +1 +4L reaches an ϵ-strong +solution z⋆ ∈ Z in O(ϵ−2/ω) iterations; that is, ⟨z − z⋆, F(z⋆)⟩ ≥ −ϵ for any z ∈ Z. +It is worth comparing (45) with another natural complexity measure, namely infz⋆∈Z +�T +t=1⟨z(t)− +z⋆, F(z(t))⟩; the latter measures how negative (external) regret can be, and has already proven +useful in certain settings that go bilinear saddle-point problems [YM22], although unlike (45), it +29 + +does not appear to be useful in characterizing time-varying bilinear saddle-point problems. In this +context, O(poly(1/ϵ)) iteration-complexity bounds can also be established whenever +• infz⋆∈Z +�T +t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −CT 1−ω for a time-invariant C > 0, or +• infz⋆∈Z +�T +t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −C �T−1 +t=1 ∥z(t+1) − z(t)∥2 +2, for a sufficiently small C > 0. +Following [YM22], identifying VIs that satisfy those relaxed conditions but not the MVI property +is an interesting direction. In particular, it is important to understand if those relaxations can shed +led light into the convergence properties of OGD in Shapley’s two-player zero-sum stochastic games. +A.2 +Proofs from Section 3.2 +In this subsection, we provide the proofs from Section 3.2, leading to our main result in Theorem 3.8. +Let us first introduce some additional notation. We let f(t) : X × Y → R be a continuously +differentiable function for any t ∈ [[T]]. We recall that in Section 3.2 it is assumed that the objective +function changes after m ∈ N (consecutive) repetitions, which is akin to the meta-learning setting. +Analogously to our setup for bilinear saddle-point problems (Section 3.1), it is assumed that Player +x is endeavoring to minimizing the objective function, while Player y is trying to maximize it. We +will denote by Reg(T) +L,x(x⋆) := �T +t=1⟨x(t) − x⋆, −u(t) +x ⟩ and Reg(T) +L,y(y⋆) := �T +t=1⟨y⋆ − y(t), u(t) +y ⟩, where +u(t) +x +:= −∇xf(x(t), y(t)) and u(t) +y +:= ∇yf(x(t), y(t)) for any t ∈ [[T]]; similar notation is used for +DReg(T) +L,x, DReg(T) +L,y. +Furthermore, we let s(T) +z += ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆) and y(t,⋆) = y(h,⋆) for +any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]]. The first important step in our analysis is that, +following the proof of Lemma 3.1, +DReg(T) +L,x(s(T) +x ) ≤ 1 +2η +H +� +h=1 +� +∥ˆx(h,1) − x(h,⋆)∥2 +2 − ∥ˆx(h,m+1) − x(h,⋆)∥2 +2 +� ++ η +T +� +t=1 +∥u(t) +x − m(t) +x ∥2 +2 +− 1 +2η +T +� +t=1 +� +∥x(t) − ˆx(t)∥2 +2 + ∥x(t) − ˆx(t+1)∥2 +2 +� +, +(47) +where ˆx(h,k) := ˆx((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]], ˆx(h,m+1) := ˆx(h+1,1) for h ∈ [[H − 1]], and +ˆx(H,m+1) := ˆx(T+1). Similarly, +DReg(T) +L,y(s(T) +y +) ≤ 1 +2η +H +� +h=1 +� +∥ˆy(h,1) − y(h,⋆)∥2 +2 − ∥ˆy(h,m+1) − y(h,⋆)∥2 +2 +� ++ η +T +� +t=1 +∥u(t) +y − m(t) +y ∥2 +2 +− 1 +2η +T +� +t=1 +� +∥y(t) − ˆy(t)∥2 +2 + ∥y(t) − ˆy(t+1)∥2 +2 +� +. +(48) +Next, we will use the following key observation, which lower bounds the sum of the players’ +(external) regrets under strong convexity-concavity. +Lemma A.15. Suppose that f : X × Y → R is a µ-strongly convex-concave function with respect to +∥ · ∥2. Then, for any Nash equilibrium z⋆ = (x⋆, y⋆) ∈ Z, +Reg(m) +L,x(x⋆) + Reg(m) +L,y (y⋆) ≥ µ +2 +m +� +t=1 +∥z(t) − z⋆∥2 +2. +30 + +Proof. First, by µ-strong convexity of f(x, ·), we have that for any time t ∈ [[m]], +⟨x(t) − x⋆, ∇xf(x(t), y(t))⟩ ≥ f(x(t), y(t)) − f(x⋆, y(t)) + µ +2 ∥x(t) − x⋆∥2 +2. +(49) +Similarly, by µ-strong concavity of f(·, y), we have that for any time t ∈ [[m]], +⟨y⋆ − y(t), ∇yf(x(t), y(t))⟩ ≥ f(x(t), y⋆) − f(x(t), y(t)) + µ +2 ∥y(t) − y⋆∥2 +2. +(50) +Further, for any Nash equilibrium (x⋆, y⋆) ∈ Z it holds that f(x(t), y⋆) ≥ f(x(t), y(t)) ≥ f(x⋆, y(t)). +Combining this fact with (49) and (50) and summing over all t ∈ [[m]] gives the statement. +In turn, this readily implies the following lower bound for the dynamic regret. +Lemma A.16. Suppose that f(h) : X × Y → R is a µ-strongly convex-concave function with respect +to ∥ · ∥2, for any h ∈ [[H]]. Consider a sequence s(T) +z += ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆) +and y(t,⋆) = y(h,⋆) for any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]]. If (x(h,⋆), y(h,⋆)) ∈ Z is a Nash +equilibrium of f(h), +DReg(T) +L,x(s(T) +x ) + DReg(T) +L,y(s(T) +y +) ≥ µ +2 +H +� +h=1 +m +� +k=1 +∥z(h,k) − z(h,⋆)∥2 +2, +where z(h,k) := z((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]]. +We next combine this with the following monotonicity property of OGD: If z⋆ is a Nash equilibrium, +∥ˆz(t) − z⋆∥2 is a decreasing function in t [Har+22, Proposition C.10]. This leads to the following +refinement of Lemma A.16. +Lemma A.17. Under the assumptions of Lemma A.16, if η ≤ +1 +2µ, +DReg(T) +L,x(s(T) +x ) + DReg(T) +L,y(s(T) +y +) + 1 +4η +T +� +t=1 +∥z(t) − ˆz(t+1)∥2 +2 ≥ µm +4 +H +� +h=1 +∥ˆz(h,m+1) − z(h,⋆)∥2 +2. +Proof. By Lemma A.16, +DReg(T) +L,x(s(T) +x ) + DReg(T) +L,y(s(T) +y +) + 1 +4η +T +� +t=1 +∥z(t) − ˆz(t+1)∥2 +2 ≥ µ +2 +H +� +h=1 +m +� +k=1 +∥z(h,k) − z(h,⋆)∥2 +2 ++ 1 +4η +T +� +t=1 +∥z(t) − ˆz(t+1)∥2 +2 +≥ µ +4 +H +� +h=1 +m +� +k=1 +∥ˆz(h,k+1) − z(h,⋆)∥2 +2 +(51) +≥ µm +4 +H +� +h=1 +∥ˆz(h,m+1) − z(h,⋆)∥2 +2, +(52) +where (51) uses that +1 +4η ≥ µ +2 along with Young’s inequality and triangle inequality, and (52) follows +from [Har+22, Proposition C.10]. +31 + +Armed with this important lemma, we are ready to establish our main result (Theorem 3.8), the +detailed version of which is given below. We first point out that a function f : X × Y → R is said +to be L-smooth if ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, where F(z) := (∇xf(x, y), −∇yf(x, y)). +Theorem A.18 (Detailed version of Theorem 3.8). Let f(h) : X × Y be a µ-strongly convex-concave +and L-smooth function, for h ∈ [[H]]. Suppose that both players employ OGD with learning rate +η = min +� +1 +8L, 1 +2µ +� +for T repetitions, where T = m × H and m ≥ +2 +ηµ. Then, +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +≤ 4D2 +Z + 8η2∥F(z(1))∥2 +2 + 8S(H) +NE + 16η2V(H) +∇f , +where S(H) +NE and V(H) +∇f are defined in (9) and (11). Thus, �T +t=1 +� +EqGap(t)(z(t)) +�2 += O(1 + S(H) +NE + +V(H) +∇f ). +Proof. Combining Lemma A.17 with (47) and (48), +0 ≤ 1 +2η +H +� +h=1 +� +∥ˆz(h,1) − z(h,⋆)∥2 +2 − 2 + ηµm +2 +∥ˆz(h,m+1) − z(h,⋆)∥2 +2 +� ++η +T +� +t=1 +∥u(t) +z − m(t) +z ∥2 +2 − 1 +4η +T +� +t=1 +� +∥z(t) − ˆz(t)∥2 +2 + ∥z(t) − ˆz(t+1)∥2 +2 +� +, +for a sequence of Nash equilibria (z(h,⋆))1≤h≤H, where we used the notation u(t) +z +:= (u(t) +x , u(t) +y ) and +m(t) +z +:= (m(t) +x , m(t) +y ). Now we bound the first term of the right-hand side above as +1 +2η +H +� +h=1 +� +∥ˆz(h,1) − z(h,⋆)∥2 +2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2 +2 +� +≤ +1 +2η∥ˆz(1,1) − z(1,⋆)∥2 +2 + 1 +2η +H−1 +� +h=1 +� +∥ˆz(h+1,1) − z(h+1,⋆)∥2 +2 − 2∥ˆz(h+1,1) − z(h,⋆)∥2 +2 +� +, +where we used the fact that m ≥ +2 +ηµ and ˆz(h,m+1) = ˆz(h+1,1), for h ∈ [[H − 1]]. Hence, continuing +from above, +1 +2η +H +� +h=1 +� +∥ˆz(h,1) − z(h,⋆)∥2 +2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2 +2 +� +≤ 1 +2η∥ˆz(1,1)−z(1,⋆)∥2 +2+ 1 +η +H−1 +� +h=1 +∥z(h+1,⋆)−z(h,⋆)∥2 +2, +since ∥ˆz(h+1,1) − z(h+1,⋆)∥2 +2 ≤ 2∥ˆz(h+1,1) − z(h,⋆)∥2 +2 + 2∥z(h,⋆) − z(h+1,⋆)∥2 +2, by the triangle inequality +and Young’s inequality. Moreover, for t ≥ 2, +∥u(t) +z −u(t−1) +z +∥2 +2 = ∥F (t)(z(t))−F (t−1)(z(t−1))∥2 +2 ≤ 2L2∥z(t)−z(t−1)∥2 +2+2∥F (t)(z(t−1))−F (t−1)(z(t−1))∥2 +2, +by L-smoothness. As a result, +T−1 +� +t=1 +∥u(t+1) +z +− u(t) +z ∥2 +2 ≤ 2L2 +T−1 +� +t=1 +∥z(t) − z(t−1)∥2 +2 + 2V(T) +∇f , +and the claimed bound on the second-order path length follows. Finally, the second claim of the +theorem follows from Claim A.9 using convexity-concavity, analogously to Theorem 3.5. +32 + +We point out that the improved prediction mechanism described in Remark A.12 can also be used +in this setting as well, resulting in the elimination of the variation measure (11) from Theorem A.18. +We conclude this subsection by pointing out an improved variation-dependent regret bound, which +follows directly from Theorem A.18 (cf. Corollary A.11). +Corollary A.19. In the setup of Theorem A.18, +Reg(T) +L,x, Reg(T) +L,y ≤ D2 +Z +η ++ 4ηL2D2 +Z + 32η3L2∥F(z(1)∥2 +2 + 32ηL2S(H) +NE + (64η3L2 + 2η)V(H) +∇f . +Thus, setting the learning rate optimally implies that Reg(T) +L,x, Reg(T) +L,y = O +�� +S(H) +NE + V(H) +∇f +� +. +A.3 +Proofs from Section 3.3 +In this subsection, we provide the proofs from Section 3.3. +A.3.1 +Potential Games +We first characterize the behavior of GD in time-varying potential games. Below we give the formal +definition of an unweighted potential game, represented in normal form. +Definition A.20 (Potential game). A game admits a potential if there exists a function Φ : +× +n +i=1 Xi → R such that for any Player i ∈ [[n]], any joint strategy profile x−i ∈×i′̸=i Xi′, and any +pair of strategies xi, x′ +i ∈ Xi, +Φ(xi, x−i) − Φ(x′ +i, x−i) = ui(xi, x−i) − ui(x′ +i, x−i). +We also recall that GD is equivalent to OGD under the prediction m(t) +x += 0 for all t. The key +ingredient in the proof of Theorem 3.10 is the following key bound on the second-order path length +of the dynamics. +Proposition A.21. Suppose that each player employs GD with a sufficiently small learning rate +η > 0 and initialization (x(1) +1 , . . . , x(1) +n ) ∈× +n +i=1 Xi. Then, +1 +2η +T +� +t=1 +n +� +i=1 +∥x(t+1) +i +− x(t) +i ∥2 +2 ≤ +T +� +t=1 +� +Φ(t)(x(t+1) +1 +, . . . , x(t+1) +n +) − Φ(t)(x(t) +1 , . . . , x(t) +n ) +� +. +(53) +This bound can be derived from [Ana+22b, Theorem 4.3]. We note that if Φ(1) = Φ(2) = · · · = +Φ(T), the right-hand side of (53) telescops, thereby implying that the second-order path-length is +bounded. More generally, the right-hand side of (53) can be upper bounded by +2Φmax + +T−1 +� +t=1 +� +Φ(t)(x(t+1) +1 +, . . . , x(t+1) +n +) − Φ(t+1)(x(t+1) +1 +, . . . , x(t+1) +n +) +� +≤ 2Φmax + V(T) +Φ , +(54) +where Φmax is an upper bound on |Φ(t)(·)| for any t ∈ [[T]], and V(T) +Φ +is the variation measure of the +potential functions we introduced in Section 3.3. Furthermore, we know that the Nash equilibrium +gap in the t-th potential game can be bounded in terms of �n +i=1 ∥x(t+1) +i +− x(t) +i ∥2 (Claim A.9). As a +result, combining this property with Proposition A.21 and (54) establishes Theorem 3.10. +33 + +A.3.2 +General-Sum Games +We next turn out attention to general-sum multi-player games using the bilinear formulation +presented in Section 3.3. To establish Property 3.11, let us first define the regret of any Player +i ∈ [[n]] as +Reg(T) +i +(¯x⋆ +i ) := +T +� +t=1 +⟨¯x⋆ +i − ¯x(t) +i , (A(t) +i )⊤µ(t)⟩, +where ¯x⋆ +i ∈ ¯Xi, so that �n +i=1 Reg(T) +i +is easily seen to be equal to the regret of the maximizing +player in (12). Further, the dynamic regret of the mediator—the minimizing player in (12)—can be +expressed as +DReg(T) +µ (µ(1,⋆), . . . , µ(T,⋆)) := +T +� +t=1 +⟨µ(t) − µ(t,⋆), +n +� +i=1 +A(t) +i ¯x(t) +i ⟩. +Property 3.11. Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]]. +Then, +DReg(T) +µ (µ(1,⋆), . . . , µ(T,⋆)) + +n +� +i=1 +Reg(T) +i +≥ 0. +Proof. We have that +DReg(T) +µ ++ +n +� +i=1 +Reg(T) +i +(¯x⋆ +i ) = +n +� +i=1 +T +� +t=1 +⟨¯x⋆ +i , (A(t) +i )⊤µ(t)⟩ − +T +� +t=1 +⟨µ(t,⋆), +n +� +i=1 +A(t) +i ¯x(t) +i ⟩. +Now for any correlated equilibrium µ(t,⋆) of the t-th game we have that ⟨µ(t,⋆), A(t) +i ¯x(t) +i ⟩ ≤ 0 for any +Player i ∈ [[n]], ¯xi ∈ ¯ +Xi, and time t ∈ [[T]], which in turn implies that − �T +t=1⟨µ(t,⋆), �n +i=1 A(t) +i ¯x(t) +i ⟩ ≥ +0. Moreover, �n +i=1 max¯x⋆ +i ∈ ¯ +Xi +�T +t=1⟨¯x⋆ +i , (A(t) +i )⊤µ(t)⟩ ≥ 0 given that, by definition, 0 ∈ ¯ +Xi. This +concludes the proof. +Next, we provide the main implication of Theorem 3.12 in the meta-learning setting, which +is similar to the meta-learning guarantee of Proposition A.13 we established earlier in two-player +zero-sum games. Below, we denote by Ξ(h,⋆) the set of correlated equilibria of the h-th game in the +meta-learning sequence. +Corollary A.22 (Meta-learning general-sum). Suppose that each player employ OGD in (12) with a +suitable learning rate η > 0 and the prediction of (43) in a meta-learning general-sum problem with +H ∈ N games, each repeated for m ∈ N consecutive iterations. Then, for an average game, +O +� +1 +ϵ2H + V(H) +CE +ϵ2H +� +(55) +iterations suffice to reach an ϵ-approximate correlated equilibrium, where +V(H) +CE := +inf +µ(h,⋆)∈Ξ(h,⋆) ∥µ(h+1,⋆) − µ(h,⋆)∥2. +34 + +In particular, in the meta-learning regime, H ≫ 1, the iteration-complexity bound (55) is +dominated by the (algorithm-independent) similarity metric of the correlated equilibria +V(H) +CE +H . +Corollary A.22 establishes significant gains when V(H) +CE +H +≪ 1. +Finally, we conclude this subsection by providing a variation-dependent regret bound in general- +sum multi-player games. To do so, we combine Corollary 3.7 with Theorem 3.12, leading to the +following guarantee. +Corollary A.23 (Regret in general-sum games). In the setup of Theorem 3.12, +Reg(T) +µ , Reg(T) +i += O +�1 +η + η +� +V(T) +CE + V(T) +A +�� +, +for any Player i ∈ [[n]]. +In particular, if one selects optimally the learning rate, Corollary A.23 implies that the individual +regret of each player is bounded by O +�� +V(T) +CE + V(T) +A +� +. We note again that there are techniques +that would allow (nearly) recovering such regret guarantees without having to know the variation +measures in advance [Zha+22]. +A.4 +Proofs from Section 3.4 +Finally, in this subsection we present the proofs omitted from Section 3.4. We begin with Proposi- +tion 3.13, the statement of which is recalled below. We first recall that a regularizer φx, 1-strongly +convex with respect to a norm ∥ · ∥, is said to be G-smooth if ∥∇φx(x) − ∇φx(x′)∥∗ ≤ G∥x − x′∥, +for all x, x′. +Proposition 3.13. Suppose that both players in a (static) two-player zero-sum game employ OMD +with a smooth regularizer. Then, DReg(T) +x , DReg(T) +y += O( +√ +T). +Proof. First, using Claim A.9, it follows that the dynamic regret DReg(T) +x +of Player x up to time T +can be bounded as +T +� +t=1 +� +max +x(t,⋆)∈X +� +⟨x(t,⋆), u(t) +x ⟩ +� +− ⟨x(t), u(t) +x ⟩ +� +≤ +T +� +t=1 +� �G +η + ∥u(t) +x ∥∗ +� +∥x(t) − ˆx(t+1)∥ + G +η ∥x(t) − ˆx(t)∥ +� +, +(56) +where G > 0 is the smoothness parameter of the regularizer, and η > 0 is the learning rate. We further +know that �T +t=1 +� +∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2� += O(1) for any instance of OMD in a two-player +zero-sum game [Ana+22b], which in turn implies that �T +t=1 +� +∥x(t) − ˆx(t)∥ + ∥x(t) − ˆx(t+1)∥ +� += +O( +√ +T) by Cauchy-Schwarz. Thus, combining with (56) we have shown that DReg(T) +x += O( +√ +T). +Similar reasoning yields that DReg(T) +y += O( +√ +T), concluding the proof. +In contrast, we next show that such a result is precluded in general-sum games. In particular, +we note that the following computational-hardness result holds beyond the online learning setting. +35 + +It should be stressed that without imposing computational or memory restrictions there are trivial +online algorithms that guarantee even O(1) dynamic regret by first exploring the payoff matrices +and then computing a Nash equilibrium; we suspect that under the memory limitations imposed in +our work, as in [DDK11], there could be unconditional information-theoretic lower bounds, but that +is left for future work. +Proposition 3.15. Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n +i=1 DReg(T) +i += Ω(T), +even if n = 2, where Ω(·) here hides polynomial factors. +Proof. We will use the fact that computing a Nash equilibrium in two-player (normal-form) games +to a sufficiently small accuracy ϵ := 1/poly is PPAD-hard [CDT09]. Indeed, suppose that there exist +polynomial-time algorithms that always guarantee that �n +i=1 DReg(T) +i +≤ ϵT, where n := 2. Then, +this implies that there exists a time t ∈ [[T]] such that +max +x(t,⋆) +1 +∈X1 +⟨x(t,⋆) +1 +, u(t) +1 ⟩ − ⟨x(t) +1 , u(t) +1 ⟩ + +max +x(t,⋆) +2 +∈X2 +⟨x(t,⋆) +2 +, u(t) +2 ⟩ − ⟨x(t) +2 , u(t) +2 ⟩ ≤ ϵ, +which in turn implies that (x(t) +1 , x(t) +2 ) is an ϵ-approximate Nash equilibrium. Further, such a time +t ∈ [[T]] can be identified in polynomial time. But this would imply that PPAD ⊆ P, concluding the +proof. +Finally, we provide the proof of Theorem 3.16, the detailed version of which is provided below. +Theorem A.24 (Detailed version of Theorem 3.16). Consider an n-player game such that ∥∇xiui(z)− +∇xiui(z′)∥2 ≤ L∥z − z′∥2, where z, z′ ∈ × +n +i=1 Xi, for any Player i ∈ [[n]]. Then, if all players +employ OGD with learning rate η > 0 it holds that +1. �n +i=1 K-DReg(T) +i += O(K√nL) for η = Θ +� +1 +L√n +� +; +2. K-DReg(T) +i += O(K3/4T 1/4n1/4√ +L), for any Player i ∈ [[n]], for η = Θ +� +K1/4 +n1/4L1/2 +� +. +Proof. First, applying Lemma 3.1 subject to the constraint that �T−1 +t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1 +gives that for any Player i ∈ [[n]], +K-DReg(T) +i +≤ D2 +Xi +2η (2K − 1) + η∥u(1) +i ∥2 +2 + η +T−1 +� +t=1 +∥u(t+1) +i +− u(t) +i ∥2 +2 − 1 +4η +T−1 +� +t=1 +∥x(t+1) +i +− x(t) +i ∥2 +2. +(57) +Further, by L-smoothness we have that +∥u(t+1) +i +− u(t) +i ∥2 +2 = ∥∇xiui(z(t+1)) − ∇xiui(z(t))∥2 +2 ≤ L2 +n +� +i=1 +∥x(t+1) +i +− x(t) +i ∥2 +2, +for any t ∈ [[T − 1]], where (x(t) +1 , . . . , x(t) +n ) = z(t) ∈× +n +i=1 Xi is the joint strategy profile at time +t. Thus, summing (57) over all i ∈ [[n]] and taking η ≤ +1 +2L√n implies that �n +i=1 K-DReg(T) +i +≤ +2K−1 +2η +�n +i=1 D2 +Xi + η �n +i=1 ∥u(1) +i ∥2 +2, yielding the first part of the statement. The second part follows +directly from (57) using the stability property of OGD: ∥x(t+1) +i +− x(t) +i ∥2 = O(η), for any time +t ∈ [[T − 1]]. +36 + +B +Experimental Examples +Finally, although the focus of this paper is theoretical, in this section we provide some illustrative +experimental examples. In particular, Appendix B.1 contains experiments on time-varying potential +games, while Appendix B.2 focuses on time-varying (two-player) zero-sum games. For simplicity, we +will be assuming that each game is represented in normal form. +B.1 +Time-Varying Potential Games +Here we consider time-varying 2-player identical-interest games. We point out that such games +are potential games (recall Definition A.20), and as such they are indeed amenable to our theory +in Section 3.3. +In our first experiment, we first sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 1000. +Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1, where A(0) := A. Here, +α > 0 is a parameter that controls the variation of the payoff matrices. In this time-varying setup, +we let each player employ (online) GD with learning rate η := 0.1. The results obtained under +different random initializations of matrices A and P are illustrated in Figure 1. +Next, we operate in the same time-varying setup but each player is now employing multiplicative +weights update (MWU), instead of gradient descent, with η := 0.1. As shown in Figure 2, while the +cumulative equilibrium gap is much larger compared to using GD (Figure 1), the dynamics still +appear to be approaching equilibria, although our theory does not cover MWU. We suspect that +theoretical results such as Theorem 3.10 should hold for MWU as well, but that has been left for +future work. +In our third experiment for identical-interest games, we again first sampled two matrices +A, P ∈ Rdx×dy, where dx = dy = 1000. Then, we defined A(t) := A(t−1) + ϵP for t ≥ 1, where +A(0) := A. Here, ϵ > 0 is the parameter intended to capture the variation of the payoff matrices. +The results obtained under different random initializations of A and P are illustrated in Figure 3. +As an aside, it is worth pointing out that this particular setting can be thought of as a game in +which the variation in the payoff matrices is controlled by another learning agent. In particular, our +theoretical results are helpful for characterizing the convergence properties of two-timescale learning +algorithms, in which the deviation of the game is controlled by a player constrained to be updating +its strategies with a much smaller learning rate. +B.2 +Time-Varying Zero-Sum Games +We next conduct experiments on time-varying bilinear saddle-point problems when players are +employing OGD. Such problems have been studied extensively in Section 3.1 from a theoretical +standpoint. +First, we sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 10; here we consider lower- +dimensional payoff matrices compared to the experiments in Appendix B.1 for convenience in the +graphical illustrations. Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1, +where A(1) := A. The results obtained under different random initializations are illustrated in +Figure 4. +37 + +0 +50 +100 +150 +200 +0 +2 +4 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0.0 +0.2 +0.4 +0.6 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +0 +1 +2 +3 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0.0 +0.2 +0.4 +0.6 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +0 +1 +2 +3 +4 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0.0 +0.2 +0.4 +0.6 +0.8 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +Iteration (t) +0 +2 +4 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +Iteration (t) +0.0 +0.2 +0.4 +0.6 +0.8 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +50 +100 +150 +200 +Iteration (t) +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +Figure 1: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest +games when both players are employing (online) GD with learning rate η := 0.1 for T := 200 iterations. +Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in +turn induces a different time-varying game. Further, each figure contains trajectories corresponding +to three different values of α ∈ {0.1, 0.2, 0.5}, but under the same initialization of A and P. As +expected, smaller values of α generally increase the equilibrium gap since the variation of the games +is more significant. Nevertheless, for all games we observe that the players are gradually approaching +equilibria. +38 + +0 +50 +100 +150 +200 +0 +100 +200 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0 +2 +4 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +20 +40 +60 +80 +100 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +0 +100 +200 +300 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0 +2 +4 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +20 +40 +60 +80 +100 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +0 +100 +200 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +0 +2 +4 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +20 +40 +60 +80 +100 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +50 +100 +150 +200 +Iterations +0 +100 +200 +300 +�t +τ=1(EG(τ))2 +0 +50 +100 +150 +200 +Iterations +0 +2 +4 +EG(t) +α = 0.5 +α = 0.2 +α = 0.1 +0 +20 +40 +60 +80 +100 +Iterations +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +Figure 2: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest +games when both players are employing (online) GD with learning rate η := 0.1 for T := 200 iterations. +Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in +turn induces a different time-varying game. Further, each figure contains trajectories corresponding +to three different values of α ∈ {0.1, 0.2, 0.5}, but under the same initialization of A and P. The +MWU dynamics still appear to be approaching equilibria, although the cumulative gap is much larger +compared to GD (Figure 1). +39 + +0 +100 +200 +300 +400 +500 +0.0 +0.5 +1.0 +�t +τ=1(EG(τ))2 +0 +100 +200 +300 +400 +500 +0.0 +0.1 +0.2 +EG(t) +ϵ = 0.001 +ϵ = 0.01 +ϵ = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +100 +200 +300 +400 +500 +0.0 +0.5 +1.0 +�t +τ=1(EG(τ))2 +0 +100 +200 +300 +400 +500 +0.0 +0.1 +0.2 +0.3 +EG(t) +ϵ = 0.001 +ϵ = 0.01 +ϵ = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +100 +200 +300 +400 +500 +0 +1 +2 +�t +τ=1(EG(τ))2 +0 +100 +200 +300 +400 +500 +0.0 +0.1 +0.2 +0.3 +EG(t) +ϵ = 0.001 +ϵ = 0.01 +ϵ = 0.1 +0 +50 +100 +150 +200 +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +0 +100 +200 +300 +400 +500 +Iterations +0 +1 +2 +3 +�t +τ=1(EG(τ))2 +0 +100 +200 +300 +400 +500 +Iterations +0.0 +0.1 +0.2 +EG(t) +ϵ = 0.001 +ϵ = 0.01 +ϵ = 0.1 +0 +50 +100 +150 +200 +Iterations +0 +2 +4 +6 +max(Reg(t) +x , Reg(t) +y ) +Figure 3: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest +games when both players are employing (online) GD with learning rate η := 0.1 for T := 500 iterations. +Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in +turn induces a different time-varying game. Further, each figure contains trajectories from three +different values of ϵ ∈ {0.1, 0.01, 0.001}, but under the same initialization of A and P. As expected, +larger values of ϵ generally increase the equilibrium gap since the variation of the games is more +significant. Yet, even for the larger value ϵ = 0.1, the dynamics are still appear to be approaching +Nash equilibria. +40 + +0 +200 +400 +600 +800 +1000 +0 +10000 +20000 +30000 +�t +τ=1(EG(τ))2 +0 +200 +400 +600 +800 +1000 +0.0 +0.2 +0.4 +0.6 +0.8 +EG(t) +α = 2 +α = 1 +α = 0.7 +0 +200 +400 +600 +800 +1000 +0 +5 +10 +15 +20 +max(Reg(t) +x , Reg(t) +y ) +0 +200 +400 +600 +800 +1000 +0 +2500 +5000 +7500 +10000 +�t +τ=1(EG(τ))2 +0 +200 +400 +600 +800 +1000 +0.0 +0.2 +0.4 +0.6 +0.8 +EG(t) +α = 2 +α = 1 +α = 0.7 +0 +200 +400 +600 +800 +1000 +0 +10 +20 +30 +40 +max(Reg(t) +x , Reg(t) +y ) +0 +200 +400 +600 +800 +1000 +0 +20000 +40000 +�t +τ=1(EG(τ))2 +0 +200 +400 +600 +800 +1000 +0.0 +0.2 +0.4 +0.6 +0.8 +EG(t) +α = 2 +α = 1 +α = 0.7 +0 +200 +400 +600 +800 +1000 +0 +10 +20 +30 +max(Reg(t) +x , Reg(t) +y ) +0 +200 +400 +600 +800 +1000 +Iteration (t) +0 +2500 +5000 +7500 +10000 +�t +τ=1(EG(τ))2 +0 +200 +400 +600 +800 +1000 +Iteration (t) +0.0 +0.2 +0.4 +0.6 +EG(t) +α = 2 +α = 1 +α = 0.7 +0 +200 +400 +600 +800 +1000 +Iteration (t) +0 +10 +20 +max(Reg(t) +x , Reg(t) +y ) +Figure 4: The equilibrium gap and the players’ regrets in 2-player time-varying zero-sum games +when both players are employing OGD with learning rate η := 0.01 and T := 1000 iterations. Each +row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn +induces a different time-varying game. Further, each figure contains trajectories from three different +values of α ∈ {0.7, 1, 2}, but under the same initialization of A and P. The OGD dynamics appear +to be approaching equilibria, albeit with a much slower rate compared to the ones observed earlier +for potential games (Figure 1). +41 + diff --git a/GtFIT4oBgHgl3EQfWyuf/content/tmp_files/load_file.txt b/GtFIT4oBgHgl3EQfWyuf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15983c603390af9ea7a9e14b1c8487820908ae5c --- /dev/null +++ b/GtFIT4oBgHgl3EQfWyuf/content/tmp_files/load_file.txt @@ -0,0 +1,1667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf,len=1666 +page_content='On the Convergence of No-Regret Learning Dynamics in Time-Varying Games Ioannis Anagnostides1, Ioannis Panageas2, Gabriele Farina3, and Tuomas Sandholm4 1,3,4Carnegie Mellon University 2University of California Irvine 4Strategy Robot, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 4Optimized Markets, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 4Strategic Machine, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' {ianagnos,gfarina,sandholm}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='edu, and ipanagea@ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='edu January 27, 2023 Abstract Most of the literature on learning in games has focused on the restrictive setting where the underlying repeated game does not change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Much less is known about the convergence of no-regret learning algorithms in dynamic multiagent settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this paper, we characterize the convergence of optimistic gradient descent (OGD) in time-varying games by drawing a strong connection with dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our framework yields sharp convergence bounds for the equilibrium gap of OGD in zero-sum games parameterized on the minimal first-order variation of the Nash equilibria and the second-order variation of the payoff matrices, subsuming known results for static games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Furthermore, we establish improved second-order variation bounds under strong convexity-concavity, as long as each game is repeated multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our results also apply to time-varying general-sum multi-player games via a bilinear formulation of correlated equilibria, which has novel implications for meta-learning and for obtaining refined variation-dependent regret bounds, addressing questions left open in prior papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, we leverage our framework to also provide new insights on dynamic regret guarantees in static games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11241v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='LG] 26 Jan 2023 1 Introduction Most of the classical results in the literate on learning in games—exemplified by, among others, the work of Hart and Mas-Colell [HM00], Foster and Vohra [FV97], and Freund and Schapire [FS99]—rest on the assumption that the underlying repeated game remains invariant throughout the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Yet, in many learning environments that is unrealistic [Duv+22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Zha+22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Car+19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' One such class is settings where the underlying game is actually changing, such as routing problems on the internet [MO11], online advertising auctions [LST16], and dynamic mechanism design [Pap+22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' DMZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Another such class consists of settings in which many similar games need to be solved [Har+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For example, one may want to solve variations of a game for the purpose of sensitivity analysis with respect to the modeling assumptions used to construct the game model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Another example is solving multiple versions of a game any one of which might be faced in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Despite the considerable interest in such dynamic multiagent environments, much less is known about the convergence of no-regret learning algorithms in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' No-regret dynamics are natural learning algorithms that have desirable convergence properties in static settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Also, the state-of-the-art algorithms for finding minimax equilibria in two-player zero-sum games are based on advanced forms of no-regret dynamics [FKS21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' BS19a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, all the superhuman milestones in poker have used them in the equilibrium-finding module of their architectures [Bow+15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' BS17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' BS19b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this paper, we seek to fill this knowledge gap by understanding properties of no-regret dy- namics in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, we primarily investigate the convergence of optimistic gradient descent (OGD) [Chi+12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' RS13] in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Unlike traditional no-regret learning algorithms, such as (online) gradient descent, OGD has been recently shown to exhibit last-iterate convergence in static (two-player) zero-sum games [Das+18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' GPD20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' COZ22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' GTG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For the more challenging scenario where the underlying game can vary in every round, a fundamental question arises: Under what conditions on the sequence of games does OGD (with high probability) approximate the sequence of Nash equilibria?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Our Results In this paper, we build a new framework that enables us to characterize the convergence of OGD in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Specifically, our first contribution is to identify natural variation measures on the sequence of games whose sublinear growth guarantees that almost all iterates of OGD are (approximate) Nash equilibria in time-varying (two-player) zero-sum games (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More precisely, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 we derive a sharp non-asymptotic characterization of the equilibrium gap of OGD as a function of the variation measures we identify: the minimal first-order variation of the Nash equilibria and the second-order variation of the payoff matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is a compelling property, in light of the multiplicity of Nash equilibria, that the variation of the Nash equilibria is measured in terms of the most favorable—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=', one that minimizes the variation—such sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Additionally, we show that our convergence bounds can be further improved by considering a variation measure that depends on the deviation of approximate Nash equilibria of the games, a measure that could be arbitrarily smaller than the one based on (even the least varying) sequence of exact Nash equilibria (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' From a technical standpoint, our analysis revolves around a new connection we draw between the convergence of OGD in time-varying games and dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, the first key observation is that dynamic regret is always nonnegative under any sequence of Nash equilibria (Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' By combining that property with a dynamic RVU bound—in the sense of Syrgkanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Syr+15]— 1 that we derive (Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1), we obtain in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 a variation-dependent bound for the second-order path length of OGD in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In turn, this leads to our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As such, we extend the regret-based framework of Anagnostides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Ana+22b] from static to time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the special case of static games, our result reduces to a tight T −1/2 rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is worth stressing that Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 is in fact more general, being intricately tied to the admission of a minimax theorem (Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3), and applies even under a sequence of approximate Nash equilibria—with slackness that gently degrades with the approximation thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Moreover, for strongly convex-concave time-varying games, we obtain a refined second-order variation bound on the sequence of Nash equilibria, as long as each game is repeated multiple times (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' this is inspired by an improved second-order bound for dynamic regret under analogous conditions due to Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a byproduct of our techniques, we point out that any no-regret learners are approaching a Nash equilibrium under strong convexity-concavity (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Those results apply even in non-strongly convex-concave settings by suitably trading-off the magnitude of a regularizer that makes the game strongly convex-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This offers significant gains in the meta-learning setting as well, wherein each game is repeated multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we extend our results to time-varying general-sum multi-player games via a bilinear formu- lation of correlated equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As such, we recover similar convergence bounds parameterized on the variation of the correlated equilibria (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' To illustrate the power of our framework, we immediately recover natural and algorithm-independent similarity measures for the meta-learning setting (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13) even in general games (Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='22), thereby addressing an open question of Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Har+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our techniques also imply new per-player regret bounds in zero-sum and general-sum games (Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='23), the latter addressing a question left open by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further parameterize the convergence of (vanilla) gradient descent in time-varying potential games in terms of the deviation of the potential functions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, building on our techniques in time-varying games, we investigate the best dynamic-regret guarantees possible in static games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Although this is a basic question, it has apparently eluded prior research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We first show that instances of optimistic mirror descent guarantee O( √ T) dynamic per-player regret (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13), matching the known rate of (online) gradient descent but for the significantly weaker notion of external regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further point out that O(log T) dynamic regret is attainable, but in a stronger two-point feedback model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In stark contrast, even obtaining sublinear dynamic regret for each player is precluded in general-sum games (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This motivates studying a relaxation of dynamic regret that constrains the number of switches in the comparator, for which we derive accelerates rates in general games (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16) by leveraging the techniques of Syrgkanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Syr+15] in conjunction with our dynamic RVU bound (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 Further Related Work Even in static (two-player) zero-sum games, the pointwise convergence of no-regret learning algo- rithms is a tenuous affair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, traditional learning dynamics within the no-regret framework, such as (online) mirror descent, may even diverge away from the equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=', see [SAF02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MPP18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Vla+20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' GVM21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Notwithstanding, the empirical frequency of no-regret learners is well-known to approach the set of Nash equilibria in zero-sum games [FS99], and the set of coarse correlated equilibria in general-sum games [HM00]—a standard relaxation of the Nash equilibrium [MV78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Aum74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Unfortunately, those classical results are of little use beyond static games, thereby offering a crucial impetus for investigating iterate-convergence in games with a time-varying component—a ubiquitous theme in many practical scenarios of interest [DMZ21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Pap+22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Ven21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Gar17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Van10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2 RG22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' PKB22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' YH15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' RJW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, there has been a considerable effort endeavoring to extend the scope of traditional game-theoretic results to the time-varying setting, approached from a variety of different stand- points [LST16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Zha+22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Car+19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MO11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MS21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Duv+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, our techniques in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 share similarities with the ones used by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22], but our primary focus is very different: Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22] were mainly interested in obtaining variation-dependent regret bounds, while our results revolve around iterate-convergence to Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We stress again that minimizing regret and approaching Nash equilibria are two inherently distinct problems, although connections have emerged [Ana+22b], and are further cultivated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Another closely related direction is on meta-learning in games [Har+22], wherein each game can be repeated for multiple iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Such considerations are motivated in part by a number of use-cases in which many “similar” games—or multiple game variations—ought to be solved [BS16], such as Poker with different stack-sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' While the meta-learning problem is a special case of our general setting, our results are strong enough to have new implications for meta-learning in games, even though the algorithms considered herein are not tailored to operate in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, although our focus is on the convergence of OGD in time-varying games, some of our results—namely, the ones formalized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7—can be viewed as part of an ongoing effort to characterize the class of variational inequalities (VIs) that are amenable to efficient algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' see [DDJ21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' CZ22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Azi+20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' BMW21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' CP04;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' DL15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MV21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' MRS20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Nou+19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Son+20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' YKH20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Das22], and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We also highlight that the techniques used to establish last-iterate convergence even in monotone (time-invariant) settings are particularly involved [GPD20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' COZ22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' GTG22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the simplicity of our framework, therefore, in the more challenging time-varying regime is a compelling aspect of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2 Preliminaries Notation We let N := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , } be the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For a number p ∈ N, we let [[p]] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For a vector w ∈ Rd, we use ∥w∥2 to represent its Euclidean norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we also overload that notation so that ∥ · ∥2 denotes the spectral norm when the argument is a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For a two-player zero-sum game, we denote by X ⊆ Rdx and Y ⊆ Rdy the strategy sets of the two players—namely, Player x and Player y, respectively—where dx, dy ∈ N represent the corresponding dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is assumed that X and Y are nonempty convex and compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For example, in the special case where X := ∆dx and Y := ∆dy—each set corresponds to a probability simplex—the game is said to be in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, we denote by DX the ℓ2-diameter of X, and by ∥X∥2 the maximum ℓ2-norm attained by a point in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We will always assume that the strategy sets remain invariant, while the payoff matrix can change in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For notational convenience, we will denote by z := (x, y) the concatenation of x and y, and by Z := X × Y the Cartesian product of X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In general n−player games, we instead use subscripts indexed by i ∈ [[n]] to specify quantities related to a player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Superscripts are typically reserved to identify the time index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, to simplify the exposition, we use the O(·) notation to suppress time-independent parameters of the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' precise statements are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Dynamic regret We operate in the usual online learning setting under full-feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Namely, at every time t ∈ N the learner decides on a strategy x(t) ∈ X, and then observes a utility x �→ ⟨x, u(t) x ⟩, for u(t) x ∈ Rdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Following Daskalakis, Deckelbaum, and Kim [DDK11], we will insist on allowing only 3 O(1) previous utilities to be stored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' this will preclude trivial exploration protocols when learning in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A strong performance benchmark in this online setting is dynamic regret, defined for a time horizon T ∈ N as follows: DReg(T) x (s(T) x ) := T � t=1 ⟨x(t,⋆) − x(t), u(t) x ⟩, (1) where s(T) x := (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) ∈ X T is the sequence of comparators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' setting x(1,⋆) = x(2,⋆) = · · · = x(T,⋆) in (1) we recover the standard notion of (external) regret (denoted simply by Reg(T) x ), which is commonly used to establish convergence of the time-average strategies in static two-player zero-sum games [FS99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' On the other hand, the more general notion of dynamic regret, introduced in (1), has been extensively used in more dynamic environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=', [Zha+20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Zha+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Jad+15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Ces+12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' HS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We also let DReg(T) x := maxs(T ) x ∈X T DReg(T) x (s(T) x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' While ensuring o(T) dynamic regret is clearly hopeless in a truly adversarial environment, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 reveals that non-trivial guarantees are possible when learning in zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Optimistic gradient descent Optimistic gradient descent (OGD) [Chi+12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' RS13] is a no-regret algorithm defined with the following update rule: x(t) := ΠX � ˆx(t) + ηm(t) x � , ˆx(t+1) := ΠX � ˆx(t) + ηu(t) x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (OGD) Here, η > 0 is the learning rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' ˆx(1) := arg minˆx∈X ∥ˆx∥2 2 represents the initialization of OGD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' m(t) x ∈ Rdx is the prediction vector at time t, and it is set as m(t) x := u(t−1) x when t ≥ 2, and m(1) x := 0dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and finally, ΠX (·) represents the Euclidean projection to the set X, which is well- defined, and can be further computed efficiently for structured sets, such as the probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For our purposes, we will posit access to a projection oracle for the set X, in which case the update rule (OGD) is efficiently implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In a multi-player n-player game, each Player i ∈ [[n]] is associated with a utility function ui :× n i=1 Xi → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We recall the following fundamental definition [Nas50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 (Approximate Nash equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A joint strategy profile (x⋆ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x⋆ n) ∈× n i=1 Xi is an ϵ-approximate Nash equilibrium (NE), for an ϵ ≥ 0, if for any Player i ∈ [[n]] and any possible deviation x′ i ∈ Xi, ui(x⋆ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x⋆ i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x⋆ n) ≥ ui(x⋆ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x⋆ n) − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 3 Convergence in Time-Varying Games In this section, we formalize our results regarding convergence in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We organize this section as follows: First, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, we build the foundations of our framework by studying the convergence of OGD in time-varying bilinear saddle-point problems, culminating in the non- asymptotic characterization of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 formalizes our improvements under strong convexity-concavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we then extend our results (in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3) to time-varying multi-player general- sum and potential games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and finally, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 concerns dynamic regret guarantees in static games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Bilinear Saddle-Point Problems We first study an online learning setting wherein two players interact in a sequence of time- varying bilinear saddle-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More precisely, we assume that in every repetition t ∈ [[T]] the players select a pair of strategies (x(t), y(t)) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, Player x receives the utility u(t) x := −A(t)y(t) ∈ Rdx, where A(t) ∈ Rdx×dy represents the payoff matrix at the t-th repetition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' similarly, Player y receives the utility u(t) y := (A(t))⊤x(t) ∈ Rdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The proofs of this subsection are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Dynamic RVU bound The first key ingredient that we need is the property of regret bounded by variation in utilities (RVU), in the sense of Syrgkanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Syr+15], but with respect to dynamic regret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' such a bound is established below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 (RVU bound for dynamic regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider any sequence of utilities (u(1) x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , u(T) x ) up to time T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The dynamic regret (1) of OGD with respect to any sequence of comparators (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) ∈ X T can be bounded by D2 X 2η + DX η T−1 � t=1 ∥x(t+1,⋆) − x(t,⋆)∥2+η T � t=1 ∥u(t) x − m(t) x ∥2 2 − 1 2η T � t=1 � ∥x(t) − ˆx(t)∥2 2 + ∥x(t) − ˆx(t+1)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (2) In the special case of external regret—x(1,⋆) = x(2,⋆) = · · · = x(T,⋆)—(2) recovers the bound for OGD of Syrgkanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Syr+15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The key takeaway from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 is that the overhead of dynamic regret in (2) grows with the first-order variation of the sequence of comparators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 we also articulate an extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 for the more general optimistic mirror descent (OMD) algorithm under a certain class of Bregman divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Having established Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, we next point out a crucial property: by selecting a sequence of Nash equilibria (recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1) as the comparators, the sum of the players’ dynamic regrets is always nonnegative: Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for s(T) x = (x(t,⋆))1≤t≤T and s(T) y = (y(t,⋆))1≤t≤T , DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) ≥ −2 T � t=1 ϵ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, if ϵ(t) = 0 for all t ∈ [[T]], we have DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (3) In fact, as we show in Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3, Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 applies even in certain (time-varying) nonconvex- nonconcave min-max optimization problems, and it is a consequence of the minimax theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 also holds for time-varying variational inequalities (VIs) that satisfy the so-called MVI property (see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For comparison, it is evident that under a static sequence of two-player zero-sum games, it holds that Reg(T) x + Reg(T) y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 5 Next, let us introduce some natural measures of the games’ variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, the first-order variation of the Nash equilibria is defined for T ≥ 2 as V(T) NE := inf z(t,⋆)∈Z(t,⋆),∀t∈[[T]] T−1 � t=1 ∥z(t+1,⋆) − z(t,⋆)∥2, (4) where Z(t,⋆) is the (nonempty) set of Nash equilibria of the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We recall that there can be a multiplicity of Nash equilibria [van91];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' as such, a compelling feature of the variation measure (4) is that it depends on the most favorable sequence of Nash equilibria—one that minimizes the first-order variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is also important to point out the well-known fact that Nash equilibria can change abruptly even under a “small” perturbation in the payoff matrix (see Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5), which is a caveat of the variation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' To address this, and in accordance with Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2, we consider a more favorable variation measure, defined as V(T) ϵ−NE := inf �T−1 � t=1 ∥z(t+1,⋆) − z(t,⋆)∥2 + C T � t=1 ϵ(t) � , for a sufficiently large parameter C > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the infimum above is subject to ϵ(t) ∈ R≥0 and z(t,⋆) ∈ Z(t,⋆) ϵ(t) for all t ∈ [[T]], where we denote by Z(t,⋆) ϵ(t) the set of ϵ(t)-approximate NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is evident that V(T) ϵ−NE ≤ V(T) NE since one can take ϵ(1) = · · · = ϵ(T) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' in fact, V(T) ϵ−NE can be arbitrarily smaller: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For any T ≥ 4, there is a sequence of T games such that V(T) NE ≥ T 2 while V(T) ϵ−NE ≤ δ, for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Moreover, we also introduce a quantity that captures the variation of the payoff matrices: V(T) A := T−1 � t=1 ∥A(t+1) − A(t)∥2 2, (5) where we recall that here ∥ · ∥2 denotes the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Unlike (4), the variation measure (5) depends on the second-order variation (of the payoff matrices), which could translate to a lower-order impact compared to (4) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=', Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We stress that while our convergence bounds will be parameterized based on (4) and (5), the underlying algorithm—namely OGD—will remain oblivious to those variation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We are ready now to establish a refined bound on the second-order path-length of OGD in time-varying zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 (Detailed version in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η ≤ 1 4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any T ∈ N, the second-order path length �T t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � can be bounded by O � 1 + V(T) ϵ−NE + V(T) A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (6) 6 It is worth noting that when the deviation of the payoff matrices is controlled by the deviation of the players’ strategies, in the sense that �T−1 t=1 ∥A(t+1)−A(t)∥2 2 ≤ W 2 �T−1 t=1 ∥z(t+1)−z(t)∥2 2 for some parameter W ∈ R>0, the variation measure V(T) A in (6)—and in the subsequent convergence bounds— can be entirely eliminated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' see Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The same, in fact, applies under an improved prediction mechanism (Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12), but that prediction is not implementable in our online learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Armed with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, we are ready to establish Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The key observation is that the Nash equilibrium gap at the t-th game can be bounded in terms of the quantity ∥z(t)− ˆz(t)∥2+∥z(t)− ˆz(t+1)∥2, which in turn allows us to use (6) to bound the cumulative (squared) Nash equilibrium gaps across the sequence of games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the aforeclaimed property was established in [Ana+22b] for static games (Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9), but readily extends to our setting as well, and in fact applies to any member of OMD under a smooth regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below, we use the notation EqGap(t)(z(t)) ∈ R≥0 to represent the Nash equilibrium gap of the joint strategy profile z(t) ∈ Z at the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 (Main result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' detailed version in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players em- ploy OGD with learning rate η = 1 4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � EqGap(t)(z(t)) �2 = O � 1 + V(T) ϵ−NE + V(T) A � , (7) where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We next state some immediate consequences of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (Item 2 below follows from (7) by Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=') Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' If at least a δ-fraction of the iterates of OGD have at least ϵ > 0 Nash equilibrium gap, then ϵ2δ ≤ O � 1 T � V(T) ϵ−NE + V(T) A + 1 �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The average Nash equilibrium gap of OGD is bounded as O �� 1 T � V(T) ϵ−NE + V(T) A + 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, in terms of asymptotic implications, if limT→+∞ V(T ) ϵ−NE T , limT→+∞ V(T ) A T = 0, then (i) for any ϵ > 0 the fraction of iterates of OGD with at least an ϵ Nash equilibrium gap converges to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and (ii) the average Nash equilibrium gap of the iterates of OGD converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the special case where V(T) ϵ−NE, V(T) A = O(1), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 recovers the T −1/2 rate of OGD in static bilinear saddle-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is also worth pointing out that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 readily extends to more general time-varying variational inequality problems as well (Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We also state below another interesting consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, which bounds each player’s individual regret parameterized based on the variation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 (Detailed version in Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the setup of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, it holds that Reg(T) x , Reg(T) y = O � 1 η + η(V(T) NE + V(T) A ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 7 The O(·) notation here is considered in the regime η ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Hence, selecting optimally the learning rate gives an O( � V(T) NE + V(T) A ) bound on the individual regret of each player;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' while that optimal value depends on the variation measures, which are not known to the learners, there are techniques that would allow bypassing this [Zha+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 can also be readily parameterized in terms of the improved variation measure V(T) ϵ−NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 we highlight certain implications of our framework on solving (static) general VIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Meta-Learning Our results also have immediate applications in the meta-learning setting [Har+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More precisely, meta-learning in games is a special case of time-varying games which consists of a sequence of H ∈ N separate games, each of which is repeated for m ∈ N consecutive rounds, so that T := m × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The central goal in meta-learning is to obtain convergence bounds parameterized by the similarity of the games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' identifying suitable similarity metrics is a central question in that line of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this context, we highlight that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 readily provides a meta-learning guarantee parameterized by the following notion of similarity between the Nash equilibria: inf z(h,⋆)∈Z(h,⋆),∀h∈[[H]] H−1 � h=1 ∥z(h+1,⋆) − z(h,⋆)∥2, (8) where Z(h,⋆) is the set of Nash equilibria of the h-th game in the meta-learning sequence,1 as well as the similarity of the payoff matrices—corresponding to the term V(T) A in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In fact, under a suitable prediction—the one used by Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Har+22]—the dependence on V(T) A can be entirely removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' see Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 for our formal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A compelling aspect of our meta-learning guarantee is that the considered algorithm is oblivious to the boundaries of the meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further provide some novel results on meta-learning in general-sum games in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 Strongly Convex-Concave Games In this subsection, we show that under additional structure we can significantly improve the variation measures established in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More precisely, we first assume that each objective function f(x, y) is µ-strongly convex with respect to x and µ-strongly concave with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our second assumption is that each game is played for multiple rounds m ∈ N, instead of only a single round;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' this is akin to the meta-learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The key insight is that, as long as m is large enough, m = Ω(1/µ), those two assumptions suffice to obtain a second-order variation bound in terms of the sequence of Nash equilibria, S(H) NE := H−1 � h=1 ∥z(h+1,⋆) − z(h,⋆)∥2 2, (9) where z(h,⋆) is a Nash equilibrium of the h-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This significantly refines the result of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, and is inspired by the improved dynamic regret bounds obtained by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below we sketch the key ideas of the improvement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the proofs are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this setting, it is assumed that Player x obtains the utility u(t) x := −∇xf(t)(x(t), y(t)) at every time t ∈ [[T]], while its regret will be denoted by Reg(T) L,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' similar notation applies for Player y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The 1In accordance to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, (8) can be refined using a sequence of approximate Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 8 first observation is that, focusing on a single (static) game, under strong convexity-concavity the sum of the players’ regrets are strongly nonnegative (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15): Reg(m) L,x(x⋆) + Reg(m) L,y (y⋆) ≥ µ 2 m � t=1 ∥z(t) − z⋆∥2 2, (10) for any Nash equilibrium z⋆ ∈ Z of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In turn, this can be cast in terms of dynamic regret over the sequence of the h games (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, combining those dynamic-regret lower bounds with a suitable RVU-type property leads to a refined second-order path length bound as long as that m = Ω(1/µ), which in turn leads to our main result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Before we present its statement, let us introduce the following measure of variation of the gradients: V(H) ∇f := H−1 � h=1 max z∈Z ∥F (h+1)(z) − F (h)(z)∥2 2, (11) where let F : z := (x, y) �→ (∇xf(x, y), −∇yf(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This variation measure is analogous to V(T) A we introduced in (5) for time-varying bilinear saddle-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 (Detailed version in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let f(h) : X × Y be a µ-strongly convex-concave and L-smooth function, for h ∈ [[H]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η = min � 1 8L, 1 2µ � for T repetitions, where T = m × H and m ≥ 2 ηµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � EqGap(t)(z(t)) �2 = O(1 + S(H) NE + V(H) ∇f ), where S(H) NE and V(H) ∇f are defined in (9) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our techniques also imply improved regret bounds in this setting, as we formalize in Corol- lary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' There is another immediate but important implication of (10): any no-regret algorithm in a (static) strongly convex-concave setting ought to be approaching the Nash equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' in contrast, this property is spectacularly false in (general) monotone settings [MPP18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let f : X × Y → R be a µ-strongly convex-concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' If players incur regrets such that Reg(T) L,x + Reg(T) L,y ≤ CT 1−ω, for some parameters C > 0 and ω ∈ (0, 1], then for any ϵ > 0 and T > � 2C µϵ2 �1/ω there is a pair of strategies z(t) ∈ Z such that ∥z(t) − z⋆∥2 ≤ ϵ, where z⋆ is a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The insights of this subsection are also of interest in general monotone settings by incorpo- rating a strongly convex regularizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' tuning its magnitude allows us to trade-off between a better approximation and the benefits of strong convexity-concavity revealed in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 General-Sum Multi-player Games Next, we turn our attention to general-sum multi-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For simplicity, in this subsection we posit that the game is represented in normal form, so that each Player i ∈ [[n]] has a finite set of available actions Ai, and Xi := ∆(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The proofs of this subsection are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 9 Potential Games First, we study the convergence of (online) gradient descent (GD) in time- varying potential games (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='20 for the formal description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 In our time-varying setup, it is assumed that each round t ∈ [[T]] corresponds to a different potential game described with a potential function Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further let d : (Φ, Φ′) �→ maxz∈×n i=1 Xi (Φ(z) − Φ′(z)), so that V(T) Φ := �T−1 t=1 d(Φ(t), Φ(t+1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we emphasize the fact that d(·, ·) is not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Analogously to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, we use EqGap(t)(z(t)) ∈ R≥0 to represent the NE gap of the joint strategy profile z(t) := (x(t) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t) n ) at the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that each player employs (online) GD with a sufficiently small learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � EqGap(t)(z(t)) �2 = O(Φmax + V(T) Φ ), where Φmax is such that |Φ(t)(·)| ≤ Φmax for any t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We refer to Appendix B for some illustrative experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' General games Unfortunately, unlike the settings considered thus far, computing Nash equilibria in general games is computationally hard [DGP08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' CDT09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Instead, learning algorithms are known to converge to relaxations of the Nash equilibrium, known as (coarse) correlated equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For our purposes, we will employ a bilinear formulation of (coarse) correlated equilibria, which dates back to the seminal work of Hart and Schmeidler [HS89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This will allow us to translate the results of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 to general multi-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Specifically, correlated equilibria3 can be expressed via a game between the n players and a medi- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Intuitively, the mediator is endeavoring to identify a correlated strategy µ ∈ Ξ := ∆ �× n i=1 Ai � for which no player has an incentive to deviate from the recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In contrast, the players are trying to optimally deviate so as to maximize their own utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More precisely, there exist matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , An, with each matrix Ai depending solely on the utility of Player i, for which the bilinear problem can be expressed as min µ∈Ξ max (¯x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=',¯xn)∈×n i=1 ¯ Xi n � i=1 µ⊤Ai ¯xi, (12) where ¯ Xi := conv(Xi, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' incorporating the 0 vector will be useful for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This zero-sum game has the property that there exists a strategy µ⋆ ∈ Ξ such that max¯xi∈ ¯ Xi(µ⋆)⊤Ai ¯xi ≤ 0, for any Player i ∈ [[n]], which corresponds to a correlated equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Before we proceed, it is important to note that the learning paradigm considered here deviates from the traditional one in that there is an additional learning agent, resulting in a less decentralized protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Yet, the dynamics induced by solving (12) via online algorithms remain uncoupled [HM00], in the sense that each player obtains feedback—corresponding to the deviation benefit—that depends solely on its own utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Now in the time-varying setting, the matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , An that capture the players’ utilities can change in each repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Crucially, we show that the structure of the induced bilinear problem (12) 2Unlike two-player zero-sum games, gradient descent is known to approach Nash equilibria in potential games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 3The following bilinear formulation applies to coarse correlated equilibria as well (with different payoff matrices), but we will focus solely on the stronger variant (CE) for the sake of exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 10 is such that there is a sequence of correlated equilibria that guarantee nonnegative dynamic regret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' this refines Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 in that only one player’s strategies suffice to guarantee nonnegativity, even if the strategies of the other player remain invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below, we denote by DReg(T) µ the dynamic regret of the min player in (12), and by Reg(T) i the regret of each Player i up to time T ∈ N, so that the regret of the max player in (12) can be expressed as �n i=1 Reg(T) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, DReg(T) µ (µ(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , µ(T,⋆)) + n � i=1 Reg(T) i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, this enables us to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 parameterized on (i) the variation of the CE V(T) CE := inf µ(t,⋆)∈Ξ(t,⋆),∀t∈[[T]] T−1 � t=1 ∥µ(t+1,⋆) − µ(t,⋆)∥2, where Ξ(t,⋆) denotes the set of CE of the t-th game, and (ii) the variation in the players’ utilities V(T) A ≤ �n i=1 �T−1 t=1 ∥A(t+1) i − A(t) i ∥2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' below, we denote by CeGap(t)(µ(t)) the CE gap of µ(t) ∈ Ξ at the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that each player employs OGD in (12) with a suitable learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � CeGap(t)(µ(t)) �2 = O(1 + V(T) CE + V(T) A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' There are further interesting implications of our framework that are worth highlighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, we obtain meta-learning guarantees for general games that depend on the (algorithm-independent) similarity of the correlated equilibria (Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' that was left as an open question by Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Har+22], where instead algorithm-dependent similarity metrics were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, by applying Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7, we derive natural variation-dependent per-player regret bounds in general games (Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='23), addressing a question left by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we suspect that obtaining such results—parameterized on the variation of the CE—are not possible without the presence of the additional player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 Dynamic Regret Bounds in Static Games Finally, in this subsection we switch gears by investigating dynamic regret guarantees when learning in static games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The proofs of this subsection are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, we point out that while traditional no-regret learning algorithms guarantee O( √ T) external regret, instances of OMD—a generalization of OGD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' see (13) in Appendix A—in fact guarantee O( √ T) dynamic regret in two-player zero-sum games, which is a much stronger performance measure: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players in a (static) two-player zero-sum game employ OMD with a smooth regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, DReg(T) x , DReg(T) y = O( √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 11 In proof, the dynamic regret for each player under OMD with a smooth regularizer can be bounded by the first-order path length of that player’s strategies, which in turn can be bounded by O( √ T) given that the second-order path length is O(1) (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In fact, Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 readily extends Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 to time-varying zero-sum games as well, implying that DReg(T) x , DReg(T) y = O �√ T(1 + V(T) ϵ−NE + V(T) A ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A question that arises from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 is whether the O( √ T) guarantee for dynamic regret of OMD can be improved in the online learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below, we point out a significant improvement to O(log T), but under a stronger two-point feedback model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' namely, we posit that in every round each player can select an additional auxiliary strategy, and each player then gets to additionally observe the utility corresponding to the auxiliary strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Notably, this is akin to how the extra-gradient method works [Hsi+19] (also cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [RS13, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2] for multi-point feedback models in the bandit setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Under two-point feedback, there exist learning algorithms that guarantee DReg(T) x , DReg(T) y = O(log T) in two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, it suffices for each player to employ OMD, but with the twist that the first strategy in each round is the time-average of OMD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the auxiliary strategy is the standard output of OMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, the dynamic regret of each player will grow as O ��T t=1 1 t � = O(log T) since the duality gap of the average strategies is decreasing with a rate of T −1 [RS13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is an interesting question whether Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='14 can be improved to O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' General-sum games In contrast, no (efficient) sublinear dynamic-regret guarantees are possible in general games: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n i=1 DReg(T) i = Ω(T), even if n = 2, where Ω(·) here hides polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, this follows since computing a Nash equilibrium to (1/poly) accuracy in two-player games is PPAD-hard [CDT09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In fact, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15 applies beyond the online learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This motivates considering a relaxation of dynamic regret, wherein the sequence of comparators is subject to the constraint �T−1 t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1, for some parameter K ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' this will be referred to as K-DReg(T) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Naturally, external regret coincides with K-DReg(T) x under K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this context, we employ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 to bound K-DReg(T) under OGD: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16 (Detailed version in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that all n players employ OGD in an L-smooth game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any K ∈ N, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' �n i=1 K-DReg(T) i = O(K√nL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' K-DReg(T) i = O(K3/4T 1/4n1/4√ L), for i ∈ [[n]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' One question that arises here is whether the per-player bound of O(K3/4T 1/4) (Item 2) can be improved to ˜O(K), where ˜O(·) hides logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The main challenge is that, even for K = 1, all known methods that obtain ˜O(1) [DFG21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' PSS21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Ana+22a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Far+22] rely on non- smooth regularizers that violate the preconditions of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1—our dynamic RVU bound that generalizes Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 beyond (squared) Euclidean regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It would also be interesting to give 12 a natural game-theoretic interpretation to the limit point of no-regret learners with K-DReg = o(T), even for a fixed K ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' for K = 1, it corresponds to the fundamental coarse correlated equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' At a superficial level, it seems to be related to the variant considered by Harrow, Natarajan, and Wu [HNW16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 4 Conclusions and Future Work In this paper, we developed a new framework for characterizing iterate-convergence of no-regret learning algorithms—primarily optimistic gradient descent (OGD)—in time-varying games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' There are many promising avenues for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Besides closing the obvious gaps we highlighted in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, it is important to characterize the behavior of no-regret learning algorithms in other fundamental multiagent settings, such as Stackelberg (security) games [Bal+15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Moreover, our results operate in the full-feedback model where each player receives feedback on all possible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Extending the scope of our framework to capture partial-feedback models as well is another interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Acknowledgements We are grateful to Vince Conitzer and Caspar Oesterheld for helpful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This material is based on work supported by the National Science Foundation under grants IIS-1901403 and CCF-1733556 and by the ARO under award W911NF2210266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' References [Ana+22a] Ioannis Anagnostides, Gabriele Farina, Christian Kroer, Chung-Wei Lee, Haipeng Luo, and Tuomas Sandholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “Uncoupled Learning Dynamics with O(log T) Swap Regret in Multiplayer Games”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: NeurIPS 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Ana+22b] Ioannis Anagnostides, Ioannis Panageas, Gabriele Farina, and Tuomas Sandholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “On Last-Iterate Convergence Beyond Zero-Sum Games”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: International Conference on Machine Learning, ICML 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proceedings of Machine Learning Research.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “Generalized monotone operators and their averaged resolvents”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 55–74.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' World Scientific, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [van91] Eric van Damme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Stability and perfection of Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 339.' metadata={'source': 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Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 732–741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+20] Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “Dynamic Regret of Convex and Smooth Functions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22] Mengxiao Zhang, Peng Zhao, Haipeng Luo, and Zhi-Hua Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “No-Regret Learning in Time-Varying Zero-Sum Games”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: International Conference on Machine Learn- ing, ICML 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proceedings of Machine Learning Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' PMLR, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 26772–26808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zin03] Martin Zinkevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' “Online Convex Programming and Generalized Infinitesimal Gradient Ascent”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In: International Conference on Machine Learning (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Washington, DC, USA, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 928–936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 18 A Omitted Proofs In this section, we provide the proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 First, we start with the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Before we proceed, it will be useful to express the update rule (OGD) in the following equivalent form: x(t) := arg max x∈X � Ψ(t) x (x) := ⟨x, m(t) x ⟩ − 1 ηBφx(x ∥ ˆx(t)) � , ˆx(t+1) := arg max ˆx∈X � ˆΨ(t) x (ˆx) := ⟨ˆx, u(t) x ⟩ − 1 ηBφx(ˆx ∥ ˆx(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (13) Here, Bφx(· ∥ ·) denotes the Bregman divergence induced by the (squared) Euclidean regularizer φx : x �→ 1 2∥x∥2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' namely, Bφx(x ∥ x′) := φ(x)−φ(x′)−⟨∇φ(x′), x−x′⟩ = 1 2∥x−x′∥2 2, for x, x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The update rule (13) for general Bregman divergences will be referred to as optimistic mirror descent (OMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Dynamic RVU Bounds We now show Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, the statement of which is recalled below for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 we provide an extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 to a broader class of regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 (RVU bound for dynamic regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider any sequence of utilities (u(1) x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , u(T) x ) up to time T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The dynamic regret (1) of OGD with respect to any sequence of comparators (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) ∈ X T can be bounded by D2 X 2η + DX η T−1 � t=1 ∥x(t+1,⋆) − x(t,⋆)∥2 + η T � t=1 ∥u(t) x − m(t) x ∥2 2 − 1 2η T � t=1 � ∥x(t) − ˆx(t)∥2 2 + ∥x(t) − ˆx(t+1)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, by (1/η)-strong convexity of the function Ψ(t) x (defined in (13)) for any time t ∈ [[T]], we have that ⟨x(t), m(t) x ⟩ − 1 2η∥x(t) − ˆx(t)∥2 2 − ⟨ˆx(t+1), m(t) x ⟩ + 1 2η∥ˆx(t+1) − ˆx(t)∥2 2 ≥ 1 2η∥x(t) − ˆx(t+1)∥2 2, (14) where we used [Sha12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Similarly, by (1/η)-strong convexity of the function ˆΨ(t) x (defined in (13)) for any time t ∈ [[T]], we have that for any comparator x(t,⋆) ∈ X, ⟨ˆx(t+1), u(t) x ⟩ − 1 2η∥ˆx(t+1) − ˆx(t)∥2 2 − ⟨x(t,⋆), u(t) x ⟩ + 1 2η∥x(t,⋆) − ˆx(t)∥2 2 ≥ 1 2η∥ˆx(t+1) − x(t,⋆)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (15) Thus, adding (14) and (15), ⟨x(t,⋆) − ˆx(t+1), u(t) x ⟩ + ⟨ˆx(t+1) − x(t), m(t) x ⟩ ≤ 1 2η � ∥ˆx(t) − x(t,⋆)∥2 2 − ∥ˆx(t+1) − x(t,⋆)∥2 2 � − 1 2η � ∥x(t) − ˆx(t)∥2 2 + ∥x(t) − ˆx(t+1)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (16) 19 We further see that ⟨x(t,⋆) − x(t), u(t) x ⟩ = ⟨x(t) − ˆx(t+1), m(t) x − u(t) x ⟩ + ⟨x(t,⋆) − ˆx(t+1), u(t) x ⟩ + ⟨ˆx(t+1) − x(t), m(t) x ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (17) Now the first term on the right-hand side can be upper bounded using the fact that, by (14) and (15), ⟨x(t) − ˆx(t+1), m(t) x − u(t) x ⟩ ≥ 1 η∥ˆx(t+1) − x(t)∥2 2 =⇒ ∥ˆx(t+1) − x(t)∥2 ≤ η∥m(t) x − u(t) x ∥2, by Cauchy-Schwarz, in turn implying that ⟨x(t) − ˆx(t+1), m(t) x − u(t) x ⟩ ≤ η∥m(t) x − u(t) x ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the proof follows by combining this bound with (16) and (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' along with the fact that T � t=1 � ∥ˆx(t) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 − ∥ˆx(t+1) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 � ≤ ∥ˆx(1) − x(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 + T−1 � t=1 � ∥ˆx(t+1) − x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 − ∥ˆx(t+1) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 � ≤ D2 X + 2DX T−1 � t=1 ∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' where the last bound follows since ∥ˆx(t+1) − x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 − ∥ˆx(t+1) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 2 ≤ 2DX ���∥ˆx(t+1) − x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 − ∥ˆx(t+1) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2 ��� ≤ 2DX ∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' where we recall that DX denotes the ℓ2-diameter of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As an aside, we remark that assuming that m(t) x := 0 and ∥u(t) x ∥2 ≤ 1 for any t ∈ [[T]], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 implies that dynamic regret can be upper bounded by O �� (1 + �T−1 t=1 ∥x(t+1,⋆) − x(t,⋆)∥2)T � , for any (bounded)—potentially adversarially selected—sequence of utilities (u(1) x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , u(T) x ), for η := � D2 X 2T + DX �T −1 t=1 ∥x(t+1,⋆)−x(t,⋆)∥2 T , which is a well-known result in online optimization [Zin03];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' while that requires setting the learning rate based on the first-order variation of the (optimal) comparators, there are standard techniques that would allow bypassing that assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we provide an extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 to the more general OMD algorithm under a broad class of regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 (Extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 beyond Euclidean regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider a 1-strongly convex continuously differentiable regularizer φ with respect to a norm ∥·∥ such that (i) ∥∇φ(x)∥∗ ≤ G for any x, and (ii) Bφx(x ∥ x′) ≤ L∥x − x′∥ for any x, x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any sequence of utilities (u(1) x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , u(T) x ) up to time T ∈ N the dynamic regret (1) of OMD with respect to any sequence of comparators (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) ∈ X T can be bounded as Bφx(x(1,⋆) ∥ ˆx(1)) η + L + 2G η T−1 � t=1 ∥x(t+1,⋆) − x(t,⋆)∥+η T � t=1 ∥u(t) x − m(t) x ∥2 ∗ − 1 2η T � t=1 � ∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 20 The proof is analogous to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, and relies on the well-known three-point identity for the Bregman divergence: Bφx(x ∥ x′) = Bφx(x ∥ x′′) + Bφx(x′′ ∥ x′) − ⟨x − x′′, ∇φ(x′) − ∇φ(x′′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (18) In particular, along with the assumptions of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 imposed on the regularizer φx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (18) implies that the term �T−1 t=1 � Bφx(x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) ∥ ˆx(t+1)) − Bφx(x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) ∥ ˆx(t+1)) � is equal to T−1 � t=1 � Bφx(x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) ∥ x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)) − ⟨x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' ∇φ(ˆx(t+1)) − ∇φ(x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆))⟩ � ≤ (L + 2G) T−1 � t=1 ∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' since Bφx(x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) ∥ x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)) ≤ L∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥ (by assumption) and ⟨x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' ∇φ(ˆx(t+1)) − ∇φ(x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆))⟩≤ ∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥∥∇φ(ˆx(t+1)) − ∇φ(x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆))∥∗ (19) ≤ � ∥∇φ(ˆx(t+1))∥∗+ ∥∇φ(x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆))∥∗ � ∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥ (20) ≤ 2G∥x(t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆) − x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='⋆)∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (21) where (19) follows from the Cauchy-Schwarz inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (20) uses the triangle inequality for the dual norm ∥ · ∥∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and (21) follows from the assumption of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 that ∥∇φ(·)∥∗ ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The rest of the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 is analogous to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, and it is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' An important question is whether Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 can be extended under any regularizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' as we explain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, this is the main obstacle to improving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 Nonnegativity of Dynamic Regret We next proceed with the proof of Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' To provide additional intuition, we first prove the following special case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the proof of Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 is then analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 (Special case of Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆))) is a Nash equilibrium of the t-th game, for any time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for s(T) x = (x(t,⋆))1≤t≤T and s(T) y = (y(t,⋆))1≤t≤T , DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let v(t) := ⟨x(t,⋆), A(t)y(t,⋆)⟩ be the value of the t-th game, for some t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, we have that v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x, A(t)y(t,⋆)⟩ for any x ∈ X, since x(t,⋆) is a best response to y(t,⋆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' similarly, v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y⟩ for any y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Hence, ⟨x(t), A(t)y(t,⋆)⟩ − ⟨x(t,⋆), A(t)y(t)⟩ ≥ 0, or equivalently, ⟨x(t,⋆), u(t) x ⟩ + ⟨y(t,⋆), u(t) y ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' But given that the game is zero-sum, it holds that ⟨x(t), u(t) x ⟩ + ⟨y(t), u(t) y ⟩ = 0, so the last inequality can be in turn cast as ⟨x(t,⋆), u(t) x ⟩ − ⟨x(t), u(t) x ⟩ + ⟨y(t,⋆), u(t) y ⟩ − ⟨y(t), u(t) y ⟩ ≥ 0, 21 for any t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, summing over all t ∈ [[T]] we have shown that DReg(T) x (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆))+ DReg(T) y (y(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , y(T,⋆)) = T � t=1 ⟨x(t,⋆), u(t) x ⟩ − ⟨x(t), u(t) x ⟩ + ⟨y(t,⋆), u(t) y ⟩ − ⟨y(t), u(t) y ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of the t-th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for s(T) x = (x(t,⋆))1≤t≤T and s(T) y = (y(t,⋆))1≤t≤T , DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) ≥ −2 T � t=1 ϵ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Given that (x(t,⋆), y(t,⋆)) ∈ Z is an ϵ(t)-approximate Nash equilibrium of the t-th game, it follows that ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x(t), A(t)y(t,⋆)⟩+ϵ(t) x and ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩−ϵ(t) y , for some ϵ(t) x , ϵ(t) y ≤ ϵ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, we have that ⟨x(t), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩ − ϵ(t) x − ϵ(t) y , or equivalently, ⟨x(t,⋆), u(t) x ⟩ + ⟨y(t,⋆), u(t) y ⟩ ≥ −ϵ(t) x − ϵ(t) y ≥ −2ϵ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, ⟨x(t,⋆), u(t) x ⟩ − ⟨x(t), u(t) x ⟩ + ⟨y(t,⋆), u(t) y ⟩ − ⟨y(t), u(t) y ⟩ ≥ −2ϵ(t), (22) for any t ∈ [[T]], and the statement follows by summing (22) over all t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In fact, as we show below (in Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3), Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 is a more general consequence of the minimax theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, for a nonlinear online learning problem, we define dynamic regret with respect to a sequence of comparators (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) ∈ X T as follows: DReg(T) x (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) := T � t=1 � u(t) x (x(t,⋆)) − u(t) x (x(t)) � , (23) where u(1) x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , u(T) x : x �→ R are the continuous utility functions observed by the learner, which could be in general nonconcave, and (x(t))1≤t≤T is the sequence of strategies produced by the learner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (23) generalizes the notion of dynamic regret (1) in online linear optimization, that is, when u(t) x : x �→ ⟨x, u(t) x ⟩, where u(t) x ∈ Rdx, for any time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that f(t) : X × Y → R is a continuous function such that for any t ∈ [[T]], min x∈X max y∈Y f(t)(x, y) = max y∈Y min x∈X f(t)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let also x(t,⋆) ∈ arg minx∈X maxy∈Y f(t)(x, y) and y(t,⋆) ∈ arg maxy∈Y minx∈X f(t)(x, y), for any t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for s(T) x = (x(t,⋆))1≤t≤T and s(T) y = (y(t,⋆))1≤t≤T , DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' By definition of dynamic regret (23), it suffices to show that f(t)(x(t), y(t,⋆)) ≥ f(t)(x(t,⋆), y(t)), for any time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, f(t)(x(t), y(t,⋆)) ≥ min x∈X f(t)(x, y(t,⋆)) (24) = max y∈Y min x∈X f(t)(x, y) (25) = min x∈X max y∈Y f(t)(x, y) (26) = max y∈Y f(t)(x(t,⋆), y) (27) ≥ f(t)(x(t,⋆), y(t)), (28) where (24) and (28) are obvious;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (25) and (27) follow from the definition of y(t,⋆) ∈ Y and x(t,⋆) ∈ X, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and (26) holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 (MVI property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property (3) can also be generalized beyond time-varying bilinear saddle-point problems to more general time-varying variational inequality (VI) problems as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let F (t) : Z → Z be the (single-valued) operator of the VI problem at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' F (t) is said to satisfy the MVI property if there exists a point z(t,⋆) ∈ Z such that ⟨z − z(t,⋆), F (t)(z)⟩ ≥ 0 for any z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For example, in the special case of a bilinear saddle-point problem, we have that F : z := (x, y) �→ (Ay, −A⊤x), and the MVI property is satisfied by virtue of Von Neumann’s minimax theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is direct to see that Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 applies to any time-varying VI with respect to the sequence (z(t,⋆))1≤t≤T as long as every operator in the sequence (F (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , F (T)) satisfies the MVI property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (Even more broadly, it suffices if almost all operators in the sequence satisfy the MVI property—in that their fraction converges to 1 as T → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=') This observation enables extending Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 beyond time-varying bilinear saddle-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 Variation of the Nash Equilibria In our next example, we point out that an arbitrarily small change in the entries of the payoff matrix can lead to a substantial deviation in the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider a 2 × 2 (two-player) zero-sum game, where X := ∆2, Y := ∆2, described by the payoff matrix A := �2δ 0 0 δ � , (29) for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, it is easy to see that the unique Nash equilibrium of this game is such that x⋆, y⋆ := ( 1 3, 2 3) ∈ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose now that the original payoff matrix (29) is perturbed to a new matrix A′ := �δ 0 0 2δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (30) The new (unique) Nash equilibrium now reads x⋆, y⋆ := ( 2 3, 1 3) ∈ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We conclude that an arbitrarily small deviation in the entries of the payoff matrix can lead to a non-trivial change in the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we leverage the simple observation of the example above to establish Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3, the statement of which is recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 23 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For any T ≥ 4, there is a sequence of T games such that V(T) NE ≥ T 2 while V(T) ϵ−NE ≤ δ, for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We consider a sequence of T games such that X, Y := ∆2, and A(t) = � A if t mod 2 = 1, A′ if t mod 2 = 0, where A, A′ are the payoff matrices defined in (29) and (30), and are parameterized by δ > 0 (Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, the exact Nash equilibria read x(t,⋆), y(t,⋆) = � ( 1 3, 2 3) if t mod 2 = 1, ( 2 3, 1 3) if t mod 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, it follows that V(T) NE := �T−1 t=1 ∥z(t+1,⋆) − z(t,⋆)∥2 = 2 3(T − 1) ≥ T 2 , for T ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In contrast, it is clear that V(T) ϵ−NE ≤ CδT, which follows by simply considering the sequence of strategies wherein both players are always selecting actions uniformly at random;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we recall that C > 0 here is the value that parameterizes V(T) ϵ−NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, taking δ := δ′ CT , for some arbitrarily small δ′ > 0, concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 Main Result Next, we proceed with the proof of our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The key ingredient is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, which bounds the second-order path length of OGD in terms of the considered variation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We first give the precise statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, and we then proceed with its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 (Detailed version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η ≤ 1 4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any time horizon T ∈ N, T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≤ 2D2 Z + 4η2L2∥Z∥2 2 + 4DZV(T) ϵ−NE + 8η2∥Z∥2 2V(T) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, for any t ≥ 2 we have that ∥u(t) x − m(t) x ∥2 2 is equal to ∥A(t)y(t) − A(t−1)y(t−1)∥2 2 ≤ 2∥A(t)(y(t) − y(t−1))∥2 2 + 2∥(A(t) − A(t−1))y(t−1)∥2 2 (31) ≤ 2∥A(t)∥2 2∥y(t) − y(t−1)∥2 2 + 2∥A(t) − A(t−1)∥2 2∥y(t−1)∥2 2 (32) ≤ 2L2∥y(t) − y(t−1)∥2 2 + 2∥Y∥2 2∥A(t) − A(t−1)∥2 2, (33) where (31) uses the triangle inequality for the norm ∥ · ∥2 along with the inequality 2ab ≤ a2 + b2 for any a, b ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (32) follows from the definition of the operator norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and (33) uses the assumption that ∥A(t)∥2 ≤ L and ∥y∥2 ≤ ∥Y∥2 for any y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A similar derivaiton shows that for t ≥ 2, ∥u(t) y − m(t) y ∥2 2 ≤ 2L2∥x(t) − x(t−1)∥2 2 + 2∥X∥2 2∥A(t) − A(t−1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (34) Further, for t = 1 we have that ∥u(1) x − m(1) x ∥2 = ∥u(1) x ∥2 = ∥ − A(1)y(1)∥2 ≤ L∥Y∥2, and ∥u(1) y − m(1) y ∥2 = ∥u(1) y ∥2 = ∥(A(1))⊤x(1)∥2 ≤ L∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we will use the following simple corollary, which follows similarly to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 24 Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For any sequence s(T) z := (z(t,⋆))1≤t≤T , the dynamic regret DReg(T) z (s(T) z ) := DReg(T) x (s(T) x ) + DReg(T) y (s(T) y ) can be bounded by D2 Z 2η + DZ η T−1 � t=1 ∥z(t+1,⋆) −z(t,⋆)∥2 +η T � t=1 ∥u(t) z −m(t) z ∥2 2 − 1 2η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , where m(t) z := (m(t) x , m(t) y ) and u(t) z := (u(t) x , u(t) y ) for any t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, combining (34) and (33) with Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 applied for the dynamic regret of both players with respect to the sequence of comparators ((x(t,⋆), y(t,⋆)))1≤t≤T yields that DReg(T) x (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) + DReg(T) y (y(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , y(T,⋆)) is upper bounded by D2 Z 2η + ηL2∥Z∥2 2 + DZ η T−1 � t=1 ∥z(t+1,⋆) − z(t,⋆)∥2+2η∥Z∥2 2V(T) A − 1 4η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , where we used the fact that 2ηL2 T � t=2 ∥z(t) − z(t−1)∥2 2 − 1 4η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≤ � 2ηL2 − 1 8η � T � t=2 ∥z(t) − z(t−1)∥2 2 ≤ 0, for η ≤ 1 4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, using the fact that DReg(T) x (x(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(T,⋆)) + DReg(T) y (y(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , y(T,⋆)) ≥ −2 �T t=1 ϵ(t) for a suitable sequence of ϵ(t)-approximate Nash equilibria (Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2)—one that attains the variation measure V(T) ϵ−NE—yields that 0 ≤ D2 Z 2η + ηL2∥Z∥2 2 + DZ η V(T) ϵ−NE + 2η∥Z∥2 2V(T) A − 1 4η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , where it suffices if the parameter C of V(T) ϵ−NE is such that 2 ≤ DZ η C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, rearranging the last displayed inequality concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we refine this theorem in time-varying games in which the deviation of the payoff matrices is bounded by the deviation of the players’ strategies, in the following formal sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η ≤ min � 1 4L, 1 8W∥Z∥ � in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2 and V(T) A ≤ W 2 �T−1 t=1 ∥z(t+1)− z(t)∥2 2, for some parameter W ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any time horizon T ∈ N, T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥ˆz(t) − ˆz(t+1)∥2 2 � ≤ 4D2 Z + 8η2L2∥Z∥2 2 + 8DZV(T) NE , where V(T) NE is defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, we have that for any η ≤ 1 4L, 0 ≤ D2 Z 2η + ηL2∥Z∥2 2 + DZ η V(T) NE + 2η∥Z∥2 2V(T) A − 1 4η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, for η ≤ 1 8W∥Z∥2 , 2η∥Z∥2 2V(T) A − 1 8η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≤ � 2η∥Z∥2 2W 2 − 1 16η � T−1 � t=1 ∥z(t+1) − z(t)∥2 2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, we have shown that 0 ≤ D2 Z 2η + ηL2∥Z∥2 2 + DZ η V(T) NE − 1 8η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , and rearranging concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, in such time-varying games it is the first-order variation term, V(T) NE , that will drive our convergence bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Now before proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, we state the connection between the equilibrium gap and the deviation of the players’ strategies � ∥z(t) − ˆz(t)∥2 + ∥z(t) − ˆz(t+1)∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, the following claim can be extracted by [Ana+22b, Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (We caution that we use a slightly different indexing for the secondary sequence (ˆx(t) i ) in the definition of OMD (13) compared to [Ana+22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=') Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that the sequences (x(t) i )1≤t≤T and (ˆx(t) i )1≤t≤T+1 are produced by OMD under a G-smooth regularizer 1-strongly convex with respect to a norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any time t ∈ [[T]] and any xi ∈ Xi, ⟨x(t) i , u(t) i ⟩ ≥ ⟨xi, u(t) i ⟩ − G η ∥ˆx(t+1) i − ˆx(t) i ∥ − ∥u(t) i ∥∗∥x(t) i − ˆx(t+1) i ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, the precise version of which is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10 (Detailed version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η = 1 4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � EqGap(t)(z(t)) �2 ≤ 2L2(4 + ∥Z∥2)2 � 2D2 Z + 4η2L2∥Z∥2 2 + 4DZV(T) ϵ−NE + 8η2∥Z∥2 2V(T) A � , where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 26 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let us first fix a time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For convenience, we denote by BR(t) x (x(t)) := maxx∈X {⟨x, u(t) x ⟩}− ⟨x(t), u(t) x ⟩, the best response gap of Player’s x strategy x(t) ∈ X, and similarly for BR(t) y (y)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' By definition, it holds that EqGap(t) := max{BR(t) x (x(t)), BR(t) y (y(t))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' By Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9, we have that BR(t) x (x(t)) ≤ 1 η∥ˆx(t+1) − ˆx(t)∥2 + ∥u(t) x ∥2∥x(t) − ˆx(t+1)∥2 (35) ≤ 4L∥ˆx(t+1) − ˆx(t)∥2 + L∥Y∥2∥x(t) − ˆx(t+1)∥2 (36) ≤ L (4 + ∥Z∥2) � ∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2 � , (37) where (35) follows from Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9 for G = 1 (since the squared Euclidean regularizer φx : x �→ 1 2∥x∥2 2) is 1-smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (36) uses the fact that η := 1 4L and ∥u(t) x ∥2 = ∥ − A(t)y(t)∥2 ≤ L∥Y∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' and (37) follows from the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A similar derivation shows that BR(t) y (y(t)) ≤ L(4 + ∥Z∥) � ∥y(t) − ˆy(t)∥2 + ∥y(t) − ˆy(t+1)∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (38) Thus, T � t=1 � EqGap(t)(z(t)) �2 = T � t=1 � max{BR(t) x (x(t)), BR(t) y (y(t))} �2 ≤ T � t=1 �� BR(t) x (x(t)) �2 + � BR(t) y (y(t)) �2� ≤ 2L2(4 + ∥Z∥2)2 T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , (39) where the last bound uses (37) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Combining (39) with Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 Variation-Dependent Regret Bounds Here we state an important implication of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 for deriving variation-dependent regret bounds in time-varying bilinear saddle-point problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Zha+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11 (Detailed version of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the setup of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4, it holds that Reg(T) x ≤ D2 X η + 8ηL2D2 Z + ηL2∥Y∥2 2 + 16η3L4∥Z∥2 2 + 16ηL2DZV(T) NE + (2η∥Y∥2 2 + 32η3L2∥Z∥2 2)V(T) A , and Reg(T) y ≤ D2 Y η + 8ηL2D2 Z + ηL2∥X∥2 2 + 16η3L4∥Z∥2 2 + 16ηL2DZV(T) NE + (2η∥X∥2 2 + 32η3L2∥Z∥2 2)V(T) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 under x(1,⋆) = · · · = x(T,⋆), we have Reg(T) x ≤ D2 X η + ηL2∥Y∥2 2 + 2ηL2 T � t=2 ∥y(t) − y(t−1)∥2 2 + 2η∥Y∥2 2 T � t=2 ∥A(t) − A(t−1)∥2 2, (40) 27 and similarly, Reg(T) y ≤ D2 Y η + ηL2∥X∥2 2 + 2ηL2 T � t=2 ∥x(t) − x(t−1)∥2 2 + 2η∥X∥2 2 T � t=2 ∥A(t) − A(t−1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (41) Now, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 we have T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≤ 2D2 Z + 4η2L2∥Z∥2 2 + 4DZV(T) NE + 8η2∥Z∥2 2V(T) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (42) Further, T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≥ T � t=1 � ∥x(t) − ˆx(t)∥2 2 + ∥x(t) − ˆx(t+1)∥2 2 � ≥ 1 2 T � t=2 ∥x(t) − x(t−1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Combining this bound with (42) and (41) gives the claimed regret bound on Reg(T) y , and a similar derivation also gives the claimed bound on Reg(T) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 Meta-Learning We next provide the implication of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 in the meta-learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We first make a remark regarding the effect of the prediction of OGD to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5, and how that relates to an assumption present in [Har+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12 (Improved predictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Throughout Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, we have considered the standard prediction m(t) x := u(t−1) x = −A(t−1)y(t−1) for t ≥ 2, and similarly for Player y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is easy to see that using the predictions m(t) x := −A(t)y(t−1) and m(t) y := (A(t))⊤x(t−1) (43) for t ≥ 1 (where z(0) := ˆz(1)) entirely removes the dependency on V(T) A on all our convergence bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' While such a prediction cannot be implemented in the standard online learning model, there are settings in which we might know the sequence of matrices in advance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the meta-learning setting offers such examples, and indeed, Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' [Har+22] use the improved prediction of (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 (Meta-learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η = 1 4L, where L := maxh∈[[H]] ∥A(h)∥2, and the prediction of (43) in a meta-learning bilinear saddle-point problem with H ∈ N games, each repeated for m ∈ N consecutive iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for an average game, � P Hϵ2 + P ′V(H) NE Hϵ2 � (44) iterations suffice to reach an ϵ-approximate Nash equilibrium, where P := 4L2(4 + ∥Z∥2)2D2 Z, P ′ := 8L2(4 + ∥Z∥)2DZ, and V(H) NE := inf z(h,⋆)∈Z(h,⋆),∀h∈[[H]] H−1 � h=1 ∥z(h+1,⋆) − z(h,⋆)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 28 The proof is a direct application of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10, where we remark that the term depending on V(T) A and the term 4η2L2∥Z∥2 2 from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10 are eliminated because of the improved prediction of Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The first term in the iteration complexity bound (44) vanishes in the meta-learning regime—as the number of games increases H ≫ 1—while the second term is proportional to V(H) NE H , a natural similarity measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (44) always recovers the m−1/2 rate, but offers significant gains if the games as similar, in the sense that V(H) NE H ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is worth noting that, unlike the similarity measure derived in [Har+22], V(H) NE H depends on the order of the games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further remark that Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 can be readily extended even if each game in the meta-learning sequence is not repeated for the same number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 General Variational Inequalities Although our main focus in this paper is on the convergence of learning algorithms in time-varying games, our techniques could also be of interest for solving (static) general variational inequality (VI) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, let F : Z → Z be a single-valued operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Solving general VIs is well-known to be computationally intractable, and so instead focus has been on identifying broad subclasses that elude those intractability barriers (see our overview in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Our framework in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 motivates introducing the following measure of complexity for a VI problem: C(F) := inf z(1,⋆),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=',z(T,⋆)∈Z T−1 � t=1 ∥z(t+1,⋆) − z(t,⋆)∥2, (45) subject to DReg(T) z (z(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , z(T,⋆)) ≥ 0 ⇐⇒ T � t=1 ⟨z(t) − z(t,⋆), F(z(t))⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (46) In words, (45) expresses the infimum first-order variation that a sequence of comparators must have in order to guarantee nonnegative dynamic regret (46);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' it is evident that (46) always admits a feasible sequence, namely s(T) z := (z(t))1≤t≤T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We note that, in accordance to our results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, one can also consider an approximate version of the complexity measure (45), which could behave much more favorably (recall Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Now in a (static) bilinear saddle-point problem, it holds that C(F) = 0 given that there exists a static comparator that guarantees nonnegativity of the dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More broadly, our techniques imply O(poly(1/ϵ)) iteration-complexity bounds for any VI problem such that C(F) ≤ CT 1−ω, for a time-independent parameter C > 0 and ω ∈ (0, 1]: Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider a variational inequality problem described with the operator F : Z → Z such that F is L-Lipschitz continuous, in the sense that ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, and C(F) ≤ CT 1−ω for C > 0 and ω ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, OGD with learning rate η = 1 4L reaches an ϵ-strong solution z⋆ ∈ Z in O(ϵ−2/ω) iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' that is, ⟨z − z⋆, F(z⋆)⟩ ≥ −ϵ for any z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' It is worth comparing (45) with another natural complexity measure, namely infz⋆∈Z �T t=1⟨z(t)− z⋆, F(z(t))⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' the latter measures how negative (external) regret can be, and has already proven useful in certain settings that go bilinear saddle-point problems [YM22], although unlike (45), it 29 does not appear to be useful in characterizing time-varying bilinear saddle-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this context, O(poly(1/ϵ)) iteration-complexity bounds can also be established whenever infz⋆∈Z �T t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −CT 1−ω for a time-invariant C > 0, or infz⋆∈Z �T t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −C �T−1 t=1 ∥z(t+1) − z(t)∥2 2, for a sufficiently small C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Following [YM22], identifying VIs that satisfy those relaxed conditions but not the MVI property is an interesting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, it is important to understand if those relaxations can shed led light into the convergence properties of OGD in Shapley’s two-player zero-sum stochastic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 Proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 In this subsection, we provide the proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2, leading to our main result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let us first introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We let f(t) : X × Y → R be a continuously differentiable function for any t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We recall that in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 it is assumed that the objective function changes after m ∈ N (consecutive) repetitions, which is akin to the meta-learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Analogously to our setup for bilinear saddle-point problems (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1), it is assumed that Player x is endeavoring to minimizing the objective function, while Player y is trying to maximize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We will denote by Reg(T) L,x(x⋆) := �T t=1⟨x(t) − x⋆, −u(t) x ⟩ and Reg(T) L,y(y⋆) := �T t=1⟨y⋆ − y(t), u(t) y ⟩, where u(t) x := −∇xf(x(t), y(t)) and u(t) y := ∇yf(x(t), y(t)) for any t ∈ [[T]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' similar notation is used for DReg(T) L,x, DReg(T) L,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Furthermore, we let s(T) z = ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆) and y(t,⋆) = y(h,⋆) for any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The first important step in our analysis is that, following the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, DReg(T) L,x(s(T) x ) ≤ 1 2η H � h=1 � ∥ˆx(h,1) − x(h,⋆)∥2 2 − ∥ˆx(h,m+1) − x(h,⋆)∥2 2 � + η T � t=1 ∥u(t) x − m(t) x ∥2 2 − 1 2η T � t=1 � ∥x(t) − ˆx(t)∥2 2 + ∥x(t) − ˆx(t+1)∥2 2 � , (47) where ˆx(h,k) := ˆx((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]], ˆx(h,m+1) := ˆx(h+1,1) for h ∈ [[H − 1]], and ˆx(H,m+1) := ˆx(T+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Similarly, DReg(T) L,y(s(T) y ) ≤ 1 2η H � h=1 � ∥ˆy(h,1) − y(h,⋆)∥2 2 − ∥ˆy(h,m+1) − y(h,⋆)∥2 2 � + η T � t=1 ∥u(t) y − m(t) y ∥2 2 − 1 2η T � t=1 � ∥y(t) − ˆy(t)∥2 2 + ∥y(t) − ˆy(t+1)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (48) Next, we will use the following key observation, which lower bounds the sum of the players’ (external) regrets under strong convexity-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that f : X × Y → R is a µ-strongly convex-concave function with respect to ∥ · ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for any Nash equilibrium z⋆ = (x⋆, y⋆) ∈ Z, Reg(m) L,x(x⋆) + Reg(m) L,y (y⋆) ≥ µ 2 m � t=1 ∥z(t) − z⋆∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, by µ-strong convexity of f(x, ·), we have that for any time t ∈ [[m]], ⟨x(t) − x⋆, ∇xf(x(t), y(t))⟩ ≥ f(x(t), y(t)) − f(x⋆, y(t)) + µ 2 ∥x(t) − x⋆∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (49) Similarly, by µ-strong concavity of f(·, y), we have that for any time t ∈ [[m]], ⟨y⋆ − y(t), ∇yf(x(t), y(t))⟩ ≥ f(x(t), y⋆) − f(x(t), y(t)) + µ 2 ∥y(t) − y⋆∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (50) Further, for any Nash equilibrium (x⋆, y⋆) ∈ Z it holds that f(x(t), y⋆) ≥ f(x(t), y(t)) ≥ f(x⋆, y(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Combining this fact with (49) and (50) and summing over all t ∈ [[m]] gives the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In turn, this readily implies the following lower bound for the dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that f(h) : X × Y → R is a µ-strongly convex-concave function with respect to ∥ · ∥2, for any h ∈ [[H]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider a sequence s(T) z = ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆) and y(t,⋆) = y(h,⋆) for any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' If (x(h,⋆), y(h,⋆)) ∈ Z is a Nash equilibrium of f(h), DReg(T) L,x(s(T) x ) + DReg(T) L,y(s(T) y ) ≥ µ 2 H � h=1 m � k=1 ∥z(h,k) − z(h,⋆)∥2 2, where z(h,k) := z((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We next combine this with the following monotonicity property of OGD: If z⋆ is a Nash equilibrium, ∥ˆz(t) − z⋆∥2 is a decreasing function in t [Har+22, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This leads to the following refinement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Under the assumptions of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16, if η ≤ 1 2µ, DReg(T) L,x(s(T) x ) + DReg(T) L,y(s(T) y ) + 1 4η T � t=1 ∥z(t) − ˆz(t+1)∥2 2 ≥ µm 4 H � h=1 ∥ˆz(h,m+1) − z(h,⋆)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16, DReg(T) L,x(s(T) x ) + DReg(T) L,y(s(T) y ) + 1 4η T � t=1 ∥z(t) − ˆz(t+1)∥2 2 ≥ µ 2 H � h=1 m � k=1 ∥z(h,k) − z(h,⋆)∥2 2 + 1 4η T � t=1 ∥z(t) − ˆz(t+1)∥2 2 ≥ µ 4 H � h=1 m � k=1 ∥ˆz(h,k+1) − z(h,⋆)∥2 2 (51) ≥ µm 4 H � h=1 ∥ˆz(h,m+1) − z(h,⋆)∥2 2, (52) where (51) uses that 1 4η ≥ µ 2 along with Young’s inequality and triangle inequality, and (52) follows from [Har+22, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 31 Armed with this important lemma, we are ready to establish our main result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8), the detailed version of which is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We first point out that a function f : X × Y → R is said to be L-smooth if ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, where F(z) := (∇xf(x, y), −∇yf(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='18 (Detailed version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Let f(h) : X × Y be a µ-strongly convex-concave and L-smooth function, for h ∈ [[H]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players employ OGD with learning rate η = min � 1 8L, 1 2µ � for T repetitions, where T = m × H and m ≥ 2 ηµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � ≤ 4D2 Z + 8η2∥F(z(1))∥2 2 + 8S(H) NE + 16η2V(H) ∇f , where S(H) NE and V(H) ∇f are defined in (9) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, �T t=1 � EqGap(t)(z(t)) �2 = O(1 + S(H) NE + V(H) ∇f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='17 with (47) and (48), 0 ≤ 1 2η H � h=1 � ∥ˆz(h,1) − z(h,⋆)∥2 2 − 2 + ηµm 2 ∥ˆz(h,m+1) − z(h,⋆)∥2 2 � +η T � t=1 ∥u(t) z − m(t) z ∥2 2 − 1 4η T � t=1 � ∥z(t) − ˆz(t)∥2 2 + ∥z(t) − ˆz(t+1)∥2 2 � , for a sequence of Nash equilibria (z(h,⋆))1≤h≤H, where we used the notation u(t) z := (u(t) x , u(t) y ) and m(t) z := (m(t) x , m(t) y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Now we bound the first term of the right-hand side above as 1 2η H � h=1 � ∥ˆz(h,1) − z(h,⋆)∥2 2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2 2 � ≤ 1 2η∥ˆz(1,1) − z(1,⋆)∥2 2 + 1 2η H−1 � h=1 � ∥ˆz(h+1,1) − z(h+1,⋆)∥2 2 − 2∥ˆz(h+1,1) − z(h,⋆)∥2 2 � , where we used the fact that m ≥ 2 ηµ and ˆz(h,m+1) = ˆz(h+1,1), for h ∈ [[H − 1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Hence, continuing from above, 1 2η H � h=1 � ∥ˆz(h,1) − z(h,⋆)∥2 2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2 2 � ≤ 1 2η∥ˆz(1,1)−z(1,⋆)∥2 2+ 1 η H−1 � h=1 ∥z(h+1,⋆)−z(h,⋆)∥2 2, since ∥ˆz(h+1,1) − z(h+1,⋆)∥2 2 ≤ 2∥ˆz(h+1,1) − z(h,⋆)∥2 2 + 2∥z(h,⋆) − z(h+1,⋆)∥2 2, by the triangle inequality and Young’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Moreover, for t ≥ 2, ∥u(t) z −u(t−1) z ∥2 2 = ∥F (t)(z(t))−F (t−1)(z(t−1))∥2 2 ≤ 2L2∥z(t)−z(t−1)∥2 2+2∥F (t)(z(t−1))−F (t−1)(z(t−1))∥2 2, by L-smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, T−1 � t=1 ∥u(t+1) z − u(t) z ∥2 2 ≤ 2L2 T−1 � t=1 ∥z(t) − z(t−1)∥2 2 + 2V(T) ∇f , and the claimed bound on the second-order path length follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, the second claim of the theorem follows from Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9 using convexity-concavity, analogously to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 32 We point out that the improved prediction mechanism described in Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12 can also be used in this setting as well, resulting in the elimination of the variation measure (11) from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We conclude this subsection by pointing out an improved variation-dependent regret bound, which follows directly from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='18 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the setup of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='18, Reg(T) L,x, Reg(T) L,y ≤ D2 Z η + 4ηL2D2 Z + 32η3L2∥F(z(1)∥2 2 + 32ηL2S(H) NE + (64η3L2 + 2η)V(H) ∇f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, setting the learning rate optimally implies that Reg(T) L,x, Reg(T) L,y = O �� S(H) NE + V(H) ∇f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 Proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 In this subsection, we provide the proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Potential Games We first characterize the behavior of GD in time-varying potential games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below we give the formal definition of an unweighted potential game, represented in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='20 (Potential game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A game admits a potential if there exists a function Φ : × n i=1 Xi → R such that for any Player i ∈ [[n]], any joint strategy profile x−i ∈×i′̸=i Xi′, and any pair of strategies xi, x′ i ∈ Xi, Φ(xi, x−i) − Φ(x′ i, x−i) = ui(xi, x−i) − ui(x′ i, x−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We also recall that GD is equivalent to OGD under the prediction m(t) x = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The key ingredient in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10 is the following key bound on the second-order path length of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that each player employs GD with a sufficiently small learning rate η > 0 and initialization (x(1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(1) n ) ∈× n i=1 Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, 1 2η T � t=1 n � i=1 ∥x(t+1) i − x(t) i ∥2 2 ≤ T � t=1 � Φ(t)(x(t+1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t+1) n ) − Φ(t)(x(t) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t) n ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (53) This bound can be derived from [Ana+22b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We note that if Φ(1) = Φ(2) = · · · = Φ(T), the right-hand side of (53) telescops, thereby implying that the second-order path-length is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' More generally, the right-hand side of (53) can be upper bounded by 2Φmax + T−1 � t=1 � Φ(t)(x(t+1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t+1) n ) − Φ(t+1)(x(t+1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t+1) n ) � ≤ 2Φmax + V(T) Φ , (54) where Φmax is an upper bound on |Φ(t)(·)| for any t ∈ [[T]], and V(T) Φ is the variation measure of the potential functions we introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Furthermore, we know that the Nash equilibrium gap in the t-th potential game can be bounded in terms of �n i=1 ∥x(t+1) i − x(t) i ∥2 (Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As a result, combining this property with Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='21 and (54) establishes Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 General-Sum Games We next turn out attention to general-sum multi-player games using the bilinear formulation presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' To establish Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11, let us first define the regret of any Player i ∈ [[n]] as Reg(T) i (¯x⋆ i ) := T � t=1 ⟨¯x⋆ i − ¯x(t) i , (A(t) i )⊤µ(t)⟩, where ¯x⋆ i ∈ ¯Xi, so that �n i=1 Reg(T) i is easily seen to be equal to the regret of the maximizing player in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, the dynamic regret of the mediator—the minimizing player in (12)—can be expressed as DReg(T) µ (µ(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , µ(T,⋆)) := T � t=1 ⟨µ(t) − µ(t,⋆), n � i=1 A(t) i ¯x(t) i ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, DReg(T) µ (µ(1,⋆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , µ(T,⋆)) + n � i=1 Reg(T) i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We have that DReg(T) µ + n � i=1 Reg(T) i (¯x⋆ i ) = n � i=1 T � t=1 ⟨¯x⋆ i , (A(t) i )⊤µ(t)⟩ − T � t=1 ⟨µ(t,⋆), n � i=1 A(t) i ¯x(t) i ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Now for any correlated equilibrium µ(t,⋆) of the t-th game we have that ⟨µ(t,⋆), A(t) i ¯x(t) i ⟩ ≤ 0 for any Player i ∈ [[n]], ¯xi ∈ ¯ Xi, and time t ∈ [[T]], which in turn implies that − �T t=1⟨µ(t,⋆), �n i=1 A(t) i ¯x(t) i ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Moreover, �n i=1 max¯x⋆ i ∈ ¯ Xi �T t=1⟨¯x⋆ i , (A(t) i )⊤µ(t)⟩ ≥ 0 given that, by definition, 0 ∈ ¯ Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we provide the main implication of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12 in the meta-learning setting, which is similar to the meta-learning guarantee of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13 we established earlier in two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Below, we denote by Ξ(h,⋆) the set of correlated equilibria of the h-th game in the meta-learning sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='22 (Meta-learning general-sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that each player employ OGD in (12) with a suitable learning rate η > 0 and the prediction of (43) in a meta-learning general-sum problem with H ∈ N games, each repeated for m ∈ N consecutive iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, for an average game, O � 1 ϵ2H + V(H) CE ϵ2H � (55) iterations suffice to reach an ϵ-approximate correlated equilibrium, where V(H) CE := inf µ(h,⋆)∈Ξ(h,⋆) ∥µ(h+1,⋆) − µ(h,⋆)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 34 In particular, in the meta-learning regime, H ≫ 1, the iteration-complexity bound (55) is dominated by the (algorithm-independent) similarity metric of the correlated equilibria V(H) CE H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='22 establishes significant gains when V(H) CE H ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, we conclude this subsection by providing a variation-dependent regret bound in general- sum multi-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' To do so, we combine Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12, leading to the following guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='23 (Regret in general-sum games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In the setup of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='12, Reg(T) µ , Reg(T) i = O �1 η + η � V(T) CE + V(T) A �� , for any Player i ∈ [[n]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, if one selects optimally the learning rate, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='23 implies that the individual regret of each player is bounded by O �� V(T) CE + V(T) A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We note again that there are techniques that would allow (nearly) recovering such regret guarantees without having to know the variation measures in advance [Zha+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 Proofs from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 Finally, in this subsection we present the proofs omitted from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We begin with Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13, the statement of which is recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We first recall that a regularizer φx, 1-strongly convex with respect to a norm ∥ · ∥, is said to be G-smooth if ∥∇φx(x) − ∇φx(x′)∥∗ ≤ G∥x − x′∥, for all x, x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Suppose that both players in a (static) two-player zero-sum game employ OMD with a smooth regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, DReg(T) x , DReg(T) y = O( √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, using Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='9, it follows that the dynamic regret DReg(T) x of Player x up to time T can be bounded as T � t=1 � max x(t,⋆)∈X � ⟨x(t,⋆), u(t) x ⟩ � − ⟨x(t), u(t) x ⟩ � ≤ T � t=1 � �G η + ∥u(t) x ∥∗ � ∥x(t) − ˆx(t+1)∥ + G η ∥x(t) − ˆx(t)∥ � , (56) where G > 0 is the smoothness parameter of the regularizer, and η > 0 is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We further know that �T t=1 � ∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2� = O(1) for any instance of OMD in a two-player zero-sum game [Ana+22b], which in turn implies that �T t=1 � ∥x(t) − ˆx(t)∥ + ∥x(t) − ˆx(t+1)∥ � = O( √ T) by Cauchy-Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, combining with (56) we have shown that DReg(T) x = O( √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Similar reasoning yields that DReg(T) y = O( √ T), concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In contrast, we next show that such a result is precluded in general-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, we note that the following computational-hardness result holds beyond the online learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 35 It should be stressed that without imposing computational or memory restrictions there are trivial online algorithms that guarantee even O(1) dynamic regret by first exploring the payoff matrices and then computing a Nash equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' we suspect that under the memory limitations imposed in our work, as in [DDK11], there could be unconditional information-theoretic lower bounds, but that is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n i=1 DReg(T) i = Ω(T), even if n = 2, where Ω(·) here hides polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We will use the fact that computing a Nash equilibrium in two-player (normal-form) games to a sufficiently small accuracy ϵ := 1/poly is PPAD-hard [CDT09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Indeed, suppose that there exist polynomial-time algorithms that always guarantee that �n i=1 DReg(T) i ≤ ϵT, where n := 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, this implies that there exists a time t ∈ [[T]] such that max x(t,⋆) 1 ∈X1 ⟨x(t,⋆) 1 , u(t) 1 ⟩ − ⟨x(t) 1 , u(t) 1 ⟩ + max x(t,⋆) 2 ∈X2 ⟨x(t,⋆) 2 , u(t) 2 ⟩ − ⟨x(t) 2 , u(t) 2 ⟩ ≤ ϵ, which in turn implies that (x(t) 1 , x(t) 2 ) is an ϵ-approximate Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, such a time t ∈ [[T]] can be identified in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' But this would imply that PPAD ⊆ P, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Finally, we provide the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16, the detailed version of which is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='24 (Detailed version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Consider an n-player game such that ∥∇xiui(z)− ∇xiui(z′)∥2 ≤ L∥z − z′∥2, where z, z′ ∈ × n i=1 Xi, for any Player i ∈ [[n]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, if all players employ OGD with learning rate η > 0 it holds that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' �n i=1 K-DReg(T) i = O(K√nL) for η = Θ � 1 L√n � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' K-DReg(T) i = O(K3/4T 1/4n1/4√ L), for any Player i ∈ [[n]], for η = Θ � K1/4 n1/4L1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 subject to the constraint that �T−1 t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1 gives that for any Player i ∈ [[n]], K-DReg(T) i ≤ D2 Xi 2η (2K − 1) + η∥u(1) i ∥2 2 + η T−1 � t=1 ∥u(t+1) i − u(t) i ∥2 2 − 1 4η T−1 � t=1 ∥x(t+1) i − x(t) i ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' (57) Further, by L-smoothness we have that ∥u(t+1) i − u(t) i ∥2 2 = ∥∇xiui(z(t+1)) − ∇xiui(z(t))∥2 2 ≤ L2 n � i=1 ∥x(t+1) i − x(t) i ∥2 2, for any t ∈ [[T − 1]], where (x(t) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' , x(t) n ) = z(t) ∈× n i=1 Xi is the joint strategy profile at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Thus, summing (57) over all i ∈ [[n]] and taking η ≤ 1 2L√n implies that �n i=1 K-DReg(T) i ≤ 2K−1 2η �n i=1 D2 Xi + η �n i=1 ∥u(1) i ∥2 2, yielding the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The second part follows directly from (57) using the stability property of OGD: ∥x(t+1) i − x(t) i ∥2 = O(η), for any time t ∈ [[T − 1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 36 B Experimental Examples Finally, although the focus of this paper is theoretical, in this section we provide some illustrative experimental examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 contains experiments on time-varying potential games, while Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 focuses on time-varying (two-player) zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' For simplicity, we will be assuming that each game is represented in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 Time-Varying Potential Games Here we consider time-varying 2-player identical-interest games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We point out that such games are potential games (recall Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='20), and as such they are indeed amenable to our theory in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In our first experiment, we first sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1, where A(0) := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Here, α > 0 is a parameter that controls the variation of the payoff matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In this time-varying setup, we let each player employ (online) GD with learning rate η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The results obtained under different random initializations of matrices A and P are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Next, we operate in the same time-varying setup but each player is now employing multiplicative weights update (MWU), instead of gradient descent, with η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As shown in Figure 2, while the cumulative equilibrium gap is much larger compared to using GD (Figure 1), the dynamics still appear to be approaching equilibria, although our theory does not cover MWU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' We suspect that theoretical results such as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='10 should hold for MWU as well, but that has been left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In our third experiment for identical-interest games, we again first sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, we defined A(t) := A(t−1) + ϵP for t ≥ 1, where A(0) := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Here, ϵ > 0 is the parameter intended to capture the variation of the payoff matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The results obtained under different random initializations of A and P are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As an aside, it is worth pointing out that this particular setting can be thought of as a game in which the variation in the payoff matrices is controlled by another learning agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' In particular, our theoretical results are helpful for characterizing the convergence properties of two-timescale learning algorithms, in which the deviation of the game is controlled by a player constrained to be updating its strategies with a much smaller learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 Time-Varying Zero-Sum Games We next conduct experiments on time-varying bilinear saddle-point problems when players are employing OGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Such problems have been studied extensively in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 from a theoretical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' First, we sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' here we consider lower- dimensional payoff matrices compared to the experiments in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 for convenience in the graphical illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1, where A(1) := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The results obtained under different random initializations are illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 37 0 50 100 150 200 0 2 4 �t τ=1(EG(τ))2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 0 1 2 3 �t τ=1(EG(τ))2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 0 1 2 3 4 �t τ=1(EG(τ))2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 Iteration (t) 0 2 4 �t τ=1(EG(τ))2 0 50 100 150 200 Iteration (t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 Iteration (t) 0 2 4 6 max(Reg(t) x , Reg(t) y ) Figure 1: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest games when both players are employing (online) GD with learning rate η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 for T := 200 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn induces a different time-varying game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, each figure contains trajectories corresponding to three different values of α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5}, but under the same initialization of A and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As expected, smaller values of α generally increase the equilibrium gap since the variation of the games is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Nevertheless, for all games we observe that the players are gradually approaching equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 38 0 50 100 150 200 0 100 200 �t τ=1(EG(τ))2 0 50 100 150 200 0 2 4 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 20 40 60 80 100 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 0 100 200 300 �t τ=1(EG(τ))2 0 50 100 150 200 0 2 4 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 20 40 60 80 100 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 0 100 200 �t τ=1(EG(τ))2 0 50 100 150 200 0 2 4 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 20 40 60 80 100 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 50 100 150 200 Iterations 0 100 200 300 �t τ=1(EG(τ))2 0 50 100 150 200 Iterations 0 2 4 EG(t) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 20 40 60 80 100 Iterations 0 2 4 6 max(Reg(t) x , Reg(t) y ) Figure 2: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest games when both players are employing (online) GD with learning rate η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 for T := 200 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn induces a different time-varying game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, each figure contains trajectories corresponding to three different values of α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5}, but under the same initialization of A and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The MWU dynamics still appear to be approaching equilibria, although the cumulative gap is much larger compared to GD (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 39 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 �t τ=1(EG(τ))2 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 EG(t) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='001 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 �t τ=1(EG(τ))2 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 EG(t) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='001 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 100 200 300 400 500 0 1 2 �t τ=1(EG(τ))2 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='3 EG(t) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='001 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 0 2 4 6 max(Reg(t) x , Reg(t) y ) 0 100 200 300 400 500 Iterations 0 1 2 3 �t τ=1(EG(τ))2 0 100 200 300 400 500 Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 EG(t) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='001 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 0 50 100 150 200 Iterations 0 2 4 6 max(Reg(t) x , Reg(t) y ) Figure 3: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest games when both players are employing (online) GD with learning rate η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1 for T := 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn induces a different time-varying game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, each figure contains trajectories from three different values of ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='001}, but under the same initialization of A and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' As expected, larger values of ϵ generally increase the equilibrium gap since the variation of the games is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Yet, even for the larger value ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='1, the dynamics are still appear to be approaching Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 40 0 200 400 600 800 1000 0 10000 20000 30000 �t τ=1(EG(τ))2 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 EG(t) α = 2 α = 1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 0 200 400 600 800 1000 0 5 10 15 20 max(Reg(t) x , Reg(t) y ) 0 200 400 600 800 1000 0 2500 5000 7500 10000 �t τ=1(EG(τ))2 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 EG(t) α = 2 α = 1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 0 200 400 600 800 1000 0 10 20 30 40 max(Reg(t) x , Reg(t) y ) 0 200 400 600 800 1000 0 20000 40000 �t τ=1(EG(τ))2 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='8 EG(t) α = 2 α = 1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 0 200 400 600 800 1000 0 10 20 30 max(Reg(t) x , Reg(t) y ) 0 200 400 600 800 1000 Iteration (t) 0 2500 5000 7500 10000 �t τ=1(EG(τ))2 0 200 400 600 800 1000 Iteration (t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='6 EG(t) α = 2 α = 1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7 0 200 400 600 800 1000 Iteration (t) 0 10 20 max(Reg(t) x , Reg(t) y ) Figure 4: The equilibrium gap and the players’ regrets in 2-player time-varying zero-sum games when both players are employing OGD with learning rate η := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='01 and T := 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn induces a different time-varying game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' Further, each figure contains trajectories from three different values of α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content='7, 1, 2}, but under the same initialization of A and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' The OGD dynamics appear to be approaching equilibria, albeit with a much slower rate compared to the ones observed earlier for potential games (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} +page_content=' 41' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFIT4oBgHgl3EQfWyuf/content/2301.11241v1.pdf'} diff --git a/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/2301.03138v1.pdf.txt b/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/2301.03138v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9485d75fbe124d3c05d5ece11bf34e88e84e7c9c --- /dev/null +++ b/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/2301.03138v1.pdf.txt @@ -0,0 +1,2168 @@ +arXiv:2301.03138v1 [math-ph] 9 Jan 2023 +Gaudin Hamiltonians on unitarizable modules over classical +Lie (super)algebras +Wan Keng Cheong and Ngau Lam +Abstract +Let L be a tensor product of unitarizable irreducible highest weight modules over +the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show +that the Gaudin Hamiltonians associated to G are diagonalizable with simple spectrum +on the space spanned by singular vectors of any fixed weight in L. In particular, we +establish the diagonalization of the Gaudin Hamiltonians, associated to any of the +orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the space +spanned by singular vectors of any fixed weight in the tensor product of infinite- +dimensional unitarizable irreducible highest weight modules. +1 +Introduction +The Gaudin model was introduced by Gaudin [G1, G2] to describe a completely integrable +quantum spin chain associated to any finite-dimensional simple Lie algebra G. Let (·, ·) +be a non-degenerate invariant bilinear form on G. Let {Ia | a = 1, . . . , d} be a basis for G +and {Ia | a = 1, . . . , d} the dual basis with respect to the bilinear form (·, ·), where d is the +dimension of G. The Casimir symmetric tensor Ω for G is defined to be Ω = �d +a=1 Ia ⊗ Ia. +Fix ℓ ∈ N with ℓ ≥ 2, and let z1, . . . , zℓ be distinct complex numbers. For any i = 1, . . . , ℓ, +the Gaudin Hamiltonian Hi of the Gaudin model associated to G is defined to be +Hi = +ℓ +� +j=1 +j̸=i +Ω(ij) +zi − zj +, +where Ω(ij) is defined as in (4.1). The Gaudin Hamiltonians H1, . . . , Hℓ act on the tensor +product M1 ⊗ · · · ⊗ Mℓ, where each Mi is a G-module, and they are mutually commuting +operators. +To find common eigenvectors for Gaudin Hamiltonians is one of the main problems +of studying the Gaudin model. Bethe ansatz method provides an explicit construction +of common eigenvectors from the solutions of the so-called Bethe ansatz equations and +proves to be effective for the special linear Lie algebra sl(2, C) (cf. [G1]). The eigenvectors +obtained by this method are called Bethe vectors. Babujian and Flume [BF] generalized +the Bethe ansatz equations to the case of any simple Lie algebra. In [FFR], Feigin, Frenkel +and Reshetikhin proposed a new approach based on Wakimoto modules over the affine +Lie algebra at the critical level. They obtained the Bethe vectors by restricting certain +invariant functionals on tensor products of Wakimoto modules and found as well that the +1 + +2 +Cheong and Lam +Bethe vectors are in close connection with the solutions of the Knizhnik–Zamolodchikov +equations [KZ] (see also [EFK]). +In this paper, we are interested in the super version of Gaudin Hamiltonians. +For +precise definitions of such Gaudin Hamiltonians, see (4.2) and (5.1). We find an interesting +connection between the Gaudin Hamiltonians associated to the (trivial) central extension +of any infinite-rank classical Lie (super)algebra of type a, c, d and the Gaudin Hamiltonians +associated to the (trivial) central extension of the corresponding Lie algebra. Our main +tool is super duality (cf. [CL2, CLW1, CLW2]), which asserts that there are equivalences +of tensor categories between the parabolic BGG categories �O of ˜g-modules, O[m] of g[m]- +modules and O[m] of g[m]-modules (cf. Theorem 2.10). The following diagrams summarize +the relationships among these categories. +�O +O[m] +O[0] +T[m] +T[0] +�O +O[m] +O[0] +T [m] +T [0] +Here the tensor functors T[m], T [m], T[0], and T [0], defined in Section 2.6, are equivalences of +tensor categories for m ∈ N. Notice that ˜g, g[m] and g[m], for m ∈ N, are Lie superalgebras +while g[0] and g[0] are Lie algebras. In other words, super duality gives equivalences of +categories between parabolic BGG categories for Lie superalgebras and the corresponding +Lie algebras. +We view the Gaudin Hamiltonians for ˜g (resp., g[m] and g[m]) as linear operators on +the tensor product of modules over ˜g (resp., g[m] and g[m]) in the category �O (resp., O[m] +and O[m]). +Instead of constructing eigenvectors for these operators directly, we apply +super duality and the work [CaL] to establish one-to-one correspondences relating the +sets of eigenvectors of the Gaudin Hamiltonians for ˜g, g[m] and g[m] for m ∈ Z+ (see +Theorem 4.7). Furthermore, by using the notion of truncation functors (see (2.18)), we +relate the eigenvectors of Gaudin Hamiltonians for g[m] and g[m] to the eigenvectors of +Gaudin Hamiltonians for their finite-rank counterparts g[m]n and g[m]n, for n ∈ N. +Let G be a finite-dimensional simple Lie algebra and Li a finite-dimensional irreducible +module over G for each i = 1, . . . , ℓ. Rybnikov shows that for generic z1, . . . , zℓ, the Gaudin +Hamiltonians are diagonalizable with simple spectrum on the space spanned by singular +vectors in L1 ⊗ · · · ⊗ Lℓ (see [MV, MTV1, MTV2, FFRy, LMV] as well). +We would like to extend Rybnikov’s result to the case of Lie (super)algebras in this +paper. We focus our attention on the finite-dimensional classical Lie (super)algebra G +x[m]n +of type x, where x denotes a fixed type among a, c, d (see Sections 2.1, 2.2 and 2.3). +Note that G +a[m]n ∼= gl(m|n), G +c[m]n ∼= spo(2m|2n) and G +d[m]n ∼= osp(2m|2n). Let L = +L1 ⊗ · · · ⊗ Lℓ, where each Li is an irreducible highest weight module over G +x[m]n. Suppose +that for x = a, each Li is a polynomial module, and for x = c, d, all highest weights of Li +lie in Q +x,I(m|n) or Q +x,II(m|n) (see Section 3.3). We show that the Gaudin Hamiltonian +Hi[m]n associated to G +x[m]n, for i = 1, . . . , ℓ, is diagonalizable with simple spectrum on +the space spanned by singular vectors in any finite direct sum of weight spaces of L (cf. +Theorem 5.5 and Theorem 5.7). We will see later that such Li’s are unitarizable modules + +Gaudin Hamiltonians on unitarizable modules +3 +over G +x[m]n with respect to a natural choice of ∗-structures on G +x[m]n (see Sections 3.2 +and 3.3). +Let us explain a little about how we obtain the diagonalization. For x = c, d (resp. +x = a), we relate each Hi[m]n to its corresponding Gaudin Hamiltonian on the tensor +product of some finite-dimensional irreducible modules over G, where G is the orthogonal +Lie algebra so(2p) or the symplectic Lie algebra sp(2p) (resp., G is the general linear +Lie algebra gl(p)) for some p ∈ N. These correspondences should give a procedure of +constructing eigenvectors of the Gaudin Hamiltonians for classical Lie (super)algebras +from the ones for the corresponding Lie algebras. +While there is a great deal of work on Gaudin models, not much appears to be known +about the super Gaudin Hamiltonians. In [MVY], Mukhin, Vicedo and Young investigate +the Gaudin Hamiltonians associated to the general linear Lie superalgebra gl(m|n) and +show that the operators are diagonalizable with simple spectrum on the space spanned +by singular vectors in the tensor product of the natural module Cm|n. In [KuM], Kulish +and Manojlovi´c explicitly construct eigenvectors for the Gaudin Hamiltonians associated +to the ortho-symplectic Lie superalgebra osp(1|2). The reader may also want to consult +[HMVY, L, LM] for related results. We hope that this paper can provide new insights into +the study of Gaudin models. +This paper is organized as follows. In Section 2, we fix notation and review some back- +ground materials on classical Lie (super)algebras and their central extensions. We also de- +fine the parabolic BGG categories �O, O[m] and O[m] associated to the Lie (super)algebras +˜g, g[m] and g[m], respectively, and discuss super duality which gives equivalences of these +categories. In Section 3, we give a brief introduction to ∗-structures and study the unita- +rizable modules which will be considered in our study of Gaudin Hamiltonians. In Section +4, we investigate Gaudin Hamiltonians associated to the central extensions of finite-rank +and infinite-rank Lie (super)algebras, and establish the one-to-one correspondences be- +tween the sets of eigenvectors (see Theorem 4.7 and Proposition 4.8). In Section 5, we +concentrate on Gaudin Hamiltonians for finite-dimensional classical Lie (super)algebras +and prove Theorem 5.5 and Theorem 5.7, which give an affirmative answer to the diago- +nalization of the operators on unitarizable modules. +Notations. Throughout the paper, N stands for the set of positive integers, Z for the set +of integers, Z∗ for the set of nonzero integers, Z+ for the set of non-negative integers, 1 +2Z +for the set of half integers and integers, and C for the set of complex numbers. All vector +spaces, algebras, tensor products, et cetera, are over C. +2 +Preliminaries +In this section, we first define the finite-rank and infinite-rank Lie (super)algebras �Gx, +Gx[m]n and G +x[m]n, where x denotes one of the three types a, c, d. We consider their central +extensions �gx, gx[m]n and gx[m]n and the parabolic BGG categories �Ox (resp., Ox[m]n and +O +x[m]n) of modules over �gx (resp., gx[m]n and gx[m]n). +We then recall the truncation +functors which relate Ox[m]∞ and O +x[m]∞ to Ox[m]n and O +x[m]n, respectively, for n ∈ N. +Finally, we describe the tensor functors T[m] and T [m] and their properties. We refer the + +4 +Cheong and Lam +readers to [CL2, Sections 2 and 3] for type a and [CLW1, Sections 2 and 3] for types c, d +for details (see also [CW, Sections 6.1 and 6.2] and [CaL, Section 2.4]). We fix m ∈ Z+ +and n ∈ N ∪ {∞} throughout this paper. +Let �V denote the superspace over C with ordered basis {vr | r ∈ 1 +2Z}. The parity of vr +is defined as follows: |vr| = ¯0 if r ∈ Z, and |vr| = ¯1 if r ∈ 1 +2 + Z. +Let gl(�V ) be the Lie superalgebra consisting of all linear endomorphisms on �V which +vanish on all but finitely many vr’s. For i, j ∈ 1 +2Z, we let Ei,j be the linear endomorphism +on �V defined by +Ei,j(vr) = δjrvi +for r ∈ 1 +2Z, +where δ is the Kronecker delta. +The Lie superalgebra gl(�V ) is spanned by Ei,j with +i, j ∈ 1 +2Z. +The Lie superalgebra gl(�V ) has a central extension, denoted by �gl(�V ), by the one- +dimensional center CK corresponding to the following 2-cocycle (cf. [CL1, p. 99]): +τ(A, B) := Str([J, A]B), +A, B ∈ gl(�V ), +(2.1) +where J = − � +r≥ 1 +2 Er,r and Str denotes the supertrace. +In fact, the cocycle τ is a +coboundary. Moreover, there is an isomorphism ι from the direct sum of Lie superalgebras +gl(�V ) ⊕ CK to �gl(�V ) defined by +ι(A) = A + Str(JA)K, +for A ∈ gl(�V ), +and +ι(K) = K. +(2.2) +Let +Jm(n) = +ß +±1 +2, ±3 +2, . . . , ±(m − 1 +2) +™ +∪ {0} ∪ { ±j | j ∈ N, j < n + 1 }, +Jm(n) = {±1, . . . , ±m} ∪ {0} ∪ +� +±(j − 1 +2) +��� j ∈ N, j < n + 1 +� +, +�J(n) = +� +r ∈ 1 +2Z +��� − n ≤ r ≤ n +� +, +J× +m(n) = Jm(n)\{0}, +J +× +m(n) = Jm(n)\{0}, +�J×(n) = �J(n))\{0}, +J+ +m(n) = { r ∈ Jm(n) | r > 0 }, +J ++ +m(n) = +� +r ∈ Jm(n) +�� r > 0 +� +, +�J+(n) = +¶ +r ∈ �J(n) +��� r > 0 +© +. +We let �V (n), Vm(n), Vm(n), �V ×(n), V × +m (n) and V +× +m (n) be the subspaces of �V with basis +{vi} indexed by �J(n), Jm(n), Jm(n), �J×(n), J× +m(n) and J +× +m(n), respectively. This gives rise +to subalgebras gl(�V (n)), gl(Vm(n)), gl(Vm(n)), gl(�V ×(n)), gl(V × +m (n)) and gl(V +× +m (n)) of the +Lie superalgebra gl(�V ). Let �b := � +r≤s,r,s∈ 1 +2Z CEr,s denote the standard Borel subalgebra +of gl(�V ). + +Gaudin Hamiltonians on unitarizable modules +5 +We will drop the symbol (n) if n = ∞. For example, Jm := Jm(∞). Define the total +orders of Jm and Jm by +. . . 0 and sgn(i) = −1 if i < 0. The bilinear form induces non- +degenerate bilinear forms on V × +m (n) and V +× +m (n). + +6 +Cheong and Lam +Let �Gc +n (resp., Gc[m]n and G +c[m]n) be the subalgebra of the Lie superalgebra gl(�V ×(n)) +(resp., gl(V × +m (n)) and gl(V +× +m (n))) which preserves the bilinear form (·|·). The Lie superal- +gebra �Gc := �Gc +∞ is spanned by the following elements (i, j ∈ Z∗ and r, s ∈ 1 +2 + Z): +Ec +i,j := −Ec +−j,−i := Ei,j − E−j,−i, +ij > 0; +Ec +i,j := Ec +−j,−i := Ei,j + E−j,−i, +ij < 0; +Ec +r,s := −Ec +−s,−r := Er,s − E−s,−r; +Ec +i,r := Ec +−r,−i := Ei,r + E−r,−i, +i > 0; +Ec +i,r := −Ec +−r,−i := Ei,r − E−r,−i, +i < 0. +The subalgebras Gc[m]n and G +c[m]n of �Gc are spanned by Ec +i,j with i, j ∈ J× +m(n) and J +× +m(n), +respectively. Note that Gc[m]n is isomorphic to osp(2m|2n) while G +c[m]n is isomorphic to +spo(2m|2n). +We let �bc := �Gc ∩ �b, bc[m]n := Gc[m]n ∩ b[m] and b +c[m]n := G +c[m]n ∩ b[m] stand for +the standard Borel subalgebras of �Gc, Gc[m]n and G +c[m]n, respectively. The corresponding +Cartan subalgebras �hc, hc[m]n and h +c[m]n have bases {Ec +i := Ei,i − E−i,−i | i ∈ +1 +2N}, +{Ec +i | i ∈ J+ +m(n)} and {Ec +i | i ∈ J ++ +m(n)}, respectively. Let {ǫi} denote the dual bases of the +Cartan subalgebras with the corresponding indices. +2.3 +Ortho-symplectic superalgebras �Gd and its subalgebras +Define a non-degenerate supersymmetric bilinear form (·|·) on �V × by +(vi|vj) = (vj|vi) = δi,−j, +i, j ∈ Z∗; +(2.6) +(vr|vs) = −(vs|vr) = sgn(r)δr,−s, +r, s ∈ 1 +2 + Z; +(2.7) +(vi|vr) = (vr|vi) = 0, +i ∈ Z∗, r ∈ 1 +2 + Z. +(2.8) +The bilinear form induces non-degenerate bilinear forms on V × +m (n) and V +× +m (n). +Let �Gd +n (resp., Gd[m]n and G +d[m]n) be the subalgebra of the Lie superalgebra gl(�V ×(n)) +(resp., gl(V × +m (n)) and gl(V +× +m (n))) which preserves the bilinear form (·|·). The Lie superal- +gebra �Gd := �Gd +∞ is spanned by the following elements (i, j ∈ Z∗ and r, s ∈ 1 +2 + Z): +Ed +i,j := −Ed +−j,−i := Ei,j − E−j,−i; +Ed +r,s := −Ed +−s,−r := Er,s − E−s,−r, +rs > 0; +Ed +r,s := Ed +−s,−r := Er,s + E−s,−r, +rs < 0; +Ed +i,r := Ed +−r,−i := Ei,r + E−r,−i, +r > 0; +Ed +i,r := −Ed +−r,−i := Ei,r − E−r,−i, +r < 0. +The subalgebras Gd[m]n and G +d[m]n of �Gd are spanned by Ed +i,j with i, j ∈ J× +m(n) and J +× +m(n), +respectively. Note that Gd[m]n is isomorphic to spo(2m|2n) while G +d[m]n is isomorphic to +osp(2m|2n). + +Gaudin Hamiltonians on unitarizable modules +7 +We let �bd := �Gd ∩ �b, bd[m]n := Gd[m]n ∩ b[m] and b +d[m]n := G +d[m]n ∩ b[m] stand for +the standard Borel subalgebras of �Gd, Gd[m]n and G +d[m]n, respectively. The corresponding +Cartan subalgebras �hd, hd[m]n and h +d[m]n have bases {Ed +i := Ei,i − E−i,−i | i ∈ 1 +2N}, +{Ed +i | i ∈ J+ +m(n)} and {Ed +i | i ∈ J ++ +m(n)}, respectively. Let {ǫi} denote the dual bases of the +Cartan subalgebras with the corresponding indices. +Define a linear automorphism ϕ of degree 1 on the superspace �V × by +ϕ(v±r) := +® v±(r− 1 +2 ), +if r ∈ N; +v±(r+ 1 +2 ), +if r ∈ 1 +2 + Z+. +(2.9) +The isomorphism ϕ induces an automorphism �ϕ on the Lie superalgebra gl(�V ×). Notice +that the supersymmetric bilinear form on �V × defined by (2.6), (2.7) and (2.8) is exactly +the bilinear form induced, via ϕ, by the skew-supersymmetric bilinear form on �V × defined +by (2.3), (2.4) and (2.5). The restriction of �ϕ to �Gc gives an isomorphism from �Gc to �Gd +and hence an isomorphism from Gc[m]n (resp., G +c[m]n) to G +d[m]n (resp., Gd[m]n). It is +clear that �ϕ preserves the corresponding Borel and Cartan subalgebras. The restrictions +of �ϕ are denoted by �ϕ as well. We summarize the results in the following lemma. +Lemma 2.1. There is an isomorphism �ϕ from �Gc to �Gd given by +�ϕ(Ec +r,s) = + + + + + + + + + + + + + + + +Ed +r− 1 +2 ,s− 1 +2, +if r, s ∈ N ∪ (−(1 +2 + Z+)); +Ed +r+ 1 +2 ,s+ 1 +2, +if r, s ∈ −N ∪ (1 +2 + Z+); +Ed +r− 1 +2 ,s+ 1 +2, +if r ∈ N ∪ (−(1 +2 + Z+)), s ∈ −N ∪ (1 +2 + Z+); +Ed +r+ 1 +2 ,s− 1 +2, +if r ∈ −N ∪ (1 +2 + Z+), s ∈ N ∪ (−(1 +2 + Z+)). +(2.10) +The restrictions of �ϕ to Gc[m]n and G +c[m]n give isomorphisms �ϕ : Gc[m]n −→ G +d[m]n and +�ϕ : G +c[m]n −→ Gd[m]n, respectively. +2.4 +Dynkin diagrams +Consider the free abelian group with basis {ǫi | i ∈ 1 +2N}. It is endowed with a symmetric +bilinear form (·, ·) defined by +(ǫr, ǫs) = (−1)2rδrs, +r, s ∈ 1 +2N. +The parity of ǫi is defined as follows: |ǫi| = 0 for i ∈ N and |ǫj| = 1 for j ∈ 1 +2 + Z+. Let +α× = ǫm − ǫ 1 +2 , +αr = ǫr − ǫr+ 1 +2, +βr = ǫr − ǫr+1, +r ∈ 1 +2N. +For x = a, c, d, the Dynkin diagrams of the Lie superalgebras �Gx +n and G +x[m]n (where +m ∈ N) together with prescribed fundamental systems are listed below ([K, Section 2.5]). +In what follows, ⃝ and � denote an even simple root and an odd isotropic simple root, +respectively. + +8 +Cheong and Lam +✡ +✠ +☛ +✟ +�Ga +n +� +� +� +� +� +· · · +α1/2 +α1 +α3/2 +αn−1 +αn−1/2 +✡ +✠ +☛ +✟ +�Gc +n +� +� +� +� +� +� +· · · +❅ +❅ +� +� +α1/2 +−ǫ1/2 − ǫ1 +α1 +α3/2 +αn−1 +αn−1/2 +✡ +✠ +☛ +✟ +�Gd +n +⃝ +� +� +� +� +· · · +=⇒ +−2ǫ1/2 +α1/2 +α1 +αn−1 +αn−1/2 +✡ +✠ +☛ +✟ +G +a[m]n +⃝ +⃝ +⃝ +⃝ +� +⃝ +⃝ +· · · +· · · +β1 +β2 +β3 +βm−1 +α× +β1/2 +βn−3/2 +✡ +✠ +☛ +✟ +G +c[m]n +⃝ +⃝ +⃝ +⃝ +� +⃝ +⃝ +=⇒ +· · · +· · · +−2ǫ1 +β1 +β2 +βm−1 +α× +β1/2 +βn−3/2 +✡ +✠ +☛ +✟ +G +d[m]n +⃝ +⃝ +⃝ +⃝ +⃝ +� +⃝ +⃝ +�� +❅❅ +· · · +· · · +β1 +−ǫ1−ǫ2 +β2 +β3 +βm−1 +α× +β1/2 +βn−3/2 +The Dynkin diagrams of the Lie algebras G +x[0]n are as follows. +✡ +✠ +☛ +✟ +G +a[0]n +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +· · · +β1/2 +β3/2 +β5/2 +βn−7/2 βn−5/2 +βn−3/2 +✡ +✠ +☛ +✟ +G +c[0]n +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +❅ +❅ +� +� +· · · +β1/2 +−ǫ1/2 − ǫ3/2 +β3/2 +β5/2 +βn−7/2 βn−5/2 +βn−3/2 +✡ +✠ +☛ +✟ +G +d[0]n +⃝ +⃝ +⃝ +⃝ +⃝ +⃝=⇒ +· · · +−2ǫ1/2 +β1/2 +β3/2 +βn−7/2 βn−5/2 +βn−3/2 +Let �Φx + +n +(resp., Φx[m]+ +n and Φ +x[m]+ +n ) denote the set of positive roots of �Gx +n (resp., Gx[m]n +and G +x[m]n). + +Gaudin Hamiltonians on unitarizable modules +9 +2.5 +Central extensions +For x = a, c, d, consider the central extension ˜gx (resp., gx[m]n and gx[m]n) of �Gx (resp., +Gx[m]n and G +x[m]n) by the one-dimensional center CK, which is inherited from the cen- +tral extension �gl(�V ) of gl(�V ) determined by the 2-cocycle (2.1). The restriction of the +isomorphism ι to �Gx ⊕ CK (resp., Gx[m]n ⊕ CK and G +x[m]n ⊕ CK) is an isomorphism +ι : �Gx ⊕ CK → ˜gx (resp. ι : Gx[m]n ⊕ CK → gx[m]n and ι : G +x[m]n ⊕ CK → gx[m]n) given +by +ι(A) = A + Str(JA)K. +(2.11) +Note that [J, A] = 0 for all A ∈ ˜ga. Thus the 2-cocycle is zero when it restricts to �Ga, +and hence +[A, B]˜ga = [A, B]�Ga, +for A, B ∈ �Ga, +where, for example, [·, ·]˜ga denotes the Lie bracket on ˜ga. For notational unity, we still +consider ˜ga, ga[m]n and ga[m]n. +Remark 2.2. Every ˜gx(resp., gx[m]n and gx[m]n)-module can be regarded as a �Gx(resp., +Gx[m]n and G +x[m]n)-module through the isomorphism (2.11). These central extensions are +convenient and conceptual for the formulation of truncation functors and super duality +described in Section 2.6 (see [CLW1, Remark 3.3] for more explanations). +We let �bx := �bx⊕CK, bx[m]n := bx[m]n⊕CK and b +x[m]n := b +x[m]n⊕CK stand for the +standard Borel subalgebras of ˜gx, gx[m]n and gx[m]n, respectively, and let ˜hx, hx[m]n and +h +x[m]n denote the Cartan subalgebras of ˜gx, gx[m]n and gx[m]n spanned by bases {K, Ex +r} +with dual bases {Λ0, ǫr} in the restricted dual ˜hx ∗, hx[m]∗ +n and h +x[m]∗ +n, where r runs over +the index sets 1 +2N, J+ +m(n) and J ++ +m(n), respectively. Here Λ0 is the element of ˜hx ∗ (resp., +hx[m]∗ +n and h +x[m]∗ +n) defined by +Λ0(K) = 1 +and +Λ0(Ex +r) = 0 +for all r ∈ 1 +2N (resp., J+ +m(n) and J ++ +m(n)). +It is easy to see that the automorphism �ϕ on gl(�V ×) induced from ϕ defined by (2.9) +can be extended to the central extension �gl(�V ×) of gl(�V ×). By Lemma 2.1, we have the +following lemma. +Lemma 2.3. The isomorphism �ϕ : �Gc −→ �Gd extends to an isomorphism from ˜gc to ˜gd, +which is also denoted by �ϕ and is given by (2.10) together with +�ϕ(K) = −K. +The restrictions of �ϕ to gc[m]n and gc[m]n give isomorphisms �ϕ : gc[m]n −→ gd[m]n and +�ϕ : gc[m]n −→ gd[m]n, respectively. +2.6 +Parabolic BGG categories and super duality +Let �lx, lx[m]n and l +x[m]n be Levi subalgebras of ˜gx, gx[m]n and gx[m]n defined by +�lx := ˜gx ∩�l, +lx[m]n := gx[m]n ∩�l, +and +l +x[m]n := gx[m]n ∩�l, + +10 +Cheong and Lam +respectively, and let �px = ˜lx +�bx, px[m]n = lx[m]n + bx[m]n and px[m]n = l +x[m]n + b +x[m]n be +the corresponding parabolic subalgebras, where �l := � +rs>0, +r,s∈ 1 +2 Z∗ CEr,s ⊕ CK. Observe that +lx[m]n ∼= ga[m]n ∼= gl(m|n) ⊕ CK +and +l +x[m]n ∼= ga[m]n ∼= gl(m|n) ⊕ CK. +Note that for x = a, �la = ˜ga, la[m]n = ga[m]n and l +a[m]n = ga[m]n. +In the remainder of the paper, we shall drop the superscript x and the +symbol ∞ if there is no ambiguity. For example, we write �G, G[m]n and G[m]n for �Gx, +Gx[m]n and G +x[m]n, and �g, g[m]n and g[m]n for �gx, gx[m]n and gx[m]n, respectively, where +x denotes a fixed type among a, c, d. Also, we write g[m] and g[m] for gx[m]∞ and gx[m]∞, +respectively. +Given a partition µ = (µ1, µ2, . . .), we denote by ℓ(µ) the length of µ and by µ′ the +conjugate partition of µ. We also denote by θ(µ) the modified Frobenius coordinates of µ: +θ(µ) := (θ(µ)1/2, θ(µ)1, θ(µ)3/2, θ(µ)2, . . .), +where +θ(µ)i−1/2 := max{µ′ +i − i + 1, 0}, +θ(µ)i := max{µi − i, 0}, +i ∈ N. +Given a partition λ = (λ1, λ2, . . .) and d ∈ C, we define +�λ := +� +r∈ 1 +2N +θ(λ)rǫr + dΛ0 ∈ ˜h∗, +(2.12) +λ[m] := +m +� +i=1 +λ′ +iǫi− 1 +2 + +� +j∈N +⟨λj − m⟩ ǫj + dΛ0 ∈ h[m]∗, +(2.13) +λ[m] := +m +� +i=1 +λiǫi + +� +j∈N +� +λ′ +j − m +� +ǫj− 1 +2 + dΛ0 ∈ h[m]∗. +(2.14) +Here ⟨r⟩ := max{r, 0}. +Let �P+(d) ⊂ ˜h∗, P[m]+(d) ⊂ h[m]∗ and P[m]+(d) ⊂ h[m]∗ denote the sets of all +weights of the forms (2.12), (2.13) and (2.14), respectively. Let �P+ = ∪d∈C�P+(d), P[m]+ = +∪d∈CP[m]+(d) and P[m]+ = ∪d∈CP[m]+(d). By definition we have bijective maps +�P+ −→ P[m]+ +�λ �→ λ[m] +and +�P+ −→ P[m]+ +�λ �→ λ[m] +(2.15) +Recall that a partition λ = (λ1, λ2, . . .) is called an (m|n)-hook partition if λm+1 ≤ n. +Let P and P(m|n) denote the set of partitions and the set of (m|n)-hook partitions, +respectively. Note that P(m|∞) = P. Given d ∈ C and λ ∈ P with λ′ ∈ P(m|n) (resp., +λ ∈ P(m|n)), we may regard λ[m] ∈ h[m]∗ +n (resp., λ[m] ∈ h[m]∗ +n) in a natural way. The +subsets of all such weights will be denoted respectively by P[m]+ +n and P[m]+ +n . +For µ ∈ �h∗, let L(˜l, µ) be the irreducible highest weight ˜l-module with highest weight +µ. We denote by ∆(˜g, µ) = Ind�g +�pL(˜l, µ) the parabolic Verma �g-module and by L(˜g, µ) the +unique irreducible quotient �g-module of ∆(˜g, µ). The modules L(l[m]n, µ) and ∆(g[m]n, µ) + +Gaudin Hamiltonians on unitarizable modules +11 +(for µ ∈ h[m]∗ +n) as well as L(l[m]n, µ) and ∆(g[m]n, µ) (for µ ∈ h[m]∗ +n) are defined anal- +ogously. We denote by L(g[m]n, µ) (resp., L(g[m]n, µ)) the unique irreducible quotient +g[m]n-module (resp., g[m]n-module) of ∆(g[m]n, µ) (resp., ∆(g[m]n, µ)). Note that for +x = a, ∆(˜ga, µ) = L(˜ga, µ) = L(˜la, µ), ∆(ga[m], µ) = L(ga[m], µ) = L(la[m], µ) and +∆(ga[m], µ) = L(ga[m], µ) = L(l +a[m], µ). +Similar to [CL2, CLW1, CLW2], let �O (resp., O[m]n and O[m]n) be the category of +˜g(resp. g[m]n and g[m]n)-modules M such that M is a semisimple �h(resp., h[m]n and +h[m]n)-module with finite-dimensional weight subspaces Mγ for γ ∈ �h∗ (resp., h[m]∗ +n and +h[m]∗ +n), satisfying +(i) M decomposes over ˜l (resp., l[m]n and l[m]n) as a direct sum of L(˜l, µ) (resp., +L(l[m]n, µ) and L(l[m]n, µ)) for µ ∈ �P+ (resp., P[m]+ +n and P[m]+ +n ). +(ii) There exist finitely many weights λ1, λ2, . . . , λk ∈ �P+ (resp., P[m]+ +n and P[m]+ +n ) +(depending on M) such that if γ is a weight in M, then λi −γ is a linear combination +of simple roots with coefficients in Z+ for some i. +The morphisms in the categories are even homomorphisms of modules, and the categories +are abelian. There is a natural Z2-gradation on each module in the categories with com- +patible action of the corresponding Lie (super)algebra to be defined below. Set +�Ξ = +� +r∈ 1 +2 N +Z+ǫr + CΛ0, +Ξ[m]n = +� +r∈J+ +m(n) +Z+ǫr + CΛ0, +(2.16) +Ξ[m]n = +� +r∈J ++ +m(n) +Z+ǫr + CΛ0. +For ε = 0 or 1 and Θ = �Ξ, Ξ[m]n or Ξ[m]n, we define +Θ(ε) := +� +µ ∈ Θ +��� +� +r∈ 1 +2 +Z+ +µ(Er) ≡ ε (mod 2) +� +, +where the summation is over all r ∈ 1 +2 + Z+ whenever µ(Er) are defined. Recall that +both l[m]n and l[m]n are isomorphic to gl(m|n) ⊕ CK. For M ∈ O[m]n (resp., O[m]n), +each weight of M is a weight of a highest weight module over l[m]n (resp., l[m]n) with +highest weight µ ∈ P[m]+ +n (resp., P[m]+ +n ) which is contained in Ξ[m]n (resp., Ξ[m]n) (see, +for example, [CW, Proposition 3.26]). By the paragraph before Theorem 6.4 in [CW], the +weights of M are contained in �Ξ for M ∈ �O. For M ∈ �O, M = M0 +� M1 is a Z2-graded +vector space such that +M0 := +� +µ∈�Ξ(0) +Mµ +and +M1 := +� +µ∈�Ξ(1) +Mµ. +(2.17) +It is clear that the Z2-gradation on M is compatible with the action of ˜g. Similarly, we may +define a Z2-gradation with compatible action of g[m]n and g[m]n on M for M ∈ O[m]n and + +12 +Cheong and Lam +O[m]n, respectively. By [CW, Theorem 3.27 and Theorem 6.4] (see also the proof of [Lus, +Theorem 6.2.2]), �O, O[m]n and O[m]n are tensor categories. In particular, �Oa, Oa[m]n and +O +a[m]n are semisimple tensor categories. Note that the Z2-gradation on M ⊗ N given by +(2.17) and the Z2-gradation on M ⊗N induced from the Z2-gradations on M and N given +by (2.17) are the same for M, N ∈ �O (resp., O[m]n and O[m]n). We summarize the results +in the following proposition. +Proposition 2.4. Let n ∈ N ∪ {∞}. Then: +(i) The weights of modules in �O (resp., O[m]n and O[m]n) are contained in �Ξ (resp., +Ξ[m]n and Ξ[m]n). +(ii) The abelian categories �O, O[m]n and O[m]n are tensor categories. +(iii) �Oa, Oa[m]n and O +a[m]n are semisimple tensor categories. +We also have the following proposition. +Proposition 2.5. Let n ∈ N ∪ {∞}. Then: +(i) The modules ∆(˜g, λ) and L(˜g, λ) lie in �O for all λ ∈ �P+. +(ii) The modules ∆(g[m]n, λ) and L(g[m]n, λ) lie in O[m]n for all λ ∈ P+[m]n. +(iii) The modules ∆(g[m]n, λ) and L(g[m]n, λ) lie in O[m]n for all λ ∈ P ++[m]n. +Proof. (i) follows from [CW, Proposition 6.7(3)]. We show (ii). For m = 0, it follows from +[CW, Proposition 6.7(1)]. For m ̸= 0, the proof is similar to that of [CW, Proposition +6.7(3)], but here we use [CW, Theorem 3.27] (valid also for n = ∞) instead of [CW, +Theorem 6.4]. The proof of (iii) is similar. +We record here an easy but useful observation, which can be seen by the description +of the weights of modules in (2.16) and is essentially [CaL, Lemma 2.3]. +Lemma 2.6. Let M, N ∈ �O (resp., O[m]n and O[m]n). Suppose that µ and γ are weights +of M and N, respectively. Then +(µ + γ)(Er) = 0 +if and only if +µ(Er) = 0 and γ(Er) = 0, +for r ∈ 1 +2N (resp., J+ +m(n) and J ++ +m(n)). +Similarly, in view of (2.16), we immediately obtain the following (cf. [CaL, Lemma +2.5]). +Lemma 2.7. Let µ, γ ∈ �Ξ. Then: +(i) µ + γ ∈ Ξ[m] if and only if µ ∈ Ξ[m] and γ ∈ Ξ[m]. +(ii) µ + γ ∈ Ξ[m] if and only if µ ∈ Ξ[m] and γ ∈ Ξ[m]. + +Gaudin Hamiltonians on unitarizable modules +13 +For 0 ≤ k < n ≤ ∞, the truncation functor trn +k : O[m]n −→ O[m]k is defined by +trn +k(M) = +� +ν∈Ξ[m]k +Mν +for M ∈ O[m]n. +(2.18) +For every f ∈ HomO[m]n(M, N), trn +k(f) is defined to be the restriction of f to trn +k(M). The +truncation functor trn +k : O[m]n −→ O[m]k can be defined in a similar way. It is clear that +trn +k and trn +k are exact functors. By Lemma 2.6, we immediately have the following lemma. +Lemma 2.8. Let 0 ≤ k < n ≤ ∞. trn +k and trn +k are tensor functors. +The following proposition will be useful later on. It may be proven in a similar way to +the proof of [CLW1, Lemma 3.2]. +Proposition 2.9. Let 0 ≤ k < n ≤ ∞ and µ ∈ P[m]+ +n . Suppose Vi = ∆(g[m]i, µ) or +L(g[m]i, µ) for i = k, n. We have +trn +k(Vn) = +® +Vk +if µ ∈ P[m]+ +k , +0 +otherwise. +Similar statement holds for trn +k. +Given M = � +γ∈˜h∗ Mγ ∈ �O, we define +T[m](M) = +� +γ∈h[m]∗ +Mγ +and +T [m](M) = +� +γ∈h[m]∗ +Mγ. +For M, N ∈ �O and f ∈ Hom�O(M, N), T[m](f) and T [m](f) are defined to be the +restrictions of f to T[m](M) and T [m](M), respectively. Note that T[m](f) : T[m](M) → +T[m](N) and T [m](f) : T [m](M) → T [m](N) are respectively a g[m]-homomorphism and a +g[m]-homomorphism. Moreover, the functors T[m] : �O → O[m] and T [m] : �O → O[m] are +exact (cf. [CW, Proposition 6.15]). +By Lemma 2.7, we have T[m](M ⊗ N) = T[m](M) ⊗ T[m](N) and T [m](M ⊗ N) = +T [m](M) ⊗ T [m](N) for all M, N ∈ �O, and so T[m] and T [m] are tensor functors. We have +the following result. +Theorem 2.10. The following statements hold: +(i) For each m ∈ Z+, T[m] : �O → O[m] is an equivalence of tensor categories. +(ii) For each m ∈ Z+, T [m] : �O → O[m] is an equivalence of tensor categories. +Moreover, T[m] and T [m] send parabolic Verma modules to parabolic Verma modules and +irreducible modules to irreducible modules. More precisely, for λ ∈ �P+, we have +T[m] +� +∆(˜g, �λ)) = ∆(g[m], λ[m]), +T[m] +� +L(˜g, �λ) +� += L(g[m], λ[m]), +T [m] +� +∆(˜g, �λ)) = ∆(g[m], λ[m]), +T [m] +� +L(˜g, �λ) +� += L(g[m], λ[m]). + +14 +Cheong and Lam +Theorem 2.10 can be proven along the lines of the proof of the super duality in [CL2, +CLW1] by making use of the techniques in [CLW2] (see particularly [CLW2, Section 7]). +The proof is omitted here. We also call Theorem 2.10 super duality. We only need that +T[m] and T [m] are tensor functors in this paper. Note that T[0] and T [0] are the functors T +and T in the degenerate case defined in [CL2, CLW1]. Theorem 2.10 also implies that the +tensor categories O[0] and O[0] for Lie algebras and the tensor categories O[m] and O[m] +for Lie superalgebras are equivalent for m ∈ N. +3 +Unitarizable G +x[m]n-modules +The notion of unitarizable modules will play an important role in our study of (super) +Gaudin Hamiltonians. In this section, we start by introducing ∗-structures on G +x[m]n and +gx[m]n. We then describe the unitarizable G +x[m]n-modules to be studied in this paper. +We first recall some basic facts about ∗-superalgebras and their unitarizable represen- +tations. A ∗-superalgebra is an associative superalgebra A together with an anti-linear +anti-involution ω : A −→ A of degree 0. A homomorphism f : (A, ω) → (A′, ω′) of ∗- +superalgebras is a homomorphism of superalgebras satisfying ω′ ◦f = f ◦ω. Let (A, ω) be +a ∗-superalgebra, and let V be a Z2-graded A-module. A Hermitian form ⟨·|·⟩ on V is said +to be contravariant if ⟨av|v′⟩ = ⟨v|ω(a)v′⟩, for all a ∈ A, v, v′ ∈ V . An A-module equipped +with a positive definite contravariant Hermitian form is called a unitarizable A-module. +A Lie superalgebra g is said to admit a ∗-structure if g is equipped with an anti-linear +anti-involution ω of degree 0. In this case, ω is also called a ∗-structure on g. A homo- +morphism f : (g, ω) → (g′, ω′) of Lie superalgebras with ∗-structures is a homomorphism +of Lie superalgebras satisfying ω′ ◦ f = f ◦ ω. Moreover, it is clear that ω is a ∗-structure +on g if and only if the natural extension of ω to the universal enveloping algebra U(g) of +g is an anti-linear anti-involution. Let (g, ω) be a Lie superalgebra with ∗-structure, and +let V be a Z2-graded g-module. A Hermitian form ⟨·|·⟩ on V is said to be contravariant if +⟨xv|v′⟩ = ⟨v|ω(x)v′⟩, for all x ∈ g, v, v′ ∈ V . A g-module equipped with a positive definite +contravariant Hermitian form is called a unitarizable g-module. Notice that a g-module +V is a unitarizable g-module if and only if V is a unitarizable U(g)-module. +3.1 +∗-structures on G +x[m]n and gx[m]n +Recall that the Lie superalgebra �gl(�V ) is the central extension of gl(�V ) with a basis +{Ei,j, K | i, j ∈ 1 +2Z}. It admits a ∗-structure ω defined by (cf. [LZ1, p. 421]) +� +i,j∈ 1 +2Z +aijEi,j �→ +� +i,j∈ 1 +2Z +(−1)[i]+[j]aijEj,i +and +K �→ K. +Here aij denotes the complex conjugate of aij ∈ C and +[i] := +® +1 +if +− i ∈ 1 +2 + Z+; +0 +if +− i ∈ 1 +2Z\(1 +2 + Z+). + +Gaudin Hamiltonians on unitarizable modules +15 +It is clear from the spanning sets (i.e., the sets of elements described in Sections 2.1, 2.2 and +2.3 together with K) of the Lie superalgebras ˜gx, gx[m]n and gx[m]n that the restrictions +of ω to these Lie superalgebras, denoted also by ω, give ∗-structures on them. +Since ω is a ∗-structure on �gl(�V ×) and �ϕ is an involution of �gl(�V ×), the map ω′ := +�ϕ ◦ ω ◦ �ϕ is a ∗-structure on �gl(�V ×). More precisely, +ω′(Er,s) = (−1)τr+τsEs,r, +for r, s ∈ 1 +2Z∗, +and +ω′(K) = K, +where +τr := +® +1 +if +− r ∈ N; +0 +if +− r ∈ 1 +2Z∗\N. +Via the isomorphism �ϕ : gc[m]n −→ gd[m]n given in Lemma 2.3, an anti-linear anti- +involution ω on gd[m]n pulls back to an anti-linear anti-involution ω′ := �ϕ−1 ◦ ω ◦ �ϕ on +gc[m]n while, via the isomorphism �ϕ−1 : gd[m]n −→ gc[m]n, an anti-linear anti-involution +ω on gc[m]n pulls back to an anti-linear anti-involution ω′ := �ϕ ◦ ω ◦ �ϕ−1 on gd[m]n. In +the other words, the map �ϕ (resp., �ϕ−1) gives an isomorphism of Lie superalgebras with +∗-structures from (gc[m]n, ω′) (resp., (gd[m]n, ω′)) to (gd[m]n, ω) (resp., (gc[m]n, ω)). Note +that the ∗-structure ω′ on gc[m]n (resp., gd[m]n) is the restriction of ω′ defined on �gl(�V ×). +Setting K = 0, the ∗-structure ω (resp., ω′) induces a ∗-structure, denoted also by +ω (resp., ω′), on gl(�V ×). For x = c, d, the restriction of ω (resp., ω′) to G +x[m]n gives a +∗-structure on G +x[m]n, denoted also by ω (resp., ω′). We have the following proposition. +Proposition 3.1. For x = c, d, the restriction of the isomorphism ι : G +x[m]n ⊕ CK −→ +gx[m]n defined by (2.11) to G +x[m]n give two monomorphisms of Lie superalgebras with +∗-structures ι : (G +x[m]n, ω) −→ (gx[m]n, ω) and ι′ : (G +x[m]n, ω′) −→ (gx[m]n, ω′). +3.2 +Unitarizable G +a[m]n-modules +Observe that the 2-cocycle (2.1) is zero when it restricts to G +a[m]n. The natural inclusion +allows us to identify G +a[m]n as a subalgebra of ga[m]n, and hence the restriction of ω on +ga[m]n to G +a[m]n is a ∗-structure on G +a[m]n, which we also denote by ω. More precisely, +ω(Ei,j) = Ej,i +for +i, j ∈ J ++ +m(n). +Recall that P(m|n) denotes the set of (m|n)-hook partitions. For λ ∈ P(m|n), we define +λ = +m +� +i=1 +λiǫi + +n +� +j=1 +⟨λ′ +j − m⟩ǫj− 1 +2 ∈ h +a[m]n. +(3.1) +Let Q +a,I(m|n) denote the set of weights of the forms (3.1). The following proposition is +well known. (See, for example [CLZ, Theorems 3.2 and 3.3] with p = q = 0.) +Proposition 3.2. For λ ∈ Q +a,I(m|n), L(G +a[m]n, λ) is a unitarizable G +a[m]n-module with +respect to the ∗-structure ω. +Remark 3.3. The modules appearing in the proposition above are exactly the irreducible +highest weight polynomial modules over G +a[m]n (see, for example, [CW, Proposition 3.26]). + +16 +Cheong and Lam +Recall λ[m] and λ[m] defined in (2.13) and (2.14), respectively. Let +Qa(m|n) := {λ[m] ∈ ha[m]∗ +n | λ′ ∈ P(m|n), d = 0}, +Q +a(m|n) := {λ[m] ∈ h +a[m]∗ +n | λ ∈ P(m|n), d = 0}. +These sets will be used in Section 5. +3.3 +Unitarizable modules over G +c[m]n and G +d[m]n +In this subsection, we will restrict our attention to x = c, d. +There are two types of +unitarizable highest weight modules over G +x[m]n corresponding to the ∗-structures ω and +ω′ defined above. +Note that the Lie superalgebra Cf (resp., Df) defined in [LZ1] is our �Gc (resp., �Gd) +while �Cf (resp., �Df) is our ˜gc (resp., ˜gd). Also, the set of the unitarizable quasi-finite irre- +ducible highest weight modules over �C (resp., �D) described in [Proposition 5.8][LZ1](resp., +[Proposition 5.9][LZ1]) are the set of unitarizable irreducible highest weight modules over +�Cf (resp., �Df). Recall that �λ is defined in (2.12). Let +�Qc := +¶ �λ ∈ �hc ∗ ��� λ1 ≤ d, λ ∈ P, d ∈ Z+ +© +, +�Qd := +� +�λ ∈ �hd ∗ ��� λ1 + λ2 ≤ 2d, λ ∈ P, d ∈ 1 +2Z+ +� +. +Reformulating the results in [LZ1] in terms of our notations, we have the following propo- +sition. +Proposition 3.4. +(i) An irreducible highest weight ˜gc-module M is unitarizable with +respect to ω if and only if M ∼= L(˜gc, ξ) for some ξ ∈ �Qc. +(ii) An irreducible highest weight ˜gd-module M is unitarizable with respect to ω if and +only if M ∼= L(˜gd, ξ) for some ξ ∈ �Qd. +Recall that λ[m] and λ[m] are defined in (2.13) and (2.14), respectively. Let +Qc(m|n) := +� +λ[m] ∈ hc[m]∗ +n +�� λ1 ≤ d, λ′ ∈ P(m|n), d ∈ Z+ +� +, +Qd(m|n) := +� +λ[m] ∈ hd[m]∗ +n +��� λ1 + λ2 ≤ 2d, λ′ ∈ P(m|n), d ∈ 1 +2Z+ +� +, +Q +c(m|n) := +� +λ[m] ∈ h +c[m]∗ +n +��� λ1 ≤ d, λ ∈ P(m|n), d ∈ Z+ +� +, +Q +d(m|n) := +� +λ[m] ∈ h +d[m]∗ +n +��� λ1 + λ2 ≤ 2d, λ ∈ P(m|n), d ∈ 1 +2Z+ +� +. +The proof of the following is straightforward. +Lemma 3.5. Let g be a Lie superalgebra with ∗-structure σ. Assume that u is a subalgebra +of g such that the restriction σ|u of σ to u is a ∗-structure on u. Let V be a unitarizable +g-module with respect to σ. If W is a u-submodule of V , then W is a unitarizable u-module +with respect to σ|u. + +Gaudin Hamiltonians on unitarizable modules +17 +The following proposition is a direct consequence of Proposition 3.4 and Lemma 3.5. +Proposition 3.6. +(i) For ξ ∈ Qc(m|n), L(gc[m]n, ξ) is a unitarizable gc[m]n-module +with respect to ω. +(ii) For ξ ∈ Qd(m|n), L(gd[m]n, ξ) is a unitarizable gd[m]n-module with respect to ω. +(iii) For ξ ∈ Q +c(m|n), L(gc[m]n, ξ) is a unitarizable gc[m]n-module with respect to ω. +(iv) For ξ ∈ Q +d(m|n), L(gd[m]n, ξ) is a unitarizable gd[m]n-module with respect to ω. +Proof. To show (iii), let λ ∈ P(m|n) and d ∈ Z+ be such that λ1 ≤ d. Then λ[m] ∈ +Q +c(m|n) and L(gc[m]n, λ[m]) = tr∞ +n (T [m](L(�gc, �λ))). By Lemma 3.5, L(gc[m]n, λ[m]) is +a unitarizable module with respect to ω. The other parts can be proven by a similar +argument. +Definition 3.7. For x = c, d, a G +x[m]n-module M is said to be a unitarizable module of +type I (resp., II) if M is unitarizable with respect to the ∗-structure ω (resp., ω′). +Let +1m|n = +m +� +i=1 +ǫi − +n +� +j=1 +ǫj− 1 +2. +Let +Q +c,I(m|n) := +� +λ − d1m|n ∈ h +c[m]∗ +n +��� λ1 ≤ d, λ ∈ P(m|n), d ∈ Z+ +� +, +Q +d,I(m|n) := +� +λ − d1m|n ∈ h +d[m]∗ +n +��� λ1 + λ2 ≤ 2d, λ ∈ P(m|n), d ∈ 1 +2Z+ +� +, +Q +c,II(m|n) := +� +λ + d1m|n ∈ h +c[m]∗ +n +��� λ′ +1 + λ′ +2 ≤ 2d, λ ∈ P(m|n), d ∈ 1 +2Z+ +� +, +Q +d,II(m|n) := +� +λ + d1m|n ∈ h +d[m]∗ +n +��� λ′ +1 ≤ d, λ ∈ P(m|n), d ∈ Z+ +� +. +Proposition 3.8. +(i) For ξ ∈ Q +c,I(m|n), L(G +c[m]n, ξ) is a unitarizable G +c[m]n-module +of type I. +(ii) For ξ ∈ Q +c,II(m|n), L(G +c[m]n, ξ) is a unitarizable G +c[m]n-module of type II. +(iii) For ξ ∈ Q +d,I(m|n), L(G +d[m]n, ξ) is a unitarizable G +d[m]n-module of type I. +(iv) For ξ ∈ Q +d,II(m|n), L(G +d[m]n, ξ) is a unitarizable G +d[m]n-module of type II. +Proof. To show (i), let λ ∈ P(m|n) and d ∈ Z+ be such that λ1 ≤ d. Note that the G +c[m]n- +module structure, induced from the monomorphism of Lie superalgebras with ∗-structures +ι : (G +c[m]n, ω) −→ (gc[m]n, ω) given in Proposition 3.1, on L(gc[m]n, λ[m]) is isomorphic to +L(G +c[m]n, λ−d1m|n). By Proposition 3.6(iii), part (i) follows. To show (ii), let λ ∈ P(m|n) +and d ∈ 1 +2Z+ be such that λ′ +1 + λ′ +2 ≤ 2d. Note that ι′ : (G +c[m]n, ω′) −→ (gc[m]n, ω′) given +in Proposition 3.1 is a monomorphism of Lie superalgebras with ∗-structures, and we have +shown that the map �ϕ given in Lemma 2.3 is an isomorphism of Lie superalgebras with + +18 +Cheong and Lam +∗-structures from (gc[m]n, ω′) to (gd[m]n, ω). The G +c[m]n-module structure induced from +�ϕ ◦ ι′ on L(gd[m]n, λ′[m]) is isomorphic to L(G +c[m]n, λ + d1m|n). By Proposition 3.6(ii), +part (ii) follows. The other parts can be proven by a similar argument. +Remark 3.9. For m = 0 and n ∈ N, G +c[m]n and G +d[m]n are Lie algebras and the highest +weights in (ii) (resp., (iv)) of the proposition above are exactly the highest weights ap- +pearing in the classification of infinite-dimensional unitarizable irreducible highest weight +modules over G +c[m]n (resp., G +d[m]n) with integral (resp., half integral and integral) values +given in [EHW, Sections 8 and 9] (see also [HLT, Theorem 2.5]). +4 +Gaudin Hamiltonians on modules over �g, g[m]n and g[m]n +In this section, we define the Casimir symmetric tensors for the Lie (super)algebras �g, g[m]n +and g[m]n of infinite and finite ranks, and then introduce the (super) Gaudin Hamilto- +nians associated to these Lie (super)algebras. Our main goal is to show that the set of +eigenvectors of each Gaudin Hamiltonian for ˜g is in one-to-one correspondence with the set +of eigenvectors of the corresponding Gaudin Hamiltonian for g[m] (resp., g[m]). Besides, +each eigenvector and its corresponding eigenvector, under the one-to-one correspondence, +has the same eigenvalue. We also show that the eigenvectors of the Gaudin Hamiltonians +for g[m] (resp, g[m]) and those of g[m]n (resp, g[m]n), for n ∈ N, are related by truncation +functors. +First of all, we have the following lemma. It is analogous to [CaL, Lemma 3.1] and +can be proven similarly. +Lemma 4.1. Let 0 ≤ k < n ≤ ∞. +If v is a weight vector of weight µ in M ∈ �O +(resp., O[m]n and O[m]n) such that µ ∈ �Ξk (resp., Ξ[m]k and Ξ[m]k), then Eβv = 0 for +all β ∈ �Φ+\�Φ+ +k (resp., Φ[m]+ +n \Φ[m]+ +k and Φ[m]+ +n \Φ[m]+ +k ) and Eiv = 0 for i > k. In +particular, for each v ∈ M and M ∈ �O, O[m] or O[m], there are only finitely many Eβ +and Ei such that Eβv ̸= 0 and Eiv ̸= 0. +Let (·, ·) denote the bilinear form on gl(�V ) defined by +(A, B) := Str(AB) +for A, B ∈ gl(�V ). +It is non-degenerate invariant even supersymmetric. The restriction of the above bilinear +form to �G (resp., G[m]n and G[m]n) is also a non-degenerate invariant even supersymmetric +bilinear form. We denote +⟨·, ·⟩ := (·, ·) +on �Ga, Ga[m]n and G +a[m]n, +and +⟨·, ·⟩ := 1 +2(·, ·) +on other cases. +It is clear that ⟨Ei, Ei⟩ = (−1)2i for any i ∈ 1 +2N. Recall that �Φ+ (resp., Φ[m]+ +n and +Φ[m]+ +n ) denote the set of positive roots of ˜g (resp., g[m]n and g[m]n). For each root β in +�Φ+ (resp., Φ[m]+ +n and Φ[m]+ +n ), we choose root vectors Eβ and Eβ of weights β and −β, +respectively, such that +⟨Eβ, Eβ⟩ = 1. + +Gaudin Hamiltonians on unitarizable modules +19 +Clearly, ⟨Eβ, Eβ⟩ = (−1)|Eβ|, where |Eβ| is the parity of Eβ. +By identifying �G (resp., Gn and Gn) as a subspace of the vector space of ˜g (resp., gn +and gn), the Casimir symmetric tensors for ˜g, g[m]n and g[m]n are defined by (cf. [CaL, +Section 3.1]) +�Ω := +� +β∈�Φ+ +(Eβ ⊗ Eβ + (−1)|Eβ|Eβ ⊗ Eβ) ++ +� +j∈ 1 +2N +� +(−1)2jEj ⊗ Ej − (K ⊗ Ej + Ej ⊗ K) +� +, +Ω[m]n := +� +β∈Φ[m]+ +n +� +Eβ ⊗ Eβ + (−1)|Eβ|Eβ ⊗ Eβ) ++ +� +j∈J+ +m(n) +� +(−1)2jEj ⊗ Ej − (K ⊗ Ej + Ej ⊗ K) +� +, +Ω[m]n := +� +β∈Φ[m]+ +n +(Eβ ⊗ Eβ + (−1)|Eβ|Eβ ⊗ Eβ) ++ +� +j∈J ++ +m(n) +� +(−1)2jEj ⊗ Ej − (K ⊗ Ej + Ej ⊗ K) +� +. +Remark 4.2. For β ∈ Φ[m]+ +n , either β ∈ �Φ+ or −β ∈ �Φ+. Assume that −β ∈ �Φ+. Then +β is an odd root. We readily see that E−β = aEβ and E−β = −a−1Eβ for some nonzero +scalar a. It follows that E−β ⊗ E−β − E−β ⊗ E−β = Eβ ⊗ Eβ − Eβ ⊗ Eβ. In other words, +Ω[m]n is a partial sum of �Ω. Similarly, Ω[m]n is a partial sum of �Ω as well. +By Lemma 4.1, the Casimir symmetric tensors �Ω, Ω[m]n and Ω[m]n are well defined +operators on M ⊗ N, for M, N ∈ �O, O[m]n and O[m]n, respectively. +Fix ℓ ∈ N with ℓ ≥ 2. For M1, . . . , Mℓ ∈ �O, let +M := M1 ⊗ · · · ⊗ Mℓ. +Let us introduce some more notation. For x ∈ ˜g (resp., g[m]n and g[m]n) and i = 1, . . . , ℓ, +let +x(i) = 1 ⊗ · · · ⊗ 1⊗ +ix ⊗1 ⊗ · · · ⊗ 1 +� +�� +� +ℓ +. +For any operator A = � +r∈I xr ⊗ yr, where xr, yr ∈ �g (resp., g[m]n and g[m]n), and for +any distinct i, j ∈ {1, . . . , ℓ}, we define +A(ij) = +� +r∈I +x(i) +r y(j) +r . +(4.1) +Then �Ω(ij) can be viewed as a linear endomorphism on M. For any i = 1, . . . , ℓ and any +distinct complex numbers z1, . . . , zℓ, the Gaudin Hamiltonian �Hi is defined by +�Hi = +ℓ +� +j=1 +j̸=i +�Ω(ij) +zi − zj +. +(4.2) + +20 +Cheong and Lam +It is a linear endomorphism on M. The Gaudin Hamiltonians Hi[m]n and H +i[m]n are +defined by replacing �Ω with Ω[m]n and Ω[m]n, respectively. +Note that they are well- +defined linear endomorphisms on M1 ⊗ · · · ⊗ Mℓ for M1, . . . , Mℓ in O[m]n and in O[m]n, +respectively. +For any N ∈ �O (resp., O[m]n and O[m]n), let +N sing := {v ∈ N | Eβv = 0 for all β ∈ �Φ+ (resp., Φ[m]+ +n and Φ[m]+ +n )} +stand for the subspace spanned by singular vectors in N, and let N sing +µ +denote the subspace +spanned by singular vectors in the weight space Nµ for any weight µ of N. Note that Nµ +is finite dimensional by definition of �O (resp., O[m]n and O[m]n). +By an argument similar to the proof of [CaL, Propositions 3.5 and 3.7], one can show +that �Hi (resp., Hi[m]n and H +i[m]n) mutually commute with each other, and they are +˜g(resp., g[m]n and g[m]n)-homomorphisms for i = 1, . . . , ℓ. We immediately see that Msing +and the finite-dimensional subspace Msing +µ +are �Hi-invariant for any weight µ of M. Thus, +we may view �Hi as a linear endomorphism on Msing +µ +. Similarly, Hi[m]n (resp., H +i[m]n) +may be viewed as a linear endomorphism on (M1 ⊗ · · · ⊗ Mℓ)sing +µ +for M1, . . . , Mℓ ∈ O[m]n +(resp., O[m]n) and any weight µ of M1 ⊗ · · · ⊗ Mℓ . +Lemma 4.3. Let N1, N2 ∈ �O, and let v ∈ N1 ⊗ N2 be a weight vector of weight µ. +(i) If µ ∈ Ξ[m], then �Ω(v) = Ω[m](v). +(ii) If µ ∈ Ξ[m], then �Ω(v) = Ω[m](v). +Proof. The proof is similar to that of [CaL, Lemma 3.11] with a slight modification. For +completeness, we include it here. We will only prove (i). +The proof of (ii) is similar. +We may assume that v = v1 ⊗ v2, where vi ∈ Ni is a weight vector of µi for i = 1, 2, +and µ1 + µ2 = µ. +For i = 1, 2, µi ∈ Ξ[m] by Lemma 2.7. +For all k ∈ +1 +2N\J+ +m and +i = 1, 2, µi(Ek) = 0, and so Ek(vi) = 0. By virtue of Remark 4.2, it remains to consider +β ∈ �Φ+ with ±β /∈ Φ[m]+. For such β, we have β(Ei) ̸= 0 for some i ∈ 1 +2N\J+ +m. It +follows that either the weight of Eβv1 or Eβv2 does not lie in �Ξ. Thus either Eβv1 = 0 or +Eβv2 = 0, and hence Eβ ⊗ Eβ(v1 ⊗ v2) = 0. Similarly, Eβ ⊗ Eβ(v1 ⊗ v2) = 0. Therefore, +�Ω(v1 ⊗ v2) = Ω[m](v1 ⊗ v2). +As a consequence, we obtain the following lemma. +Lemma 4.4. Let M1, . . . , Mℓ ∈ �O, and let v ∈ M1 ⊗ · · · ⊗ Mℓ be a weight vector of weight +µ. +(i) If µ ∈ Ξ[m], then �Hiv = Hi[m](v) for all i = 1, · · · , ℓ. +(ii) If µ ∈ Ξ[m], then �Hiv = H +i[m](v) for all i = 1, · · · , ℓ. +We would like to ask whether the eigenvectors of Hi, Hi[m] and H +i[m] are related. To +answer the question, we need the following proposition. Recall the bijections �λ ↔ λ[m] ↔ +λ[m] in (2.15). + +Gaudin Hamiltonians on unitarizable modules +21 +Proposition 4.5. Let M ∈ �O, and let �µ ∈ �P+ be a weight of M. Then: +(i) There exists A ∈ U(˜l) such that the map t�µ +[m] : Msing +�µ +→ T[m](M)sing +µ[m], defined by +t�µ +[m](v) = Av for v ∈ Msing +�µ +, is a linear isomorphism. +(ii) There exists ¯A ∈ U(˜l) such that the map t�µ +[m] : Msing +�µ +→ T [m](M)sing +µ[m], defined by +t�µ +[m](v) = ¯Av for v ∈ Msing +µ +, is a linear isomorphism. +Proof. We will only prove (i). The proof of (ii) is similar. Note that there is a linear +isomorphism +Hom�O (∆(˜g, �µ), M) −→ Msing +�µ +ϕ �→ ϕ(v�µ) +where v�µ is a highest weight vector of ∆(˜g, �µ). +On the other hand, there exists A ∈ U(˜l) such that vµ[m] := Av�µ is a highest weight +vector of ∆(g[m], µ[m]) = T[m](∆(˜g, �µ)). In fact, A is a product of elements in ˜l corre- +sponding to a sequence of odd reflections (see [CL2, Section 3.1] and [CLW1, Section 4] +for details). Similarly, the map +HomO[m] +�∆(g[m], µ[m]), T[m](M)� −→ T[m](M)sing +µ[m] +φ �→ +φ(vµ[m]) +is a linear isomorphism. By Theorem 2.10, we have +Hom�O (∆(˜g, �µ), M) ∼= HomO[m] +� +∆(g[m], µ[m]), T[m](M) +� +, +and hence Msing +µ +∼= T[m](M)sing +µ[m]. We may also see that any vector v ∈ Msing +µ +corresponds +to Av ∈ T[m](M)sing +µ[m] under the isomorphism, which shows that the isomorphism is indeed +the map t�µ +[m] as stated. +Remark 4.6. +(i) The elements A and ¯A in Proposition 4.5 depend only on the weight �µ, +but not on the module M. +(ii) There exist B, ¯B ∈ U(˜l) such that the inverses of t�µ +[m] and t�µ +[m] are given respectively +by (t�µ +[m])−1(v) = Bv and (t�µ +[m])−1(w) = ¯Bw for any v ∈ T[m](M)sing +µ[m] and w ∈ +T [m](M)sing +µ[m]. Again B and B are products of elements in˜l corresponding to sequences +of odd reflections. +Theorem 4.7. For M1, . . . , Mℓ ∈ �O, let M = M1 ⊗ · · · ⊗ Mℓ. Suppose that v ∈ Msing +�µ +with �µ ∈ �P+. For any m ∈ Z+, let vm = t�µ +[m](v) and vm = t�µ +[m](v). For each i = 1, . . . , ℓ, +we have: + +22 +Cheong and Lam +(i) v is an eigenvector of �Hi with eigenvalue c if and only if vm is an eigenvector of +Hi[m] with eigenvalue c. +Moreover, �Hi is diagonalizable on Msing +�µ +if and only if Hi[m] is diagonalizable on +T[m](M)sing +µ[m]. In this case, they have the same spectrum. +(ii) v is an eigenvector of �Hi with eigenvalue c if and only if vm is an eigenvector of +H +i[m] with eigenvalue c. +Moreover, �Hi is diagonalizable on Msing +�µ +if and only if H +i[m] is diagonalizable on +T [m](M)sing +µ[m]. In this case, they have the same spectrum. +Proof. We will only prove (i). The proof of (ii) is similar. We know that vm = Av for +some A ∈ U(˜l). Suppose �Hiv = cv for some c ∈ C. By Lemma 4.4 together with the fact +that Av is a vector of weight µ[m] ∈ Ξ[m], we have +Hi[m](Av) = �Hi(Av). +As �Hi is a �g-homomorphism, it follows that +Hi[m](vm) = A �Hi(v) = A(cv) = cvm. +Conversely, suppose Hi[m](vm) = cvm for some c ∈ C. By �Hi being a �g-homomorphism +and Lemma 4.4 again, we have +A �Hiv = �Hi(Av) = Hi[m](Av) = cvm. +By Remark 4.6, A has an inverse, and we deduce that �Hi(v) = cv. This proves the first +part of (i). The second part is a direct consequence of the first part. +Proposition 4.8. +(i) For M1, . . . , Mℓ ∈ O[m], let M = M1 ⊗ · · · ⊗ Mℓ. Suppose that +v ∈ Msing +µ +with µ ∈ P[m]+ +n . Then for each i = 1, . . . , ℓ, v is an eigenvector of Hi[m] +with eigenvalue c if and only if v is an eigenvector of Hi[m]n with eigenvalue c. +Moreover, Hi[m] is diagonalizable on Msing +µ +if and only if Hi[m]n is diagonalizable +on tr∞ +n (M)sing +µ +. In this case, they have the same spectrum. +(ii) For M1, . . . , Mℓ ∈ O[m], let M = M1 ⊗ · · · ⊗ Mℓ. Suppose that v ∈ Msing +µ +with +µ ∈ P[m]+ +n . For each i = 1, . . . , ℓ, v is an eigenvector of H +i[m] with eigenvalue c if +and only if v is an eigenvector of H +i[m]n with eigenvalue c. +Moreover, H +i[m] is diagonalizable on Msing +µ +if and only if H +i[m]n is diagonalizable +on tr∞ +n (M)sing +µ +. In this case, they have the same spectrum. +Proof. We will only prove (i). The proof of (ii) is similar. Note that tr∞ +n (M)sing +µ += Msing +µ +for µ ∈ P[m]+ +n . By Lemma 4.1, we have +Hi[m](w) = Hi[m]n(w), +for all w ∈ Msing +µ +. +The first part of (i) follows. The second part of (i) clearly follows from the first part. + +Gaudin Hamiltonians on unitarizable modules +23 +5 +Gaudin Hamiltonians on modules over G[m]n +In this section, we consider Gaudin Hamiltonians for finite-dimensional Lie (super)algebras. +We relate the Gaudin Hamiltonians for G[m]n (resp, G[m]n) to those for g[m]n (resp, +g[m]n) for n ∈ N. Furthermore, we study the Gaudin Hamiltonians on the tensor product +of unitarizable irreducile highest weight modules and give an affirmative answer to the +diagonalization of these operators. +Let us fix ℓ ∈ N with ℓ ≥ 2. For n ∈ N, the Casimir symmetric tensors for G[m]n and +G[m]n are defined by (cf. [CaL, Section 3.4]) +˚Ω[m]n = +� +β∈Φ[m]+ +n +� +Eβ ⊗ Eβ + (−1)|Eβ|Eβ ⊗ Eβ) + +� +j∈J+ +m(n) +(−1)2jEj ⊗ Ej, +˚Ω[m]n = +� +β∈Φ[m]+ +n +(Eβ ⊗ Eβ + (−1)|Eβ|Eβ ⊗ Eβ) + +� +j∈J ++ +m(n) +(−1)2jEj ⊗ Ej. +Clearly, ˚Ω[m]n and ˚Ω[m]n lie in U(G[m]n) ⊗ U(G[m]n) and U(G[m]n) ⊗ U(G[m]n), respec- +tively. +For any i = 1, . . . , ℓ and any distinct complex numbers z1, . . . , zn, the Gaudin Hamil- +tonians Hi[m]n and Hi[m]n are defined by +Hi[m]n = +ℓ +� +j=1 +j̸=i +˚Ω[m]n +(ij) +zi − zj +and +Hi[m]n = +ℓ +� +j=1 +j̸=i +˚Ω[m]n +(ij) +zi − zj +. +(5.1) +From now on, we fix ξ1, . . . , ξℓ ∈ h[m]∗ +n (resp., ξ1, . . . , ξℓ ∈ h[m]∗ +n), put ξ := (ξ1, . . . , ξℓ) +(resp., ξ := (ξ1, . . . , ξℓ)). We define +L(G[m]n, ξ) := L(G[m]n, ξ1) ⊗ · · · ⊗ L(G[m]n, ξℓ) +and +L(G[m]n, ξ) := L(G[m]n, ξ1) ⊗ · · · ⊗ L(G[m]n, ξℓ). +Setting K = 0 to the cases involving central extensions, the Gaudin Hamiltonians +Hi[m]n on L(G[m]n, ξ) (resp., Hi[m]n on L(G[m]n, ξ)) mutually commute with each other, +and they are G[m]n(resp., G[m]n)-homomorphisms. It is also evident that for any weight µ +of L(G[m]n, ξ) (resp., L(G[m]n, ξ)), the subspace L(G[m]n, ξ)sing +µ +(resp., L(G[m]n, ξ)sing +µ +)) is +Hi[m]n-invariant (resp., Hi[m]n-invariant). Here and below, N sing stands for the subspace +spanned by singular vectors in N with respect to the standard Borel subalgebra and N sing +µ +for the subspace spanned by singular vectors in the weight space Nµ for any G[m]n(resp., +G[m]n)-module N and any weight µ of N. +Each gx[m]n(resp., gx[m]n)-module can be regarded as a Gx[m]n(resp., G +x[m]n)-module +through the homomorphism ι defined by (2.11). It is clear that the set of singular vectors +in a module M regarded as a gx[m]n(resp., gx[m]n)-module equals the set of singular vectors +in the module M regarded as a Gx[m]n(resp., G +x[m]n)-module. + +24 +Cheong and Lam +Proposition 5.1. +(i) For n ∈ N and i = 1, . . . , ℓ, let Li be an irreducible highest weight +gx[m]n-module with highest weight in Qx(m|n) and let L = L1 ⊗ · · · ⊗ Lℓ. Suppose +that µ = � +j∈J+ +m(n) ajǫj + dΛ0 is a weight of L. Let ˆµ = � +j∈J+ +m(n) ajǫj + d1m|n. +Then for each i = 1, . . . , ℓ, Hi[m]n is diagonalizable on Lsing +µ +if and only if Hi[m]n is +diagonalizable on Lsing +ˆµ +. Moreover, Hi[m]n has simple spectrum on the space spanned +by singular vectors in a finite direct sum of weight spaces of L if and only if Hi[m]n +has simple spectrum on the space spanned by singular vectors in the finite direct sum +of the corresponding weight spaces. +(ii) For n ∈ N and i = 1, . . . , ℓ, let Li be an irreducible highest weight gx[m]n-module +with highest weight in Q +x(m|n), and let L = L1 ⊗ · · · ⊗ Lℓ. +Suppose that µ = +� +j∈J ++ +m(n) ajǫj + dΛ0 is a weight of L. Let ˆµ = � +j∈J ++ +m(n) ajǫj − d1m|n. Then for each +i = 1, . . . , ℓ, H +i[m]n is diagonalizable on Lsing +µ +if and only if Hi[m]n is diagonalizable +on Lsing +ˆµ +. Moreover, H +i[m]n has simple spectrum on the space spanned by singular +vectors in a finite direct sum of weight spaces of L if and only if Hi[m]n has simple +spectrum on the space spanned by singular vectors in the finite direct sum of the +corresponding weight spaces. +Proof. Note that ι ⊗ ι�˚Ω[m]n +� = Ω[m]n − (m − n)K ⊗ K. Therefore we have +Hi[m]n(v) = Hi[m]n(v) − (m − n) +ℓ +� +j=1 +j̸=i +K(i)K(j) +zi − zj +(v). +The last term on the right hand side is a fixed scalar times v. This implies (i). The proof +of (ii) is similar. +The following theorem follows from the Main Corollary of Rybnikov [Ry]. +Theorem 5.2. Let n ∈ N and ξ1, . . . , ξℓ ∈ h[0]∗ +n dominant integral weights for Gx[0]n. For +generic z1, . . . , zℓ, the Gaudin Hamiltonian Hi[0]n is diagonalizable with simple spectrum +on the space L(Gx[0]n, ξ)sing for all i = 1, . . . , ℓ. +Remark 5.3. The results in [Ry] involve only simple Lie algebras. +The Lie algebra +Ga[0]n ∼= gl(n) is, however, not semisimple. +The case where x = a is still true since +the irreducible highest weight modules over gl(n) coincides with those over sl(n), and +each Gaudin Hamiltonian for gl(n) minus that for sl(n) is a scalar multiple of the identity +operator on the space L(Ga[0]n, ξ)sing. +Corollary 5.4. Let ξ1, . . . , ξℓ ∈ Qx(0|∞). For each i = 1, . . . , ℓ and generic z1, . . . , zℓ, the +Gaudin Hamiltonian Hi[0] is diagonalizable with simple spectrum on the space spanned by +singular vectors in any finite direct sum of weight spaces of L(gx[0], ξ1) ⊗ · · · ⊗ L(gx[0], ξℓ). +Proof. Note that tr∞ +n (L(gx[0], ξi), regarded as a Gx[0]n-module through the homomorphism +ι defined by (2.11), is an irreducible module with dominant integral highest weight. Given +a finite number of weight spaces of L(gx[0], ξi), we can choose n large enough such that +tr∞ +n (L(gx[0], ξi) contains the given weight spaces. Now the corollary follows from Proposi- +tion 4.8, Proposition 5.1 and Theorem 5.2. + +Gaudin Hamiltonians on unitarizable modules +25 +Theorem 5.5. Let n ∈ N and ξ1, . . . , ξℓ ∈ Q +x,I(m|n). For each i = 1, . . . , ℓ and generic +z1, . . . , zℓ, the Gaudin Hamiltonian Hi[m]n is diagonalizable with simple spectrum on the +space spanned by singular vectors in any finite direct sum of weight spaces of L(G +x[m]n, ξ). +In particular, for x = a, i = 1, . . . , ℓ and generic z1, . . . , zℓ, the Gaudin Hamiltonian +Hi[m]n is diagonalizable with simple spectrum on L(G +a[m]n, ξ)sing. +Proof. The theorem follows from Proposition 5.1(ii), Proposition 4.8(ii), Theorem 4.7(ii) +and the case where m = 0 of Theorem 4.7(i), and Corollary 5.4 together with an explicit +description of the highest weights involved. The second part follows from the fact that +L(G +a[m]n, ξ) is finite-dimensional. +Remark 5.6. +(i) For x = a and each L(G +a[m]n, ξi) being the natural module Cm|n, the +corresponding result in Theorem 5.5 has been obtained by Mukhin, Vicedo and Young +(cf. [MVY]). +(ii) We have G +c[0]n ∼= so(2n) and G +d[0]n ∼= sp(2n). The weights ξ1, . . . , ξℓ in Theorem 5.5 +are highest weights of infinite-dimensional unitarizable irreducible highest weight +modules (see Remark 3.9). +(iii) Theorem 5.5 is also valid for the ortho-symplectic Lie superalgebra osp(2m + 1|2n) +if ξ1, . . . , ξℓ are the highest weights such that for each i = 1, . . . , ℓ, ξi = λi − di +1m|n +for some λi ∈ P(m|n) and di ∈ Z+ satisfying ℓ(λi) ≤ di. The proof is identical and +is omitted here. +Theorem 5.7. Let x = c, d, n ∈ N and ξ1, . . . , ξℓ ∈ Q +x,II(m|n). For each i = 1, . . . , ℓ +and generic z1, . . . , zℓ, the Gaudin Hamiltonian Hi[m]n is diagonalizable with simple spec- +trum on the space spanned by singular vectors in any finite direct sum of weight spaces of +L(G +x[m]n, ξ). +Proof. Each G +c[m]n(resp., G +d[m]n)-module L(G +c[m]n, ξi) (resp., L(G +d[m]n, ξi)) can be re- +garded as a Gd[m]n(resp., Gc[m]n)-module through the isomorphism �ϕ defined by (2.10). +Now the theorem follows from Proposition 5.1(i), Proposition 4.8(i), Theorem 4.7(i) and +its special case where m = 0, and Corollary 5.4 together with an explicit description of +the highest weights involved. +We anticipate that the results of this paper may provide an approach of constructing +common eigenvectors of the Gaudin Hamiltonians associated to the Lie (super)algebra +G +x[m]n from common eigenvectors of the Gaudin Hamiltonians associated to the Lie algebra +Gx[0]k for some k ∈ N. Let us explain in more detail. For each i = 1, . . . , ℓ, consider +the Gaudin Hamiltonian Hi[m]n on the weight space L(G +x[m]n, ξ)sing +µ +for each weight µ. +Suppose that the weights ξ1, . . . , ξℓ all lie in Q +x,I(m|n) or Q +x,II(m|n). +The arguments +in Sections 4 and 5 allow us to construct an eigenvector of the linear operator from an +eigenvector of the Gaudin Hamiltonian for the Lie algebra Gx[0]k, for some k ∈ N, on the +space spanned by singular vectors in L1 ⊗ · · · ⊗ Lℓ, where Li is some finite-dimensional +irreducible Gx[0]k-module for each i = 1, . . . , ℓ. An explicit construction would be to apply +a sequence of certain odd reflections to the Bethe vectors of Hi[0]k, which are constructed + +26 +Cheong and Lam +by Bethe ansatz method. +These odd reflections depend on the weight µ and can be +determined explicitly. However, one can expect that the resulting eigenvectors might take +complicated forms. It would be interesting to know whether this procedure is directly +related to any other known method of constructing eigenvectors. +Acknowledgments. The first author was partially supported by Ministry of Science and +Technology grant 110-2115-M-006-006 of Taiwan. The second author was partially sup- +ported by Ministry of Science and Technology grant 109-2115-M-006-019-MY3 of Taiwan. +References +[BF] +H. M. Babujian and R. Flume, Off-shell Bethe ansatz equation for Gaudin mag- +nets and solutions of Knizhnik-Zamolodchikov equations, Modern Phys. Lett. A +9 (1994), no. 22, 2029–2039. +[CaL] +B. Cao and N. Lam, Solutions of super Knizhnik-Zamolodchikov equations, Lett. +Math. Phys. 110 (2020), no. 7, 1799–1834. +[CL1] +S.-J. Cheng and N. Lam, Infinite-dimensional Lie superalgebras and Hook Schur +functions, Commun. Math. Phys. 238 (2003), 95–118. +[CL2] +S.-J. Cheng and N. Lam, Irreducible characters of general linear superalgebra +and super duality, Commun. Math. Phys. 280 (2010), 645–672. +[CLW1] +S.-J. Cheng, N. Lam and W. Wang, Super duality and irreducible characters of +ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), 189–224. +[CLW2] +S.-J. Cheng, N. Lam and W. Wang, +Super duality for general linear Lie su- +peralgebras and applications in Recent Developments in Lie Algebras, Groups +and Representation Theory, Proc. Sympos. Pure Math. 86, Amer. Math. Soc., +Providence, 2012, 113–136. +[CLZ] +S.-J. Cheng, N. Lam and R. Zhang, Character formula for infinite-dimensional +unitarizable modules of the general linear superalgebra, J. Algebra 273 (2004), +no. 2, 780–805. +[CW] +S.-J. Cheng and W. Wang, Dualities and representations of Lie superalgebras, +Graduate Studies in Mathematics 144, American Mathematical Society, 2012. +[EHW] +T. J. Enright, R. Howe and N. R. Wallach, A classification of unitary highest +weight modules, in Representation theory of reductive group, Birkh¨auser, Boston, +1983, 97–143. +[EFK] +P. I. Etingof, I. B. Frenkel and A. A. Kirillov, Jr., Lectures on representation +theory and Knizhnik-Zamolodchikov equations, Mathematical surveys and mono- +graphs 58, American Mathematical Society, 1998. +[FFR] +B. Feigin, E. Frenkel and N. Reshetikhin, Gaudin model, Bethe ansatz and crit- +ical level, Comm. Math. Phys. 166 (1994), no. 1, 27–62. +[FFRy] +B. Feigin, E. Frenkel and L. Rybnikov, Opers with irregular singularity and +spectra of the shift of argument subalgebra, Duke Math. J. 155 (2010), no. 2, +337–363. + +Gaudin Hamiltonians on unitarizable modules +27 +[G1] +M. Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin, J. Physique +37 (1976), no. 10, 1089–1098. +[G2] +M. Gaudin, La fonction d’onde de Bethe, Collection du Commissariat a‘ +l’E’nergie Atomique: Se’rie Scientifique, Masson, Paris, 1983. +[HMVY] C. Huang, E. Mukhin, B. Vicedo and C. Young, The solutions of glM|N Bethe +ansatz equation and rational pseudodifferential operators, Selecta Math. (N.S.) +25 (2019), no. 4, Paper No. 52, 34 pp. +[HLT] +P.-Y. Huang, N. Lam and T.-M. To, Super duality and homology of unitarizable +modules of Lie algebras, Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, 45–63. +[K] +V. G. Kac, Lie Superalgebras, Adv. Math. 16 (1977), 8–96. +[KZ] +V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino +model in two dimensions., Nucl. Phys. B 247 (1984), 83–103. +[KuM] +P. P. Kulish and N. Manojlovi´c, Bethe vectors of the osp(1|2) Gaudin model, +Lett. Math. Phys. 55 (2001), no. 1, 77–95. +[LZ1] +N. Lam and R. Zhang, Quasi-finite modules for Lie superalgebras of infinite rank, +Trans. Amer. Math. Soc. 358 (2006), no. 1, 403–439. +[LZ2] +N. Lam and R. Zhang, u-cohomology formula for unitarizable modules over gen- +eral linear superalgebras, J. Algebra 327 (2011), 50–70. +[L] +K. Lu, Completeness of Bethe ansatz for Gaudin models associated with gl(1|1), +Nuclear Phys. B 980 (2022), Paper No. 115790, 23 pp. +[LM] +K. Lu and E. Mukhin, Bethe Ansatz equations for orthosymplectic Lie superalge- +bras and self-dual superspaces, Ann. Henri Poincar´e 22 (2021), no. 12, 4087–4130. +[LMV] +K. Lu, E. Mukhin and A. Varchenko, On the Gaudin model associated to Lie +algebras of classical types, J. Math. Phys. 57 (2016), no. 10, 101703, 23 pp. +[Lus] +G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, +Birkh¨auser, Boston, 1993. +[MTV1] +E. Mukhin, V. Tarasov and A. Varchenko, Schubert calculus and representations +of the general linear group, J. Amer. Math. Soc. 22 (2009), no. 4, 909–940. +[MTV2] +E. Mukhin, V. Tarasov and A. Varchenko, On separation of variables and com- +pleteness of the Bethe ansatz for quantum glN Gaudin model, Glasg. Math. J. +51 (2009), no. A, 137–145. +[MV] +E. Mukhin and A. Varchenko, Norm of a Bethe vector and the Hessian of the +master function, Compos. Math. 141 (2005), no. 4, 1012–1028. +[MVY] +E. Mukhin, B. Vicedo and C. Young, Gaudin models for gl(m|n), J. Math. 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IMRN (2020), no. 22, 8766–8785. +Department of Mathematics, National Cheng Kung University, Tainan 701401, Taiwan +E-mail address: keng@ncku.edu.tw +Department of Mathematics, National Cheng Kung University, Tainan 701401, Taiwan +E-mail address: nlam@ncku.edu.tw + diff --git a/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/load_file.txt b/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe252b46edd6f08b1fa15feef761b7037a978bc7 --- /dev/null +++ b/J9E1T4oBgHgl3EQfYgQR/content/tmp_files/load_file.txt @@ -0,0 +1,1401 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf,len=1400 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='03138v1 [math-ph] 9 Jan 2023 Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras Wan Keng Cheong and Ngau Lam Abstract Let L be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We show that the Gaudin Hamiltonians associated to G are diagonalizable with simple spectrum on the space spanned by singular vectors of any fixed weight in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the space spanned by singular vectors of any fixed weight in the tensor product of infinite- dimensional unitarizable irreducible highest weight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' 1 Introduction The Gaudin model was introduced by Gaudin [G1, G2] to describe a completely integrable quantum spin chain associated to any finite-dimensional simple Lie algebra G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let (·, ·) be a non-degenerate invariant bilinear form on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let {Ia | a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , d} be a basis for G and {Ia | a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , d} the dual basis with respect to the bilinear form (·, ·), where d is the dimension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The Casimir symmetric tensor Ω for G is defined to be Ω = �d a=1 Ia ⊗ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Fix ℓ ∈ N with ℓ ≥ 2, and let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , zℓ be distinct complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' For any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , ℓ, the Gaudin Hamiltonian Hi of the Gaudin model associated to G is defined to be Hi = ℓ � j=1 j̸=i Ω(ij) zi − zj , where Ω(ij) is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The Gaudin Hamiltonians H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , Hℓ act on the tensor product M1 ⊗ · · · ⊗ Mℓ, where each Mi is a G-module, and they are mutually commuting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' To find common eigenvectors for Gaudin Hamiltonians is one of the main problems of studying the Gaudin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Bethe ansatz method provides an explicit construction of common eigenvectors from the solutions of the so-called Bethe ansatz equations and proves to be effective for the special linear Lie algebra sl(2, C) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' [G1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The eigenvectors obtained by this method are called Bethe vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Babujian and Flume [BF] generalized the Bethe ansatz equations to the case of any simple Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In [FFR], Feigin, Frenkel and Reshetikhin proposed a new approach based on Wakimoto modules over the affine Lie algebra at the critical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' They obtained the Bethe vectors by restricting certain invariant functionals on tensor products of Wakimoto modules and found as well that the 1 2 Cheong and Lam Bethe vectors are in close connection with the solutions of the Knizhnik–Zamolodchikov equations [KZ] (see also [EFK]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In this paper, we are interested in the super version of Gaudin Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' For precise definitions of such Gaudin Hamiltonians, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We find an interesting connection between the Gaudin Hamiltonians associated to the (trivial) central extension of any infinite-rank classical Lie (super)algebra of type a, c, d and the Gaudin Hamiltonians associated to the (trivial) central extension of the corresponding Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Our main tool is super duality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' [CL2, CLW1, CLW2]), which asserts that there are equivalences of tensor categories between the parabolic BGG categories �O of ˜g-modules, O[m] of g[m]- modules and O[m] of g[m]-modules (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The following diagrams summarize the relationships among these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' �O O[m] O[0] T[m] T[0] �O O[m] O[0] T [m] T [0] Here the tensor functors T[m], T [m], T[0], and T [0], defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='6, are equivalences of tensor categories for m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Notice that ˜g, g[m] and g[m], for m ∈ N, are Lie superalgebras while g[0] and g[0] are Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In other words, super duality gives equivalences of categories between parabolic BGG categories for Lie superalgebras and the corresponding Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We view the Gaudin Hamiltonians for ˜g (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', g[m] and g[m]) as linear operators on the tensor product of modules over ˜g (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', g[m] and g[m]) in the category �O (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', O[m] and O[m]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Instead of constructing eigenvectors for these operators directly, we apply super duality and the work [CaL] to establish one-to-one correspondences relating the sets of eigenvectors of the Gaudin Hamiltonians for ˜g, g[m] and g[m] for m ∈ Z+ (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Furthermore, by using the notion of truncation functors (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='18)), we relate the eigenvectors of Gaudin Hamiltonians for g[m] and g[m] to the eigenvectors of Gaudin Hamiltonians for their finite-rank counterparts g[m]n and g[m]n, for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let G be a finite-dimensional simple Lie algebra and Li a finite-dimensional irreducible module over G for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Rybnikov shows that for generic z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , zℓ, the Gaudin Hamiltonians are diagonalizable with simple spectrum on the space spanned by singular vectors in L1 ⊗ · · · ⊗ Lℓ (see [MV, MTV1, MTV2, FFRy, LMV] as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We would like to extend Rybnikov’s result to the case of Lie (super)algebras in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We focus our attention on the finite-dimensional classical Lie (super)algebra G x[m]n of type x, where x denotes a fixed type among a, c, d (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Note that G a[m]n ∼= gl(m|n), G c[m]n ∼= spo(2m|2n) and G d[m]n ∼= osp(2m|2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let L = L1 ⊗ · · · ⊗ Lℓ, where each Li is an irreducible highest weight module over G x[m]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Suppose that for x = a, each Li is a polynomial module, and for x = c, d, all highest weights of Li lie in Q x,I(m|n) or Q x,II(m|n) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We show that the Gaudin Hamiltonian Hi[m]n associated to G x[m]n, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , ℓ, is diagonalizable with simple spectrum on the space spanned by singular vectors in any finite direct sum of weight spaces of L (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='5 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We will see later that such Li’s are unitarizable modules Gaudin Hamiltonians on unitarizable modules 3 over G x[m]n with respect to a natural choice of ∗-structures on G x[m]n (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let us explain a little about how we obtain the diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' For x = c, d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' x = a), we relate each Hi[m]n to its corresponding Gaudin Hamiltonian on the tensor product of some finite-dimensional irreducible modules over G, where G is the orthogonal Lie algebra so(2p) or the symplectic Lie algebra sp(2p) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', G is the general linear Lie algebra gl(p)) for some p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' These correspondences should give a procedure of constructing eigenvectors of the Gaudin Hamiltonians for classical Lie (super)algebras from the ones for the corresponding Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' While there is a great deal of work on Gaudin models, not much appears to be known about the super Gaudin Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In [MVY], Mukhin, Vicedo and Young investigate the Gaudin Hamiltonians associated to the general linear Lie superalgebra gl(m|n) and show that the operators are diagonalizable with simple spectrum on the space spanned by singular vectors in the tensor product of the natural module Cm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In [KuM], Kulish and Manojlovi´c explicitly construct eigenvectors for the Gaudin Hamiltonians associated to the ortho-symplectic Lie superalgebra osp(1|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The reader may also want to consult [HMVY, L, LM] for related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We hope that this paper can provide new insights into the study of Gaudin models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In Section 2, we fix notation and review some back- ground materials on classical Lie (super)algebras and their central extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We also de- fine the parabolic BGG categories �O, O[m] and O[m] associated to the Lie (super)algebras ˜g, g[m] and g[m], respectively, and discuss super duality which gives equivalences of these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In Section 3, we give a brief introduction to ∗-structures and study the unita- rizable modules which will be considered in our study of Gaudin Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In Section 4, we investigate Gaudin Hamiltonians associated to the central extensions of finite-rank and infinite-rank Lie (super)algebras, and establish the one-to-one correspondences be- tween the sets of eigenvectors (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='7 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In Section 5, we concentrate on Gaudin Hamiltonians for finite-dimensional classical Lie (super)algebras and prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='5 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='7, which give an affirmative answer to the diago- nalization of the operators on unitarizable modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Throughout the paper, N stands for the set of positive integers, Z for the set of integers, Z∗ for the set of nonzero integers, Z+ for the set of non-negative integers, 1 2Z for the set of half integers and integers, and C for the set of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' All vector spaces, algebras, tensor products, et cetera, are over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' 2 Preliminaries In this section, we first define the finite-rank and infinite-rank Lie (super)algebras �Gx, Gx[m]n and G x[m]n, where x denotes one of the three types a, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We consider their central extensions �gx, gx[m]n and gx[m]n and the parabolic BGG categories �Ox (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', Ox[m]n and O x[m]n) of modules over �gx (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=', gx[m]n and gx[m]n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We then recall the truncation functors which relate Ox[m]∞ and O x[m]∞ to Ox[m]n and O x[m]n, respectively, for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Finally, we describe the tensor functors T[m] and T [m] and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We refer the 4 Cheong and Lam readers to [CL2, Sections 2 and 3] for type a and [CLW1, Sections 2 and 3] for types c, d for details (see also [CW, Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='2] and [CaL, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We fix m ∈ Z+ and n ∈ N ∪ {∞} throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let �V denote the superspace over C with ordered basis {vr | r ∈ 1 2Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The parity of vr is defined as follows: |vr| = ¯0 if r ∈ Z, and |vr| = ¯1 if r ∈ 1 2 + Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let gl(�V ) be the Lie superalgebra consisting of all linear endomorphisms on �V which vanish on all but finitely many vr’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' For i, j ∈ 1 2Z, we let Ei,j be the linear endomorphism on �V defined by Ei,j(vr) = δjrvi for r ∈ 1 2Z, where δ is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The Lie superalgebra gl(�V ) is spanned by Ei,j with i, j ∈ 1 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' The Lie superalgebra gl(�V ) has a central extension, denoted by �gl(�V ), by the one- dimensional center CK corresponding to the following 2-cocycle (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' [CL1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' 99]): τ(A, B) := Str([J, A]B), A, B ∈ gl(�V ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='1) where J = − � r≥ 1 2 Er,r and Str denotes the supertrace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' In fact, the cocycle τ is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Moreover, there is an isomorphism ι from the direct sum of Lie superalgebras gl(�V ) ⊕ CK to �gl(�V ) defined by ι(A) = A + Str(JA)K, for A ∈ gl(�V ), and ι(K) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content='2) Let Jm(n) = ß ±1 2, ±3 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , ±(m − 1 2) ™ ∪ {0} ∪ { ±j | j ∈ N, j < n + 1 }, Jm(n) = {±1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' , ±m} ∪ {0} ∪ � ±(j − 1 2) ��� j ∈ N, j < n + 1 � , �J(n) = � r ∈ 1 2Z ��� − n ≤ r ≤ n � , J× m(n) = Jm(n)\\{0}, J × m(n) = Jm(n)\\{0}, �J×(n) = �J(n))\\{0}, J+ m(n) = { r ∈ Jm(n) | r > 0 }, J + m(n) = � r ∈ Jm(n) �� r > 0 � , �J+(n) = ¶ r ∈ �J(n) ��� r > 0 © .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' We let �V (n), Vm(n), Vm(n), �V ×(n), V × m (n) and V × m (n) be the subspaces of �V with basis {vi} indexed by �J(n), Jm(n), Jm(n), �J×(n), J× m(n) and J × m(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' This gives rise to subalgebras gl(�V (n)), gl(Vm(n)), gl(Vm(n)), gl(�V ×(n)), gl(V × m (n)) and gl(V × m (n)) of the Lie superalgebra gl(�V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Let �b := � r≤s,r,s∈ 1 2Z CEr,s denote the standard Borel subalgebra of gl(�V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Gaudin Hamiltonians on unitarizable modules 5 We will drop the symbol (n) if n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' For example, Jm := Jm(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' Define the total orders of Jm and Jm by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfYgQR/content/2301.03138v1.pdf'} +page_content=' 1. However, observational studies have remained inconclusive about the fraction beyond +redshift unity because, to date, only two clusters in this redshift regime have been investigated. One +shows a much lower fraction than the mean value at low redshift3, whereas the other possesses a fraction +similar to the low-redshift value4. Here we report an ICL study of ten galaxy clusters at 1 ≲ z ≲ 2 based +on deep infrared imaging data. Contrary to the leading theories, our study finds that ICL is already +abundant at z ≳ 1, with a mean ICL fraction of approximately 17%. Moreover, no significant correlation +between cluster mass and ICL fraction or between ICL colour and cluster-centric radius is observed. +Our findings suggest that gradual stripping can no longer be the dominant mechanism of ICL formation. +Instead, our study supports the scenario wherein the dominant ICL production occurs in tandem with +the formation and growth of the brightest cluster galaxies and/or through the accretion of preprocessed +stray stars. +Main +Intracluster light (ICL) is predominantly distributed in the central region of the cluster, in most +cases around the brightest cluster galaxy (BCG) out to several hundred kilo-parsecs3-6. Some studies +reported that significant ICL is also found around intermediate and massive satellites7,8. We detected +ICL around the BCGs of ten galaxy clusters at z ≳ 1with the Wide Field Camera 3 (WFC3) near-infrared +imager on board Hubble Space Telescope (HST) (Fig. 1). In most cases, a clear surface brightness (SB) +profile is obtained out to approximately 200 kpc, where it approaches the surface brightness limit μ ≈ +28 mag arcses-2 (Fig. 2). The exception is the result for JKCS041, which is the highest redshift (z = 1.8) +target in our sample (Extended Data Table 1). Its SB profile approached the limit at around 100 kpc. +Figure 2 shows that, overall, the SB profiles are well described by a superposition of two or three multi- +Sérsic components convolved with the instrument point spread function (PSF). Regardless of the +number of components, the outermost component is predominantly responsible for the shape of the +outer part of the SB profile, which is assumed to characterize the ICL here, whereas all inner + +components (one component if the total number of components is two) are considered to represent the +BCG profile. Most clusters in our sample show no significant gradients in their SB colour profiles and +their colours are in good agreement with those of the reddest cluster members. Exceptions are found for +SpARCS1049 and IDCS1426, which possess a clear negative gradient, with the colour difference +between the BCG- and ICL-dominant regions being around 1 mag (Fig. 3). IDCS1426 and +SpARCS1049 are the second (z = 1.75) and third (z = 1.71) highest redshift clusters in our sample. +When converted to the rest-frame B and V mags, the BCG colours span the 0.5 < B − V < 0.8 range, +which overlaps the theoretical distribution9. Combining spectroscopic and photometric member +selections, we measured the BCG + ICL and ICL fractions (fBCG+ICL and fICL) using an aperture of +r = 0.5 Mpc (Extended Data Table 2). The mean BCG fraction is approximately 4.5%, which is well +bracketed by the values in previous studies3,8,10,11. Figure 4 shows that the mean ICL fraction of our +sample is similar to that of the low-redshift sample in the literature1,3,5,8,10-21. +One potential difficulty for the interpretation of Fig. 4 is the diversity of the methodology in +the previous studies. We investigated the impacts of the following two factors: aperture size and ICL +definition. The results compiled in Fig. 4 are based on apertures ranging from 100 kpc to 1.7 Mpc. We +verified that there is no correlation between aperture size and ICL fraction in the published result. +Moreover, the mean aperture size in the literature is 0.58 Mpc, which is similar to our choice of 0.5 +Mpc. Finally, when we repeated the analysis using the subsample that used the aperture sizes between +0.35 and 0.65 Mpc, the result remained unchanged. The ICL community is aware that the results from +the traditional SB cut (SBC) method can differ systematically from those obtained by the new +multicomponent decomposition method22 to which our approach belongs. To address the issue, we +divided the literature sample into the SBC and multicomponent decomposition subsamples. Our +regression based on the latter shows that the slope is still consistent with zero at the 2σ level. +We find that dwarf galaxies fainter than our detection limit do not bias our ICL fractions high. +To investigate the impact of sources fainter than our detection limit, we carried out image simulations +by randomly distributing dwarf galaxies, whose number is estimated by fitting a Schechter luminosity +function to the detected source distribution and computing the difference between the best-fit luminosity +function and observed distribution. We considered two types of radial distributions. The first is a +uniform distribution across the field. The second is the distribution that follows a Navarro–Frenk–White +profile23. In the first case, the ICL fraction is unchanged because adding a uniform dwarf galaxy +distribution is equivalent to elevating the sky level by the same degree simultaneously across the entire +field. In the second case, the dwarf galaxies are mostly concentrated near the BCG. Although this +certainly would lead to the overestimation of the BCG luminosity, the impact on the ICL luminosity +was negligible. + +We rule out the possibility that unmasked galaxy light might artificially increase the ICL +fraction. In our analysis, we employed a moderate-sized mask and later applied a correction factor to +obtain the result effectively measured with the full mask (see Methods). This correction scheme was +verified to be accurate, leading to only an approximately 0.02% difference in the average ICL fraction +(Extended Data Table 2). +One may argue that the ten clusters in our sample correspond to the most massive population +at high redshift and thus should not be compared directly with the low-redshift clusters. Although four +of our ten clusters may potentially belong to extremely massive (around 10 +15M☉) populations in the 1 +≲ z ≲ 2 universe, the masses of the remaining six clusters span the range 2–6 × 1014M☉ (Extended +Data Table 3). We found that the cluster masses do not correlate with the ICL fractions for our sample +(Extended Data Fig. 5). Although theoretical studies +1,2,24–27 remain inconclusive about the fICL–mass +correlation, observational studies +22,28,29 agree that there is no correlation. This lack of the fICL–mass +correlation is also supported in the study where the sample is limited to a narrow redshift range of z < +0.0722. Hence, we do not attribute the absence of the fICL–redshift correlation to a selection effect. +The dominant ICL production mechanism is still unknown, although the current consensus is +that merger, stripping and preprocessing are the three important candidates22,30–32. The ICL fraction is +an important observable sensitive to the timescale of the ICL formation, and its evolution with redshift +can be used to discriminate between competing theories regarding the dominant ICL production +mechanism. A strong evolution +1,2,10,28,33 favours a gradual process through stripping, whereas the +opposite +5,29 supports the scenario wherein the dominant ICL production happened at high redshifts. The +absence of the apparent evolution of the ICL fraction in the 0 ≲ z ≲ 2 redshift regime in the current +study contracts the current leading theories1,2, which predict that the mean ICL fraction decreases to a +negligible level (less than 5%) at z = 1.5 (Fig. 4). Therefore, the most straightforward interpretation of +the current finding is that the dominant ICL formation and its evolution with redshift can occur not +through gradual stripping, but in tandem with the BCG formation and growth, and/or through the +accretion of preprocessed stray stars. +Together with the ICL fraction, another traditional but still critical method to discriminate +between competing theories on the dominant mechanism of ICL production is to investigate the ICL +stellar population with its colour and compare the results with those of the cluster galaxies, including +the BCG. For instance, if major mergers with the BCG are the dominant mechanism, no significant +colour difference between BCG and ICL is expected. On the other hand, if the ICL is formed by a more +gradual process such as stellar stripping, we expect that the ICL would be bluer than the BCG, or a + +nega- tive gradient would be present in the radial colour profile. In this case, matching the colour +between the ICL and cluster galaxies can constrain the progenitors of the ICL. Previous observational +studies in general support the presence of negative gradients, although exceptions are not +uncommon4,19,34,35, which implies that the gradient may depend on the particular assembly history of +individual clusters9. The absence of the ICL colour gradient in most of the cases in our sample indicates +that gradual stripping is not likely to be the dominant mechanism of ICL production within the 1 ≲ z ≲ +2 epoch. It is possible that occasional major mergers can potentially mix the intracluster stars and flatten +the ICL colour profile even in the case where ICL production through stripping is dominant. However, +as the ICL colours in the flat gradient cases are in good agreement with those of the reddest cluster +members, our observation cannot reconcile with this scenario. + + + + +Fig. 1 | BCG + ICL images of our ten z ≳ 1 clusters. The images were created by masking out every +discrete source detected by SExtractor except for the BCGs. Here the result is based on an expansion +coefficient of 2. We also visually scanned the result and applied additional manual masking for the +objects that SExtractor failed to identify. Green dashed lines show the SMA = 200 kpc ellipses, whose +ellipticities and position angles are determined by AutoProf. Throughout the paper, we assume a flat Λ- +dominated cold dark matter cosmology characterized by h = 0.7 and Ωm,0 = 1 − ΩΛ,0 = 0.3, where h, +Ωm,0 and ΩΛ,0 represent the dimensionless Hubble, matter density, and dark energy density parameters +at present day, respectively. + +XMM1229/E160W +SPT2106/E140W +M001142/F140W +RDCS1252/E160W +100 kpc +100 kpc +100 kpc +100 kpc +M001014/F140W +SPT0205/F140W +XMM2235/F160W +SpARCS1049/F160W +100 kpc +100 kpc +100 kpc +100 kpc +DCS1426/F140W +JKCS041/F160W +22 +24 +26 +28 +μ (mag arcsec-2) +100 kpc +100 kpc +Fig. 2 | BCG + ICL radial profiles. Data points are the observed surface brightness from the elliptical +bins. The errors are computed from quadratic sums of the background level error and shot noise. With +dashed and solid lines, we show our best-fit multi-Sérsic component model. Magenta (cyan) dashed +lines are the innermost (outermost) components. When the target requires three components, we use +orange lines to represent the middle component. Red solid lines illustrate the summation of all +components. The legends of each panel show the target name, filter type, χ2 value and best-fit Sérsic +indices. + +167 +167 +XMM1229/F105W +. +XMM1229/F160W +. +SPT2106/F105W +. +SPT2106/F140W +18 - +Fitting,x= 4.15 +Fitting, x = 1.59 +18- +Fitting,x2= 2.66 +Fitting, x = 3.04 +20 +BCG, n = 2.3 ± 0.6 +BCG, n = 3.0 ± 0.4 +arcsec-2) +20 +BCG, n = 0.9 ± 0.1 +BCG, n = 1.4 ± 0.1 +ICL, n = 0.9 ± 0.2 +ICL, n = 0.7 ± 0.1 +0:0:00 +ICL, n = 1.4 ± 0.4 +ICL, n = 1.9 ± 0.7 +22 +22 +24 +(mag +24 +26 - +8 +26 +28 +28 +30 +30 - +100 +101 +102 +100 +101 +102 +100 +101 +102 +100 +101 +102 +16 - +. +MOO1142/F105W +MOO1142/F140W +16 +RDCS1252/F105W +RDCS1252/F160W +18 +Fitting, x = 1.47 +Fitting, x = 1.87 +18 +Fitting, x = 1.46 +Fiting, x = 3.25 +BCG1, n = 0.9 ± 0.3 +BCG1, n = 0.9 ± 0.3 +BCG1, n = 0.9 ± 0.2 +BCG1, n = 0.3 ±0.1 +20 +BCG2, n = 1.4 ± 0.1 +BCG2, n = 1.5 ± 0.1 +arcsec +20 +BCG2, n = 0.4 ± 0.2 +BCG2, n = 0.5 ± 0.2 +22 +ICL, n = 1.8 ± 0.5 +ICL, n = 2.1 ± 0.5 +22 +ICL, n = 1.0 ± 0.2 +ICL, n = 0.9 ± 0.3 +24 +(mag : +24 +8 +26 - +8 +26 - +28 +28 - +30 +30 - +100 +101 +102 +100 +101 +102 +100 +101 +102 +100 +101 +102 +167 +MO01014/F105W +MO01014/F140W +167 +. +SPT0205/F105W +SPT0205/F140W +18 - +Fitting, x = 0.77 +Fitting, x = 1.72 +18 - +Fitting, x = 0.85 +Fitting, x = 1.17 +BCG1, n = 0.9± 0.5 +BCG1, n = 1.5± 0.4 +BCG, n = 0.6 ± 0.3 +BCG, n = 0.7 ± 0.3 +arcsec-2) +20 +BCG2, n = 0.6 ± 0.4 +BCG2, n = 0.9 ± 0.4 +arcsec-2) +20 +ICL,n = 3.1 ± 0.4 +ICL, n = 3.7 ± 0.2 +22 +ICL, n = 1.9 ± 0.4 +ICL, n = 1.2 ± 0.4 +(mag +24 +(mag +24 +26 - +SB +26 +28 +28 +30 +30 - +100 +101 +102 +100 +101 +102 +100 +101 +102 +100 +101 +102 +167 +167 +. +XMM2235/F105W +XMM2235/F160W +. +SpARCS1049/F105W +SpARCS1049/F160W +18 +Fitting, x = 4.12 +- Fiting, x2 = 1.59 +18 +Fitting, x = 1.10 +Fitting, x2 = 1.19 +BCG, n = 2.0 ± 0.2 +BCG, n = 1.4 ± 0.2 +BCG1, n = 1.0 ±0.1 +BCG1, n = 1.1 ± 0.1 +arcsec-2) +20 +ICL, n = 1.9± 0.3 +ICL, n = 1.2 ± 0.4 +arcsec-2) +20 +BCG2, n = 0.6 ± 0.1 +BCG2, n = 0.3 ±0.1 +22 +ICL,n = 0.8 ±0.3 +ICL, n = 0.7 ± 0.3 +22 +24 +(mag +24 +26 - +8 +26 +28 +28 +30 +30 - +100 +101 +102 +100 +101 +102 +100 +101 +102 +100 +101 +102 +167 +. +IDCS1426/F105W +IDCS1426/F140W +167 +. +JKCS041/F105W +JKCS041/F160W +18- +Fiting, x = 1.27 +Fitting, x = 3.39 +18 +Fitting, x2 = 0.68 +Fitting, x = 0.68 +BCG1, n = 0.6± 0.3 +BCG1, n = 1.4 ± 0.4 +BCG, n = 1.8 ± 0.4 +BCG, n = 1.3 ± 0.2 +arcsec-2) +20 +BCG2, n = 1.3 ±0.1 +BCG2, n = 1.3 ±0.0 +g arcsec-2) +20 +ICL,n = 1.2 ± 0.5 +ICL, n = 1.8 ± 0.4 +ICL, n = 2.2 ± 0.2 +22 +ICL, n = 2.2 ± 0.1 +22 +24 + (mag +24 +26 - +26 +28 +28 +30 - +30 - +100 +101 +102 +100 +101 +102 +100 +101 +102 +100 +101 +102 +SMA (kpc) +SMA (kpc) +SMA (kpc) +SMA (kpc)Fig. 3 | BCG + ICL radial colour profiles. Black solid lines are the observed colour. The dark grey +shades represent the 68% uncertainty. The scale radius of each component is shown with the same +colour scheme used in Fig. 2. The red data point is a representative mean value in each subregion. The +orange stars indicate the colours and positions of spectroscopic member galaxies. The light grey shade +indicates the radial extent of the BCG measured by SExtractor. + + + +2 - +Colour +0- +1 +Representative value +Representative value +100 +101 +102 +100 +101 +102 +2- +Colour +0. +0 +Representativevalue +米 +Representativevalue +100 +101 +102 +100 +101 +102 +2 - +Colour +0- +MO01014F105W-F140W +0 +米 +Representativevalue +Representativevalue +100 +101 +102 +100 +101 +102 +2 - +2 +Colour +0. +XMM2235 +0 +SpARCS1049 +F105W-F160W +105W-F160W +米 +Representativevalue +米 +Representativevalue +100 +101 +102 +100 +101 +102 +2- +Colour +0. +0 +Representativevalue +Representativevalue +100 +101 +102 +100 +101 +102 +SMA (kpc) +SMA (kpc) +Fig. 4 | ICL fraction evolution. Filled red circles are the current results based on the r = 0.5 Mpc +aperture. We extrapolated the best-fit Sérsic profiles to the same aperture to estimate the total ICL flux. +When we avoided the extrapolation and performed integration only within the range where the SB +profile is above the detection limit, the resulting ICL fraction is reduced by about 1.9% on average. The +mean aperture size of the literature sample is 0.58 Mpc. The dashed line and pink shade show the best- +fit linear regression and its 68% uncertainty, respectively. We weighted all data points equally and +adjusted them in such a way that the reduced χ +2 value becomes unity. The comparison between our +high-redshift and the literature low-redshift samples shows that there is no significant evolution of the +ICL fraction with redshift in observation, which contradicts the current theoretical prediction +1 (grey). +Although here we displayed the theoretical model that estimates the ICL fraction based on the SBC at +26.0 mag arcsec-2, similarly steep evolutions are obtained even when different ICL definitions such as +binding energy criteria are used. + + + +50 +11j +[12] +13) +40 +[19] +[21] +This study +30 +ICL fraction (%) +20 +10 +0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.6 +2.0 +zMethods +Target selection +We searched the Mikulski Archive for Space Telescope for the WFC3-near-infrared imaging +programs that have observed z ≳ 1 clusters in at least two filters with the surface brightness limit μ ≈ +28 mag arcsec-2. We excluded the targets if they do not possess any distinct BCGs, or very bright +stars/foreground galaxies are present near the cluster centres. The search resulted in a sample of ten +galaxy clusters at 1 ≲ z ≲ 2 with a minimum (maximum) redshift of 0.98 (1.803). Extended Data Table +1 summarizes our target selection, including the redshift, coordinate, program number, surface +brightness limit and so on. Although F105W exists for all ten clusters, either F140W or F160W is +available for each cluster. +Reduction pipeline optimized for ICL measurement +The common data reduction procedure recommended in the HST Data Handbook +34 works well +if one is interested in discrete astronomical sources such as stars and galaxies. However, when we are +looking for signals from diffuse components whose surface brightness is within a subpercentage of the +sky brightness and slowly varies across the detector, additional care is needed. Our reduction begins +with the flat-fielded (FLT) images processed by the Space Telescope Science Institute calwf3 tool36, +which removes most instrumental signatures of WFC3, except for geometric distortion. We visually +inspected these FLT images and manually masked out any remaining artefacts such as satellite/asteroid +trails. We ran the TweakReg package +37 for astrometric calibration by finding common astronomical +sources. The FLT images are already flatfielded with the default Space Telescope Science Institute +composite flats, which are claimed to be accurate within less than 0.5%, except for the region within +128 pixels of the detector edge34. This claim has been verified by independently constructing residual +flats utilizing large WFC3 survey programs4. We also investigated the impact of the residual flat in the +final mosaic by performing drizzling as if we were stacking science frames and found that the dithered +residual flats would cause at most around 0.4% errors, which is already negligible compared to other +sources of errors (for example, background determination). In this study, we applied these residual flats +to our FLT data to further reduce the residual flat errors. The application of the aforementioned residual +flatfielding cannot remove large-scale sky gradients arising from intrinsic sky gradient, detector +persistence, internal reflection and so on. We removed the sky gradient by fitting a first-order +polynomial plane (F(x,y) = ax + by + c) to the object-masked residual-flatfielded image and subtracting +the result. The subtraction result was visually inspected, and we discarded the frame if the first-order +polynomial plane could not adequately describe the sky gradient. To create a final deep mosaic where + +ICL is measured, careful and consistent sky subtraction from each exposure is required. In typical +ground-based data reduction designed for non-ICL-related studies, position-dependent sky +estimation/subtraction is routinely performed after astronomical objects are masked out. Although this +scheme may provide cosmetically ‘good’ results when different frames are combined together, the +inevitable consequence is sky oversubtraction in the region where non-discrete astronomical +components are dominant. Therefore, in this study, we determined only the global sky level (that is, a +single constant) for each frame and subtracted it. Because individual exposures observed with dithers +cover non-identical areas around the target, it is necessary to choose the same physical sky region that +is present in common for all exposures. To accomplish this, we set up the largest possible annulus that +is approximately centred on the BCG and is observed by all exposures. An illustration of this common +annulus is presented in Extended Data Fig. 1. +Discrete astronomical sources in this annulus were detected using SExtractor +38 with the +settings of DETECT_MINAREA = 5 and DETECT_ THRESHOLD = 2. As a matter of course, the +resulting segmentation map fails to include the contributions from the faint diffuse wings of the objects. +To address this, we gradually expanded the segmentation maps and investigated the resulting +background level change as a function of the mask size. One should not increase the segmentation map +by expanding the boundaries at the same rate for all objects. That is, the segmentation boundaries of +compact objects should expand slower than those of extended objects. Thus, we used SExtractor’s half- +light radius rh and scaled the expansion with it. The width (w) of the expansion band is determined by +w = ce rh, where ce is the expansion coefficient. We stopped the segmentation map expansion at ce = 6, +beyond which the sky estimation converged (Extended Data Fig. 2). Note that this exposure-by- +exposure sky level estimation is performed on the individual drizzled images to minimize the impact of +the geo- metric distortion. We created the final mosaic using AstroDrizzle +33 with its sky subtraction +option turned off. The output pixel scale is set to 0.05 arcsec. We used the Gaussian kernel for drizzling. +Extended Data Fig. 3 schematically summarizes our data reduction pipeline. +Object masking for ICL measurement +To characterize ICL from the mosaic image, it is necessary to mask out light from discrete +objects. In the background level estimation above, we found that the choice ce = 6 was sufficient. +However, this large expansion coefficient cannot be used in the central region of the cluster because +doing so would leave very few pixels there, resulting in too large statistical errors exceeding the +systematic error caused by the incomplete masking (Extended Data Fig. 2). Therefore, for ICL +measurement from the final stack, we chose to employ ce = 2 for object masking and apply a correction +factor to obtain the result effectively measured with the full mask (ce = 6). The correction factor is + +derived by comparing multiple ICL profiles measured with different ce values. We masked out every +discrete source detected by SExtractor except for the BCG. We also visually scanned the result and +applied manual masking for the objects that SExtractor failed to identify (Fig. 1). +To assess the validity of our statistical correction scheme, we repeated our analysis with the +full (ce = 6) masking. This is supposed to generate results with better accuracy at the expense of +precision (that is, smaller systematic errors and larger statistical errors). However, if any large +systematic discrepancy from our fiducial measurement (for example, statistically corrected result after +the use of ce = 2) is found, this indicates that the aforementioned correction scheme is problematic. The +mean of the differences is close to zero (about 0.02%, Extended Data Table 2), which verifies that our +correction scheme with the use of the moderate masking expansion (ce = 2) is robust. +Radial profile measurement with elliptical binning +As the BCG + ICL isophotes are elliptical, the use of circular binning would spread the BCG- +ICL transition over multiple bins. Thus, we measured the radial profile of the BCG + ICL using an +elliptical bin- ning scheme. To determine the ellipticity and position angle (PA) of the ellipse, we used +the AutoProf package39. With both detection image and masking map as inputs, AutoProf calculates +ellipticity and PA based on isophotal fitting and Fourier analysis. +Two sets of outputs are generated by AutoProf. One is a series of ellipticity and PA values, +which vary with radius. The other is a single pair of ellipticity and PA, which represents the global +shape. The former is useful when one’s interest is the radius-dependent isophotal shape of high signal- +to-noise-ratio objects. In this study, where our scientific interest is faint diffuse light, we use the second +set of outputs. +We defined a series of semi-major axes (SMA) with a logarithmic scale and measured the +surface brightness at each radial bin. We applied the 3σ clipping method to minimize the impact of the +outliers and adopt the median as the representative surface brightness of the bin. This also reduces the +effects of any potential unidentified substructures within each elliptical annulus. The total error of the +surface brightness estimate is computed as the quadratic sum of the 1σ photon noise and sky estimation +(background level) uncertainty. As mentioned above, the residual flat error is negligible and hence is +not included here. The latter dominates our error budget. +Multicomponent decomposition + +The traditional method for measuring ICL is to define an SBC and char- acterize the light +component fainter than the threshold. As the choice of the threshold is arbitrary, it is difficult to use the +method to compare results from different studies. In particular, because the current redshift regime (1 +≲ z ≲ 2) is considerably different from those of the previous studies, one cannot objectively characterize +the ICL properties based on this traditional approach. +In this study, we decompose the BCG + ICL profile into multiple Sérsic components. A Sérsic +profile +40 is defined as: +𝐼(𝑟) = 𝐼! exp +−𝑏" ./ +# +#!0 +$/" +− 123, +where r, re, n and Ie are the radius, half right radius, Sérsic index and intensity at the half right radius, +respectively. In this equation, bn is the constant that is not independent and is solely determined by n. +Then, the BCG + ICL profile S(r) is modelled as a superposition of PSF-convolved multiple +Sérsic components Im(r): S(r)=PSF(r)⁎∑mIm(r), where PSF(r) is the PSF radial profile and the symbol +‘*’ represents the convolution. The PSF radial profile was constructed by combining the core from +observed stellar images and the wing from the TinyTim +41 result. If the PSF correction procedure is +omitted, the mean ICL fraction increases by around 1.5% (Extended Data Table 2). +How do we determine the total number of Sérsic components for each cluster? A model based +on more components has higher degrees of freedom and leads to smaller residuals. However, the +drawback is overfitting. About 70% of elliptical galaxies are reported to require more than one Sérsic +component to adequately describe their profiles +42–44 In this study, we limit the maximum number of +Sérsic components for the description of the BCG + ICL profile to three. +To determine the optimal number of Sérsic components, we use two criteria. The first is the +Bayes factor K45, defined as follows: +𝐾 = +&((|*") +&((|*#), +where p(D|Mk) is the probability of the data (D) given the model Mk. As P(D|Mk) cannot be computed +directly, in practice K is evaluated as follows: +𝐾 = +&(*"|()&(*#) +&(*#|()&(*") = +&(*"|() +&(*#|(), + +where we assume the equality between the two priors p(M1) and p(M2). The Bayes factor K informs us +of how the first model M1 is preferred over the second M2. Its outstanding advantage is that K inherently +penalizes the model according to its degrees of freedom based on first principles. The second criterion +is the number of inflexion points in the derivative of the SB radial profile. Naturally, the existence of l +inflexion points implies a preference towards a model with l + 1 components. After investigation of our +ten clusters with the above two criteria, we find that for the entire sample (1) the Bayes factor K between +the best and second-best models is ln(K) ≳ 2, (2) the optimal number of Sérsic components inferred +from the Bayes factor agrees with the result from the inflexion point analysis and (3) the SB profiles +require either two or three components for optimal decomposition. +Estimation of the total cluster luminosity +One of the key requirements in measuring the ICL fraction is a robust selection of the cluster +member galaxies. In this study, the first step towards this goal is the compilation of the results from +previous spec- troscopic studies (see Extended Data Table 3). We used this spectroscopic cluster +member catalogue to define the initial red sequence locus from the colour–magnitude diagram. By +iteratively applying linear regression to the red sequence and selecting objects within 0.5 mag from the +best-fit line, we built up the second-stage cluster member catalogue. In this iteration, we removed stars +using the CLASS_STAR value reported by SExtractor and the objects either brighter than the BCG or +fainter than F105W = 26 mag. This second-stage cluster member catalogue needed to be improved +because the distribution of the object distance from the best-fit red sequence line was asymmetric (the +blue side was blended with the neighbouring blue cloud). Thus, we fitted a double Gaussian model to +the distribution (see the bottom panel of Extended Data Fig. 4). The new centre and 1σ width of the red +sequence were used to update the intercept and width of our previous best-fit linear regression result +and the final cluster member catalogue was obtained (see the top panel of Extended Data Fig. 4). +Although we employed a sophisticated procedure for selection of cluster members, inevitably the +method is designed to select only red members, except for the blue spectroscopic members. If the +contribution from the blue members is large, our ICL fraction would be overestimated. However, we +argue that the overestimation, if any, would not be significant because (1) in most cases the +spectroscopic catalogue includes the brightest blue cluster members (Extended Data Table 3), (2) even +the brightest blue cluster members are found to be still a few magnitudes fainter than the BCG and other +brightest red members and (3) some fraction of galaxies in our red sequence catalogue are non-cluster +members. We verified this claim by utilizing the publicly available photometric redshift catalogue for +SPT0205 (ref. 46), which includes the blue cluster member candidates. When we repeated the +measurement of the ICL fraction with it, the resulting ICL fraction shifted by only 1%. + +The total luminosity is estimated as follows. First, we masked out non-member galaxies/stars +from our imaging data. The total masked area is non-negligible and simply assigning zero flux to the +area would lead to substantial underestimation of the total luminosity. Thus, we filled the masked +regions with the predicted flux from our best-fit multi-Sérsic model. Finally, the total luminosity is +computed by the summation of the pixel values of the resulting image, which is comprised of the flux +from the BCG, ICL and cluster members. +Impact of the red sequence selection criteria +Although we took care to robustly define the locus of the red sequence through somewhat +sophisticated iteration, two of the remaining ambiguities worth further investigation are the faint-end +limit and the width of the red sequence. To examine the dependence of the ICL fraction on these +selection criteria, we considered the following three additional cases: +• Test A: the magnitude limit decreased to 24th mag +• Test B: the magnitude limit increased to 28th mag +• Test C: the width increased to 1.5σ +We list the test results for individual clusters in Extended Data Table 3. In the case of Test A, +the average (maximum) increase in ICL fraction is found to be approximately 1.9% (approximately +3.8%). Test B shows that the average (maximum) decrease in ICL fraction is approximately 0.9% +(approximately 1.6%). Finally, Test C gives an average (maximum) decrease of approximately 2.9% +(approximately 5.8%). In summary, our ICL fraction measurements are not sensitive to the selection +criteria tested here. +ICL fraction versus mass correlation +Together with the ICL fraction evolution with redshift, the ICL community has also been +investigating the correlation between ICL fraction and halo mass. Under the assumption that massive +halos represent older populations, a strong correlation would imply a significant time evolution. +Extended Data Fig. 5 shows the ICL fraction versus mass relation for our sample. The masses come +from weak lensing studies (Extended Data Table 3). No significant correlation is observed for our +sample. + + + +Online content Any methods, additional references, Nature Portfolio reporting summaries, source +data, extended data, supplementary information, acknowledgements, peer review information, details +of author contributions and competing interests, and statements of data and code availability are +available at https://doi.org/10.1038/s41586-022-05396-4. +Data availability The raw HST near-infrared imaging data used for the current study are publicly +available. The processed imaging data are available on the github repository at +https://github.com/Hyungjin-Joo/High_z_ICL. Source data are provided with this paper. +Code availability An exhaustive repository of code for our custom data processing and analyses +reported in this manuscript are available on the github repository at https://github.com/Hyungjin- +Joo/High_z_ICL. +Acknowledgements This study is based on observations created with NASA/ESA Hubble Space +Telescope and downloaded from the Mikulski Archive for Space Telescope at the Space Telescope +Science Institute. The current research is supported by the National Research Foundation of Korea +under programme 2022R1A2C1003130 and the Yonsei Future-Leading Research Initiative programme. +Author contributions M.J.J. conceived, designed and supervised the project. M.J.J. and H.J. analysed +the Hubble Space Telescope imaging data, developed the pipeline, interpreted the results and wrote the +manuscript. + + +Extended Data Fig. 1 | Definition of common sky areas. (A) Exposure map for the single-frame +image. (B) Same as (C) except that it is for the mosaic image. (C) Science image for single frame. (D) +Same as (C) except that it is for the mosaic image. The pink circular region in (A) is the region that is +observed in common by all contributing frames. (B) shows how this common region is positioned in +one of the input frames. As the central region of this circle is likely to be heavily influenced by the ICL, +we excluded the central region and instead defined the annulus shown in (C) and (D) to estimate the +background level. + + + +(A) +exposuretime along blue horizontal line +(B) +line +horizontal +blue +along +exposuretime +exposuretimealongblueverticalline +exposuretimealongblueverticalline +C +Extended Data Fig. 2 | Masking size growth and impacts on background level. (A), (B) and (C) +illustrate our scheme for masking size growth from the original to the ce = 2 and ce = 6 cases. Note that +we exhaust pixels for ICL measurement at ce = 6. In (D), we show how the background level (green) +changes as we vary the masking size using the expansion coefficient for a single exposure. We observe +that at ce ≳ 6 the measurement converges (red). The black solid line indicates the result when instead +we use a 3σ clipping algorithm without considering the diffuse wings of the astronomical objects. The +yellow line shows the surface brightness level measured at each ce. (E) is the same as the left except +that the measurement is from the final deep stack. Solid lines indicate the median value and shaded +regions show the 68% uncertainty. As the image is deeper, the number of pixels discarded (masked out) +at the same ce value is much greater. + +Original_mask +Ce = 2.0 +Ce = 6.0 +(A) +(B) +(C) +1.30 +(D) +Sky +All Light Sources +sky value of this methid +SB [count / s / pixe/2] +1.25 +sky value of 3-sigma clipping +1.20 +1.15 +2 +4 +6 +8 +10 +12 +Ce +(E) +Sky +0.0 +All Light Sources +[z/axid / s / sunolas +2.5 +5.0 +7.5 +10.0 +2 +4 +6 +8 +10 +12 +Ce +Extended Data Fig. 3 | Schematic diagram of our ICL-oriented data reduction. Dark grey +rectangles show the steps where external packages are used, while light grey rectangles illustrate our +custom procedures. Parallelograms represent the input/output data. + + + +Reject +Start +no +Astrometric +Residual +Fitting +*flt.fits +Calibration +Plane fitting +Flat Fielding +Success? +(TweakReg) +yes +*flt_plf.fits +Segmentation +Expanded +Source Detection +Map +ExpandingMask +Mask +(SExtractor) +(For masking) +Map +Image Drizzling +High Overlapped +Overlapped +Drizzled Image +SkyEstimation +(Astrodrizzle) +Region Selection +WCS +Sky Level +Image Drizzling +Final +?ReDrizzledImage +Sky Subtraction +End +(Astrodrizzle) +Drizzled Image +Extended Data Fig. 4 | Red sequence selection scheme. Here we display the case for SPT2106. (A) +Colour–magnitude diagram. Black dots are all sources detected by SExtractor. The red dots represent +the spectroscopic members, whereas the orange dots are our red sequence candidates. The BCG is +indicated with a red star. The red dashed line shows the final, best-fit red sequence. The dot-dashed +lines bracket the 68% distribution. (B) Distribution of the F105W < 26 object distances from the best- +fit red sequence. The green line shows the best-fit double Gaussian models. The yellow line illustrates +a single Gaussian component, which represents the distribution of the red sequence candidates. + + + +(A) +0.75 +F105W - F140W (Auto) +0.50 +0.25 +Red sequence +1 s.t.d +0.00 +All +BCG +-0.25 +Spec. Member +Member Candidate +-0.50 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +(B) +F105W (ISO) +Line +0.50 +d Sequence +0.25 +0.00 +Distance from Red +PDF +-0.25 +DoubleGauss +Gauss +-0.50 +Red Sequence +-0.75 +1 s.t.d +All +1.00 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Number of Galaxies +Extended Data Fig. 5 | Comparison between ICL fraction and cluster mass. The mass comes from +weak lensing studies. No significant correlation between ICL fraction and mass is found. + + + +35 +30 - +25 +. fraction [%] +20 +T +15 +ICL +10 +T +T +0 +0.5 +1.0 +1.5 +2.0 +2.5 +M200[Mo] +1e15Extended Data Table 1 | Target List +Target +(Short Name) +Redshift +(z) +R.A +Dec +HST Proposal +ID +SB limit [mag / arcs2] +(Exposure Time [s]) +F150W +F140W +F160W +XDCP J1229+0151 +(XMM1229) +0.98 +12:29:28 ++01:51:34 +12501 +29.62 +(1,311.7) +- +29.12 +(1,111.7) +SPT-CL J2106-5844 +(SPT2106) +1.1312 +21:06:05 +-58:44:42 +13677, 14327 +28.26 +(12,567.7) +28.59 +(12,771.6) +- +MOO J1142_1529 +(MOO1142) +1.19 +11:42:46 ++15:27:14 +14327 +27.96 +(6,283.8) +28.26 +(6,983.8) +- +RDCS J1252-2927 +(RDCS1252) +1.237 +12:52:57 +-29:27:15 +12501 +28.26 +(1,211.7) +- +29.00 +(1,211.7) +MOO J1014+0038 +(MOO1014) +1.24 +10:14:08 ++00:38:26 +13677, 14327 +28.47 +(18,255.5) +28.63 +(17,810.1) +- +SPT-CL J0205-5829 +(SPT0205) +1.322 +02:05:46 +-58:29:06 +13677, 14327 +28.66 +(23,007.2) +28.84 +(25,052.9) +- +XDCP J2235-2557 +(XMM2235) +1.39 +22:35:21 +-25:57:25 +12501 +28.18 +(1,211.7) +- +27.85 +(1,211.7) +SpARCS J1049+5640 +(SpARCS1049) +1.71 +10:49:22 ++56:40:34 +13677, 13747 +28.52 +(8,543.3) +- +28.28 +(9,237.4) +IDCS J1426.5+3508 +(IDCS1426) +1.75 +14:26:33 ++35:05:24 +12203, 13677, +14327 +30.56 +(10,972.4) +30.79 +(11,225.4) +- +JKCS041 +1.803 +05:26:44 ++04:41:37 +12927 +27.73 +(2,670.6) +- +27.99 +(4,509.4) + + + + +Extended Data Table 2 | ICL fractions and impact of various systematics +Name +Filter +fBCG+ICL [%] +fICL [%] +Red Sequence Selection Criteria +Unmasked +Wings +No PSF +Test A +Test B +Test C + + +(1) +(2) +(3) +(4) +(5) +(6) +(7) +XMM1229 +F105W +13.6$%.' +(%.) +11.7$%.* +(%.* +0.8 +-0.3 +-0.8 +-0.2 +2.5 + +F160W +14.2$%.+ +(%.+ +11.3$%.' +(%.+ +0.7 +-0.3 +-0.7 +-0.2 +2.8 +SPT2106 +F105W +18.9$,.- +(,.- +14.2$,.. +(,.. +0.1 +-0.9 +-2.8 +0.5 +0.3 + +F140W +21.4$/.% +(/./ +14.5$/.+ +(/., +0.1 +-0.8 +-2.7 +0.3 +0.6 +MOO1142 +F105W +21.1$+.% +(+.+ +15.9$+.+ +(+.+ +0.9 +-0.4 +-3.0 +0.8 +-0.7 + +F140W +24.6$+.+ +(+.' +16.8$+.* +(+.* +1.2 +-0.7 +-4.3 +0.9 +3.7 +RDCS1252 +F105W +27.9$%.* +(%.. +21.7$,., +(,./ +3.8 +-1.3 +-3.9 +-4.7 +0.8 + +F160W +29.1$,./ +(,.' +24.1$,.0 +(/., +2.8 +-0.9 +-3.1 +-3.5 +-0.1 +MOO1014 +F105W +22.5$'.+ +('.* +20.4$'.' +('.* +1.4 +-0.5 +-1.9 +1.1 +-0.5 + +F140W +17.8$/.. +(/.0 +15.2$+.% +(+., +1.4 +-0.5 +-1.8 +0.1 +5.0 +SPT0205 +F105W +17.8$,.0 +(/.+ +16.9$/.% +(/.+ +1.9 +-0.5 +-3.2 +0.1 +-3.2 + +F160W +20.7$%.. +(%.0 +19.6$%.* +(%.. +2.0 +-0.5 +-3.3 +0.2 +-4.2 +XMM2235 +F105W +23.0$/.0 +(+.* +20.7$/.. +(+.) +3.0 +-1.4 +-2.6 +2.9 +0.3 + +F160W +25.8$/.0 +(/.0 +22.9$/.0 +(+.* +3.3 +-1.6 +-2.7 +2.9 +2.5 +SpARCS1049 +F105W +27.3$/.* +('.+ +19.7$/.* +('.+ +2.4 +-1.2 +-5.8 +-1.4 +4.5 + +F160W +26.6$/.* +('.% +13.3$/.0 +(+.. +2.5 +-1.1 +-5.2 +-2.0 +9.7 +IDCS1426 +F105W +22.8$,., +(,./ +19.7$,., +(,./ +2.2 +-0.7 +-3.5 +0.3 +-1.0 + +F140W +28.4$%.. +(%.. +22.2$%.0 +(%.0 +1.9 +-0.7 +-3.5 +-0.8 +3.2 +JKCS041 +F105W +7.5$%.0 +(,.- +5.7$,./ +(,.+ +1.7 +-1.0 +-1.4 +1.2 +1.6 + +F160W +16.6$%.0 +(%.. +12.3$/.% +(,.+ +3.1 +-1.5 +-2.4 +1.9 +0.4 +Average +- +21.38 +16.89 +1.86 +-0.89 +-2.93 +0.02 +1.46 +(1) BCG + ICL fraction. (2) Fiducial ICL fraction. (3) Change when the magnitude limit decreased to +24th mag. (4) Change when magnitude limit increased to 28th mag. (5) Change when the width of the +red sequence increased to 1.5σ. (6) Change when the expansion coefficient increased to ce = 6. (7) +Change when the PSF effect is neglected. For (1) and (2), we quote measurements at r = 0.5 Mpc. + + + +Extended Data Table 3 | Weak lensing mass, the number of spectroscopic member galaxies and +their references +Name +Weak +Lensing +Mass (10$,𝑀⊙) +Number +of +Spectroscopic +Members +Spectroscopic +Catalogue +Reference +XMM1229 +2.04./.12 +3$.,1 +17 +47 +SPT2106 +14.9.4.5, +3,.$1 +31 +46 +MOO1142 +5.69.$.,4 +3$.56 +8 +48 +RDCS1252 +15.7.7.6/ +37.,/ +22 +49 +MOO1014 +3.35.$./2 +37./5 +7 +50 +SPT0205 +2.00./.25 +3/.82 +21 +46 +XMM2235 +14.6.,.82 +31.26 +10 +51 +SpARCS1049 +3.50.$.7/ +3$.7/ +11 +52 +IDCS1426 +3.65.$.65 +37.5/ +6 +53 +JKCS041 +13.1.5.8/ +31.12 +17 +54 + + + + +References +1. Rudick, C. S., Mihos, J. 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Stellar mass fractions of a +sample of high-redshift infrared-selected clusters. Astrophys. J. 878, 72–84 (2019). +51. Santos, J. S. et al. Dust-obscured star formation in the outskirts of XMMU J2235.3-2557, a massive +galaxy cluster at z = 1.4. Mon. Not. R. Astron. Soc. 433, 1287–1299 (2013). +52. Webb, T. M. A. et al. The star formation history of BCGs to z = 1.8 from the SpARCS/SWIRE +Survey: evidence for significant in situ star formation at high redshift. Astrophys. J. 814, 96–107 +(2015). + +53. Stanford, S. A. et al. IDCS J1426.5+3508: discovery of a massive, infrared-selected galaxy cluster +at z = 1.75. Astrophys. J. 753, 164–171 (2012). +54. Newman, A. B. et al. Spectroscopic confirmation of the rich z = 1.80 galaxy cluster JKCS041 using +the WFC3 grism: environmental trends in the ages and structure of quiescent galaxies. Astrophys. +J. 788, 51–76 (2014). + diff --git a/JdAzT4oBgHgl3EQfj_0H/content/tmp_files/load_file.txt b/JdAzT4oBgHgl3EQfj_0H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a8fb7c4b662bbff4f94286f1d33721e812e05cd --- /dev/null +++ b/JdAzT4oBgHgl3EQfj_0H/content/tmp_files/load_file.txt @@ -0,0 +1,1354 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf,len=1353 +page_content='Intracluster light is already abundant at redshift beyond unity Hyungjin Joo1 & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' James Jee1,2 1Department of Astronomy, Yonsei University, Seoul, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2Department of Physics and Astronomy, University of California, Davis, Davis, CA, USA Volume 613 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1038/s41586-022-0536-4 Received: 8 June 2022 Accepted: 28 September 2022 Intracluster light (ICL) is diffuse light from stars that are gravitationally bound not to individual member galaxies, but to the halo of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Leading theories1,2 predict that the ICL fraction, defined by the ratio of the ICL to the total light, rapidly decreases with increasing redshift, to the level of a few per cent at z > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, observational studies have remained inconclusive about the fraction beyond redshift unity because, to date, only two clusters in this redshift regime have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' One shows a much lower fraction than the mean value at low redshift3, whereas the other possesses a fraction similar to the low-redshift value4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Here we report an ICL study of ten galaxy clusters at 1 ≲ z ≲ 2 based on deep infrared imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Contrary to the leading theories, our study finds that ICL is already abundant at z ≳ 1, with a mean ICL fraction of approximately 17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Moreover, no significant correlation between cluster mass and ICL fraction or between ICL colour and cluster-centric radius is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Our findings suggest that gradual stripping can no longer be the dominant mechanism of ICL formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Instead, our study supports the scenario wherein the dominant ICL production occurs in tandem with the formation and growth of the brightest cluster galaxies and/or through the accretion of preprocessed stray stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Main Intracluster light (ICL) is predominantly distributed in the central region of the cluster, in most cases around the brightest cluster galaxy (BCG) out to several hundred kilo-parsecs3-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Some studies reported that significant ICL is also found around intermediate and massive satellites7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We detected ICL around the BCGs of ten galaxy clusters at z ≳ 1with the Wide Field Camera 3 (WFC3) near-infrared imager on board Hubble Space Telescope (HST) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In most cases, a clear surface brightness (SB) profile is obtained out to approximately 200 kpc, where it approaches the surface brightness limit μ ≈ 28 mag arcses-2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The exception is the result for JKCS041, which is the highest redshift (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8) target in our sample (Extended Data Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Its SB profile approached the limit at around 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Figure 2 shows that, overall, the SB profiles are well described by a superposition of two or three multi- Sérsic components convolved with the instrument point spread function (PSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Regardless of the number of components, the outermost component is predominantly responsible for the shape of the outer part of the SB profile, which is assumed to characterize the ICL here, whereas all inner components (one component if the total number of components is two) are considered to represent the BCG profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Most clusters in our sample show no significant gradients in their SB colour profiles and their colours are in good agreement with those of the reddest cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Exceptions are found for SpARCS1049 and IDCS1426, which possess a clear negative gradient, with the colour difference between the BCG- and ICL-dominant regions being around 1 mag (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' IDCS1426 and SpARCS1049 are the second (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='75) and third (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='71) highest redshift clusters in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' When converted to the rest-frame B and V mags, the BCG colours span the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 < B − V < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 range, which overlaps the theoretical distribution9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Combining spectroscopic and photometric member selections, we measured the BCG + ICL and ICL fractions (fBCG+ICL and fICL) using an aperture of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 Mpc (Extended Data Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The mean BCG fraction is approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5%, which is well bracketed by the values in previous studies3,8,10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Figure 4 shows that the mean ICL fraction of our sample is similar to that of the low-redshift sample in the literature1,3,5,8,10-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' One potential difficulty for the interpretation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4 is the diversity of the methodology in the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We investigated the impacts of the following two factors: aperture size and ICL definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The results compiled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4 are based on apertures ranging from 100 kpc to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We verified that there is no correlation between aperture size and ICL fraction in the published result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Moreover, the mean aperture size in the literature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='58 Mpc, which is similar to our choice of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Finally, when we repeated the analysis using the subsample that used the aperture sizes between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='65 Mpc, the result remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The ICL community is aware that the results from the traditional SB cut (SBC) method can differ systematically from those obtained by the new multicomponent decomposition method22 to which our approach belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To address the issue, we divided the literature sample into the SBC and multicomponent decomposition subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Our regression based on the latter shows that the slope is still consistent with zero at the 2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We find that dwarf galaxies fainter than our detection limit do not bias our ICL fractions high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To investigate the impact of sources fainter than our detection limit, we carried out image simulations by randomly distributing dwarf galaxies, whose number is estimated by fitting a Schechter luminosity function to the detected source distribution and computing the difference between the best-fit luminosity function and observed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We considered two types of radial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The first is a uniform distribution across the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The second is the distribution that follows a Navarro–Frenk–White profile23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In the first case, the ICL fraction is unchanged because adding a uniform dwarf galaxy distribution is equivalent to elevating the sky level by the same degree simultaneously across the entire field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In the second case, the dwarf galaxies are mostly concentrated near the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although this certainly would lead to the overestimation of the BCG luminosity, the impact on the ICL luminosity was negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We rule out the possibility that unmasked galaxy light might artificially increase the ICL fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In our analysis, we employed a moderate-sized mask and later applied a correction factor to obtain the result effectively measured with the full mask (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This correction scheme was verified to be accurate, leading to only an approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='02% difference in the average ICL fraction (Extended Data Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' One may argue that the ten clusters in our sample correspond to the most massive population at high redshift and thus should not be compared directly with the low-redshift clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although four of our ten clusters may potentially belong to extremely massive (around 10 15M☉) populations in the 1 ≲ z ≲ 2 universe, the masses of the remaining six clusters span the range 2–6 × 1014M☉ (Extended Data Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We found that the cluster masses do not correlate with the ICL fractions for our sample (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although theoretical studies 1,2,24–27 remain inconclusive about the fICL–mass correlation, observational studies 22,28,29 agree that there is no correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This lack of the fICL–mass correlation is also supported in the study where the sample is limited to a narrow redshift range of z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Hence, we do not attribute the absence of the fICL–redshift correlation to a selection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The dominant ICL production mechanism is still unknown, although the current consensus is that merger, stripping and preprocessing are the three important candidates22,30–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The ICL fraction is an important observable sensitive to the timescale of the ICL formation, and its evolution with redshift can be used to discriminate between competing theories regarding the dominant ICL production mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A strong evolution 1,2,10,28,33 favours a gradual process through stripping, whereas the opposite 5,29 supports the scenario wherein the dominant ICL production happened at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The absence of the apparent evolution of the ICL fraction in the 0 ≲ z ≲ 2 redshift regime in the current study contracts the current leading theories1,2, which predict that the mean ICL fraction decreases to a negligible level (less than 5%) at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Therefore, the most straightforward interpretation of the current finding is that the dominant ICL formation and its evolution with redshift can occur not through gradual stripping, but in tandem with the BCG formation and growth, and/or through the accretion of preprocessed stray stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Together with the ICL fraction, another traditional but still critical method to discriminate between competing theories on the dominant mechanism of ICL production is to investigate the ICL stellar population with its colour and compare the results with those of the cluster galaxies, including the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' For instance, if major mergers with the BCG are the dominant mechanism, no significant colour difference between BCG and ICL is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' On the other hand, if the ICL is formed by a more gradual process such as stellar stripping, we expect that the ICL would be bluer than the BCG, or a nega- tive gradient would be present in the radial colour profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this case, matching the colour between the ICL and cluster galaxies can constrain the progenitors of the ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Previous observational studies in general support the presence of negative gradients, although exceptions are not uncommon4,19,34,35, which implies that the gradient may depend on the particular assembly history of individual clusters9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The absence of the ICL colour gradient in most of the cases in our sample indicates that gradual stripping is not likely to be the dominant mechanism of ICL production within the 1 ≲ z ≲ 2 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' It is possible that occasional major mergers can potentially mix the intracluster stars and flatten the ICL colour profile even in the case where ICL production through stripping is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, as the ICL colours in the flat gradient cases are in good agreement with those of the reddest cluster members, our observation cannot reconcile with this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 1 | BCG + ICL images of our ten z ≳ 1 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The images were created by masking out every discrete source detected by SExtractor except for the BCGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Here the result is based on an expansion coefficient of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We also visually scanned the result and applied additional manual masking for the objects that SExtractor failed to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Green dashed lines show the SMA = 200 kpc ellipses, whose ellipticities and position angles are determined by AutoProf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Throughout the paper, we assume a flat Λ- dominated cold dark matter cosmology characterized by h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 and Ωm,0 = 1 − ΩΛ,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3, where h, Ωm,0 and ΩΛ,0 represent the dimensionless Hubble, matter density, and dark energy density parameters at present day, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' XMM1229/E160W SPT2106/E140W M001142/F140W RDCS1252/E160W 100 kpc 100 kpc 100 kpc 100 kpc M001014/F140W SPT0205/F140W XMM2235/F160W SpARCS1049/F160W 100 kpc 100 kpc 100 kpc 100 kpc DCS1426/F140W JKCS041/F160W 22 24 26 28 μ (mag arcsec-2) 100 kpc 100 kpc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2 | BCG + ICL radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Data points are the observed surface brightness from the elliptical bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The errors are computed from quadratic sums of the background level error and shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' With dashed and solid lines, we show our best-fit multi-Sérsic component model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Magenta (cyan) dashed lines are the innermost (outermost) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' When the target requires three components, we use orange lines to represent the middle component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Red solid lines illustrate the summation of all components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The legends of each panel show the target name, filter type, χ2 value and best-fit Sérsic indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 167 167 XMM1229/F105W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' XMM1229/F160W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' SPT2106/F105W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' SPT2106/F140W 18 - Fitting,x= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='15 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='59 18- Fitting,x2= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='66 Fitting, x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='04 20 BCG, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 BCG, n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 arcsec-2) 20 BCG, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 BCG, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 ICL, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ICL, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 0:0:00 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 22 22 24 (mag 24 26 - 8 26 28 28 30 30 - 100 101 102 100 101 102 100 101 102 100 101 102 16 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' MOO1142/F105W MOO1142/F140W 16 RDCS1252/F105W RDCS1252/F160W 18 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='47 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='87 18 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='46 Fiting, x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 20 BCG2, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 BCG2, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 arcsec 20 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 22 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 ICL, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 22 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ICL, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 24 (mag : 24 8 26 - 8 26 - 28 28 - 30 30 - 100 101 102 100 101 102 100 101 102 100 101 102 167 MO01014/F105W MO01014/F140W 167 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' SPT0205/F105W SPT0205/F140W 18 - Fitting, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='77 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='72 18 - Fitting, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='85 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='17 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 BCG1, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 BCG, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 BCG, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 arcsec-2) 20 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 arcsec-2) 20 ICL,n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ICL, n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 22 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 (mag 24 (mag 24 26 - SB 26 28 28 30 30 - 100 101 102 100 101 102 100 101 102 100 101 102 167 167 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' XMM2235/F105W XMM2235/F160W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' SpARCS1049/F105W SpARCS1049/F160W 18 Fitting, x = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='12 Fiting, x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='59 18 Fitting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='10 Fitting, x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='19 BCG, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 BCG, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 BCG1, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 BCG1, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 arcsec-2) 20 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 arcsec-2) 20 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 BCG2, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 22 ICL,n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ICL, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 22 24 (mag 24 26 - 8 26 28 28 30 30 - 100 101 102 100 101 102 100 101 102 100 101 102 167 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' IDCS1426/F105W IDCS1426/F140W 167 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' JKCS041/F105W JKCS041/F160W 18- Fiting, x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='27 Fitting, x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='39 18 Fitting, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='68 Fitting, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='68 BCG1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 BCG1, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 BCG, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 BCG, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 arcsec-2) 20 BCG2, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 BCG2, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 g arcsec-2) 20 ICL,n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 ICL, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 ICL, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 22 ICL, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 22 24 (mag 24 26 - 26 28 28 30 - 30 - 100 101 102 100 101 102 100 101 102 100 101 102 SMA (kpc) SMA (kpc) SMA (kpc) SMA (kpc)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 3 | BCG + ICL radial colour profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Black solid lines are the observed colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The dark grey shades represent the 68% uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The scale radius of each component is shown with the same colour scheme used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The red data point is a representative mean value in each subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The orange stars indicate the colours and positions of spectroscopic member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The light grey shade indicates the radial extent of the BCG measured by SExtractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2 - Colour 0- 1 Representative value Representative value 100 101 102 100 101 102 2- Colour 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 0 Representativevalue 米 Representativevalue 100 101 102 100 101 102 2 - Colour 0- MO01014F105W-F140W 0 米 Representativevalue Representativevalue 100 101 102 100 101 102 2 - 2 Colour 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' XMM2235 0 SpARCS1049 F105W-F160W 105W-F160W 米 Representativevalue 米 Representativevalue 100 101 102 100 101 102 2- Colour 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 0 Representativevalue Representativevalue 100 101 102 100 101 102 SMA (kpc) SMA (kpc) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4 | ICL fraction evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Filled red circles are the current results based on the r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 Mpc aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We extrapolated the best-fit Sérsic profiles to the same aperture to estimate the total ICL flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' When we avoided the extrapolation and performed integration only within the range where the SB profile is above the detection limit, the resulting ICL fraction is reduced by about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The mean aperture size of the literature sample is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='58 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The dashed line and pink shade show the best- fit linear regression and its 68% uncertainty, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We weighted all data points equally and adjusted them in such a way that the reduced χ 2 value becomes unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The comparison between our high-redshift and the literature low-redshift samples shows that there is no significant evolution of the ICL fraction with redshift in observation, which contradicts the current theoretical prediction 1 (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although here we displayed the theoretical model that estimates the ICL fraction based on the SBC at 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 mag arcsec-2, similarly steep evolutions are obtained even when different ICL definitions such as binding energy criteria are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 50 11j [12] 13) 40 [19] [21] This study 30 ICL fraction (%) 20 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 zMethods Target selection We searched the Mikulski Archive for Space Telescope for the WFC3-near-infrared imaging programs that have observed z ≳ 1 clusters in at least two filters with the surface brightness limit μ ≈ 28 mag arcsec-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We excluded the targets if they do not possess any distinct BCGs, or very bright stars/foreground galaxies are present near the cluster centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The search resulted in a sample of ten galaxy clusters at 1 ≲ z ≲ 2 with a minimum (maximum) redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='98 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='803).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Extended Data Table 1 summarizes our target selection, including the redshift, coordinate, program number, surface brightness limit and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although F105W exists for all ten clusters, either F140W or F160W is available for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Reduction pipeline optimized for ICL measurement The common data reduction procedure recommended in the HST Data Handbook 34 works well if one is interested in discrete astronomical sources such as stars and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, when we are looking for signals from diffuse components whose surface brightness is within a subpercentage of the sky brightness and slowly varies across the detector, additional care is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Our reduction begins with the flat-fielded (FLT) images processed by the Space Telescope Science Institute calwf3 tool36, which removes most instrumental signatures of WFC3, except for geometric distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We visually inspected these FLT images and manually masked out any remaining artefacts such as satellite/asteroid trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We ran the TweakReg package 37 for astrometric calibration by finding common astronomical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The FLT images are already flatfielded with the default Space Telescope Science Institute composite flats, which are claimed to be accurate within less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5%, except for the region within 128 pixels of the detector edge34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This claim has been verified by independently constructing residual flats utilizing large WFC3 survey programs4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We also investigated the impact of the residual flat in the final mosaic by performing drizzling as if we were stacking science frames and found that the dithered residual flats would cause at most around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4% errors, which is already negligible compared to other sources of errors (for example, background determination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this study, we applied these residual flats to our FLT data to further reduce the residual flat errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The application of the aforementioned residual flatfielding cannot remove large-scale sky gradients arising from intrinsic sky gradient, detector persistence, internal reflection and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We removed the sky gradient by fitting a first-order polynomial plane (F(x,y) = ax + by + c) to the object-masked residual-flatfielded image and subtracting the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The subtraction result was visually inspected, and we discarded the frame if the first-order polynomial plane could not adequately describe the sky gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To create a final deep mosaic where ICL is measured, careful and consistent sky subtraction from each exposure is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In typical ground-based data reduction designed for non-ICL-related studies, position-dependent sky estimation/subtraction is routinely performed after astronomical objects are masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although this scheme may provide cosmetically ‘good’ results when different frames are combined together, the inevitable consequence is sky oversubtraction in the region where non-discrete astronomical components are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Therefore, in this study, we determined only the global sky level (that is, a single constant) for each frame and subtracted it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Because individual exposures observed with dithers cover non-identical areas around the target, it is necessary to choose the same physical sky region that is present in common for all exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To accomplish this, we set up the largest possible annulus that is approximately centred on the BCG and is observed by all exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' An illustration of this common annulus is presented in Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Discrete astronomical sources in this annulus were detected using SExtractor 38 with the settings of DETECT_MINAREA = 5 and DETECT_ THRESHOLD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As a matter of course, the resulting segmentation map fails to include the contributions from the faint diffuse wings of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To address this, we gradually expanded the segmentation maps and investigated the resulting background level change as a function of the mask size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' One should not increase the segmentation map by expanding the boundaries at the same rate for all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' That is, the segmentation boundaries of compact objects should expand slower than those of extended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Thus, we used SExtractor’s half- light radius rh and scaled the expansion with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The width (w) of the expansion band is determined by w = ce rh, where ce is the expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We stopped the segmentation map expansion at ce = 6, beyond which the sky estimation converged (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Note that this exposure-by- exposure sky level estimation is performed on the individual drizzled images to minimize the impact of the geo- metric distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We created the final mosaic using AstroDrizzle 33 with its sky subtraction option turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The output pixel scale is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='05 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We used the Gaussian kernel for drizzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 3 schematically summarizes our data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Object masking for ICL measurement To characterize ICL from the mosaic image, it is necessary to mask out light from discrete objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In the background level estimation above, we found that the choice ce = 6 was sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, this large expansion coefficient cannot be used in the central region of the cluster because doing so would leave very few pixels there, resulting in too large statistical errors exceeding the systematic error caused by the incomplete masking (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Therefore, for ICL measurement from the final stack, we chose to employ ce = 2 for object masking and apply a correction factor to obtain the result effectively measured with the full mask (ce = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The correction factor is derived by comparing multiple ICL profiles measured with different ce values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We masked out every discrete source detected by SExtractor except for the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We also visually scanned the result and applied manual masking for the objects that SExtractor failed to identify (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To assess the validity of our statistical correction scheme, we repeated our analysis with the full (ce = 6) masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This is supposed to generate results with better accuracy at the expense of precision (that is, smaller systematic errors and larger statistical errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, if any large systematic discrepancy from our fiducial measurement (for example, statistically corrected result after the use of ce = 2) is found, this indicates that the aforementioned correction scheme is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The mean of the differences is close to zero (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='02%, Extended Data Table 2), which verifies that our correction scheme with the use of the moderate masking expansion (ce = 2) is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Radial profile measurement with elliptical binning As the BCG + ICL isophotes are elliptical, the use of circular binning would spread the BCG- ICL transition over multiple bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Thus, we measured the radial profile of the BCG + ICL using an elliptical bin- ning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To determine the ellipticity and position angle (PA) of the ellipse, we used the AutoProf package39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' With both detection image and masking map as inputs, AutoProf calculates ellipticity and PA based on isophotal fitting and Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Two sets of outputs are generated by AutoProf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' One is a series of ellipticity and PA values, which vary with radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The other is a single pair of ellipticity and PA, which represents the global shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The former is useful when one’s interest is the radius-dependent isophotal shape of high signal- to-noise-ratio objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this study, where our scientific interest is faint diffuse light, we use the second set of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We defined a series of semi-major axes (SMA) with a logarithmic scale and measured the surface brightness at each radial bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We applied the 3σ clipping method to minimize the impact of the outliers and adopt the median as the representative surface brightness of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This also reduces the effects of any potential unidentified substructures within each elliptical annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The total error of the surface brightness estimate is computed as the quadratic sum of the 1σ photon noise and sky estimation (background level) uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As mentioned above, the residual flat error is negligible and hence is not included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The latter dominates our error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Multicomponent decomposition The traditional method for measuring ICL is to define an SBC and char- acterize the light component fainter than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As the choice of the threshold is arbitrary, it is difficult to use the method to compare results from different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In particular, because the current redshift regime (1 ≲ z ≲ 2) is considerably different from those of the previous studies, one cannot objectively characterize the ICL properties based on this traditional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this study, we decompose the BCG + ICL profile into multiple Sérsic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A Sérsic profile 40 is defined as: 𝐼(𝑟) = 𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' exp +−𝑏" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='/ # #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 $/" − 123, where r, re, n and Ie are the radius, half right radius, Sérsic index and intensity at the half right radius, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this equation, bn is the constant that is not independent and is solely determined by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Then, the BCG + ICL profile S(r) is modelled as a superposition of PSF-convolved multiple Sérsic components Im(r): S(r)=PSF(r)⁎∑mIm(r), where PSF(r) is the PSF radial profile and the symbol ‘*’ represents the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The PSF radial profile was constructed by combining the core from observed stellar images and the wing from the TinyTim 41 result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' If the PSF correction procedure is omitted, the mean ICL fraction increases by around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5% (Extended Data Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' How do we determine the total number of Sérsic components for each cluster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A model based on more components has higher degrees of freedom and leads to smaller residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, the drawback is overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' About 70% of elliptical galaxies are reported to require more than one Sérsic component to adequately describe their profiles 42–44 In this study, we limit the maximum number of Sérsic components for the description of the BCG + ICL profile to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To determine the optimal number of Sérsic components, we use two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The first is the Bayes factor K45, defined as follows: 𝐾 = &((|*") &((|*#), where p(D|Mk) is the probability of the data (D) given the model Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As P(D|Mk) cannot be computed directly, in practice K is evaluated as follows: 𝐾 = &(*"|()&(*#) &(*#|()&(*") = &(*"|() &(*#|(), where we assume the equality between the two priors p(M1) and p(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The Bayes factor K informs us of how the first model M1 is preferred over the second M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Its outstanding advantage is that K inherently penalizes the model according to its degrees of freedom based on first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The second criterion is the number of inflexion points in the derivative of the SB radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Naturally, the existence of l inflexion points implies a preference towards a model with l + 1 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' After investigation of our ten clusters with the above two criteria, we find that for the entire sample (1) the Bayes factor K between the best and second-best models is ln(K) ≳ 2, (2) the optimal number of Sérsic components inferred from the Bayes factor agrees with the result from the inflexion point analysis and (3) the SB profiles require either two or three components for optimal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Estimation of the total cluster luminosity One of the key requirements in measuring the ICL fraction is a robust selection of the cluster member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this study, the first step towards this goal is the compilation of the results from previous spec- troscopic studies (see Extended Data Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We used this spectroscopic cluster member catalogue to define the initial red sequence locus from the colour–magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' By iteratively applying linear regression to the red sequence and selecting objects within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 mag from the best-fit line, we built up the second-stage cluster member catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In this iteration, we removed stars using the CLASS_STAR value reported by SExtractor and the objects either brighter than the BCG or fainter than F105W = 26 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' This second-stage cluster member catalogue needed to be improved because the distribution of the object distance from the best-fit red sequence line was asymmetric (the blue side was blended with the neighbouring blue cloud).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Thus, we fitted a double Gaussian model to the distribution (see the bottom panel of Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The new centre and 1σ width of the red sequence were used to update the intercept and width of our previous best-fit linear regression result and the final cluster member catalogue was obtained (see the top panel of Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Although we employed a sophisticated procedure for selection of cluster members, inevitably the method is designed to select only red members, except for the blue spectroscopic members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' If the contribution from the blue members is large, our ICL fraction would be overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' However, we argue that the overestimation, if any, would not be significant because (1) in most cases the spectroscopic catalogue includes the brightest blue cluster members (Extended Data Table 3), (2) even the brightest blue cluster members are found to be still a few magnitudes fainter than the BCG and other brightest red members and (3) some fraction of galaxies in our red sequence catalogue are non-cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We verified this claim by utilizing the publicly available photometric redshift catalogue for SPT0205 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 46), which includes the blue cluster member candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' When we repeated the measurement of the ICL fraction with it, the resulting ICL fraction shifted by only 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The total luminosity is estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' First, we masked out non-member galaxies/stars from our imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The total masked area is non-negligible and simply assigning zero flux to the area would lead to substantial underestimation of the total luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Thus, we filled the masked regions with the predicted flux from our best-fit multi-Sérsic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Finally, the total luminosity is computed by the summation of the pixel values of the resulting image, which is comprised of the flux from the BCG, ICL and cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Impact of the red sequence selection criteria Although we took care to robustly define the locus of the red sequence through somewhat sophisticated iteration, two of the remaining ambiguities worth further investigation are the faint-end limit and the width of the red sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' To examine the dependence of the ICL fraction on these selection criteria, we considered the following three additional cases: Test A: the magnitude limit decreased to 24th mag Test B: the magnitude limit increased to 28th mag Test C: the width increased to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5σ We list the test results for individual clusters in Extended Data Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In the case of Test A, the average (maximum) increase in ICL fraction is found to be approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9% (approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Test B shows that the average (maximum) decrease in ICL fraction is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9% (approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Finally, Test C gives an average (maximum) decrease of approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9% (approximately 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In summary, our ICL fraction measurements are not sensitive to the selection criteria tested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' ICL fraction versus mass correlation Together with the ICL fraction evolution with redshift, the ICL community has also been investigating the correlation between ICL fraction and halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Under the assumption that massive halos represent older populations, a strong correlation would imply a significant time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 5 shows the ICL fraction versus mass relation for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The masses come from weak lensing studies (Extended Data Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' No significant correlation is observed for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Online content Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information, details of author contributions and competing interests, and statements of data and code availability are available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1038/s41586-022-05396-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Data availability The raw HST near-infrared imaging data used for the current study are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The processed imaging data are available on the github repository at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='com/Hyungjin-Joo/High_z_ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Source data are provided with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Code availability An exhaustive repository of code for our custom data processing and analyses reported in this manuscript are available on the github repository at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='com/Hyungjin- Joo/High_z_ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Acknowledgements This study is based on observations created with NASA/ESA Hubble Space Telescope and downloaded from the Mikulski Archive for Space Telescope at the Space Telescope Science Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The current research is supported by the National Research Foundation of Korea under programme 2022R1A2C1003130 and the Yonsei Future-Leading Research Initiative programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Author contributions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' conceived, designed and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' analysed the Hubble Space Telescope imaging data, developed the pipeline, interpreted the results and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 1 | Definition of common sky areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (A) Exposure map for the single-frame image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (B) Same as (C) except that it is for the mosaic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (C) Science image for single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (D) Same as (C) except that it is for the mosaic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The pink circular region in (A) is the region that is observed in common by all contributing frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (B) shows how this common region is positioned in one of the input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As the central region of this circle is likely to be heavily influenced by the ICL, we excluded the central region and instead defined the annulus shown in (C) and (D) to estimate the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (A) exposuretime along blue horizontal line (B) line horizontal blue along exposuretime exposuretimealongblueverticalline exposuretimealongblueverticalline C Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2 | Masking size growth and impacts on background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (A), (B) and (C) illustrate our scheme for masking size growth from the original to the ce = 2 and ce = 6 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Note that we exhaust pixels for ICL measurement at ce = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' In (D), we show how the background level (green) changes as we vary the masking size using the expansion coefficient for a single exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' We observe that at ce ≳ 6 the measurement converges (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The black solid line indicates the result when instead we use a 3σ clipping algorithm without considering the diffuse wings of the astronomical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The yellow line shows the surface brightness level measured at each ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (E) is the same as the left except that the measurement is from the final deep stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Solid lines indicate the median value and shaded regions show the 68% uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' As the image is deeper, the number of pixels discarded (masked out) at the same ce value is much greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Original_mask Ce = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 Ce = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 (A) (B) (C) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='30 (D) Sky All Light Sources sky value of this methid SB [count / s / pixe/2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 sky value of 3-sigma clipping 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='15 2 4 6 8 10 12 Ce (E) Sky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 All Light Sources [z/axid / s / sunolas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 2 4 6 8 10 12 Ce Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 3 | Schematic diagram of our ICL-oriented data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Dark grey rectangles show the steps where external packages are used, while light grey rectangles illustrate our custom procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Parallelograms represent the input/output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Reject Start no Astrometric Residual Fitting flt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='fits Calibration Plane fitting Flat Fielding Success?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (TweakReg) yes flt_plf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='fits Segmentation Expanded Source Detection Map ExpandingMask Mask (SExtractor) (For masking) Map Image Drizzling High Overlapped Overlapped Drizzled Image SkyEstimation (Astrodrizzle) Region Selection WCS Sky Level Image Drizzling Final ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='ReDrizzledImage Sky Subtraction End (Astrodrizzle) Drizzled Image Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4 | Red sequence selection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Here we display the case for SPT2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (A) Colour–magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Black dots are all sources detected by SExtractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The red dots represent the spectroscopic members, whereas the orange dots are our red sequence candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The BCG is indicated with a red star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The red dashed line shows the final, best-fit red sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The dot-dashed lines bracket the 68% distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (B) Distribution of the F105W < 26 object distances from the best- fit red sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The green line shows the best-fit double Gaussian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The yellow line illustrates a single Gaussian component, which represents the distribution of the red sequence candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='75 F105W - F140W (Auto) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 Red sequence 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='00 All BCG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Member Member Candidate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='50 18 19 20 21 22 23 24 25 26 27 (B) F105W (ISO) Line 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='50 d Sequence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='00 Distance from Red PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='25 DoubleGauss Gauss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='50 Red Sequence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='75 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='d All 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 Number of Galaxies Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 5 | Comparison between ICL fraction and cluster mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The mass comes from weak lensing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' No significant correlation between ICL fraction and mass is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 35 30 - 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' fraction [%] 20 T 15 ICL 10 T T 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 M200[Mo] 1e15Extended Data Table 1 | Target List Target (Short Name) Redshift (z) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='A Dec HST Proposal ID SB limit [mag / arcs2] (Exposure Time [s]) F150W F140W F160W XDCP J1229+0151 (XMM1229) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='98 12:29:28 +01:51:34 12501 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='62 (1,311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='12 (1,111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7) SPT-CL J2106-5844 (SPT2106) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1312 21:06:05 58:44:42 13677, 14327 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='26 (12,567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='59 (12,771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='6) MOO J1142_1529 (MOO1142) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='19 11:42:46 +15:27:14 14327 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='96 (6,283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='26 (6,983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8) RDCS J1252-2927 (RDCS1252) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='237 12:52:57 29:27:15 12501 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='26 (1,211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='00 (1,211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='7) MOO J1014+0038 (MOO1014) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='24 10:14:08 +00:38:26 13677, 14327 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='47 (18,255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='63 (17,810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1) SPT-CL J0205-5829 (SPT0205) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='322 02:05:46 58:29:06 13677, 14327 28.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='12 17 54 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Rudick, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Mihos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & McBride, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The quantity of intracluster light: comparing theoretical and observational measurement techniques using simulated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 732, 48–64 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Contini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', De Lucia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Villalobos, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Bogani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' On the formation and physical properties of the intracluster light in hierarchical galaxy formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 437, 3787– 3802 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Burke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Collins, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Stott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Hilton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Measurement of the intracluster at z ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 425, 2058–2068 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Ko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Jee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Evidence for the existence of abundant intracluster light at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 862, 95–103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Montes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Trujillo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Intracluster light at the Frontier - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The Frontier Fields Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 474, 917–932 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' DeMaio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The growth of brightest cluster galaxies and intracluster light over the past 10 billion years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 491, 3751–3759 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Gonzalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Galaxy cluster baryon fractions revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 778, 14–29 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Presotto, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Intracluster light properties in the CLASH-VLT cluster MACS J1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='2- 0847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 565, A126 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Contini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Yi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Kang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Theoretical predictions of colors and metallicity of the intracluster light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 871, 24–33 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Burke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Hilton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Collins, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Coevolution of brightest cluster galaxies and intracluster light using CLASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 449, 2353–2367 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Morishita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Characterizing intracluster light in the Hubble Frontier Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 846, 139–151 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Almao-Martinez, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Blakeslee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Specific frequencies and luminosity profiles of cluster galaxies and intracluster light in Abell 1689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 849, 6–24 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Ellien, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The complex case of MACS J0717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5+6745 and its extended filament: intra-cluster light, galaxy luminosity function, and galaxy orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 628, A34 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Feldmeier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Intracluster planetary nebulae in the Virgo Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Luminosity of the intracluster light and tests of the spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 475, 2853–2869 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 16.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 475, 648–675 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Murante, G.' metadata={'source': 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+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Purcell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Bullock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Zentner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Shredded galaxies as the source of diffuse intrahalo light on varying scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 666, 20–33 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Furnell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The growth of intracluster light in XCS-HSC galaxy clusters from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='1 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 502, 2419–2437 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Guennou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Intracluster light in clusters of galaxies at redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='4 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 537, A64 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Contini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' On the origin and evolution of the intra-cluster light: a brief review of the most recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' MDPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 9, 60 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Murante, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The importance of mergers for the origin of intracluster stars in cosmological simulations of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 377, 2–16 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Contini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=', Yi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Kang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The different growth pathways of brightest cluster galaxies and intracluster light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 479, 932–944 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Tang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' An investigation of intracluster light evolution using cosmological hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 859, 85–97 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' DeMaio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' On the origin of the intracluster light in massive galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 448, 1162–1177 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' DeMaio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Lost but not forgotten: intracluster light in galaxy groups and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 474, 3009–3031 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Sahu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' WFC3 Data Handbook v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='5 (STScI, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Hoffmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The DrizzlePac Handbook v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='0 (STScI, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Bertin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' & Arnouts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' SExtractor: software for source extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 117, 393– 404 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Stone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' AutoProf - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' An automated non-parametric light profile pipeline for modern galaxy surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 508, 1870–1887 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Sérsic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Boletin de la Asociacion Argentina de Astronomia La Plata Argentina 6, 41–43 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Krist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Tiny Tim: an HST PSF simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astronomical Data Analysis Software and Systems II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 52, 536 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Oser, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The two phases of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 725, 2312–2323 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The Carnegie-Irvine Galaxy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The three-component structure of nearby elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 766, 47 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The Carnegie-Irvine Galaxy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' A method to determine the average mass ratio of mergers that built massive elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 821, 114–133 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Gill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Bayesian Methods: A Social Behavioral Science Approch 2nd edn (CRC, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Balogh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The GOGREEN and GCLASS surveys: first data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 500, 358–387 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Balogh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The GOGREEN and GCLASS surveys: first data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 500, 358–387 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Santos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Multiwavelength observations of a rich galaxy cluster at z ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The HST/ACS colour-magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 501, 49–60 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Gongalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' The massive and distant clusters of WISE Survey: MOO J1142+1527, a 10 15 M⊙ galaxy cluster at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 812, L40 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Demarco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' VLT and ACS observations of RDCS J1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='9-2927: dynamical structure and galaxy populations in a massive cluster at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content='237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 663, 164–182 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfj_0H/content/2301.01523v1.pdf'} +page_content=' 50.' metadata={'source': 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clustering with machine +learning in CAMELS +Ana Maria Delgado,1★ Daniel Anglés-Alcázar,2,3 Leander Thiele,4 Michelle Ntampaka,5,6 Shivam Pandey,7,8 +Kai Lehman,9,10 Lars Hernquist,1 Rachel S. Somerville,3 Shy Genel,3 Francisco Villaescusa-Navarro3 +1Center for Astrophysics, Harvard and Smithsonian, 60 Garden Street, Cambridge, MA, 02138 USA +2Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046, Storrs, CT, 06269, USA +3Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA +4Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ, 08544, USA +5Data Science Mission Office, Space Telescope Science Institute, Baltimore, MD, 21218, USA +6Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, 21218, USA +7Department of Physics, Colombia University, New York, NY, USA +8Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104, USA +9Institute for Astronomy, University of Hawai’i, 2680 Woodlawn drive, Honolulu, HI, 96822, USA +10Universitäts-Sternwarter München, Fakultät für Physik, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679, München, Germany +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Extracting information from the total matter power spectrum with the precision needed for upcoming large cosmological surveys +requires unraveling the complex effects of galaxy formation processes on the distribution of matter. In this work, we investigate +the impact of baryonic physics on matter clustering at 𝑧 = 0 using a large library of power spectra from the Cosmology and +Astrophysics with MachinELearningSimulations(CAMELS)project,containingthousandsof (25 ℎ−1Mpc)3 volumerealizations +with varying cosmology, initial random field, stellar and AGN feedback strength, sub-grid model implementation, and (magneto)- +hydrodynamics methods. We show that baryonic physics can profoundly affect matter clustering on scales 𝑘 ≳ 0.1 ℎ Mpc−1 and the +magnitude of this effect is highly dependent on the details of the galaxy formation implementation and variations of cosmological +and astrophysical parameters. Increasing AGN feedback strength decreases halo baryon fractions and yields generally stronger +suppression of power relative to N-body simulations, while stronger stellar feedback often results in weaker overall effects by +suppressing black hole growth and therefore the impact of AGN feedback. We find a broad correlation between mean baryon +fraction of massive halos (𝑀200c > 1013.5 M⊙/ℎ) and suppression of matter clustering but with significant scatter compared +to previous work owing to wider exploration of feedback parameters and cosmic variance effects. We show that a random +forest regressor trained on the baryon content and abundance of halos across the full mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 +can predict the effect of galaxy formation on the matter power spectrum on scales 𝑘 = 0.5–20 ℎ Mpc−1, providing access to +cosmological information in the highly non-linear regime. +Key words: galaxies: halos, clustering – cosmology: large-scale structure of Universe, theory – machine learning: random forest +– methods:numerical +1 INTRODUCTION +The field of cosmology has many exciting endeavors to look forward +to within the next decade. With the arrival of enormous photomet- +ric and spectroscopic galaxy redshift survey missions such as DESI +(DESI Collaboration et al. 2016), the Nancy Roman Space Telescope +(Spergel et al. 2015), Euclid (Laureijs et al. 2011) and the Vera Rubin +Observatory (LSST Science Collaboration et al. 2009), the commu- +nity will have the opportunity to tackle many ambitious goals, such as +mapping the distribution of matter and the large scale structure of the +★ E-mail: ana_maria.delgado@cfa.harvard.edu (AMD) +Universe, measuring cosmological parameters to percent-level preci- +sion, and constraining the sum of neutrino masses. An important step +in fully realizing the statistical power of these upcoming surveys is +to model the matter power spectrum and other summary statistics to +∼1% precision down to scales as small as 𝑘 = 10 ℎ Mpc−1 (Huterer & +Takada 2005; Laureijs 2009; Hearin et al. 2012). However, previous +studies have shown that complex galaxy formation processes involv- +ing feedback from massive stars and active galactic nuclei (AGN) can +suppress power relative to dark matter-only simulations out to large +scales (van Daalen et al. 2011; van Daalen et al. 2020; Chisari et al. +2018; Gebhardt et al. 2023). Galactic winds driven by supernovae +and AGN-driven outflows can eject a large amount of material from +© 2021 The Authors +arXiv:2301.02231v1 [astro-ph.GA] 5 Jan 2023 + +2 +Delgado et al. +the center of galaxies out to large distances (Anglés-Alcázar et al. +2017b; Borrow et al. 2020; Hafen et al. 2020; Wright et al. 2020; +Ayromlou et al. 2022; Mitchell & Schaye 2022; Sorini et al. 2022) +and the resulting suppression of power by feedback creates signif- +icant biases when attempting to constrain cosmological parameters +(Semboloni et al. 2011; Zentner et al. 2013; Chisari et al. 2019). +Several approaches to addressing the suppression of matter clus- +tering caused by baryonic physics have been devised. Cosmological +hydrodynamic simulations provide the most direct method to under- +stand the impact of baryonic effects on the distribution and clustering +of matter (Vogelsberger et al. 2014b; Hellwing et al. 2016; Tenneti +et al. 2015; Springel et al. 2018; Chisari et al. 2019). Modern cos- +mological large-volume simulations such as Horizon-AGN (Dubois +et al. 2014), Eagle (Schaye et al. 2015), IllustrisTNG (Pillepich et al. +2018b; Springel et al. 2018; Nelson et al. 2018; Naiman et al. 2018; +Marinacci et al. 2018), and SIMBA (Davé et al. 2019) produce galax- +ies that broadly match observations in properties such as the stellar +mass function and the bimodality in galaxy colors. Comparing the +power spectrum of hydrodynamic simulations with those of their +phase-matched, collisonless N-body, dark matter-only simulations +allows us to measure how baryonic feedback suppresses the cluster- +ing of matter. However, many key feedback processes remain poorly +understood and most current models require extensive tuning of free +parameters to match observations, limiting their predictive power +(Somerville & Davé 2015). Higher resolution cosmological “zoom- +in” simulations can reduce subgrid model uncertainties (e.g., Agertz +& Kravtsov 2016; Hopkins et al. 2018; Anglés-Alcázar et al. 2021), +but at the expense of modeling volumes that are too small for many +cosmological applications. +More flexible approaches to address the impact of baryonic physics +using analytic models include: modifying the “halo model” (Seljak +2000; Semboloni et al. 2013; Fedeli 2014; Mead et al. 2015) us- +ing observational constraints and simulation results as the basis for +parameterizing the transfer of power produced by the presence of +baryons (Mohammed & Seljak 2014; Schneider & Teyssier 2015), +and mitigating the presence of baryons altogether by marginalizing +over the parameters of effective models (Semboloni et al. 2011) or +over the principle components in linear combinations of observables +that are most strongly affected by baryonic effects (Eifler et al. 2015; +Kitching et al. 2016). However, the success of these techniques rely +heavily on the flexibility of the models to capture the true underlying +distribution of matter (McCarthy et al. 2017) and they are limited +by assumptions about halo bias relative to the linear density field, +smooth halo profiles neglecting substructure, and uncertainties in +the spatial and redshift dependence of baryonic effects (Chisari et al. +2019). Alternatively, power spectra produced by a large number of +cosmological simulations with varying cosmologies and feedback +parameters can be used to inform semi-analytic models attempting +to mitigate the effects of baryons, characterize the theoretical uncer- +tainties in galaxy formation, and marginalize over feedback effects. +van Daalen et al. (2011) employed a suite of 50 cosmological +hydrodynamic simulations from the OWLS project (Schaye et al. +2010) to study the effects of different baryonic processes on the mat- +ter power spectrum over a range of scales and, more recently, van +Daalen et al. (2020) (henceforth vDMS) included additional simula- +tions from the cosmo-OWLS (Le Brun et al. 2014) and BAHAMAS +(McCarthy et al. 2017) projects to produce a library of 92 matter +power spectra from simulations with varying subgrid models and +feedback strengths. Relating the effects of galaxy formation physics +to the suppression of power, vDMS proposed that it is possible to +predict the fractional impact of baryons on the clustering of matter, +𝑃hydro/𝑃DM, given only the mean baryon fraction of massive halos +(𝑀halo ∼ 1014 M⊙), where 𝑃hydro and 𝑃DM are the matter power +spectra from hydrodynamic simulations and their corresponding N- +body simulations, respectively. Importantly, the empirical vDMS re- +lation between baryon fraction and power suppression is satisfied +by a variety of simulations with different galaxy formation imple- +mentations, including the Horizon-AGN, EAGLE, and IllustrisTNG +simulations, which opens the possibility to accurately correct dark +matter only power spectra based on observational constraints on gas +fractions in massive halos. However, this relation is valid only on +large scales, 𝑘 ≤ 1 ℎ Mpc−1, and the still limited number of differ- +ent feedback implementations and cosmologies represented in the +vDMS library of matter power spectra may not be representative of +a broader range of plausible galaxy formation models. +In this work, we use 2,000+ cosmological hydrodynamic simu- +lations and their corresponding collisionless (N-body) simulations +from the Cosmology and Astrophysics with MachinE Learning Sim- +ulations (CAMELS1) project (Villaescusa-Navarro et al. 2021c) to +examine the impact of baryonic physics on matter clustering using +the largest library of power spectra available including variations of +cosmological and feedback parameters. In recent related work using +CAMELS, Nicola et al. (2022) trained a neural network on thou- +sands of electron density auto-power spectra from large scales down +to 𝑘 = 10 ℎ Mpc−1, breaking the baryon-cosmology degeneracy and +providing tight constraints on the total matter density Ωm and the +mean baryon fraction in intermediate-mass halos while marginaliz- +ing over uncertainties in galaxy formation physics implementations. +Here, we significantly expand upon the work of vDMS and investigate +how supernova and AGN feedback affect the mean baryon fraction +across a range of halo masses (1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014) and +the resulting impact on the matter power spectrum. Furthermore, +we take advantage of the design of CAMELS for machine learning +and train a random forest regressor to predict the relative difference +between the matter clustering in hydrodynamical and N-body simu- +lations on scales 𝑘 = 0.5–20 ℎ Mpc−1 given the mean baryon fraction +of halos across a broad range of halo masses. We thus demonstrate +that we are able to extract valuable information from lower mass +halos and predict the suppression of power all the way to the highly +non-linear regime. The work presented here is complementary to +Pandey et al. (2023), which show that information about the impact +of baryonic effects on the matter power spectrum can be extracted us- +ing the tSZ signals from low-mass halos, and include related results +utilizing the suite of CAMELS produced with the Astrid simulation. +The layout of this paper is as follows: In section 2 we describe +the simulations and halo selection, define our variables, and describe +our machine learning methods. In sections 3 and 4 we present our +results. Finally, in section 5 we provide a summary and discussion of +our work. +2 METHODS +2.1 Simulations: CAMELS +The CAMELS project (Villaescusa-Navarro et al. 2021c) contains +thousands of state-of-the-art (magneto-)hydrodynamic simulations +and their corresponding N-body simulations. In this work, we focus +on the simulation suites produced with the IllustrisTNG (Pillepich +et al. 2018b; Springel et al. 2018; Nelson et al. 2018; Naiman +et al. 2018; Marinacci et al. 2018) and SIMBA (Davé et al. 2019) +galaxy formation models that are part of the CAMELS public data +1 https://www.camel-simulations.org +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +3 +release2 (Villaescusa-Navarro et al. 2022), providing us with two +independent feedback model implementations and thousands of pa- +rameter variations. Each simulation is a periodic box of length +𝐿box = 25 ℎ−1Mpc containing 2563 resolution elements with mass +resolution of 6.49 × 107 (Ωm − Ωb)/0.251 ℎ−1M⊙ for dark matter +and 1.27 × 107 ℎ−1M⊙ for baryons. This is the same resolution as +the original SIMBA simulation and similar to that of the original +TNG300-1 simulation of IllustrisTNG. +The initial conditions of CAMELS simulations were generated at +𝑧 = 127 using second order Lagrangian perturbation theory assuming +that the initial power spectra of dark matter and gas are the same +and equal to that of total matter. Each of the CAMELS simulations +contains 34 snapshots from redshifts 𝑧 = 6 down to 𝑧 = 0; in this +work we focus on 𝑧 = 0. In addition to the initial random phases, +each simulation is specified by two cosmological parameters and +four astrophysical (feedback) parameters which are varied across the +individual realizations. In the case of cosmological parameters, we +vary: +• Ωm: the fraction of the Universe made up of ordinary and dark +matter varies in the range Ωm ∈ [0.1, 0.5] while keeping Ωb = 0.049 +constant. +• 𝜎8: the variance of the spatial fluctuations of total matter on +8 Mpc ℎ−1 scales is varied in the range 𝜎8 ∈ [0.6, 1.0]. +In the case of astrophysical parameters, the fiducial values are de- +fined by the stellar and AGN feedback models of the corresponding +original IllustrisTNG and SIMBA simulations. The fiducial astro- +physical parameters are assigned a value 𝐴 = 1.0 and then varied +across realizations by multiplying by an amplitude factor 𝐴 in order +to increase/decrease the amount of feedback. However, we emphasize +that the stellar and AGN feedback prescriptions differ substantially +between IllustrisTNG and SIMBA and the corresponding parameter +variations in CAMELS have a different definition in each model, +which we briefly describe below. +2.1.1 IllustrisTNG +The IllustrisTNG model (also referred to as “TNG”; Pillepich et al. +2018b; Springel et al. 2018; Nelson et al. 2018; Naiman et al. 2018; +Marinacci et al. 2018) is implemented in the AREPO hydrodynam- +ics code (Springel 2010; Weinberger et al. 2020), which utilizes a +hybrid tree/particle-mesh scheme to solve for gravitational interac- +tions and an unstructured, moving mesh to solve the equations of +hydrodynamics. Compared to the galaxy formation model of its pre- +decessor Illustris (Vogelsberger et al. 2014a,b; Genel et al. 2014), +the galaxy formation model in IllustrisTNG has updated implemen- +tations of AGN feedback (Weinberger et al. 2017) and galactic winds +(Pillepich et al. 2018a), and incorporates magnetic fields (Pakmor +et al. 2014). +The stellar feedback parameter variations in the CAMELS-TNG +simulations introduce 𝐴SN1 to control the total energy injection rate +in galactic winds per unit star formation (𝐴SN1 ∈ [0.25, 4.0]) and +𝐴SN2 to vary the galactic wind speed (𝐴SN2 ∈ [0.5, 2.0]). The AGN +feedback parameter variations pertain to the low-accretion, kinetic- +mode black hole feedback, where 𝐴AGN1 varies the feedback en- +ergy per unit black hole accretion rate (𝐴AGN1 ∈ [0.25, 4.0]) and +𝐴AGN2 varies the burstiness and effective ejection speed (𝐴AGN2 ∈ +[0.5, 2.0]). +2 https://camels.readthedocs.io +2.1.2 SIMBA +The SIMBA galaxy formation model (Davé et al. 2019) is imple- +mented in the GIZMO meshless finite mass hydrodynamics code +(Hopkins 2015, 2017). Relative to its predecessor MUFASA (Davé +et al. 2016), SIMBA includes a black hole model based on gravi- +tational torque accretion and two-mode kinetic feedback (Anglés- +Alcázar et al. 2017a), galactic winds with mass-loading and velocity +scalings derived from the FIRE zoom-in simulations (Muratov et al. +2015; Anglés-Alcázar et al. 2017b), and a model for the creation and +destruction of dust (Li et al. 2019). +The stellar feedback parameter variations in the CAMELS-SIMBA +simulations introduce 𝐴SN1 to control the mass loading factor of +galactic winds and 𝐴SN2 to control the wind speed. The AGN feed- +back parameter variations introduce 𝐴AGN1 to change the total mo- +mentum flux of either quasar-mode winds or radio-mode jets, while +𝐴AGN2 controls the maximum velocity of gas ejected by jets. These +parameters are varied over the same range as in IllustrisTNG, with +𝐴SN1 = 𝐴SN2 = 𝐴AGN1 = 𝐴AGN2 = 1 corresponding to the fiducial +model. +2.1.3 Simulation sets in CAMELS +We take advantage of the following simulation sets in CAMELS: +• Latin Hypercube (“LH”) set: 1,000 realizations each con- +taining different initial conditions and different values of the six +aforementioned parameters. The LH set is the main training set in +this work. +• 1 Parameter (“1P”) set: 66 realizations using the same initial +conditions and further divided into six subsets of 11 realizations +where only the value of one parameter is varied while the other five +parameters are held constant. In this work we make use of the 1P +sets to study how a single cosmological or feedback parameter can +affect halo baryon fractions and the suppression of the matter power +spectrum. +• Cosmic Variance (“CV”) set: 27 realizations with different ini- +tial conditions while the values of all six parameters are held constant. +The CV set is used to evaluate the impact of cosmic variance on any +of the quantities that we measure from the simulations. +We refer the reader to Villaescusa-Navarro et al. (2021c) for further +details about CAMELS, the parameter variations, and the simulation +sets available. +2.2 Halo Selection +We identify halos in CAMELS using the AMIGA Halo Finder (AHF; +Knollmann & Knebe 2011). AHF uses an adaptive mesh to locate +halo centers, calculate the gravitational potential of the halo and it- +eratively remove unbound particles (particles whose velocities are +greater than the escape velocity at a given radius) from within the +boundary of the halo. We refer the reader to (Knollmann & Knebe +(2011)) for a full description and implementation of AHF. We select +halos with masses 𝑀halo ≥ 1010ℎ−1M⊙ using a virial radius defini- +tion of 200c (i.e., 200 times the critical density of the Universe). +2.3 Matter power spectra and halo baryon fractions +We use a library of 4,000+ total matter power spectra from CAMELS +(Villaescusa-Navarro et al. 2021c, 2022). For each simulation, the +MNRAS 000, 1–18 (2021) + +4 +Delgado et al. +matter power spectrum is computed by assigning particle masses +(dark matter, gas, stars, and black holes) to a regular grid with 5123 +voxels. The grid is then Fourier transformed and the power spectrum +is computed by averaging over 𝑘-bins with an equal width to the +fundamental frequency, 𝑘F = 2𝜋/𝐿, where 𝐿 = 25ℎ−1Mpc. We then +compute the relative difference between the total matter power spec- +trum of hydrodynamical and phase-matched N-body simulations, +which we refer to as the “suppression of matter power spectrum” and +define as: +Δ𝑃 +𝑃DM += +𝑃hydro − 𝑃DM +𝑃DM +, +(1) +where 𝑃DM is the matter power spectrum of the N-body simulation +and 𝑃hydro is that of its corresponding hydrodynamical simulation. +We compute the baryon fraction of a given halo as: +𝑓bar = 𝑀star + 𝑀gas +𝑀halo +, +(2) +where 𝑀star and 𝑀gas are the total stellar mass and gas mass of the +halo and 𝑀halo is the virial mass of the halo corresponding to 𝑅200c. +We further calculate the mean baryon fraction within a given halo +mass range in each simulation as: +𝑓 bar = 1 +𝑛 +𝑛 +∑︁ +𝑖=1 +𝑓bari/ Ωb +Ωm +, +(3) +where 𝑓bar is defined in Equation 2, subscript 𝑖 is the 𝑖th halo and 𝑛 +the total number of halos in a given mass range, and following vDMS +we normalize by Ωb/Ωm in order to account for the differences in +cosmology for different simulations. +2.4 Machine Learning +A supervised machine learning algorithm trains a model by provid- +ing a subset of data, referred to as the training set, including input +variables (henceforth called “features”) and output variables (hence- +forth called “target”). The goal is for the algorithm to use the training +set to learn the relation between the features and the target. The +trained model is then used to predict the target for a different subset +of features referred to as the test set. +In this work we use the random forest regressor algorithm from the +publicly available package Scikit-Learn (Pedregosa et al. 2011). +A random forest (RF) is an ensemble machine learning method that +can be used for both classification and regression problems. The +algorithm works by constructing a “forest” from a user specified +number of decision trees and using the mean of the predictions from +those trees as output. This method has three key advantages: 1) little +hyper-parameter tuning is required, 2) it is computationally efficient, +and 3) its ensemble characteristic lessens over fitting. Furthermore, +the RF algorithm provides us with some interpretability by way of +the “feature importance” attribute, with a ranking of features based +on their importance as predictor variables. +We use the following metrics for scoring the predictive perfor- +mance of the RF: +R2(y, ˆy) = 1 − +�n +i=1(yi − ˆyi)2 +�n +i=1(yi − y)2 , +(4) +RMSE(𝑦, ˆ𝑦) = +√︄�𝑛 +i=1(𝑦i − ˆ𝑦i) +𝑛 +, +(5) +where 𝑦i are the given target values, ˆ𝑦i are the RF predicted target +values, and 𝑦 is the mean of 𝑦i. The R2 score provides the proportion +of the target variable that is predictable by the given features. Because +the R2 outputs a score between 0.0–1.0, it provides comparable in- +formation about performance when comparing various experiments. +The RMSE scores, on the other hand, are based on the target value +range. Therefore, in order to account for the range in target values +across multiple experiments, we normalize our RMSE scores by the +Interquartile range (IQR): +𝐼𝑄𝑅 = 𝑄3 − 𝑄1, +(6) +where 𝑄3 is the 3rd quartile (75th percentile) of a given set and 𝑄1 +is the first quartile (25th percentile) of the set. In other words, we +normalize the RMSE by the middle 50% dispersion of the target +values as RMSE/IQR. +2.4.1 Features and Targets +Using the thousands of realizations in the CAMELS LH simulation +sets, we train random forest regressors to predict the suppression of +the matter power spectrum Δ𝑃/𝑃DM at a range of scales based on the +baryon fraction and abundance of halos. We construct the following +features for each realization: +• 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1): the mean baryon fraction of +massive halos, those with masses greater than 1013.5 M⊙ℎ−1 in each +simulation. +• 𝑓 bar(𝑀j +halo); array of mean halo baryon fraction, binned by +halo mass in the range [1010 − 1015] M⊙ℎ−1. +• 𝑁j +halo: the number of halos per mass bin. +Our target are the Δ𝑃/𝑃DM values for each realization at five +different 𝑘-values: 𝑘 = [0.5, 1, 5, 10, 20] ℎ Mpc−1. +2.4.2 Robustness of random forest predictions +One inherent benefit of CAMELS is that we are able to test the effects +of feedback model implementation by way of its TNG and SIMBA +simulations sets. We create an 80% / 20% train/test split of the LH +simulations and perform the following experiments using either TNG +or SIMBA: +• Train on 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) to predict Δ𝑃/𝑃DM at +𝑘 = [0.5, 1, 5, 10, 20] ℎ Mpc−1. We perform this experiment at +𝑘 = 0.5 with the intent of comparing our results with vDMS. +• Train on 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) and 𝑁halo for high-mass +halos to predict Δ𝑃/𝑃DM at 𝑘 = [0.5, 1, 5, 10, 20] ℎ Mpc−1. +• Train on 𝑓 bar(𝑀j +halo) and 𝑁j +halo to predict Δ𝑃/𝑃DM at 𝑘 = +[0.5, 1, 5, 10, 20] ℎ Mpc−1. +We are further able to determine how well our algorithm can +marginalize over subgrid physics model by performing “two-model” +experiments where we train on the entire LH set of one of the feed- +back implementations and test on the entire LH set of the other, i.e., +training on SIMBA and testing on TNG and viceversa. For these tests +of robustness we use the same setup that produced the best results +from the above experiments. +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +5 +3 IMPACT OF COSMOLOGICAL AND FEEDBACK +PARAMETER VARIATIONS +In this section we examine the impact of cosmological and baryonic +feedback parameter variations on the matter power spectrum and the +baryon fraction of halos of different masses, exploring also the con- +nection between the suppression of the total matter power spectrum +and the mean baryon fraction of massive halos. We perform this +analysis for both the TNG and SIMBA galaxy formation models. +3.1 Matter power spectra +We use the total matter power spectra from the 1P simulations to +examine how cosmological and feedback parameters affect the clus- +tering of matter at various scales. For each run in the 1P set, described +in section 2.1, we measure the fractional impact of baryons on the +total matter power spectrum, 𝑃hydro/𝑃DM. If baryonic physics has +no effect on matter clustering, 𝑃hydro/𝑃DM should be of order unity. +However, if baryonic physics suppresses the clustering of matter com- +pared to dark-matter only simulations, usually by way of feedback +ejecting gas out to large distances, 𝑃hydro/𝑃DM should fall below +unity on a range of scales. +Fig. 1 shows 𝑃hydro(𝑘)/𝑃DM(𝑘) as a function of wave number +𝑘 for the 1P simulations, where each spectrum is color coded by +the value of each parameter variation. We notice two overall trends +that are roughly independent of cosmological or feedback parame- +ters. The first is the general “scoop” shape of 𝑃hydro/𝑃DM, which is +consistent with previous works (van Daalen et al. (2011); Hellwing +et al. (2016); Peters et al. (2018); Chisari et al. (2018); Springel +et al. (2018); vDMS). This shape conveys agreement between matter +clustering in hydrodynamical and dark matter-only simulations on +large scales (𝑘 ≲ 0.1 ℎ Mpc−1) while at intermediate scales there is +suppression of power by baryonic feedback (𝑃hydro/𝑃DM < 1) and +at small scales (𝑘 ≳ 40 ℎ Mpc−1) there is enhanced, as opposed to +suppressed, clustering relative to dark matter owing to gas dissipa- +tive processes (𝑃hydro/𝑃DM > 1). The second overall trend is that the +SIMBA galaxy formation model (solid lines) tends to suppress power +on intermediate scales more strongly compared to the TNG galaxy +formation model (dashed lines) while driving a steeper increase in +small-scale clustering (𝑘 ⪅ 30 ℎ Mpc−1). +We now analyze in more detail how each parameter affects mat- +ter clustering by comparing 𝑃hydro(𝑘)/𝑃DM(𝑘) between the fiducial +models of TNG and SIMBA (shown in red) and that of the individual +parameter variations: +• Cosmological parameters: The top two panels in Fig. 1 show +the sensitivity of the total matter power spectrum to Ωm and 𝜎8 for +a fixed galaxy formation model. We see a strong dependence of +𝑃hydro/𝑃DM on the value of Ωm both in TNG and SIMBA. As Ωm +decreases (at fixed Ωb), there is a greater suppression of power on +intermediate scales. This can be understood as a consequence of +baryons contributing a higher fraction of the total matter content +making feedback more efficient at pushing gas out of halos and +distributing matter on larger scales, in agreement with the analysis +of large-scale baryon spread in Gebhardt et al. (2023). In contrast, +we identify weaker trends for 𝜎8, with significant scatter. +• Supernova feedback parameters: The middle two panels in +Fig. 1 show the impact of changing the stellar feedback parameters +𝐴SN1 and 𝐴SN2 respectively, which control the mass loading and +velocity of galactic winds, on matter clustering. Both panels show +somewhat counterintuitive effects of stellar feedback. Increasing +𝐴SN1 in SIMBA reduces (rather than enhances) the suppression +of power on small scales (𝑘 ≳ 10 ℎ Mpc−1) and increasing 𝐴SN2 +further increases 𝑃hydro/𝑃DM over the full range of scales. This +can be understood as a consequence of the nonlinear interplay +between stellar and AGN feedback, where stronger stellar feedback +suppresses black hole growth and results in weaker effective impact +of AGN feedback on matter clustering (van Daalen et al. 2011; +Gebhardt et al. 2023). The TNG model shows rather different trends, +with reduced suppression of power on scales 𝑘 ≲ 10–20 ℎ Mpc−1 but +enhanced suppression of power on smaller scales when increasing +𝐴SN1 and 𝐴SN2. These results are consistent with the analysis of +electron power spectra in CAMELS by Nicola et al. (2022), which +highlights the sensitivity of predicted baryonic effects on galaxy +formation implementation. +• AGN feedback parameters: The bottom two panels of Fig. 1 show +the impact of varying AGN feedback efficiency on matter clustering. +In this case, there are clear systematic trends for stronger suppression +of power when increasing both 𝐴AGN1 and 𝐴AGN2 for both galaxy +formation models (TNG and SIMBA). The sensitivity of 𝑃hydro/𝑃DM +to 𝐴AGN1 is weaker given its range of variation, with no more than +10% difference relative to the fiducial model. In contrast, the matter +power spectrum in SIMBA displays a strong sensitivity to the AGN +jet speed, 𝐴AGN2, with strong suppression of power across scales, +reaching 𝑃hydro/𝑃DM ∼ 0.6 at 𝑘 ∼ 10 ℎ Mpc−1 with jets twice as +fast relative to the fiducial model. This results are also consistent +with previous findings for electron power spectra Nicola et al. (2022) +and the impact of large scale jets on cosmological baryon spread +(Gebhardt et al. 2023). +3.2 Halo baryon fraction +In Fig. 2 we use again the specialized CAMELS 1P simulation sets to +analyze the impact of individual cosmological and feedback param- +eter variations on the average halo baryon fraction as a function of +halo mass, 𝑓 bar(𝑀halo), where we consider logarithmically-spaced +halo mass bins in the range 1010–1013 M⊙ℎ−1. We notice two main +trends roughly independent of cosmological or feedback parameters +when comparing the fiducial realizations (indicated in red) for the +TNG (dashed lines) and SIMBA (solid lines) models. The first being +that peak of the halo baryon fraction occurs at ∼ 1012 M⊙ℎ−1. We +notice a drop in mean baryon fraction as halos exceed this mass range, +when powerful feedback process can expel material out of the halo. +We note, however, that at very high mass halos, we expect feedback +to be less efficient at expelling material and for there to be another +rise in mean baryon fraction. The second main trend is that SIMBA +has overall lower 𝑓 bar(𝑀halo) compared to TNG, with the fiducial +models reaching their peak at 𝑓 bar∼ 0.5 for SIMBA and 𝑓 bar∼ 0.7 +for TNG. +We now analyze in more detail how each CAMELS parameter +variation affects halo baryon fractions, keeping in mind that the +definitions of feedback parameters are not the same for TNG and +SIMBA: +• Cosmological parameters: The top two panels of Fig. 2 show +the sensitivity of 𝑓 bar(𝑀halo) to our cosmological parameters Ωm +and 𝜎8. Halo baryon fractions appear to be more sensitive to +cosmology in SIMBA compared to TNG. However, both galaxy +formation models predict qualitatively similar trends, with lower +𝑓 bar(𝑀halo) when increasing Ωm and 𝜎8 across a range of halo +masses. This trend may seem trivial for Ωm since we hold the value of +Ωb constant in all CAMELS simulations, implying that the average +MNRAS 000, 1–18 (2021) + +6 +Delgado et al. +100 +101 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Phydro/PDM +Ωm +SIMBA fiducial +IllustrisTNG fiducial +100 +101 +σ8 +100 +101 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Phydro/PDM +ASN1 +100 +101 +ASN2 +100 +101 +k [h Mpc−1] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Phydro/PDM +AAGN1 +100 +101 +k [h Mpc−1] +AAGN2 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1 +2 +3 +4 +1 +2 +3 +4 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Figure 1. The effect of baryonic physics on matter clustering for different cosmological and feedback parameter variations. Each panel shows the ratio of total +matter power spectrum in hydrodynamic simulations to that of the corresponding dark matter-only simulations (𝑃hydro/𝑃DM) as a function of wave number 𝑘 +when varying a single parameter in the CAMELS 1P sets. Lines of different colors indicate the value of each parameter variation, and red lines indicate the +fiducial model for TNG (dashed lines) and SIMBA (solid lines). Variations in feedback model, as well as in feedback amplitude, result in variation in total matter +clustering. +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +7 +cosmic baryon fraction decreases with higher Ωm and so should +the corresponding halo baryon fractions. However, 𝑓 bar(𝑀halo) is +normalized by Ωb/Ωm for each simulation (Equation 3), removing +the trivial effect of varying Ωm at fixed Ωb. The impact of increasing +Ωm on 𝑓 bar(𝑀halo) is thus a reflection of the effective efficiency of +feedback when changing the amount of baryons relative to the dark +matter gravitational potential, and this effect seems more prominent +in lower mass halos for both TNG and SIMBA. Interestingly, the +baryon fraction decreases systematically at all halo masses when +increasing 𝜎8, while the suppression of power does not seem to +follow a clear trend with 𝜎8. +• Supernova feedback parameters: The middle two panels in +Fig. 2 show the impact of changing 𝐴SN1 and 𝐴SN2 on halo baryon +fractions. As for the power spectra, varying the mass loading of +galactic winds (𝐴SN1) has a different effect in each galaxy formation +model. We might intuitively expect that as 𝐴SN1 increases, more +gas would be ejected out of galaxies resulting in lower 𝑓 bar(𝑀halo). +However, we only see this behavior in TNG for halos with mass +𝑀halo ≲ 1012 M⊙ℎ−1, while the baryon fraction of higher mass +halos increases with 𝐴SN1 owing to the suppression of AGN +feedback. This reversed trend with 𝐴SN1 can explain its impact on +𝑃hydro/𝑃DM for TNG, with an enhancement of power on interme- +diate scales coinciding with the highest gas fraction in high mass +halos with 𝐴SN1 = 4. Meanwhile, SIMBA shows systematically +higher baryon fractions when increasing 𝐴SN1 across the full halo +mass range, indicating a different non-linear coupling of stellar and +AGN feedback compared to TNG, which can explain the reduced +suppression of power seen in Fig. 1. On the other hand, increasing +the speed of galactic winds (𝐴SN2) results in systematically lower +𝑓 bar(𝑀halo) values for both SIMBA and TNG, but in this case lower +baryon fractions correlate with less suppression of matter clustering +in SIMBA on all scales. +• AGN feedback parameters: The bottom two panels of Fig. 2 +show the sensitivity of 𝑓 bar(𝑀halo) to changes in AGN feedback effi- +ciency. Halo baryon fractions are significantly reduced by increasing +the kinetic mode black hole feedback efficiency 𝐴AGN1 in TNG +in the intermediate halo mass range 𝑀halo = 1011–1012.5M⊙ℎ−1, +while the burstiness parameter 𝐴AGN2 has a stronger effect reducing +𝑓 bar(𝑀halo) in higher mass halos. In both cases, the decrease in halo +baryon fraction with higher AGN feedback efficiency correlates with +stronger suppression of matter clustering. Similarly, increasing the +AGN jet speed in SIMBA (𝐴AGN2) drives an overall reduction of +halo baryon fractions and increased suppression of matter clustering +on all scales shown in Fig. 1, corresponding to more efficient spread +of baryons on large scales relative to the TNG model (Tillman et al. +2022; Gebhardt et al. 2023). However, the effect of increasing the +momentum flux 𝐴AGN1 in SIMBA seems more complex, driving an +increase in baryon fraction in high mass halos (possibly due to black +hole self-regulation) but stronger suppression of power in the matter +power spectrum. +3.3 Suppression of matter power spectrum as a function of +baryon fraction +Using a suite of matter power spectra from hydrodynamical and dark +matter only simulations, vDMS found a tight relation between the +suppression of the matter power spectrum (Δ𝑃/𝑃DM; defined in §2.3) +in the linear regime and the average baryon fraction ( 𝑓 bar) of high +mass halos (∼ 1014 M⊙ℎ−1). In this study, we use the LH simulation +sets of CAMELS, described in §2.1, to investigate this relationship by +examining how it is affected by cosmological and baryonic feedback +parameters over a broader range of model variations. +Fig. 3 and Fig. 4 show Δ𝑃/𝑃DM ≡ (𝑃hydro−𝑃DM)/𝑃DM evaluated +at 𝑘 = 0.5 ℎ Mpc−1 as a function of 𝑓 bar for the TNG and SIMBA LH +sets, respectively. We note that due to the small simulated volumes in +CAMELS there are not enough halos of mass ∼ 1014 M⊙ℎ−1 to make +a direct comparison to the results of vDMS, and we therefore evaluate +𝑓 bar for halos with mass > 1013.5 M⊙ℎ−1. Each panel reproduces +the same data but color coded by the corresponding parameter value +for each of the six parameters that are simultaneously varied in the +TNG and SIMBA LH sets. This allows us to examine how individual +cosmological and feedback parameters affect the relation between +Δ𝑃/𝑃DM and 𝑓 bar, which we compare to the fitting function derived +by vDMS (their equation 5) for baryon fractions calculated using the +200c virial definition, which we henceforth refer to as the “vDMS +model” and indicate by the blue solid line and gray shaded region. +Lastly, we overlay the results for the CV sets of TNG and SIMBA +(described in §2.1) as the red triangles in the top center panel of +each figure in order to examine the effect of cosmic variance on this +relation. +Our CAMELS results in Figs. 3 and 4 reveal a good qualitative +agreement with the general trend found in vDMS: the suppression of +the matter power spectrum increases as the average baryon fraction in +massive halos decreases. While the SIMBA LH set probes a range of +Δ𝑃/𝑃DM and 𝑓 bar values significantly larger than the TNG LH set, as +expected from Figs. 1 and 2, both models roughly follow the vDMS +trend, suggesting that 𝑓 bar in massive halos can be used to infer +the redistribution of baryons over large scales regardless of galaxy +formation model implementation. However, we find considerable +spread in Δ𝑃/𝑃DM at fixed 𝑓 bar compared to vDMS, which can be +attributed to the broader range of parameter variations explored in +CAMELS but also to the impact of cosmic variance: +• Cosmological parameters: The left two panels in Figs. 3 +and 4 explore the dependence of the Δ𝑃/𝑃DM– 𝑓 bar relation on +Ωm and 𝜎8. For both TNG and SIMBA, there is a clear trend of +higher Δ𝑃/𝑃DM (i.e. less suppression of power) at fixed 𝑓 bar for +higher values of Ωm. This implies that the same impact on the +total matter power spectrum (at 𝑘 = 0.5 ℎ Mpc−1) can be predicted +by simulations that yield different halo baryon fractions, in this +case as a consequence of the different response of feedback to +changes in Ωm at fixed Ωb. There is also a visible, albeit less +pronounced, trend for 𝜎8, where Δ𝑃/𝑃DM becomes more negative +(i.e. stronger suppression of power) at fixed 𝑓 bar for higher values of +𝜎8. Overall, the non-linear response of the fiducial galaxy formation +model to variations in cosmology appears to explain a significant +fraction of the scatter in the vDMS relation seen for SIMBA and TNG. +• Supernova feedback parameters: The middle panels of Figs. 3 +and 4 explore the dependence of the vDMS relation on systematic +variations of the mass loading factor and speed of galactic winds +driven by stellar feedback (parameters 𝐴SN1 and 𝐴SN2, respec- +tively). In TNG, there is indication for simulations clustering around +Δ𝑃/𝑃DM ∼ 0 and 𝑓 bar∼ 1 for higher values of 𝐴SN1 and 𝐴SN2, +corresponding to weaker overall impact of feedback owing to the +suppression of black hole growth and therefore AGN feedback. +Decreasing the strength of stellar feedback tends to yield more +negative Δ𝑃/𝑃DM values and correspondingly lower 𝑓 bar (i.e., +stronger impact), displacing simulations roughly along the vDMS +relation but with increasing scatter. We find qualitatively similar +trends in SIMBA for variations in 𝐴SN2 while no clear dependence +MNRAS 000, 1–18 (2021) + +8 +Delgado et al. +1010 +1011 +1012 +1013 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f bar +Ωm +SIMBA fiducial +IllustrisTNG fiducial +1010 +1011 +1012 +1013 +σ8 +1010 +1011 +1012 +1013 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f bar +ASN1 +1010 +1011 +1012 +1013 +ASN2 +1010 +1011 +1012 +1013 +Mhalo [h−1 M⊙] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f bar +AAGN1 +1010 +1011 +1012 +1013 +Mhalo [h−1 M⊙] +AAGN2 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Figure 2. The effect of each parameter variation on the mean halo baryon fraction as a function of halo mass, 𝑓 bar(Mhalo). As in Fig. 1, the color bar for each +panel indicates the value of the corresponding parameter variation while all other parameters are held constant. The red lines indicate the fiducial parameters +for TNG (dashed lines) and SIMBA (solid lines), with their peak baryon fraction occurring at ≲ 1012 M⊙ℎ−1 in both fiducial models. The 𝑓 bar values and their +halo mass dependence differ substantially between galaxy formation implementations and model parameter variations. +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +9 +on 𝐴SN1, with overall larger scatter in the vDMS relation compared +to TNG. +• AGN feedback parameters: The right two panels of Figs. 3 +and 4 show the impact of AGN feedback parameter variations in the +vDMS relation. In this case, we find a general trend for simulations +with stronger AGN feedback to populate the lower-left region of the +Δ𝑃/𝑃DM– 𝑓 bar plane, indicating more efficient evacuation of gas +from halos and stronger suppression of matter clustering. In both +TNG and SIMBA, this trend is more apparent for 𝐴AGN2 compared +to 𝐴AGN1, as expected from Figs. 1 and 2. +• Cosmic variance: In order to examine how cosmic variance +affects the predicted variation of Δ𝑃/𝑃DM as a function of 𝑓 bar, +the top middle panel of Figs. 3 and 4 overlay the results from the +CAMELS CV sets corresponding to 27 realizations of the fiducial +TNG and SIMBA models using different initial conditions (red tri- +angles). For TNG, the CV set yields roughly similar range in average +baryon fraction of massive halos as the entire LH set, indicating that +stochastic variations owing to the small CAMELS volumes and cor- +respondingly low number of massive halos play an important role. +Nonetheless, the TNG CV simulations roughly follow the vDMS re- +lation. The SIMBA CV set also yields a wide range of Δ𝑃/𝑃DM and +𝑓 bar values, but in this case suggesting a systematic offset relative to +the vDMS model. +We can quantify the impact of cosmic variance on the predicted +suppression of matter clustering as the root mean square variation in +Δ𝑃/𝑃DM relative to the mean: +𝛿cv ≡ 𝜎cv +|𝑝cv| , +(7) +with 𝑝cv ≡ Δ𝑃/𝑃DM for the CV set and +𝜎2 +cv = 1 +n +n +∑︁ +i=1 +� +pi +cv − pcv +�2 +, +(8) +where 𝑛 = 27 realizations and 𝑝cv represents the average of 𝑝cv +over the CV set. Evaluating Eq. 7 for 𝑘 = 0.5 ℎ Mpc−1 gives 𝛿cv = +0.436 for TNG and 𝛿cv = 0.211 for SIMBA, indicating that there is +considerable variation due to cosmic variance alone. +The considerable spread of CAMELS predictions relative to the +vDMS model shown in this section provides motivation for the ma- +chine learning experiments described in section 2.4. Given the larger +data set in CAMELS with broader variations in feedback and cos- +mology compared to previous libraries of power spectra, it is possible +that the vDMS model relating halo baryon fraction and suppression +of matter clustering is not general enough to include every plau- +sible feedback model. For example, we later examine the original +SIMBA model against the vDMS relation and find that SIMBA does +not fall within 1% of the vDMS fit, as shown in Fig.11. However, it +is also possible that having smaller volumes which are significantly +affected by cosmic variance as compared to the data set in vDMS, +along with the lack of halos of mass ∼ 1014 M⊙ℎ−1, may explain +the disagreement between our results and the vDMS model. These +results motivate us to explore the relation between Δ𝑃/𝑃DM and +halo baryon fraction with a machine learning approach, where we +can extract information from a broader halo mass range to improve +the accuracy of predictions for the impact of baryonic physics on the +total matter power spectrum. +4 ESTIMATING THE IMPACT OF FEEDBACK ON +MATTER CLUSTERING WITH MACHINE LEARNING +A major goal of this work is to show that machine learning can be +used to extract information from the full range of halo masses in order +to estimate the suppression of the matter power spectrum by baryonic +processes all the way to the non-linear regime. In this section, we +discuss the results of training a random forest regressor (RF) to +estimate the impact of feedback on the clustering of matter using +the LH simulation sets in CAMELS, which vary simultaneously +cosmological and feedback parameters (§2.1). The general setup of +our experiments is described in §2.4. +4.1 Extracting information across the halo mass range with +random forest regression +Fig. 5 shows the results from training a RF regressor on differ- +ent input features to estimate the suppression of power Δ𝑃/𝑃DM at +𝑘 = 0.5 ℎ Mpc−1. We begin by training a RF with 𝑓 bar(𝑀halo > +1013.5 M⊙ℎ−1) as the only training feature, in analogy with the +information used by the vDMS fitting function. Halos of mass +∼ 1013.5 M⊙ℎ−1 are only available for ∼700 out of 1,000 LH re- +alizations for each of TNG and SIMBA, limiting the size of the +training set. In this first experiment, the RF is only able to predict +∼45% and ∼60% of the variation of Δ𝑃/𝑃DM in TNG and SIMBA, +respectively, with the predicted versus true values of Δ𝑃/𝑃DM shown +by the green data points in the left panels of Fig. 5. Next, we add +the number of high mass halos 𝑁halo corresponding to the measured +𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) as an additional input feature, with +results indicated by the orange data points. In this case, the RF pre- +dictions improved by ∼10% in TNG and ∼16% for SIMBA. We then +incorporate information from halos across the full mass range by +introducing the baryon fraction 𝑓 bar(𝑀j +halo) and the corresponding +number of halos 𝑁j +halo within each halo mass bin (see §2.4), with +results shown by the blue data points. In this case, we can use the full +LH sets of CAMELS for training and testing since we are not limited +by the availability of high mass halos. With these additional features +using information from a range of halo masses, the RF predicted +∼70% of the variation in Δ𝑃/𝑃DM at 𝑘 = 0.5 ℎ Mpc−1 for both TNG +and SIMBA, significantly improving upon the original results. +As stated in section 2.4, one advantage of the RF is that it provides +some level of interpretability by means of the “feature importance” +attribute. The right panels of Fig. 5 display the relative importance +assigned to each feature by the trained RF on the test set for TNG (top) +and SIMBA (bottom). Interestingly, the RF ranked 𝑓 bar in halos with +mass 𝑀halo = 1010.5–1011.0 M⊙ℎ−1 as the most important feature +to predict the suppression of power Δ𝑃/𝑃DM at 𝑘 = 0.5 ℎ Mpc−1 in +TNG. Furthermore, we see that the RF ranked several features across +the mass ranges as important predictors of Δ𝑃/𝑃DM in SIMBA. +These feature importance results reveal that the RF was able to extract +valuable information across a range of halo masses. +4.2 Random forest predictions in the highly non-linear regime +In the previous subsection, we have established that training the +RF on features from a range of halo masses improves the pre- +dictions for Δ𝑃/𝑃DM in the linear regime compared to using +𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) as a training feature alone. We now ex- +ploit the same methodology to extend our predictions into the highly +non-linear regime by repeating our experiments at a range of scales, +predicting Δ𝑃/𝑃DM at 𝑘 = [1, 5, 10, 20] ℎ Mpc−1. Fig. 6 provides a +MNRAS 000, 1–18 (2021) + +10 +Delgado et al. +−0.10 +−0.05 +0.00 +∆P/PDM +Ωm +vDMS model +ASN1 +CV set +AAGN1 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +−0.10 +−0.05 +0.00 +∆P/PDM +σ8 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +ASN2 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +AAGN2 +0.15 +0.25 +0.35 +0.45 +0.65 +0.75 +0.85 +0.95 +0.5 +1.5 +2.5 +3.5 +0.5 +1.5 +2.5 +3.5 +0.6 +1.0 +1.4 +1.8 +0.6 +1.0 +1.4 +1.8 +IllustrisTNG | k = 0.5 h Mpc−1 +Figure 3. Suppression of the matter power spectrum, Δ𝑃/𝑃DM, as a function of mean baryon fraction of high mass halos, 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1), for +the TNG LH simulation set. The blue line is the fitting function of vDMS for a halo definition of 200c (200 times the critical density of the Universe), with the +gray shaded region indicating 1% variation in Δ𝑃/𝑃DM. Each panel shows the same data points color coded by the value of each of the six parameters varied +simultaneously in the LH set. The middle top panel shows additional results from the CV simulation set, where all six parameters are constant and only the +initial conditions are varied. We find that Δ𝑃/𝑃DM increases for higher values of 𝑓 bar, meaning that there is less suppression of the matter power spectrum in +simulations where feedback is less effective at removing gas from halos. Data points fall generally along the vDMS model, with large scatter owing to broad +parameter variations and cosmic variance. +−0.10 +−0.05 +0.00 +∆P/PDM +Ωm +vDMS model +ASN1 +CV set +AAGN1 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +−0.10 +−0.05 +0.00 +∆P/PDM +σ8 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +ASN2 +0.0 +0.5 +1.0 +f bar(Mhalo > 1013.5) +AAGN2 +0.15 +0.25 +0.35 +0.45 +0.65 +0.75 +0.85 +0.95 +0.5 +1.5 +2.5 +3.5 +0.5 +1.5 +2.5 +3.5 +0.6 +1.0 +1.4 +1.8 +0.6 +1.0 +1.4 +1.8 +SIMBA | k = 0.5 h Mpc−1 +Figure 4. Same as Fig. 3 but for the SIMBA LH simulation set, which produces a much broader range of variation in Δ𝑃/𝑃DM and 𝑓 bar compared to the TNG +LH set. SIMBA also follows the general trend of the vDMS model but with considerably more scatter and some indication for a systematic offset relative to the +vDMS relation. +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +11 +−0.10 +−0.08 +−0.06 +−0.04 +−0.02 +0.00 +TRUE ∆P/PDM +−0.10 +−0.08 +−0.06 +−0.04 +−0.02 +0.00 +PREDICTED ∆P/PDM +IllustrisTNG +k = 0.5 | f bar(Mhalo > 1013.5) +k = 0.5 | f bar(Mhalo > 1013.5) & Nhalo +k = 0.5 | f bar(Mj +halo) & Nj +halo +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1 M⊙] +0.0 +0.1 +0.2 +0.3 +0.4 +IMPORTANCE +k = 0.5 [h Mpc−1] +f bar(Mj +halo) +Nj +halo +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +TRUE ∆P/PDM +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +PREDICTED ∆P/PDM +SIMBA +k = 0.5 | f bar(Mhalo > 1013.5) +k = 0.5 | f bar(Mhalo > 1013.5) & Nhalo +k = 0.5 | f bar(Mj +halo) & Nj +halo +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1 M⊙] +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +IMPORTANCE +k = 0.5 [h Mpc−1] +f bar(Mj +halo) +Nj +halo +Figure 5. Results from different Random Forest experiments. We create an 80/20 train/test split of the LH data sets to predict Δ𝑃/𝑃DM at 𝑘 = 0.5 for TNG +(top) and SIMBA (bottom). The left panels show the predicted target values compared to the true target values as given by the test sets in CAMELS, where the +red line indicates a perfect one-to-one relation. The green data points correspond to predictions by a RF trained only on the baryon fraction of massive halos, +𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1), the orange data points show results for a RF trained on 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) as well as the number 𝑁halo of massive +halos, while results from training a RF on 𝑓 bar(𝑀j +halo) and 𝑁 j +halo for halo mass bins 𝑗 spanning the full mass range are shown in blue. The right panels show +the feature importances corresponding to the blue data points in the left panel (training on 𝑓 bar(𝑀j +halo) and 𝑁 j +halo), indicating the relative rank ordering of +importance (from 0.0 to 1.0) given to each training feature by the RF. Predictions improve by providing training data across the full range of halo masses. +summary of performance scores for these experiments using the LH +simulation sets for TNG (squares) and SIMBA (circles). We also per- +formed additional experiments using the two LH data sets combined, +which we refer to as “TNG+SIMBA” (diamonds). Descriptions of the +scoring metrics can be found in §2.4. The top panels in Fig. 6 show +the R2 scores and the bottom two panels show the RMSE scores nor- +malized by the interquartile range (IQR). We normalize the RMSE by +the IQR in order to account for the variation in the range of Δ𝑃/𝑃DM, +which depends on the 𝑘 value; the range of Δ𝑃/𝑃DM increases as +we move to non-linear regimes. In addition to presenting results at a +range of scales for each training set, Fig. 6 compares the predictions +based on 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) alone (right panels) versus +providing the baryon fraction 𝑓 bar(𝑀j +halo) in different halo mass bins +(left panels). Results based on baryon fractions alone are shown in +orange while results that also incorporate the corresponding number +of halos 𝑁j +halo are shown in blue. +Training on 𝑓 bar(𝑀j +halo) and 𝑁j +halo improved the prediction of +Δ𝑃/𝑃DM for both TNG and SIMBA on all scales 𝑘 = 0.5– +20 ℎ Mpc−1 as measured by the R2 and RMSE/IQR scores (Fig. 6). +For TNG, we achieved the highest R2 score of all experiments +at 𝑘 = 5 ℎ Mpc−1, with R2 = 0.923 (consistent with the lowest +RMSE/IQR score). In other words, the RF was able to account for +approximately 92% of the variation in the suppression of the matter +power spectrum due to feedback using 𝑓 bar(𝑀j +halo) and 𝑁j +halo as +training features. This represents ∼40% improvement over train- +ing on 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) at 𝑘 = 0.5 ℎ Mpc−1 and +≳20% improvement over training on 𝑓 bar(𝑀j +halo) and 𝑁j +halo at +𝑘 = 0.5 ℎ Mpc−1. The trained RF also shows very good performance +down to smaller scales, with R2 = 0.85–0.9 at 𝑘 = 10–20 ℎ Mpc−1 +when training simultaneously on the baryon fraction and number of +halos in different mass bins. Similar results are obtained for SIMBA, +also performing better at 𝑘 = 5–20 ℎ Mpc−1 compared to larger +scales, with R2 scores slightly lower than TNG: R2 ≈ 0.8 − 0.85 at +𝑘 = 5–20 ℎ Mpc−1. Remarkably, while the vDMS model can only +predict Δ𝑃/𝑃DM for a given average baryon fraction of massive halos +on large scales 𝑘 < 1 ℎ Mpc−1, our RF regressor performs better on +scales where the impact of feedback on the matter power spectrum +becomes the highest. The strongest suppression of power occurs at +roughly 𝑘 = 10 ℎ Mpc−1 in SIMBA and 𝑘 = 20 ℎ Mpc−1 in TNG for +their fiducial models, and the RF is able to account for ∼80-85% of +the Δ𝑃/𝑃DM variation on these scales. +Fig. 7 illustrates in more detail the predicted results versus true +values of Δ𝑃/𝑃DM when training a RF regressor on different input +MNRAS 000, 1–18 (2021) + +12 +Delgado et al. +k [h Mpc−1] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +R2 score +Mj +halo +f bar +f bar & Nhalos +f bar +f bar & Nhalos +k [h Mpc−1] +Mhalo > 1013.5 [h−1 M⊙] +100 +101 +k [h Mpc−1] +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +RMSE/IQR +IllustrisTNG +SIMBA +TNG + SIMBA +100 +101 +k [h Mpc−1] +Figure 6. Performance scores for all RF experiments predicting Δ𝑃/𝑃DM at five different scales, 𝑘 = [0.5, 1, 5, 10, 20] ℎ Mpc−1. The orange dashed lines +connect data points showing scores for a RF trained on average baryon fractions 𝑓 bar, while the blue solid lines connect those where both 𝑓 bar and the number +of halos 𝑁halo are used as training features. We show results for three different data sets: the TNG LH set (squares), the SIMBA LH set (circles), and the two +LH sets combined, labeled “TNG+SIMBA” (diamonds). Left panels correspond to training on features from a range of halo masses, 𝑓 bar(𝑀j +halo), while the +right panels correspond to training on high-mass halos only, 𝑀halo > 1013.5 M⊙ℎ−1, as described in section 2.4. We report R2 scores (top) and RMSE scores +normalized by the interquartile range (IQR) of the respective data set (bottom); higher R2 scores and lower RMSE/IQR scores convey an improved performance. +We obtain higher scores when training on 𝑓 bar(𝑀j +halo) and 𝑁 j +halo for a range of halo masses at all scales compared to training on high-mass halos only, and the +highest scores occur in the highly non-linear regime at 𝑘 ∼ 5–10 ℎ Mpc−1, meaning that important information can be extracted from a range of halo masses in +the non-linear regime. +features at 𝑘 = 5 ℎ Mpc−1 for the TNG (top) and SIMBA (bottom) +LH sets. The highest R2 score was obtained at 𝑘 = 5 ℎ Mpc−1 for +TNG using 𝑓 bar(𝑀j +halo) and 𝑁j +halo as training features, which cor- +responds to the tighter distribution of blue data points along the +one-to-one line of perfect prediction in the top left panel, with re- +duced scatter compared to the prediction based on high mass halos +alone (orange data points). We find similar trends for SIMBA, with +an apparent increase in scatter relative to TNG as expected from the +lower R2 scores. Interestingly, the feature importance analysis (right +panels) indicates that the number of halos in the low mass range +𝑀halo = 1010.5–1011 M⊙ is contributing significantly to improve the +Δ𝑃/𝑃DM predictions. +4.3 Interpretation of feature importances +Comparing the feature importances shown in Figs. 5 and 7 for +𝑘 = 0.5 ℎ Mpc−1 and 𝑘 = 5 ℎ Mpc−1, respectively, it appears that the +most informative input features vary with scale. We further investi- +gate the physical properties that inform the prediction of Δ𝑃/𝑃DM by +the RF in Fig. 8, where we provide a summary of the relative im- +portance of the 𝑓 bar(𝑀j +halo) features (diamonds) and 𝑁j +halo features +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +13 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +TRUE ∆P/PDM +−0.7 +−0.6 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +PREDICTED ∆P/PDM +IllustrisTNG +k = 0.5 | f bar(Mhalo > 1013.5) +k = 5.0 | f bar(Mhalo > 1013.5) & Nhalo +k = 5.0 | f bar(Mj +halo) & Nj +halo +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1] M⊙ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +IMPORTANCE +k = 5.0 [h Mpc−1] +f bar(Mj +halo) +Nj +halo +−0.6 +−0.4 +−0.2 +0.0 +0.2 +TRUE ∆P/PDM +−0.7 +−0.6 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +PREDICTED ∆P/PDM +SIMBA +k = 0.5 | f bar(Mhalo > 1013.5) +k = 5.0 | f bar(Mhalo > 1013.5) & Nhalo +k = 5.0 | f bar(Mj +halo) & Nj +halo +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1 M⊙] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +IMPORTANCE +k = 5.0 [h Mpc−1] +f bar(Mj +halo) +Nj +halo +Figure 7. Similar to Fig. 5 but for results at 𝑘 = 5.0 ℎ Mpc−1. The left panels show the predicted versus true Δ𝑃/𝑃DM values for RF regressors trained on +𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) and 𝑁halo (orange) and trained on 𝑓 bar(𝑀j +halo) and 𝑁 j +halo (blue), both at 𝑘 = 5 ℎ Mpc−1. For comparison, we also show the +results using only 𝑓 bar(𝑀halo > 1013.5 M⊙ℎ−1) at 𝑘 = 0.5 ℎ Mpc−1 as in Fig. 5 (green). The right panels show the feature importances when training on +𝑓 bar(𝑀j +halo) and 𝑁 j +halo. Going to non-linear regimes the range of the target value Δ𝑃/𝑃DM increases, enabling more accurate predictions. The highest ranked +feature at 𝑘 = 5.0 ℎ Mpc−1 is 𝑁 j +halo at 𝑀halo = 1010.5–1011 M⊙ℎ−1, meaning that the number of low-mass halos is highly informative for estimating the power +at 5Mpc length scales. +(squares) corresponding to different halo mass bins when predicting +Δ𝑃/𝑃DM at a given scale 𝑘. The ranking of features for each 𝑘-value +is indicated by the color scale, which shows the log of the fractional +importance assigned by the RF. The most important ranked feature +at each 𝑘-value is further outlined in red. While the RF appears to be +learning from the baryon fractions and abundances of halos across the +halo mass range, there are some interesting trends that are worth not- +ing. The least informative features for both TNG and SIMBA corre- +spond to the most massive halos (𝑀halo ∼ 1014 M⊙ℎ−1), which is in +contrast to earlier work identifying the baryon fraction in groups and +clusters as a primary predictor of Δ𝑃/𝑃DM on scales 𝑘 < 1 ℎ Mpc−1 +(vDMS). Unsurprisingly, the small simulated volumes in CAMELS +contain a small number of massive halos, which are thus not opti- +mal as predictors of Δ𝑃/𝑃DM. In contrast, the RF assigns significant +importance to 𝑓 bar(𝑀j +halo) and 𝑁j +halo in low- to intermediate-mass +halos in the range 𝑀halo ∼ 1010.5–1012.5 M⊙ℎ−1 for all 𝑘-values +analyzed. +Interestingly, the number of halos 𝑁j +halo in the mass range +𝑀halo ∼ 1010.5–1011 M⊙ℎ−1 is among the top features identified +by the RF across different scales (see also Fig 7). We explore further +the significance of this feature in Fig. 9, where we show Δ𝑃/𝑃DM at +𝑘 = 5 ℎ Mpc−1 as a function of 𝑁j +halo in this halo mass range for the +LH sets of TNG (left) and SIMBA (right). We find that there is a +clear correlation between Δ𝑃/𝑃DM and the number of low-mass ha- +los for both galaxy formation models, as expected given that 𝑁j +halo is +identified by the RF as one of the most predictive features. Physically, +a plausible explanation for this correlation is that 𝑁j +halo for low mass +halos is a strong tracer of Ωm, as indicated by the color scale, and +Ωm itself is one of the main parameters driving large variations in +Δ𝑃/𝑃DM at all 𝑘-values in CAMELS (see Figs. 1, 3, and 4). This is +consistent with Pandey et al. (2023), which find a simple model is also +able to capture information about Δ𝑃/𝑃DM given 𝑓 bar in low-mass +halos and Ωm. While increasing the value of Ωm increases system- +atically the number of halos at all masses (e.g., Villaescusa-Navarro +et al. 2021c), we note that the correlation between Δ𝑃/𝑃DM and +𝑁j +halo worsens for higher halo mass bins, which are thus often as- +signed lower feature importance by the RF. This can be explained by +the number of halos in higher mass bins being more sensitive to cos- +mic variance and therefore not as good predictors of cosmology for +small simulated volumes. On the other hand, the lowest halo mass +bin considered here, 𝑀halo < 1010.5 M⊙ℎ−1, becomes unresolved +in terms of the minimum number of dark matter particles per halo +for the higher Ωm values in CAMELS. We therefore conclude that +𝑁j +halo in the halo mass bin 𝑀halo ∼ 1010.5–1011 M⊙ℎ−1 is the least +MNRAS 000, 1–18 (2021) + +14 +Delgado et al. +0.5 +1.0 +5.0 +10.0 +20.0 +k [h Mpc−1] +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1 M⊙] +IllustrisTNG +f bar(Mj +halo) +Nj +halo +−5 +−4 +−3 +−2 +−1 +log10 IMPORTANCE +0.5 +1.0 +5.0 +10.0 +20.0 +k [h Mpc−1] +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +13.5 +14.0 +14.5 +log10Mhalo [h−1 M⊙] +SIMBA +f bar(Mj +halo) +Nj +halo +−4 +−3 +−2 +−1 +log10 IMPORTANCE +Figure +8. +Feature +importance +summary +for +RF +models +trained +on +𝑓 bar(𝑀j +halo) +and +𝑁 j +halo +to +predict +Δ𝑃/𝑃DM +at +𝑘 += +[0.5, 1, 5, 10, 20] ℎ Mpc−1 for the TNG (top) and SIMBA (bottom) +models. For each halo mass bin and 𝑘-value, the relative importance of +the corresponding 𝑓 bar(𝑀j +halo) (diamonds) and 𝑁 j +halo (squares) features is +indicated by the color scale. The highest ranked feature for each 𝑘-value +is outlined in red. There is no clear trend of importance in features across +scales, suggesting feature importance results are specific to each scale. +sensitive to cosmic variance and it is a strong predictor of Δ𝑃/𝑃DM in +part because 𝑁j +halo informs the RF about the variation of Ωm in a +large number of well resolved halos. +4.4 Marginalizing over galaxy formation physics +A key advantage of CAMELS over more standard cosmological sim- +ulations performed with a single fiducial galaxy formation model +is the ability to train machine learning algorithms to learn funda- +mental properties of galaxies and the Universe while marginalizing +over uncertainties in subgrid physics (e.g., Villaescusa-Navarro et al. +2021b; Nicola et al. 2022; Perez et al. 2022; Shao et al. 2022b,a; +Villaescusa-Navarro et al. 2021a; Villanueva-Domingo et al. 2022). +When training a RF on the LH simulation set of either TNG or +SIMBA to predict Δ𝑃/𝑃DM given halo baryon fractions as input +features, we are at the same time marginalizing over uncertainties +in physical processes represented by the parameter variations intro- +duced in a given galaxy formation model. +However, evaluating the robustness of the ML model to uncer- +tainties in galaxy formation physics should also consider different +implementations and not just variations of parameters within a given +subgrid physics implementation. We thus perform a more stringent +test of robustness by training the RF on the full LH set of one galaxy +formation model (either SIMBA or TNG) and then testing on the +full LH set of the other model. Fig. 10 shows the predicted versus +true values of Δ𝑃/𝑃DM at 𝑘 = 5 ℎ Mpc−1 when training a RF using +𝑓 bar(𝑀j +halo) and 𝑁j +halo from SIMBA and testing on TNG (top) and +when training on TNG and testing on SIMBA (bottom). The top +panel of Fig. 10 shows that the RF trained on SIMBA can explain +∼80% of the variation of Δ𝑃/𝑃DM at 𝑘 = 5 ℎ Mpc−1 when tested +on TNG, suggesting that the RF has found a relation between halo +baryon fractions and suppression of matter clustering which is rel- +atively robust to galaxy formation model implementation. On the +other hand, the bottom panel of Fig. 10 shows that the RF trained +on TNG is less robust when tested on SIMBA, and can only explain +∼70% of the variation in Δ𝑃/𝑃DM predicted by the SIMBA model. +In this case, we can see that the predicted Δ𝑃/𝑃DM is clearly biased +high (i.e., less negative) when training on TNG and predicting on +SIMBA, implying that the inferred suppression of matter clustering +is under-predicted given the halo baryon fractions in SIMBA and the +connection to Δ𝑃/𝑃DM learned from TNG. As expected, we see a +bias in the opposite direction when training on SIMBA and predict- +ing on TNG (top panel), over-predicting the suppression of power at +𝑘 = 5 ℎ Mpc−1. +As seen in Figs. 3 and 4, the overall range of variation in +Δ𝑃/𝑃DM is significantly larger in the SIMBA LH set compared +to the TNG LH set. Given that the RF cannot predict values outside +of the range of the training data, this can explain why the RF trained +on TNG is less robust relative to galaxy formation implementation +and shows a stronger bias when tested on SIMBA. The biased estima- +tion of Δ𝑃/𝑃DM can thus be partially attributed to the RF learning +the limits of the range of variation in the training data. Nonetheless, +our results suggest that the TNG and SIMBA models may predict +different Δ𝑃/𝑃DM even when implementing parameters that yield +similar 𝑓 bar(𝑀j +halo), implying a non-unique relation between halo +baryon fractions and impact on matter clustering. +5 SUMMARY AND DISCUSSION +In this paper, we have investigated how baryonic physics affects +the clustering of matter relative to N-body simulations and its re- +lation to the baryonic content of halos using thousands of cos- +mological hydrodynamic simulations from the CAMELS project +(Villaescusa-Navarro et al. 2021c). In the first part of the paper, +we examined how variations of individual cosmological parameters +(Ωm and 𝜎8) and feedback parameters (controlling the efficiency +of large-scale outflows driven by SNe and AGN) impact the total +matter power spectrum, the mean halo baryon fraction as a function +of halo mass, and the connection between the suppression of clus- +tering Δ𝑃/𝑃DM and the mean baryon fraction of massive halos. In +the second part of the paper, we have presented a set of machine +learning experiments training a random forest (RF) regressor on fea- +tures including the baryon fraction and abundance of halos across +the mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 to predict the sup- +pression of power Δ𝑃/𝑃DM from linear to highly non-linear scales +𝑘 = 0.5–20 ℎ Mpc−1. Throughout the paper, we have made use of the +CAMELS simulation suites performed with the TNG (Pillepich et al. +2018b; Springel et al. 2018; Nelson et al. 2018; Naiman et al. 2018; +Marinacci et al. 2018)and SIMBA (Davé et al. 2019) models to un- +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +15 +500 +1000 +1500 +Nj +halo +−0.6 +−0.4 +−0.2 +0.0 +0.2 +∆P/PDM +k = 5.0 h Mpc−1 +Mj +halo(1010.5 − 1011.0) h−1 M⊙ +IllustrisTNG +500 +1000 +1500 +Nj +halo +SIMBA +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Ωm +Figure 9. Correlation between power spectrum suppression Δ𝑃/𝑃DM and the number 𝑁 j +halo of halos in the mass range 𝑀j +halo = 1010.5–1011 ℎ−1 M⊙ℎ−1 at +𝑘 = 5 ℎ Mpc−1 for TNG (left) and SIMBA (right), color coded by Ωm. The number of low mass halos 𝑁 j +halo (highest ranked feature at 𝑘 = 5 ℎ Mpc−1) is a +strong tracer of Ωm in CAMELS and a good predictor of Δ𝑃/𝑃DM. +−0.6 +−0.4 +−0.2 +0.0 +TRUE ∆P/PDM +−0.7 +−0.6 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +PREDICTED ∆P/PDM +Train : SIMBA +Test : IllustrisTNG +k = 5.0 [h Mpc−1] +R2 = 0.814 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +TRUE ∆P/PDM +−0.7 +−0.6 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +PREDICTED ∆P/PDM +Train : IllustrisTNG +Test : SIMBA +k = 5.0 [h Mpc−1] +R2 = 0.698 +Figure 10. Similar to Fig. 7 but training a RF on the entire LH set of one galaxy +formation model and testing on the entire LH set of the other for both TNG +and SIMBA. We note higher prediction score when training on SIMBA and +testing on TNG (𝑅2 = 0.814) compared to training on TNG and predicting +on SIMBA (𝑅2 = 0.698), which can be attributed to SIMBA’s wider range of +Δ𝑃/𝑃DM values. The RF tends to under-predict Δ𝑃/𝑃DM when trained on +SIMBA and over-predict Δ𝑃/𝑃DM when trained on TNG due to variations +in feedback models which suggests that care must be made if applying this +model to data where feedback is not precisely known. +derstand the dependence of results and the robustness of the trained +machine learning models to changes in the specific galaxy formation +physics implementation. Our main findings can be summarized as +follows: +• In agreement with previous work (e.g., van Daalen et al. 2011; +Chisari et al. 2018, 2019; Villaescusa-Navarro et al. 2021c), we +find that baryonic physics can profoundly affect the total matter +power spectrum all the way to scales 𝑘 < 0.5 ℎ Mpc−1, and the +magnitude of this effect is highly dependent on the details of the +galaxy formation implementation and variations of cosmological +and astrophysical parameters. +• The suppression of power |Δ𝑃/𝑃DM| increases systematically +with decreasing Ωm at fixed Ωb, with baryons contributing a higher +fraction of the total matter content and feedback more efficiently +spreading matter over larger scales relative to N-body simulations. +Varying 𝜎8 at fixed galaxy formation physics does not drive +systematic variations in Δ𝑃/𝑃DM when measured on the small +(25 ℎ−1Mpc)3 volumes simulated in CAMELS. +• Increasing AGN feedback efficiency generally drives higher +suppression of matter clustering, in agreement with previous work +(e.g.; vDMS; Nicola et al. 2022), with the strongest effect seen for +high-speed jets in SIMBA which are able to spread a substantial +amount of baryons over scales of several Mpc (Borrow et al. 2020; +Gebhardt et al. 2023). The qualitative effect of stellar feedback on +matter clustering is more dependent on galaxy formation model, +which can either suppress or enhance power on different scales +depending on the interplay between stellar and AGN feedback. +Stronger stellar feedback often results in weaker overall suppression +of matter clustering by suppressing black hole growth and therefore +the effective efficiency of AGN feedback (van Daalen et al. 2011; +Nicola et al. 2022; Gebhardt et al. 2023). +• Halo baryon fractions 𝑓 bar are very sensitive to galaxy forma- +tion model, with TNG producing systematically more baryon-rich +halos compared to SIMBA for a broad range of parameter variations. +Higher AGN feedback efficiency generally decreases halo baryon +fractions, but the extent of the effect and the affected halo mass range +depend on model details. Increasing the strength of stellar feedback +MNRAS 000, 1–18 (2021) + +16 +Delgado et al. +can either decrease or increase the baryon fraction depending on the +non-linear coupling of stellar feedback and black hole growth. Halo +baryon fractions are also very sensitive to changes in cosmology. +Increasing Ωm (at fixed Ωb) or 𝜎8 systematically decreases the +baryon fraction of halos (normalized by Ωb/Ωm), indicating a +non-trivial response of feedback to changes in the amount of baryons +relative to dark matter and the growth history of halos. +• We find a broad correlation between the amount of suppression +of the matter power spectrum Δ𝑃/𝑃DM and the baryon fraction +of massive halos 𝑓 bar, indicating that the feedback mechanisms +responsible for evacuating gas from massive halos also dominate +the impact of baryonic effects on matter clustering. These results +are in broad agreement with vDMS, but the thousands of sim- +ulations in CAMELS produce significantly larger scatter in the +Δ𝑃/𝑃DM– 𝑓 bar relation. Cosmic variance alone can significantly +affect the matter power spectrum on our (25 ℎ−1Mpc)3 simulated +volumes, but the complex trends seen for the impact of individual +cosmological and feedback parameter variations on Δ𝑃/𝑃DM and +𝑓 bar suggest that the vDMS model predicting Δ𝑃/𝑃DM given +only 𝑓 bar for massive halos is not general enough to include every +plausible feedback model. +• Predicting the impact on matter clustering based only on +the mean baryon fraction of massive halos using the vDMS +Δ𝑃/𝑃DM– 𝑓 bar relation is not possible given the broad range of +galaxy formation models and the impact of cosmic variance in +CAMELS. However, we have demonstrated that a RF regressor +trained on CAMELS is able to extract information from halos +across the full mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 to +estimate the suppression of the matter power spectrum on scales +𝑘 = 0.5–20 ℎ Mpc−1. We are thus not only extracting information +from low-mass halos but also predicting Δ𝑃/𝑃DM in the highly +non-linear regime, significantly extending the range of scales +𝑘 < 1 ℎ Mpc−1 where the vDMS model can be applied. +• Using the mean halo baryon fraction and abundance in different +halo mass bins as input features, the RF regressor was able to account +for ∼80–85% of the Δ𝑃/𝑃DM variation occurring on scales 𝑘 = 10– +20 ℎ Mpc−1 where the impact of feedback on the matter power spec- +trum becomes the highest. At 𝑘 = 5.0 ℎMpc−1, our best model was +able to explain ∼92% of the variance in the suppression of power due +to feedback when training on the TNG model. However, the same +model can only explain ∼70% of the variation in Δ𝑃/𝑃DM and tends +to under-predict the suppression of matter clustering when tested on +the SIMBA simulations, indicating that the RF is only moderately +robust relative to changes in the underlying galaxy formation im- +plementation. Training on SIMBA increases the robustness of the +model owing to its larger range of variation in Δ𝑃/𝑃DM compared to +TNG, but in this case the RF tends to over-predict |Δ𝑃/𝑃DM| when +tested on TNG. These results suggest that the lack of a universal +relation between halo baryon fractions and impact on matter clus- +tering and emphasize the need to construct models that are robust +against assumptions in baryonic physics (e.g., Villaescusa-Navarro +et al. 2021b; Nicola et al. 2022; Shao et al. 2022a). +A unique advantage of CAMELS relative to previous work is that it +performs simulations for different baryonic physics implementations +and a broad range of cosmological and feedback parameter varia- +tions, providing a data-set sufficiently large to train machine learning +algorithms for a variety of applications (Villaescusa-Navarro et al. +2021c). However, an important limitation of CAMELS is the small +volume of each simulation realization, 𝐿box = 25 ℎ−1Mpc, with +important implications for this work. Given the small box sizes, +the matter power spectrum is sensitive to the specific initial condi- +tions in each realization, and the impact of baryonic effects further +depends on stochastic processes related to feedback operating on +a limited number of massive halos. As a result, cosmic variance +represents a challenge to infer the suppression of matter clustering +Δ𝑃/𝑃DM given only the baryon fraction of massive halos. Previ- +ous works in CAMELS have devised strategies to correct for the +noise introduced by cosmic variance. When training a neural net- +work on electron density auto-power spectra to predict Ωm, Nicola +et al. (2022) constructed a cosmic variance parameter based on the +distribution of halo masses in each realization, improving the pre- +dictions significantly when introduced as an additional training fea- +ture. Thiele et al. (2022) used spectral distortion measurements to +constrain baryonic feedback and applied a correction factor to the +Compton-y distortion by comparing expected values from a simple +halo model evaluated for the halo mass function in each CAMELS +simulation compared to that of a standard halo mass function. In +our RF experiments, introducing the number of halos in each mass +bin 𝑁j +halo as input features (i.e., basically the halo mass function) +improves the accuracy of the predictions significantly. The number +of low-mass halos was one of the most predictive features identified +by the RF, which can be understood as a strong tracer of Ωm, but +𝑁j +halo also helped reduced cosmic variance effects, in agreement to +previous works. +The small simulated volumes in CAMELS also complicate a di- +rect comparison between our results and vDMS, where in addition +to cosmic variance affecting Δ𝑃/𝑃DM we also lack halos massive +enough to evaluate the mean baryon fraction under the same con- +ditions. Based on a suite of simulations of substantially larger vol- +umes, vDMS identified a tight relationship between mean baryon +fraction in massive halos and baryonic power spectrum suppression +at 𝑘 < 1 ℎ Mpc−1. While some degree of correlation between these +quantities is expected, as we find here, the extremely small scatter in +the observed relationship was remarkable given the variety of galaxy +formation models compared by vDMS. Since we observed some hints +at a deviation from the vDMS relationship in CAMELS, particularly +for the parameter variations based on the SIMBA model, we have +added the original 100 ℎ−1Mpc SIMBA volume to the vDMS rela- +tion in Fig. 11. Here, we show the new SIMBA data points as circles, +while the measurements considered in vDMS are shown as triangles. +We have verified our pipeline by running it on IllustrisTNG-300, +achieving excellent agreement with the vDMS measurements. As +can be seen, SIMBA constitutes a considerable outlier and does not +fall within the 1% interval around the vDMS fit. The only other +simulation scattering that far is Illustris which, however, does not +reproduce the observed baryon fraction. It is known that the feed- +back prescription in SIMBA is unique in its ability to re-distribute +baryons across large scales (Borrow et al. 2020), which could explain +the observed deviation from the vDMS relation. However, SIMBA +generally does not do worse in reproducing observational relation- +ships than the other major hydrodynamic simulations. Thus, these +results suggest the possibility that the vDMS relation only holds in +a sub-space of simulations while there exists at least one dimension +along which deviations occur. +6 ACKNOWLEDGMENTS +We wish to thank Dylan Nelson for helpful comments. +DAA acknowledges support by NSF grants AST-2009687 and +MNRAS 000, 1–18 (2021) + +Matter clustering in CAMELS +17 +10 +3 +10 +2 +10 +1 +100 +P(k)/PDM(k) +k = 0.2 h/Mpc +k = 0.3 h/Mpc +k = 0.4 h/Mpc +10 +3 +10 +2 +10 +1 +100 +P(k)/PDM(k) +k = 0.5 h/Mpc +SIMBA100 +van Daalen+2020 fits +(intervals=1%) +k = 0.6 h/Mpc +k = 0.7 h/Mpc +0.2 +0.4 +0.6 +0.8 +1.0 +fbar(r < R , M +1014M +)/( +b/ +m) +10 +3 +10 +2 +10 +1 +100 +P(k)/PDM(k) +k = 0.8 h/Mpc +0.2 +0.4 +0.6 +0.8 +1.0 +fbar(r < R , M +1014M +)/( +b/ +m) +k = 0.9 h/Mpc +0.2 +0.4 +0.6 +0.8 +1.0 +fbar(r < R , M +1014M +)/( +b/ +m) +k = 1.0 h/Mpc +van Daalen+2020: +Cosmo-OWLS +BAHAMAS +EAGLE +Illustris +Illustris TNG100 +Illustris TNG300 +Horizon-AGN +L > 100 Mpc/h +L +100 Mpc/h += 500c += 200c +Figure 11. Comparison of the power spectrum suppression in the original SIMBA simulation with the results from vDMS. The SIMBA data points are circles, +while the measurements from vDMS are reproduced as triangles. We also show the fitting functions from vDMS as dashed lines as well as 1% range of variations. +200c and 500c mass definitions are shown in different colors. The vertical error bars have been estimated by splitting the 100 ℎ−1Mpc SIMBA simulation into 8 +sub-volumes. Statistical error bars on the horizontal axis are small (∼ 10−3 as estimated using jack-knife), but there are somewhat larger systematic errors from +the halo finding (∼ 10−2, as estimated by running the analysis with FOF and Rockstar). 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G., Schaye J., 2020, Monthly Notices of the +Royal Astronomical Society, 491, 2424 +MNRAS 000, 1–18 (2021) + diff --git a/NNE0T4oBgHgl3EQfTADA/content/tmp_files/load_file.txt b/NNE0T4oBgHgl3EQfTADA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..acff5406593b8e611170efce12778dfe2ce2fa19 --- /dev/null +++ b/NNE0T4oBgHgl3EQfTADA/content/tmp_files/load_file.txt @@ -0,0 +1,1601 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf,len=1600 +page_content='MNRAS 000, 1–18 (2021) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 Predicting the impact of feedback on matter clustering with machine learning in CAMELS Ana Maria Delgado,1★ Daniel Anglés-Alcázar,2,3 Leander Thiele,4 Michelle Ntampaka,5,6 Shivam Pandey,7,8 Kai Lehman,9,10 Lars 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Extracting information from the total matter power spectrum with the precision needed for upcoming large cosmological surveys requires unraveling the complex effects of galaxy formation processes on the distribution of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this work, we investigate the impact of baryonic physics on matter clustering at 𝑧 = 0 using a large library of power spectra from the Cosmology and Astrophysics with MachinELearningSimulations(CAMELS)project,containingthousandsof (25 ℎ−1Mpc)3 volumerealizations with varying cosmology, initial random field, stellar and AGN feedback strength, sub-grid model implementation, and (magneto)- hydrodynamics methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We show that baryonic physics can profoundly affect matter clustering on scales 𝑘 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 ℎ Mpc−1 and the magnitude of this effect is highly dependent on the details of the galaxy formation implementation and variations of cosmological and astrophysical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Increasing AGN feedback strength decreases halo baryon fractions and yields generally stronger suppression of power relative to N-body simulations, while stronger stellar feedback often results in weaker overall effects by suppressing black hole growth and therefore the impact of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find a broad correlation between mean baryon fraction of massive halos (𝑀200c > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙/ℎ) and suppression of matter clustering but with significant scatter compared to previous work owing to wider exploration of feedback parameters and cosmic variance effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We show that a random forest regressor trained on the baryon content and abundance of halos across the full mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 can predict the effect of galaxy formation on the matter power spectrum on scales 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–20 ℎ Mpc−1, providing access to cosmological information in the highly non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Key words: galaxies: halos, clustering – cosmology: large-scale structure of Universe, theory – machine learning: random forest – methods:numerical 1 INTRODUCTION The field of cosmology has many exciting endeavors to look forward to within the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' With the arrival of enormous photomet- ric and spectroscopic galaxy redshift survey missions such as DESI (DESI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2016), the Nancy Roman Space Telescope (Spergel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015), Euclid (Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011) and the Vera Rubin Observatory (LSST Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2009), the commu- nity will have the opportunity to tackle many ambitious goals, such as mapping the distribution of matter and the large scale structure of the ★ E-mail: ana_maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='delgado@cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='edu (AMD) Universe, measuring cosmological parameters to percent-level preci- sion, and constraining the sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' An important step in fully realizing the statistical power of these upcoming surveys is to model the matter power spectrum and other summary statistics to ∼1% precision down to scales as small as 𝑘 = 10 ℎ Mpc−1 (Huterer & Takada 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Laureijs 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, previous studies have shown that complex galaxy formation processes involv- ing feedback from massive stars and active galactic nuclei (AGN) can suppress power relative to dark matter-only simulations out to large scales (van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Galactic winds driven by supernovae and AGN-driven outflows can eject a large amount of material from © 2021 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='02231v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='GA] 5 Jan 2023 2 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' the center of galaxies out to large distances (Anglés-Alcázar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Borrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Hafen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Ayromlou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Mitchell & Schaye 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Sorini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022) and the resulting suppression of power by feedback creates signif- icant biases when attempting to constrain cosmological parameters (Semboloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Zentner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Several approaches to addressing the suppression of matter clus- tering caused by baryonic physics have been devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Cosmological hydrodynamic simulations provide the most direct method to under- stand the impact of baryonic effects on the distribution and clustering of matter (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Hellwing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Tenneti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Modern cos- mological large-volume simulations such as Horizon-AGN (Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014), Eagle (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015), IllustrisTNG (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018), and SIMBA (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019) produce galax- ies that broadly match observations in properties such as the stellar mass function and the bimodality in galaxy colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Comparing the power spectrum of hydrodynamic simulations with those of their phase-matched, collisonless N-body, dark matter-only simulations allows us to measure how baryonic feedback suppresses the cluster- ing of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, many key feedback processes remain poorly understood and most current models require extensive tuning of free parameters to match observations, limiting their predictive power (Somerville & Davé 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Higher resolution cosmological “zoom- in” simulations can reduce subgrid model uncertainties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', Agertz & Kravtsov 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Anglés-Alcázar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021), but at the expense of modeling volumes that are too small for many cosmological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' More flexible approaches to address the impact of baryonic physics using analytic models include: modifying the “halo model” (Seljak 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Semboloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fedeli 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Mead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015) us- ing observational constraints and simulation results as the basis for parameterizing the transfer of power produced by the presence of baryons (Mohammed & Seljak 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Schneider & Teyssier 2015), and mitigating the presence of baryons altogether by marginalizing over the parameters of effective models (Semboloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011) or over the principle components in linear combinations of observables that are most strongly affected by baryonic effects (Eifler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Kitching et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, the success of these techniques rely heavily on the flexibility of the models to capture the true underlying distribution of matter (McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017) and they are limited by assumptions about halo bias relative to the linear density field, smooth halo profiles neglecting substructure, and uncertainties in the spatial and redshift dependence of baryonic effects (Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Alternatively, power spectra produced by a large number of cosmological simulations with varying cosmologies and feedback parameters can be used to inform semi-analytic models attempting to mitigate the effects of baryons, characterize the theoretical uncer- tainties in galaxy formation, and marginalize over feedback effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2011) employed a suite of 50 cosmological hydrodynamic simulations from the OWLS project (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2010) to study the effects of different baryonic processes on the mat- ter power spectrum over a range of scales and, more recently, van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2020) (henceforth vDMS) included additional simula- tions from the cosmo-OWLS (Le Brun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014) and BAHAMAS (McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017) projects to produce a library of 92 matter power spectra from simulations with varying subgrid models and feedback strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Relating the effects of galaxy formation physics to the suppression of power, vDMS proposed that it is possible to predict the fractional impact of baryons on the clustering of matter, 𝑃hydro/𝑃DM, given only the mean baryon fraction of massive halos (𝑀halo ∼ 1014 M⊙), where 𝑃hydro and 𝑃DM are the matter power spectra from hydrodynamic simulations and their corresponding N- body simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Importantly, the empirical vDMS re- lation between baryon fraction and power suppression is satisfied by a variety of simulations with different galaxy formation imple- mentations, including the Horizon-AGN, EAGLE, and IllustrisTNG simulations, which opens the possibility to accurately correct dark matter only power spectra based on observational constraints on gas fractions in massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, this relation is valid only on large scales, 𝑘 ≤ 1 ℎ Mpc−1, and the still limited number of differ- ent feedback implementations and cosmologies represented in the vDMS library of matter power spectra may not be representative of a broader range of plausible galaxy formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this work, we use 2,000+ cosmological hydrodynamic simu- lations and their corresponding collisionless (N-body) simulations from the Cosmology and Astrophysics with MachinE Learning Sim- ulations (CAMELS1) project (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c) to examine the impact of baryonic physics on matter clustering using the largest library of power spectra available including variations of cosmological and feedback parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In recent related work using CAMELS, Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2022) trained a neural network on thou- sands of electron density auto-power spectra from large scales down to 𝑘 = 10 ℎ Mpc−1, breaking the baryon-cosmology degeneracy and providing tight constraints on the total matter density Ωm and the mean baryon fraction in intermediate-mass halos while marginaliz- ing over uncertainties in galaxy formation physics implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Here, we significantly expand upon the work of vDMS and investigate how supernova and AGN feedback affect the mean baryon fraction across a range of halo masses (1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014) and the resulting impact on the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Furthermore, we take advantage of the design of CAMELS for machine learning and train a random forest regressor to predict the relative difference between the matter clustering in hydrodynamical and N-body simu- lations on scales 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–20 ℎ Mpc−1 given the mean baryon fraction of halos across a broad range of halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We thus demonstrate that we are able to extract valuable information from lower mass halos and predict the suppression of power all the way to the highly non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The work presented here is complementary to Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2023), which show that information about the impact of baryonic effects on the matter power spectrum can be extracted us- ing the tSZ signals from low-mass halos, and include related results utilizing the suite of CAMELS produced with the Astrid simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The layout of this paper is as follows: In section 2 we describe the simulations and halo selection, define our variables, and describe our machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In sections 3 and 4 we present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Finally, in section 5 we provide a summary and discussion of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 Simulations: CAMELS The CAMELS project (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c) contains thousands of state-of-the-art (magneto-)hydrodynamic simulations and their corresponding N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this work, we focus on the simulation suites produced with the IllustrisTNG (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018) and SIMBA (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019) galaxy formation models that are part of the CAMELS public data 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='camel-simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='org MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 3 release2 (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022), providing us with two independent feedback model implementations and thousands of pa- rameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Each simulation is a periodic box of length 𝐿box = 25 ℎ−1Mpc containing 2563 resolution elements with mass resolution of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='49 × 107 (Ωm − Ωb)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='251 ℎ−1M⊙ for dark matter and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='27 × 107 ℎ−1M⊙ for baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This is the same resolution as the original SIMBA simulation and similar to that of the original TNG300-1 simulation of IllustrisTNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The initial conditions of CAMELS simulations were generated at 𝑧 = 127 using second order Lagrangian perturbation theory assuming that the initial power spectra of dark matter and gas are the same and equal to that of total matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Each of the CAMELS simulations contains 34 snapshots from redshifts 𝑧 = 6 down to 𝑧 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' in this work we focus on 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In addition to the initial random phases, each simulation is specified by two cosmological parameters and four astrophysical (feedback) parameters which are varied across the individual realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In the case of cosmological parameters, we vary: Ωm: the fraction of the Universe made up of ordinary and dark matter varies in the range Ωm ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5] while keeping Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='049 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 𝜎8: the variance of the spatial fluctuations of total matter on 8 Mpc ℎ−1 scales is varied in the range 𝜎8 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In the case of astrophysical parameters, the fiducial values are de- fined by the stellar and AGN feedback models of the corresponding original IllustrisTNG and SIMBA simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The fiducial astro- physical parameters are assigned a value 𝐴 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 and then varied across realizations by multiplying by an amplitude factor 𝐴 in order to increase/decrease the amount of feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, we emphasize that the stellar and AGN feedback prescriptions differ substantially between IllustrisTNG and SIMBA and the corresponding parameter variations in CAMELS have a different definition in each model, which we briefly describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 IllustrisTNG The IllustrisTNG model (also referred to as “TNG”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018) is implemented in the AREPO hydrodynam- ics code (Springel 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020), which utilizes a hybrid tree/particle-mesh scheme to solve for gravitational interac- tions and an unstructured, moving mesh to solve the equations of hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Compared to the galaxy formation model of its pre- decessor Illustris (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014), the galaxy formation model in IllustrisTNG has updated implemen- tations of AGN feedback (Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017) and galactic winds (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018a), and incorporates magnetic fields (Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The stellar feedback parameter variations in the CAMELS-TNG simulations introduce 𝐴SN1 to control the total energy injection rate in galactic winds per unit star formation (𝐴SN1 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0]) and 𝐴SN2 to vary the galactic wind speed (𝐴SN2 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The AGN feedback parameter variations pertain to the low-accretion, kinetic- mode black hole feedback, where 𝐴AGN1 varies the feedback en- ergy per unit black hole accretion rate (𝐴AGN1 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0]) and 𝐴AGN2 varies the burstiness and effective ejection speed (𝐴AGN2 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 https://camels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='io 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 SIMBA The SIMBA galaxy formation model (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019) is imple- mented in the GIZMO meshless finite mass hydrodynamics code (Hopkins 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Relative to its predecessor MUFASA (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2016), SIMBA includes a black hole model based on gravi- tational torque accretion and two-mode kinetic feedback (Anglés- Alcázar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017a), galactic winds with mass-loading and velocity scalings derived from the FIRE zoom-in simulations (Muratov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Anglés-Alcázar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2017b), and a model for the creation and destruction of dust (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The stellar feedback parameter variations in the CAMELS-SIMBA simulations introduce 𝐴SN1 to control the mass loading factor of galactic winds and 𝐴SN2 to control the wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The AGN feed- back parameter variations introduce 𝐴AGN1 to change the total mo- mentum flux of either quasar-mode winds or radio-mode jets, while 𝐴AGN2 controls the maximum velocity of gas ejected by jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These parameters are varied over the same range as in IllustrisTNG, with 𝐴SN1 = 𝐴SN2 = 𝐴AGN1 = 𝐴AGN2 = 1 corresponding to the fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 Simulation sets in CAMELS We take advantage of the following simulation sets in CAMELS: Latin Hypercube (“LH”) set: 1,000 realizations each con- taining different initial conditions and different values of the six aforementioned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The LH set is the main training set in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 Parameter (“1P”) set: 66 realizations using the same initial conditions and further divided into six subsets of 11 realizations where only the value of one parameter is varied while the other five parameters are held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this work we make use of the 1P sets to study how a single cosmological or feedback parameter can affect halo baryon fractions and the suppression of the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Cosmic Variance (“CV”) set: 27 realizations with different ini- tial conditions while the values of all six parameters are held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The CV set is used to evaluate the impact of cosmic variance on any of the quantities that we measure from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We refer the reader to Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2021c) for further details about CAMELS, the parameter variations, and the simulation sets available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 Halo Selection We identify halos in CAMELS using the AMIGA Halo Finder (AHF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Knollmann & Knebe 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' AHF uses an adaptive mesh to locate halo centers, calculate the gravitational potential of the halo and it- eratively remove unbound particles (particles whose velocities are greater than the escape velocity at a given radius) from within the boundary of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We refer the reader to (Knollmann & Knebe (2011)) for a full description and implementation of AHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We select halos with masses 𝑀halo ≥ 1010ℎ−1M⊙ using a virial radius defini- tion of 200c (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', 200 times the critical density of the Universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 Matter power spectra and halo baryon fractions We use a library of 4,000+ total matter power spectra from CAMELS (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For each simulation, the MNRAS 000, 1–18 (2021) 4 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' matter power spectrum is computed by assigning particle masses (dark matter, gas, stars, and black holes) to a regular grid with 5123 voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The grid is then Fourier transformed and the power spectrum is computed by averaging over 𝑘-bins with an equal width to the fundamental frequency, 𝑘F = 2𝜋/𝐿, where 𝐿 = 25ℎ−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We then compute the relative difference between the total matter power spec- trum of hydrodynamical and phase-matched N-body simulations, which we refer to as the “suppression of matter power spectrum” and define as: Δ𝑃 𝑃DM = 𝑃hydro − 𝑃DM 𝑃DM , (1) where 𝑃DM is the matter power spectrum of the N-body simulation and 𝑃hydro is that of its corresponding hydrodynamical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We compute the baryon fraction of a given halo as: 𝑓bar = 𝑀star + 𝑀gas 𝑀halo , (2) where 𝑀star and 𝑀gas are the total stellar mass and gas mass of the halo and 𝑀halo is the virial mass of the halo corresponding to 𝑅200c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We further calculate the mean baryon fraction within a given halo mass range in each simulation as: 𝑓 bar = 1 𝑛 𝑛 ∑︁ 𝑖=1 𝑓bari/ Ωb Ωm , (3) where 𝑓bar is defined in Equation 2, subscript 𝑖 is the 𝑖th halo and 𝑛 the total number of halos in a given mass range, and following vDMS we normalize by Ωb/Ωm in order to account for the differences in cosmology for different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 Machine Learning A supervised machine learning algorithm trains a model by provid- ing a subset of data, referred to as the training set, including input variables (henceforth called “features”) and output variables (hence- forth called “target”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The goal is for the algorithm to use the training set to learn the relation between the features and the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The trained model is then used to predict the target for a different subset of features referred to as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this work we use the random forest regressor algorithm from the publicly available package Scikit-Learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' A random forest (RF) is an ensemble machine learning method that can be used for both classification and regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The algorithm works by constructing a “forest” from a user specified number of decision trees and using the mean of the predictions from those trees as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This method has three key advantages: 1) little hyper-parameter tuning is required, 2) it is computationally efficient, and 3) its ensemble characteristic lessens over fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Furthermore, the RF algorithm provides us with some interpretability by way of the “feature importance” attribute, with a ranking of features based on their importance as predictor variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We use the following metrics for scoring the predictive perfor- mance of the RF: R2(y, ˆy) = 1 − �n i=1(yi − ˆyi)2 �n i=1(yi − y)2 , (4) RMSE(𝑦, ˆ𝑦) = √︄�𝑛 i=1(𝑦i − ˆ𝑦i) 𝑛 , (5) where 𝑦i are the given target values, ˆ𝑦i are the RF predicted target values, and 𝑦 is the mean of 𝑦i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The R2 score provides the proportion of the target variable that is predictable by the given features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Because the R2 outputs a score between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0, it provides comparable in- formation about performance when comparing various experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The RMSE scores, on the other hand, are based on the target value range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Therefore, in order to account for the range in target values across multiple experiments, we normalize our RMSE scores by the Interquartile range (IQR): 𝐼𝑄𝑅 = 𝑄3 − 𝑄1, (6) where 𝑄3 is the 3rd quartile (75th percentile) of a given set and 𝑄1 is the first quartile (25th percentile) of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In other words, we normalize the RMSE by the middle 50% dispersion of the target values as RMSE/IQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 Features and Targets Using the thousands of realizations in the CAMELS LH simulation sets, we train random forest regressors to predict the suppression of the matter power spectrum Δ𝑃/𝑃DM at a range of scales based on the baryon fraction and abundance of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We construct the following features for each realization: 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1): the mean baryon fraction of massive halos, those with masses greater than 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1 in each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 𝑓 bar(𝑀j halo);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' array of mean halo baryon fraction, binned by halo mass in the range [1010 − 1015] M⊙ℎ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 𝑁j halo: the number of halos per mass bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Our target are the Δ𝑃/𝑃DM values for each realization at five different 𝑘-values: 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 Robustness of random forest predictions One inherent benefit of CAMELS is that we are able to test the effects of feedback model implementation by way of its TNG and SIMBA simulations sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We create an 80% / 20% train/test split of the LH simulations and perform the following experiments using either TNG or SIMBA: Train on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) to predict Δ𝑃/𝑃DM at 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We perform this experiment at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 with the intent of comparing our results with vDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Train on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) and 𝑁halo for high-mass halos to predict Δ𝑃/𝑃DM at 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Train on 𝑓 bar(𝑀j halo) and 𝑁j halo to predict Δ𝑃/𝑃DM at 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We are further able to determine how well our algorithm can marginalize over subgrid physics model by performing “two-model” experiments where we train on the entire LH set of one of the feed- back implementations and test on the entire LH set of the other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', training on SIMBA and testing on TNG and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For these tests of robustness we use the same setup that produced the best results from the above experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 5 3 IMPACT OF COSMOLOGICAL AND FEEDBACK PARAMETER VARIATIONS In this section we examine the impact of cosmological and baryonic feedback parameter variations on the matter power spectrum and the baryon fraction of halos of different masses, exploring also the con- nection between the suppression of the total matter power spectrum and the mean baryon fraction of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We perform this analysis for both the TNG and SIMBA galaxy formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 Matter power spectra We use the total matter power spectra from the 1P simulations to examine how cosmological and feedback parameters affect the clus- tering of matter at various scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For each run in the 1P set, described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1, we measure the fractional impact of baryons on the total matter power spectrum, 𝑃hydro/𝑃DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' If baryonic physics has no effect on matter clustering, 𝑃hydro/𝑃DM should be of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, if baryonic physics suppresses the clustering of matter com- pared to dark-matter only simulations, usually by way of feedback ejecting gas out to large distances, 𝑃hydro/𝑃DM should fall below unity on a range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 shows 𝑃hydro(𝑘)/𝑃DM(𝑘) as a function of wave number 𝑘 for the 1P simulations, where each spectrum is color coded by the value of each parameter variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We notice two overall trends that are roughly independent of cosmological or feedback parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The first is the general “scoop” shape of 𝑃hydro/𝑃DM, which is consistent with previous works (van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Hellwing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' vDMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This shape conveys agreement between matter clustering in hydrodynamical and dark matter-only simulations on large scales (𝑘 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 ℎ Mpc−1) while at intermediate scales there is suppression of power by baryonic feedback (𝑃hydro/𝑃DM < 1) and at small scales (𝑘 ≳ 40 ℎ Mpc−1) there is enhanced, as opposed to suppressed, clustering relative to dark matter owing to gas dissipa- tive processes (𝑃hydro/𝑃DM > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The second overall trend is that the SIMBA galaxy formation model (solid lines) tends to suppress power on intermediate scales more strongly compared to the TNG galaxy formation model (dashed lines) while driving a steeper increase in small-scale clustering (𝑘 ⪅ 30 ℎ Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We now analyze in more detail how each parameter affects mat- ter clustering by comparing 𝑃hydro(𝑘)/𝑃DM(𝑘) between the fiducial models of TNG and SIMBA (shown in red) and that of the individual parameter variations: Cosmological parameters: The top two panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 show the sensitivity of the total matter power spectrum to Ωm and 𝜎8 for a fixed galaxy formation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We see a strong dependence of 𝑃hydro/𝑃DM on the value of Ωm both in TNG and SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As Ωm decreases (at fixed Ωb), there is a greater suppression of power on intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This can be understood as a consequence of baryons contributing a higher fraction of the total matter content making feedback more efficient at pushing gas out of halos and distributing matter on larger scales, in agreement with the analysis of large-scale baryon spread in Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In contrast, we identify weaker trends for 𝜎8, with significant scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Supernova feedback parameters: The middle two panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 show the impact of changing the stellar feedback parameters 𝐴SN1 and 𝐴SN2 respectively, which control the mass loading and velocity of galactic winds, on matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Both panels show somewhat counterintuitive effects of stellar feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Increasing 𝐴SN1 in SIMBA reduces (rather than enhances) the suppression of power on small scales (𝑘 ≳ 10 ℎ Mpc−1) and increasing 𝐴SN2 further increases 𝑃hydro/𝑃DM over the full range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This can be understood as a consequence of the nonlinear interplay between stellar and AGN feedback, where stronger stellar feedback suppresses black hole growth and results in weaker effective impact of AGN feedback on matter clustering (van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The TNG model shows rather different trends, with reduced suppression of power on scales 𝑘 ≲ 10–20 ℎ Mpc−1 but enhanced suppression of power on smaller scales when increasing 𝐴SN1 and 𝐴SN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These results are consistent with the analysis of electron power spectra in CAMELS by Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2022), which highlights the sensitivity of predicted baryonic effects on galaxy formation implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' AGN feedback parameters: The bottom two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 show the impact of varying AGN feedback efficiency on matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this case, there are clear systematic trends for stronger suppression of power when increasing both 𝐴AGN1 and 𝐴AGN2 for both galaxy formation models (TNG and SIMBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The sensitivity of 𝑃hydro/𝑃DM to 𝐴AGN1 is weaker given its range of variation, with no more than 10% difference relative to the fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In contrast, the matter power spectrum in SIMBA displays a strong sensitivity to the AGN jet speed, 𝐴AGN2, with strong suppression of power across scales, reaching 𝑃hydro/𝑃DM ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 at 𝑘 ∼ 10 ℎ Mpc−1 with jets twice as fast relative to the fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This results are also consistent with previous findings for electron power spectra Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2022) and the impact of large scale jets on cosmological baryon spread (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 Halo baryon fraction In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 we use again the specialized CAMELS 1P simulation sets to analyze the impact of individual cosmological and feedback param- eter variations on the average halo baryon fraction as a function of halo mass, 𝑓 bar(𝑀halo), where we consider logarithmically-spaced halo mass bins in the range 1010–1013 M⊙ℎ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We notice two main trends roughly independent of cosmological or feedback parameters when comparing the fiducial realizations (indicated in red) for the TNG (dashed lines) and SIMBA (solid lines) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The first being that peak of the halo baryon fraction occurs at ∼ 1012 M⊙ℎ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We notice a drop in mean baryon fraction as halos exceed this mass range, when powerful feedback process can expel material out of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We note, however, that at very high mass halos, we expect feedback to be less efficient at expelling material and for there to be another rise in mean baryon fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The second main trend is that SIMBA has overall lower 𝑓 bar(𝑀halo) compared to TNG, with the fiducial models reaching their peak at 𝑓 bar∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 for SIMBA and 𝑓 bar∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 for TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We now analyze in more detail how each CAMELS parameter variation affects halo baryon fractions, keeping in mind that the definitions of feedback parameters are not the same for TNG and SIMBA: Cosmological parameters: The top two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 show the sensitivity of 𝑓 bar(𝑀halo) to our cosmological parameters Ωm and 𝜎8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Halo baryon fractions appear to be more sensitive to cosmology in SIMBA compared to TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, both galaxy formation models predict qualitatively similar trends, with lower 𝑓 bar(𝑀halo) when increasing Ωm and 𝜎8 across a range of halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This trend may seem trivial for Ωm since we hold the value of Ωb constant in all CAMELS simulations, implying that the average MNRAS 000, 1–18 (2021) 6 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 Phydro/PDM Ωm SIMBA fiducial IllustrisTNG fiducial 100 101 σ8 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 Phydro/PDM ASN1 100 101 ASN2 100 101 k [h Mpc−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 Phydro/PDM AAGN1 100 101 k [h Mpc−1] AAGN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1 2 3 4 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The effect of baryonic physics on matter clustering for different cosmological and feedback parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Each panel shows the ratio of total matter power spectrum in hydrodynamic simulations to that of the corresponding dark matter-only simulations (𝑃hydro/𝑃DM) as a function of wave number 𝑘 when varying a single parameter in the CAMELS 1P sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Lines of different colors indicate the value of each parameter variation, and red lines indicate the fiducial model for TNG (dashed lines) and SIMBA (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Variations in feedback model, as well as in feedback amplitude, result in variation in total matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 7 cosmic baryon fraction decreases with higher Ωm and so should the corresponding halo baryon fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, 𝑓 bar(𝑀halo) is normalized by Ωb/Ωm for each simulation (Equation 3), removing the trivial effect of varying Ωm at fixed Ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The impact of increasing Ωm on 𝑓 bar(𝑀halo) is thus a reflection of the effective efficiency of feedback when changing the amount of baryons relative to the dark matter gravitational potential, and this effect seems more prominent in lower mass halos for both TNG and SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Interestingly, the baryon fraction decreases systematically at all halo masses when increasing 𝜎8, while the suppression of power does not seem to follow a clear trend with 𝜎8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Supernova feedback parameters: The middle two panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 show the impact of changing 𝐴SN1 and 𝐴SN2 on halo baryon fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As for the power spectra, varying the mass loading of galactic winds (𝐴SN1) has a different effect in each galaxy formation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We might intuitively expect that as 𝐴SN1 increases, more gas would be ejected out of galaxies resulting in lower 𝑓 bar(𝑀halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, we only see this behavior in TNG for halos with mass 𝑀halo ≲ 1012 M⊙ℎ−1, while the baryon fraction of higher mass halos increases with 𝐴SN1 owing to the suppression of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This reversed trend with 𝐴SN1 can explain its impact on 𝑃hydro/𝑃DM for TNG, with an enhancement of power on interme- diate scales coinciding with the highest gas fraction in high mass halos with 𝐴SN1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Meanwhile, SIMBA shows systematically higher baryon fractions when increasing 𝐴SN1 across the full halo mass range, indicating a different non-linear coupling of stellar and AGN feedback compared to TNG, which can explain the reduced suppression of power seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' On the other hand, increasing the speed of galactic winds (𝐴SN2) results in systematically lower 𝑓 bar(𝑀halo) values for both SIMBA and TNG, but in this case lower baryon fractions correlate with less suppression of matter clustering in SIMBA on all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' AGN feedback parameters: The bottom two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2 show the sensitivity of 𝑓 bar(𝑀halo) to changes in AGN feedback effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Halo baryon fractions are significantly reduced by increasing the kinetic mode black hole feedback efficiency 𝐴AGN1 in TNG in the intermediate halo mass range 𝑀halo = 1011–1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5M⊙ℎ−1, while the burstiness parameter 𝐴AGN2 has a stronger effect reducing 𝑓 bar(𝑀halo) in higher mass halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In both cases, the decrease in halo baryon fraction with higher AGN feedback efficiency correlates with stronger suppression of matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Similarly, increasing the AGN jet speed in SIMBA (𝐴AGN2) drives an overall reduction of halo baryon fractions and increased suppression of matter clustering on all scales shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1, corresponding to more efficient spread of baryons on large scales relative to the TNG model (Tillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, the effect of increasing the momentum flux 𝐴AGN1 in SIMBA seems more complex, driving an increase in baryon fraction in high mass halos (possibly due to black hole self-regulation) but stronger suppression of power in the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 Suppression of matter power spectrum as a function of baryon fraction Using a suite of matter power spectra from hydrodynamical and dark matter only simulations, vDMS found a tight relation between the suppression of the matter power spectrum (Δ𝑃/𝑃DM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3) in the linear regime and the average baryon fraction ( 𝑓 bar) of high mass halos (∼ 1014 M⊙ℎ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this study, we use the LH simulation sets of CAMELS, described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1, to investigate this relationship by examining how it is affected by cosmological and baryonic feedback parameters over a broader range of model variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4 show Δ𝑃/𝑃DM ≡ (𝑃hydro−𝑃DM)/𝑃DM evaluated at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 as a function of 𝑓 bar for the TNG and SIMBA LH sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We note that due to the small simulated volumes in CAMELS there are not enough halos of mass ∼ 1014 M⊙ℎ−1 to make a direct comparison to the results of vDMS, and we therefore evaluate 𝑓 bar for halos with mass > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Each panel reproduces the same data but color coded by the corresponding parameter value for each of the six parameters that are simultaneously varied in the TNG and SIMBA LH sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This allows us to examine how individual cosmological and feedback parameters affect the relation between Δ𝑃/𝑃DM and 𝑓 bar, which we compare to the fitting function derived by vDMS (their equation 5) for baryon fractions calculated using the 200c virial definition, which we henceforth refer to as the “vDMS model” and indicate by the blue solid line and gray shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Lastly, we overlay the results for the CV sets of TNG and SIMBA (described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1) as the red triangles in the top center panel of each figure in order to examine the effect of cosmic variance on this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Our CAMELS results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4 reveal a good qualitative agreement with the general trend found in vDMS: the suppression of the matter power spectrum increases as the average baryon fraction in massive halos decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' While the SIMBA LH set probes a range of Δ𝑃/𝑃DM and 𝑓 bar values significantly larger than the TNG LH set, as expected from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 and 2, both models roughly follow the vDMS trend, suggesting that 𝑓 bar in massive halos can be used to infer the redistribution of baryons over large scales regardless of galaxy formation model implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, we find considerable spread in Δ𝑃/𝑃DM at fixed 𝑓 bar compared to vDMS, which can be attributed to the broader range of parameter variations explored in CAMELS but also to the impact of cosmic variance: Cosmological parameters: The left two panels in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4 explore the dependence of the Δ𝑃/𝑃DM– 𝑓 bar relation on Ωm and 𝜎8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For both TNG and SIMBA, there is a clear trend of higher Δ𝑃/𝑃DM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' less suppression of power) at fixed 𝑓 bar for higher values of Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This implies that the same impact on the total matter power spectrum (at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1) can be predicted by simulations that yield different halo baryon fractions, in this case as a consequence of the different response of feedback to changes in Ωm at fixed Ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' There is also a visible, albeit less pronounced, trend for 𝜎8, where Δ𝑃/𝑃DM becomes more negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' stronger suppression of power) at fixed 𝑓 bar for higher values of 𝜎8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Overall, the non-linear response of the fiducial galaxy formation model to variations in cosmology appears to explain a significant fraction of the scatter in the vDMS relation seen for SIMBA and TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Supernova feedback parameters: The middle panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4 explore the dependence of the vDMS relation on systematic variations of the mass loading factor and speed of galactic winds driven by stellar feedback (parameters 𝐴SN1 and 𝐴SN2, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In TNG, there is indication for simulations clustering around Δ𝑃/𝑃DM ∼ 0 and 𝑓 bar∼ 1 for higher values of 𝐴SN1 and 𝐴SN2, corresponding to weaker overall impact of feedback owing to the suppression of black hole growth and therefore AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Decreasing the strength of stellar feedback tends to yield more negative Δ𝑃/𝑃DM values and correspondingly lower 𝑓 bar (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', stronger impact), displacing simulations roughly along the vDMS relation but with increasing scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find qualitatively similar trends in SIMBA for variations in 𝐴SN2 while no clear dependence MNRAS 000, 1–18 (2021) 8 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1010 1011 1012 1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar Ωm SIMBA fiducial IllustrisTNG fiducial 1010 1011 1012 1013 σ8 1010 1011 1012 1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar ASN1 1010 1011 1012 1013 ASN2 1010 1011 1012 1013 Mhalo [h−1 M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar AAGN1 1010 1011 1012 1013 Mhalo [h−1 M⊙] AAGN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='15 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The effect of each parameter variation on the mean halo baryon fraction as a function of halo mass, 𝑓 bar(Mhalo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1, the color bar for each panel indicates the value of the corresponding parameter variation while all other parameters are held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The red lines indicate the fiducial parameters for TNG (dashed lines) and SIMBA (solid lines), with their peak baryon fraction occurring at ≲ 1012 M⊙ℎ−1 in both fiducial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The 𝑓 bar values and their halo mass dependence differ substantially between galaxy formation implementations and model parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 9 on 𝐴SN1, with overall larger scatter in the vDMS relation compared to TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' AGN feedback parameters: The right two panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4 show the impact of AGN feedback parameter variations in the vDMS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this case, we find a general trend for simulations with stronger AGN feedback to populate the lower-left region of the Δ𝑃/𝑃DM– 𝑓 bar plane, indicating more efficient evacuation of gas from halos and stronger suppression of matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In both TNG and SIMBA, this trend is more apparent for 𝐴AGN2 compared to 𝐴AGN1, as expected from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Cosmic variance: In order to examine how cosmic variance affects the predicted variation of Δ𝑃/𝑃DM as a function of 𝑓 bar, the top middle panel of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4 overlay the results from the CAMELS CV sets corresponding to 27 realizations of the fiducial TNG and SIMBA models using different initial conditions (red tri- angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For TNG, the CV set yields roughly similar range in average baryon fraction of massive halos as the entire LH set, indicating that stochastic variations owing to the small CAMELS volumes and cor- respondingly low number of massive halos play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nonetheless, the TNG CV simulations roughly follow the vDMS re- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The SIMBA CV set also yields a wide range of Δ𝑃/𝑃DM and 𝑓 bar values, but in this case suggesting a systematic offset relative to the vDMS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We can quantify the impact of cosmic variance on the predicted suppression of matter clustering as the root mean square variation in Δ𝑃/𝑃DM relative to the mean: 𝛿cv ≡ 𝜎cv |𝑝cv| , (7) with 𝑝cv ≡ Δ𝑃/𝑃DM for the CV set and 𝜎2 cv = 1 n n ∑︁ i=1 � pi cv − pcv �2 , (8) where 𝑛 = 27 realizations and 𝑝cv represents the average of 𝑝cv over the CV set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 7 for 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 gives 𝛿cv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='436 for TNG and 𝛿cv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='211 for SIMBA, indicating that there is considerable variation due to cosmic variance alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The considerable spread of CAMELS predictions relative to the vDMS model shown in this section provides motivation for the ma- chine learning experiments described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Given the larger data set in CAMELS with broader variations in feedback and cos- mology compared to previous libraries of power spectra, it is possible that the vDMS model relating halo baryon fraction and suppression of matter clustering is not general enough to include every plau- sible feedback model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For example, we later examine the original SIMBA model against the vDMS relation and find that SIMBA does not fall within 1% of the vDMS fit, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, it is also possible that having smaller volumes which are significantly affected by cosmic variance as compared to the data set in vDMS, along with the lack of halos of mass ∼ 1014 M⊙ℎ−1, may explain the disagreement between our results and the vDMS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These results motivate us to explore the relation between Δ𝑃/𝑃DM and halo baryon fraction with a machine learning approach, where we can extract information from a broader halo mass range to improve the accuracy of predictions for the impact of baryonic physics on the total matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4 ESTIMATING THE IMPACT OF FEEDBACK ON MATTER CLUSTERING WITH MACHINE LEARNING A major goal of this work is to show that machine learning can be used to extract information from the full range of halo masses in order to estimate the suppression of the matter power spectrum by baryonic processes all the way to the non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this section, we discuss the results of training a random forest regressor (RF) to estimate the impact of feedback on the clustering of matter using the LH simulation sets in CAMELS, which vary simultaneously cosmological and feedback parameters (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The general setup of our experiments is described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 Extracting information across the halo mass range with random forest regression Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 shows the results from training a RF regressor on differ- ent input features to estimate the suppression of power Δ𝑃/𝑃DM at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We begin by training a RF with 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) as the only training feature, in analogy with the information used by the vDMS fitting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Halos of mass ∼ 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1 are only available for ∼700 out of 1,000 LH re- alizations for each of TNG and SIMBA, limiting the size of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this first experiment, the RF is only able to predict ∼45% and ∼60% of the variation of Δ𝑃/𝑃DM in TNG and SIMBA, respectively, with the predicted versus true values of Δ𝑃/𝑃DM shown by the green data points in the left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Next, we add the number of high mass halos 𝑁halo corresponding to the measured 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) as an additional input feature, with results indicated by the orange data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this case, the RF pre- dictions improved by ∼10% in TNG and ∼16% for SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We then incorporate information from halos across the full mass range by introducing the baryon fraction 𝑓 bar(𝑀j halo) and the corresponding number of halos 𝑁j halo within each halo mass bin (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4), with results shown by the blue data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this case, we can use the full LH sets of CAMELS for training and testing since we are not limited by the availability of high mass halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' With these additional features using information from a range of halo masses, the RF predicted ∼70% of the variation in Δ𝑃/𝑃DM at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 for both TNG and SIMBA, significantly improving upon the original results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As stated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4, one advantage of the RF is that it provides some level of interpretability by means of the “feature importance” attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 display the relative importance assigned to each feature by the trained RF on the test set for TNG (top) and SIMBA (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Interestingly, the RF ranked 𝑓 bar in halos with mass 𝑀halo = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 M⊙ℎ−1 as the most important feature to predict the suppression of power Δ𝑃/𝑃DM at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 in TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Furthermore, we see that the RF ranked several features across the mass ranges as important predictors of Δ𝑃/𝑃DM in SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These feature importance results reveal that the RF was able to extract valuable information across a range of halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 Random forest predictions in the highly non-linear regime In the previous subsection, we have established that training the RF on features from a range of halo masses improves the pre- dictions for Δ𝑃/𝑃DM in the linear regime compared to using 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) as a training feature alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We now ex- ploit the same methodology to extend our predictions into the highly non-linear regime by repeating our experiments at a range of scales, predicting Δ𝑃/𝑃DM at 𝑘 = [1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 6 provides a MNRAS 000, 1–18 (2021) 10 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 ∆P/PDM Ωm vDMS model ASN1 CV set AAGN1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 ∆P/PDM σ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) ASN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) AAGN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 IllustrisTNG | k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 h Mpc−1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Suppression of the matter power spectrum, Δ𝑃/𝑃DM, as a function of mean baryon fraction of high mass halos, 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1), for the TNG LH simulation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The blue line is the fitting function of vDMS for a halo definition of 200c (200 times the critical density of the Universe), with the gray shaded region indicating 1% variation in Δ𝑃/𝑃DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Each panel shows the same data points color coded by the value of each of the six parameters varied simultaneously in the LH set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The middle top panel shows additional results from the CV simulation set, where all six parameters are constant and only the initial conditions are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find that Δ𝑃/𝑃DM increases for higher values of 𝑓 bar, meaning that there is less suppression of the matter power spectrum in simulations where feedback is less effective at removing gas from halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Data points fall generally along the vDMS model, with large scatter owing to broad parameter variations and cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 ∆P/PDM Ωm vDMS model ASN1 CV set AAGN1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 ∆P/PDM σ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) ASN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) AAGN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='35 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 SIMBA | k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 h Mpc−1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 but for the SIMBA LH simulation set, which produces a much broader range of variation in Δ𝑃/𝑃DM and 𝑓 bar compared to the TNG LH set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' SIMBA also follows the general trend of the vDMS model but with considerably more scatter and some indication for a systematic offset relative to the vDMS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 PREDICTED ∆P/PDM IllustrisTNG k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) & Nhalo k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mj halo) & Nj halo 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1 M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 IMPORTANCE k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 [h Mpc−1] f bar(Mj halo) Nj halo −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 PREDICTED ∆P/PDM SIMBA k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) & Nhalo k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mj halo) & Nj halo 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1 M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='30 IMPORTANCE k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 [h Mpc−1] f bar(Mj halo) Nj halo Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Results from different Random Forest experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We create an 80/20 train/test split of the LH data sets to predict Δ𝑃/𝑃DM at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 for TNG (top) and SIMBA (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The left panels show the predicted target values compared to the true target values as given by the test sets in CAMELS, where the red line indicates a perfect one-to-one relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The green data points correspond to predictions by a RF trained only on the baryon fraction of massive halos, 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1), the orange data points show results for a RF trained on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) as well as the number 𝑁halo of massive halos, while results from training a RF on 𝑓 bar(𝑀j halo) and 𝑁 j halo for halo mass bins 𝑗 spanning the full mass range are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The right panels show the feature importances corresponding to the blue data points in the left panel (training on 𝑓 bar(𝑀j halo) and 𝑁 j halo), indicating the relative rank ordering of importance (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0) given to each training feature by the RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Predictions improve by providing training data across the full range of halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' summary of performance scores for these experiments using the LH simulation sets for TNG (squares) and SIMBA (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We also per- formed additional experiments using the two LH data sets combined, which we refer to as “TNG+SIMBA” (diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Descriptions of the scoring metrics can be found in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The top panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 6 show the R2 scores and the bottom two panels show the RMSE scores nor- malized by the interquartile range (IQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We normalize the RMSE by the IQR in order to account for the variation in the range of Δ𝑃/𝑃DM, which depends on the 𝑘 value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' the range of Δ𝑃/𝑃DM increases as we move to non-linear regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In addition to presenting results at a range of scales for each training set, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 6 compares the predictions based on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) alone (right panels) versus providing the baryon fraction 𝑓 bar(𝑀j halo) in different halo mass bins (left panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Results based on baryon fractions alone are shown in orange while results that also incorporate the corresponding number of halos 𝑁j halo are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Training on 𝑓 bar(𝑀j halo) and 𝑁j halo improved the prediction of Δ𝑃/𝑃DM for both TNG and SIMBA on all scales 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5– 20 ℎ Mpc−1 as measured by the R2 and RMSE/IQR scores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For TNG, we achieved the highest R2 score of all experiments at 𝑘 = 5 ℎ Mpc−1, with R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='923 (consistent with the lowest RMSE/IQR score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In other words, the RF was able to account for approximately 92% of the variation in the suppression of the matter power spectrum due to feedback using 𝑓 bar(𝑀j halo) and 𝑁j halo as training features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This represents ∼40% improvement over train- ing on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 and ≳20% improvement over training on 𝑓 bar(𝑀j halo) and 𝑁j halo at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The trained RF also shows very good performance down to smaller scales, with R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='85–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='9 at 𝑘 = 10–20 ℎ Mpc−1 when training simultaneously on the baryon fraction and number of halos in different mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Similar results are obtained for SIMBA, also performing better at 𝑘 = 5–20 ℎ Mpc−1 compared to larger scales, with R2 scores slightly lower than TNG: R2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='85 at 𝑘 = 5–20 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Remarkably, while the vDMS model can only predict Δ𝑃/𝑃DM for a given average baryon fraction of massive halos on large scales 𝑘 < 1 ℎ Mpc−1, our RF regressor performs better on scales where the impact of feedback on the matter power spectrum becomes the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The strongest suppression of power occurs at roughly 𝑘 = 10 ℎ Mpc−1 in SIMBA and 𝑘 = 20 ℎ Mpc−1 in TNG for their fiducial models, and the RF is able to account for ∼80-85% of the Δ𝑃/𝑃DM variation on these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 7 illustrates in more detail the predicted results versus true values of Δ𝑃/𝑃DM when training a RF regressor on different input MNRAS 000, 1–18 (2021) 12 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' k [h Mpc−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 R2 score Mj halo f bar f bar & Nhalos f bar f bar & Nhalos k [h Mpc−1] Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 [h−1 M⊙] 100 101 k [h Mpc−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 RMSE/IQR IllustrisTNG SIMBA TNG + SIMBA 100 101 k [h Mpc−1] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Performance scores for all RF experiments predicting Δ𝑃/𝑃DM at five different scales, 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The orange dashed lines connect data points showing scores for a RF trained on average baryon fractions 𝑓 bar, while the blue solid lines connect those where both 𝑓 bar and the number of halos 𝑁halo are used as training features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We show results for three different data sets: the TNG LH set (squares), the SIMBA LH set (circles), and the two LH sets combined, labeled “TNG+SIMBA” (diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Left panels correspond to training on features from a range of halo masses, 𝑓 bar(𝑀j halo), while the right panels correspond to training on high-mass halos only, 𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1, as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We report R2 scores (top) and RMSE scores normalized by the interquartile range (IQR) of the respective data set (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' higher R2 scores and lower RMSE/IQR scores convey an improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We obtain higher scores when training on 𝑓 bar(𝑀j halo) and 𝑁 j halo for a range of halo masses at all scales compared to training on high-mass halos only, and the highest scores occur in the highly non-linear regime at 𝑘 ∼ 5–10 ℎ Mpc−1, meaning that important information can be extracted from a range of halo masses in the non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' features at 𝑘 = 5 ℎ Mpc−1 for the TNG (top) and SIMBA (bottom) LH sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The highest R2 score was obtained at 𝑘 = 5 ℎ Mpc−1 for TNG using 𝑓 bar(𝑀j halo) and 𝑁j halo as training features, which cor- responds to the tighter distribution of blue data points along the one-to-one line of perfect prediction in the top left panel, with re- duced scatter compared to the prediction based on high mass halos alone (orange data points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find similar trends for SIMBA, with an apparent increase in scatter relative to TNG as expected from the lower R2 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Interestingly, the feature importance analysis (right panels) indicates that the number of halos in the low mass range 𝑀halo = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011 M⊙ is contributing significantly to improve the Δ𝑃/𝑃DM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 Interpretation of feature importances Comparing the feature importances shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 and 7 for 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 and 𝑘 = 5 ℎ Mpc−1, respectively, it appears that the most informative input features vary with scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We further investi- gate the physical properties that inform the prediction of Δ𝑃/𝑃DM by the RF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 8, where we provide a summary of the relative im- portance of the 𝑓 bar(𝑀j halo) features (diamonds) and 𝑁j halo features MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 PREDICTED ∆P/PDM IllustrisTNG k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) & Nhalo k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 | f bar(Mj halo) & Nj halo 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1] M⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 IMPORTANCE k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 [h Mpc−1] f bar(Mj halo) Nj halo −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 PREDICTED ∆P/PDM SIMBA k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 | f bar(Mhalo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5) & Nhalo k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 | f bar(Mj halo) & Nj halo 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1 M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 IMPORTANCE k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 [h Mpc−1] f bar(Mj halo) Nj halo Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 but for results at 𝑘 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The left panels show the predicted versus true Δ𝑃/𝑃DM values for RF regressors trained on 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) and 𝑁halo (orange) and trained on 𝑓 bar(𝑀j halo) and 𝑁 j halo (blue), both at 𝑘 = 5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For comparison, we also show the results using only 𝑓 bar(𝑀halo > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1) at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The right panels show the feature importances when training on 𝑓 bar(𝑀j halo) and 𝑁 j halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Going to non-linear regimes the range of the target value Δ𝑃/𝑃DM increases, enabling more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The highest ranked feature at 𝑘 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 ℎ Mpc−1 is 𝑁 j halo at 𝑀halo = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011 M⊙ℎ−1, meaning that the number of low-mass halos is highly informative for estimating the power at 5Mpc length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (squares) corresponding to different halo mass bins when predicting Δ𝑃/𝑃DM at a given scale 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The ranking of features for each 𝑘-value is indicated by the color scale, which shows the log of the fractional importance assigned by the RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The most important ranked feature at each 𝑘-value is further outlined in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' While the RF appears to be learning from the baryon fractions and abundances of halos across the halo mass range, there are some interesting trends that are worth not- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The least informative features for both TNG and SIMBA corre- spond to the most massive halos (𝑀halo ∼ 1014 M⊙ℎ−1), which is in contrast to earlier work identifying the baryon fraction in groups and clusters as a primary predictor of Δ𝑃/𝑃DM on scales 𝑘 < 1 ℎ Mpc−1 (vDMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Unsurprisingly, the small simulated volumes in CAMELS contain a small number of massive halos, which are thus not opti- mal as predictors of Δ𝑃/𝑃DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In contrast, the RF assigns significant importance to 𝑓 bar(𝑀j halo) and 𝑁j halo in low- to intermediate-mass halos in the range 𝑀halo ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1 for all 𝑘-values analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Interestingly, the number of halos 𝑁j halo in the mass range 𝑀halo ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011 M⊙ℎ−1 is among the top features identified by the RF across different scales (see also Fig 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We explore further the significance of this feature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 9, where we show Δ𝑃/𝑃DM at 𝑘 = 5 ℎ Mpc−1 as a function of 𝑁j halo in this halo mass range for the LH sets of TNG (left) and SIMBA (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find that there is a clear correlation between Δ𝑃/𝑃DM and the number of low-mass ha- los for both galaxy formation models, as expected given that 𝑁j halo is identified by the RF as one of the most predictive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Physically, a plausible explanation for this correlation is that 𝑁j halo for low mass halos is a strong tracer of Ωm, as indicated by the color scale, and Ωm itself is one of the main parameters driving large variations in Δ𝑃/𝑃DM at all 𝑘-values in CAMELS (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 1, 3, and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This is consistent with Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2023), which find a simple model is also able to capture information about Δ𝑃/𝑃DM given 𝑓 bar in low-mass halos and Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' While increasing the value of Ωm increases system- atically the number of halos at all masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c), we note that the correlation between Δ𝑃/𝑃DM and 𝑁j halo worsens for higher halo mass bins, which are thus often as- signed lower feature importance by the RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' This can be explained by the number of halos in higher mass bins being more sensitive to cos- mic variance and therefore not as good predictors of cosmology for small simulated volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' On the other hand, the lowest halo mass bin considered here, 𝑀halo < 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 M⊙ℎ−1, becomes unresolved in terms of the minimum number of dark matter particles per halo for the higher Ωm values in CAMELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We therefore conclude that 𝑁j halo in the halo mass bin 𝑀halo ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011 M⊙ℎ−1 is the least MNRAS 000, 1–18 (2021) 14 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 k [h Mpc−1] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1 M⊙] IllustrisTNG f bar(Mj halo) Nj halo −5 −4 −3 −2 −1 log10 IMPORTANCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 k [h Mpc−1] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 log10Mhalo [h−1 M⊙] SIMBA f bar(Mj halo) Nj halo −4 −3 −2 −1 log10 IMPORTANCE Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Feature importance summary for RF models trained on 𝑓 bar(𝑀j halo) and 𝑁 j halo to predict Δ𝑃/𝑃DM at 𝑘 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5, 1, 5, 10, 20] ℎ Mpc−1 for the TNG (top) and SIMBA (bottom) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' For each halo mass bin and 𝑘-value, the relative importance of the corresponding 𝑓 bar(𝑀j halo) (diamonds) and 𝑁 j halo (squares) features is indicated by the color scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The highest ranked feature for each 𝑘-value is outlined in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' There is no clear trend of importance in features across scales, suggesting feature importance results are specific to each scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' sensitive to cosmic variance and it is a strong predictor of Δ𝑃/𝑃DM in part because 𝑁j halo informs the RF about the variation of Ωm in a large number of well resolved halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 Marginalizing over galaxy formation physics A key advantage of CAMELS over more standard cosmological sim- ulations performed with a single fiducial galaxy formation model is the ability to train machine learning algorithms to learn funda- mental properties of galaxies and the Universe while marginalizing over uncertainties in subgrid physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Villanueva-Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' When training a RF on the LH simulation set of either TNG or SIMBA to predict Δ𝑃/𝑃DM given halo baryon fractions as input features, we are at the same time marginalizing over uncertainties in physical processes represented by the parameter variations intro- duced in a given galaxy formation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, evaluating the robustness of the ML model to uncer- tainties in galaxy formation physics should also consider different implementations and not just variations of parameters within a given subgrid physics implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We thus perform a more stringent test of robustness by training the RF on the full LH set of one galaxy formation model (either SIMBA or TNG) and then testing on the full LH set of the other model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 10 shows the predicted versus true values of Δ𝑃/𝑃DM at 𝑘 = 5 ℎ Mpc−1 when training a RF using 𝑓 bar(𝑀j halo) and 𝑁j halo from SIMBA and testing on TNG (top) and when training on TNG and testing on SIMBA (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 10 shows that the RF trained on SIMBA can explain ∼80% of the variation of Δ𝑃/𝑃DM at 𝑘 = 5 ℎ Mpc−1 when tested on TNG, suggesting that the RF has found a relation between halo baryon fractions and suppression of matter clustering which is rel- atively robust to galaxy formation model implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' On the other hand, the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 10 shows that the RF trained on TNG is less robust when tested on SIMBA, and can only explain ∼70% of the variation in Δ𝑃/𝑃DM predicted by the SIMBA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In this case, we can see that the predicted Δ𝑃/𝑃DM is clearly biased high (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', less negative) when training on TNG and predicting on SIMBA, implying that the inferred suppression of matter clustering is under-predicted given the halo baryon fractions in SIMBA and the connection to Δ𝑃/𝑃DM learned from TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As expected, we see a bias in the opposite direction when training on SIMBA and predict- ing on TNG (top panel), over-predicting the suppression of power at 𝑘 = 5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 3 and 4, the overall range of variation in Δ𝑃/𝑃DM is significantly larger in the SIMBA LH set compared to the TNG LH set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Given that the RF cannot predict values outside of the range of the training data, this can explain why the RF trained on TNG is less robust relative to galaxy formation implementation and shows a stronger bias when tested on SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The biased estima- tion of Δ𝑃/𝑃DM can thus be partially attributed to the RF learning the limits of the range of variation in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nonetheless, our results suggest that the TNG and SIMBA models may predict different Δ𝑃/𝑃DM even when implementing parameters that yield similar 𝑓 bar(𝑀j halo), implying a non-unique relation between halo baryon fractions and impact on matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 5 SUMMARY AND DISCUSSION In this paper, we have investigated how baryonic physics affects the clustering of matter relative to N-body simulations and its re- lation to the baryonic content of halos using thousands of cos- mological hydrodynamic simulations from the CAMELS project (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In the first part of the paper, we examined how variations of individual cosmological parameters (Ωm and 𝜎8) and feedback parameters (controlling the efficiency of large-scale outflows driven by SNe and AGN) impact the total matter power spectrum, the mean halo baryon fraction as a function of halo mass, and the connection between the suppression of clus- tering Δ𝑃/𝑃DM and the mean baryon fraction of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In the second part of the paper, we have presented a set of machine learning experiments training a random forest (RF) regressor on fea- tures including the baryon fraction and abundance of halos across the mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 to predict the sup- pression of power Δ𝑃/𝑃DM from linear to highly non-linear scales 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–20 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Throughout the paper, we have made use of the CAMELS simulation suites performed with the TNG (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018)and SIMBA (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2019) models to un- MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 15 500 1000 1500 Nj halo −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 ∆P/PDM k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 h Mpc−1 Mj halo(1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 − 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0) h−1 M⊙ IllustrisTNG 500 1000 1500 Nj halo SIMBA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='45 Ωm Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Correlation between power spectrum suppression Δ𝑃/𝑃DM and the number 𝑁 j halo of halos in the mass range 𝑀j halo = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–1011 ℎ−1 M⊙ℎ−1 at 𝑘 = 5 ℎ Mpc−1 for TNG (left) and SIMBA (right), color coded by Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The number of low mass halos 𝑁 j halo (highest ranked feature at 𝑘 = 5 ℎ Mpc−1) is a strong tracer of Ωm in CAMELS and a good predictor of Δ𝑃/𝑃DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 PREDICTED ∆P/PDM Train : SIMBA Test : IllustrisTNG k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 [h Mpc−1] R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='814 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 TRUE ∆P/PDM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 PREDICTED ∆P/PDM Train : IllustrisTNG Test : SIMBA k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 [h Mpc−1] R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='698 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 7 but training a RF on the entire LH set of one galaxy formation model and testing on the entire LH set of the other for both TNG and SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We note higher prediction score when training on SIMBA and testing on TNG (𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='814) compared to training on TNG and predicting on SIMBA (𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='698), which can be attributed to SIMBA’s wider range of Δ𝑃/𝑃DM values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The RF tends to under-predict Δ𝑃/𝑃DM when trained on SIMBA and over-predict Δ𝑃/𝑃DM when trained on TNG due to variations in feedback models which suggests that care must be made if applying this model to data where feedback is not precisely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' derstand the dependence of results and the robustness of the trained machine learning models to changes in the specific galaxy formation physics implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Our main findings can be summarized as follows: In agreement with previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c), we find that baryonic physics can profoundly affect the total matter power spectrum all the way to scales 𝑘 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 ℎ Mpc−1, and the magnitude of this effect is highly dependent on the details of the galaxy formation implementation and variations of cosmological and astrophysical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The suppression of power |Δ𝑃/𝑃DM| increases systematically with decreasing Ωm at fixed Ωb, with baryons contributing a higher fraction of the total matter content and feedback more efficiently spreading matter over larger scales relative to N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Varying 𝜎8 at fixed galaxy formation physics does not drive systematic variations in Δ𝑃/𝑃DM when measured on the small (25 ℎ−1Mpc)3 volumes simulated in CAMELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Increasing AGN feedback efficiency generally drives higher suppression of matter clustering, in agreement with previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' vDMS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022), with the strongest effect seen for high-speed jets in SIMBA which are able to spread a substantial amount of baryons over scales of several Mpc (Borrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The qualitative effect of stellar feedback on matter clustering is more dependent on galaxy formation model, which can either suppress or enhance power on different scales depending on the interplay between stellar and AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Stronger stellar feedback often results in weaker overall suppression of matter clustering by suppressing black hole growth and therefore the effective efficiency of AGN feedback (van Daalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Halo baryon fractions 𝑓 bar are very sensitive to galaxy forma- tion model, with TNG producing systematically more baryon-rich halos compared to SIMBA for a broad range of parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Higher AGN feedback efficiency generally decreases halo baryon fractions, but the extent of the effect and the affected halo mass range depend on model details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Increasing the strength of stellar feedback MNRAS 000, 1–18 (2021) 16 Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' can either decrease or increase the baryon fraction depending on the non-linear coupling of stellar feedback and black hole growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Halo baryon fractions are also very sensitive to changes in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Increasing Ωm (at fixed Ωb) or 𝜎8 systematically decreases the baryon fraction of halos (normalized by Ωb/Ωm), indicating a non-trivial response of feedback to changes in the amount of baryons relative to dark matter and the growth history of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We find a broad correlation between the amount of suppression of the matter power spectrum Δ𝑃/𝑃DM and the baryon fraction of massive halos 𝑓 bar, indicating that the feedback mechanisms responsible for evacuating gas from massive halos also dominate the impact of baryonic effects on matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These results are in broad agreement with vDMS, but the thousands of sim- ulations in CAMELS produce significantly larger scatter in the Δ𝑃/𝑃DM– 𝑓 bar relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Cosmic variance alone can significantly affect the matter power spectrum on our (25 ℎ−1Mpc)3 simulated volumes, but the complex trends seen for the impact of individual cosmological and feedback parameter variations on Δ𝑃/𝑃DM and 𝑓 bar suggest that the vDMS model predicting Δ𝑃/𝑃DM given only 𝑓 bar for massive halos is not general enough to include every plausible feedback model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Predicting the impact on matter clustering based only on the mean baryon fraction of massive halos using the vDMS Δ𝑃/𝑃DM– 𝑓 bar relation is not possible given the broad range of galaxy formation models and the impact of cosmic variance in CAMELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, we have demonstrated that a RF regressor trained on CAMELS is able to extract information from halos across the full mass range 1010 ≤ 𝑀halo/M⊙ ℎ−1 < 1014 to estimate the suppression of the matter power spectrum on scales 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5–20 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We are thus not only extracting information from low-mass halos but also predicting Δ𝑃/𝑃DM in the highly non-linear regime, significantly extending the range of scales 𝑘 < 1 ℎ Mpc−1 where the vDMS model can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Using the mean halo baryon fraction and abundance in different halo mass bins as input features, the RF regressor was able to account for ∼80–85% of the Δ𝑃/𝑃DM variation occurring on scales 𝑘 = 10– 20 ℎ Mpc−1 where the impact of feedback on the matter power spec- trum becomes the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' At 𝑘 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 ℎMpc−1, our best model was able to explain ∼92% of the variance in the suppression of power due to feedback when training on the TNG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, the same model can only explain ∼70% of the variation in Δ𝑃/𝑃DM and tends to under-predict the suppression of matter clustering when tested on the SIMBA simulations, indicating that the RF is only moderately robust relative to changes in the underlying galaxy formation im- plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Training on SIMBA increases the robustness of the model owing to its larger range of variation in Δ𝑃/𝑃DM compared to TNG, but in this case the RF tends to over-predict |Δ𝑃/𝑃DM| when tested on TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' These results suggest that the lack of a universal relation between halo baryon fractions and impact on matter clus- tering and emphasize the need to construct models that are robust against assumptions in baryonic physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' A unique advantage of CAMELS relative to previous work is that it performs simulations for different baryonic physics implementations and a broad range of cosmological and feedback parameter varia- tions, providing a data-set sufficiently large to train machine learning algorithms for a variety of applications (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, an important limitation of CAMELS is the small volume of each simulation realization, 𝐿box = 25 ℎ−1Mpc, with important implications for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Given the small box sizes, the matter power spectrum is sensitive to the specific initial condi- tions in each realization, and the impact of baryonic effects further depends on stochastic processes related to feedback operating on a limited number of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As a result, cosmic variance represents a challenge to infer the suppression of matter clustering Δ𝑃/𝑃DM given only the baryon fraction of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Previ- ous works in CAMELS have devised strategies to correct for the noise introduced by cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' When training a neural net- work on electron density auto-power spectra to predict Ωm, Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2022) constructed a cosmic variance parameter based on the distribution of halo masses in each realization, improving the pre- dictions significantly when introduced as an additional training fea- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Thiele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' (2022) used spectral distortion measurements to constrain baryonic feedback and applied a correction factor to the Compton-y distortion by comparing expected values from a simple halo model evaluated for the halo mass function in each CAMELS simulation compared to that of a standard halo mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' In our RF experiments, introducing the number of halos in each mass bin 𝑁j halo as input features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=', basically the halo mass function) improves the accuracy of the predictions significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The number of low-mass halos was one of the most predictive features identified by the RF, which can be understood as a strong tracer of Ωm, but 𝑁j halo also helped reduced cosmic variance effects, in agreement to previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The small simulated volumes in CAMELS also complicate a di- rect comparison between our results and vDMS, where in addition to cosmic variance affecting Δ𝑃/𝑃DM we also lack halos massive enough to evaluate the mean baryon fraction under the same con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Based on a suite of simulations of substantially larger vol- umes, vDMS identified a tight relationship between mean baryon fraction in massive halos and baryonic power spectrum suppression at 𝑘 < 1 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' While some degree of correlation between these quantities is expected, as we find here, the extremely small scatter in the observed relationship was remarkable given the variety of galaxy formation models compared by vDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Since we observed some hints at a deviation from the vDMS relationship in CAMELS, particularly for the parameter variations based on the SIMBA model, we have added the original 100 ℎ−1Mpc SIMBA volume to the vDMS rela- tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Here, we show the new SIMBA data points as circles, while the measurements considered in vDMS are shown as triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We have verified our pipeline by running it on IllustrisTNG-300, achieving excellent agreement with the vDMS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' As can be seen, SIMBA constitutes a considerable outlier and does not fall within the 1% interval around the vDMS fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The only other simulation scattering that far is Illustris which, however, does not reproduce the observed baryon fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' It is known that the feed- back prescription in SIMBA is unique in its ability to re-distribute baryons across large scales (Borrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2020), which could explain the observed deviation from the vDMS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' However, SIMBA generally does not do worse in reproducing observational relation- ships than the other major hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Thus, these results suggest the possibility that the vDMS relation only holds in a sub-space of simulations while there exists at least one dimension along which deviations occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 6 ACKNOWLEDGMENTS We wish to thank Dylan Nelson for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' DAA acknowledges support by NSF grants AST-2009687 and MNRAS 000, 1–18 (2021) Matter clustering in CAMELS 17 10 3 10 2 10 1 100 P(k)/PDM(k) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 h/Mpc k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='3 h/Mpc k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 h/Mpc 10 3 10 2 10 1 100 P(k)/PDM(k) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='5 h/Mpc SIMBA100 van Daalen+2020 fits (intervals=1%) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 h/Mpc k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='7 h/Mpc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 fbar(r < R , M 1014M )/( b/ m) 10 3 10 2 10 1 100 P(k)/PDM(k) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 h/Mpc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 fbar(r < R , M 1014M )/( b/ m) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='9 h/Mpc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 fbar(r < R , M 1014M )/( b/ m) k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='0 h/Mpc van Daalen+2020: Cosmo-OWLS BAHAMAS EAGLE Illustris Illustris TNG100 Illustris TNG300 Horizon-AGN L > 100 Mpc/h L 100 Mpc/h = 500c = 200c Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Comparison of the power spectrum suppression in the original SIMBA simulation with the results from vDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The SIMBA data points are circles, while the measurements from vDMS are reproduced as triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' We also show the fitting functions from vDMS as dashed lines as well as 1% range of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 200c and 500c mass definitions are shown in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' The vertical error bars have been estimated by splitting the 100 ℎ−1Mpc SIMBA simulation into 8 sub-volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' Statistical error bars on the horizontal axis are small (∼ 10−3 as estimated using jack-knife), but there are somewhat larger systematic errors from the halo finding (∼ 10−2, as estimated by running the analysis with FOF and Rockstar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' SIMBA appears to deviate from the best-fit relation of vDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' AST-2108944, CXO grant TM2-23006X, and Simons Foundation award CCA-1018464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 7 DATA AVAILABILITY The simulations used in this work are part of the CAMELS public data release (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' 2022) and are available at https://camels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content='io/en/latest/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQfTADA/content/2301.02231v1.pdf'} +page_content=' REFERENCES Agertz O.' metadata={'source': 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a/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/2301.01540v1.pdf.txt b/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/2301.01540v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d20fceb8ddee321b613653040219efa9635eacb9 --- /dev/null +++ b/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/2301.01540v1.pdf.txt @@ -0,0 +1,3739 @@ +arXiv:2301.01540v1 [math.PR] 4 Jan 2023 +GAUSSIAN APPROXIMATION FOR THE MOVING AVERAGED +MODULUS WAVELET TRANSFORM AND ITS VARIANTS +GI-REN LIU, YUAN-CHUNG SHEU, AND HAU-TIENG WU +Abstract. The moving average of the complex modulus of the analytic wavelet trans- +form provides a robust time-scale representation for signals to small time shifts and +deformation. In this work, we derive the Wiener chaos expansion of this represen- +tation for stationary Gaussian processes by the Malliavin calculus and combinatorial +techniques. The expansion allows us to obtain a lower bound for the Wasserstein dis- +tance between the time-scale representations of two long-range dependent Gaussian +processes in terms of Hurst indices. Moreover, we apply the expansion to establish an +upper bound for the smooth Wasserstein distance and the Kolmogorov distance be- +tween the distributions of a random vector derived from the time-scale representation +and its normal counterpart. It is worth mentioning that the expansion consists of infi- +nite Wiener chaos and the projection coefficients converge to zero slowly as the order +of the Wiener chaos increases, and we provide a rational-decay upper bound for these +distribution distances, the rate of which depends on the nonlinear transformation of +the amplitude of the complex wavelet coefficients. +1. Introduction +The wavelet transform (WT) is one of the most useful tools in signal processing [15, 30]. +It performs a scale decomposition of an input signal X by convolving it with a family +of functions {ψj}j∈Z, which are generated by dilating a mother wavelet ψ in the way +ψj(t) = 2−jψ(2−jt), where t ∈ R. The WT of X, denoted by {W[j]X(t)}t∈R,j∈Z, where +W[j]X(t) := X ⋆ ψj(t), provides both the magnitude and phase information of X in the +time-scale domain when the mother wavelet ψ is complex-valued. This information has +been applied to analyze the heart rate variability [39], detect the seizure through the +Electroencephalography (EEG) signals [17], and prove the existence of intermittency in +the local field potentials recorded from patients with Parkinson’s disease [42]. The WT +also serves as a feature extractor in the field of machine learning [10, 24, 25, 45, 47], +particularly when the invariance to small temporal shifts is desired. In light of the fact +that a small temporal shift of X produces a phase shift on {W[j]X(t)}t∈R,j∈Z, usually +the modulus of the wavelet coefficients (i.e., {UA[j]X}j∈Z := {A(|W[j]X(t)|)}t∈R,j∈Z +with A(r) = r) and its variants (e.g., A(r) = r2 [6] and A(r) = ln(r) [7, 21]) are +used [19, Theorem 1]. In order to make the features stable to actions of small shifts +2020 Mathematics Subject Classification. Primary 60G60, 60H05, 62M15; Secondary 35K15. +Key words and phrases. +analytic wavelet transform; complex modulus; Wiener-Itˆo decomposition; +Gaussian approximation; Stein’s method; Malliavin calculus; smooth Wasserstein distance. +1 + +2 +G. R. LIU, Y. C. SHEU, AND H. T. WU +and deformations to X, practitioners consider SA +J [j], which comes from convolving the +output of UA[j] with a low-pass filter φJ [31]: +(1) +SA +J [j]X(t) := +� +UA[j]X +� +⋆ φJ(t), +where J is an integer used to control the width of the low-pass filter φJ through +φJ(t) = 2−Jφ +� +2−Jt +� +and φ is a real-valued function, usually chosen to be the father +wavelet associated with ψ. In the work [31], {SA +J [j]}j∈Z is called the first-order scatter- +ing transform and its higher-order counterparts +� +UA[jn] · · ·UA[j2]UA[j1]X +� +⋆ φJ, where +n ∈ N and j1, . . . , jn ∈ Z, are introduced to improve the mathematical understanding of +convolutional neural networks. +In the presence of noise and interferences, the input signal X is usually viewed as +a random process. +In [3, 4, 11, 32], some properties of random processes, such as +stationarity and self-similarity, are characterized by corresponding properties of their +wavelet transform. Especially, for stationary random process X, which may be deformed +by a stationarity-breaking operator, how to use the instantaneous wavelet spectrum (i.e., +E +� +UA[j]X(t) +� +with A(r) = r2) to determine the second-order statistical properties of X +was discussed in [22, 34]. In [41], the authors considered a discrete-time analog of the +transformation W[j]X +Zj,t = +�� +k∈Z +vi,j(γjt − k)X(k) +� +i=1,2,...,d +, +(2) +where d is a fixed positive integer corresponding to the number of WT coefficients we are +interested in, {γj}j∈N is a divergent sequence of positive integers corresponding to the +scale, e.g., γj = 2j, t ∈ Z corresponding to the time, vi,j : Z → R corresponding to the +discretized wavelet satisfies � +k∈Z +v2 +i,j(k) < ∞, and {X(k)}k∈Z is a sequence of independent +and identically distributed (i.i.d.) real-valued random variables. When j → ∞, they +proved that the sequence of d-dimensional vectors +n−1/2 +j +nj−1 +� +t=0 +� +Z2 +j,t − E +� +Z2 +j,t +�� +, +(3) +converges to a d-dimensional normal random vector in distribution sense under some +conditions on vi,j and width of the averaging nj. The i.i.d. random sequence {X(k)}k∈Z +in (2) was extended to long-range dependent Gaussian sequences in [35] as well as sub- +ordinated Gaussian sequences with long-term or short-term memory in [13, 43]. Due to +the difference in the strength of dependence of the sequence X, the limiting theorem for +the wavelet coefficients in [13] was proved by the so-called large scale reduction principle, +while the central limit theorem for (3) in [43, Theorem 1] was proved by showing that +the sequence {Z2 +j,t −E +� +Z2 +j,t +� +}t is strong mixing. Especially, when the sequence X is gen- +erated from a Hermite polynomial of a stationary Gaussian sequence with long memory, +the authors of [14] showed that after suitable renormalization, (3) converges either to +a Gaussian process or to a Rosenblatt process when j → ∞ depending on the order of + +3 +Hermite polynomials. For the discrete wavelet packet decomposition, which performs +the discrete WT iteratively without taking account of any nonlinear activation functions, +of stationary random processes, [2] proves that the outputs of decomposition converge +in distribution to white Gaussian processes when the resolution level of decomposition +increases by showing that the third and higher-order cumulants of the outputs converge +to zero. In [9], the authors considered a transformation similar to (1) as follows +2j/2|X ⋆ ψ| ⋆ ψj(t), t ∈ R. +(4) +When X is a fractional Brownian motion, the authors of [9] applied a central limit +theorem for sums of locally dependent random variables [8] to show that the marginal +distribution of the random process defined in (4) converges to a complex normal distribu- +tion when j → ∞ [9, Lemma 3.3]. The limit of (4) was also studied in our previous work +[27] in the case when X is a stationary Gaussian process with long-range or short-range +dependence and ψ is a real-valued wavelet. +In this work, we analyze the smooth Wasserstein distance and the Kolmogorov distance +between the finite-dimensional distributions of the centralized SA +J [j]X and its Gaussian +counterpart, which is a normal random vector with the same covariance structure as +SA +J [j]X, in the case when X is a stationary Gaussian process and ψ is an analytic wavelet. +According to our literature survey, there is no previous work on establishing the spectral +representation of the modulus, or more general transform, of WT of Gaussian processes. +It is indispensable for calculating the convolution UA[j]X ⋆ φJ. The first contribution of +this work is to provide spectral representations for UA[j]X and SA +J [j]X for homogeneous +or logarithmic functions A. The representation is expressed as a linear combination +of finite or infinite orthogonal Wiener chaos, depending on function A. +The second +contribution of this work is to provide a quantitative central limit theorem for SA +J [j]X. +We note that the quantitative central limit theorem for random processes expressed as +a linear combination of finite Wiener chaos has been well analyzed [36, 37, 38], but it is +unclear how to deal with the case when there are infinite Wiener chaos. +To achieve these goals, we apply the Malliavin calculus and combinatorial techniques +to analyze the complex modulus of the wavelet coefficients W[j]X. This part has never +been considered in existing works, including our previous work [26] in which the wavelet +ψ is real-valued and A(r) = r2. It is worth mentioning that the orthogonal expansion +of SA +J [j]X is an infinite sum of Wiener chaos, particularly for the practical cases A(r) = +r and A(r) = ln(r). +In order to apply the general upper bound for the Gaussian +approximation error in [36, Proposition 3.7] and [37, Theorem 6.1.2 and (6.3.2)] to the +case of infinite sum of Wiener chaos, we simplify the complexity of the expression of the +upper bound and provide a more concise expression. From the more concise expression +for the upper bound (Proposition 1), we obtain upper bounds for the smooth Wasserstein +distance and the Kolmogorov distance between the laws of the centralized SA +J [j]X and +its Gaussian counterpart. The upper bounds depend on the window length of the moving +average and the function A (Theorem 2 and Corollary 2). + +4 +G. R. LIU, Y. C. SHEU, AND H. T. WU +The rest of the paper is organized as follows. In Section 2, we summarize necessary +material for WT when the wavelet is analytic and present some preliminaries about the +Wiener-Itˆo integrals and the Malliavin calculus. In Section 3, we state our main results, +including Theorems 1 and 2. The proofs of our main results and some technical lemmas +are given in the appendix. +2. Preliminaries +2.1. Wavelet transform with an analytic wavelet. Let ψ be a complex-valued +function in L1(R) ∩ L2(R) defined through two real-valued mother wavelets ψR and ψI +as follows +ψ(t) = ψR(t) + iψI(t), t ∈ R. +The function ψ is called an analytic wavelet if ψR and ψI meet the Hilbert pair require- +ment +(5) +ψI(t) = 1 +π +� ∞ +−∞ +ψR(τ) +t − τ dτ. +Given a signal X : R → R, the family of analytic wavelets +ψj(t) = 1 +2j ψ +� t +2j +� +, j ∈ Z, +defines the WT of X [23] through the convolution of ψj with the input X: +W[j]X(t) = X ⋆ ψj(t) = +� +R +X(s)ψj(t − s)ds, j ∈ Z. +The magnitude information returned by the analytic wavelet transform +UA [j] X :=A(|W[j]X(t)|), j ∈ Z, +(6) +describes the envelope of wavelet coefficients of X, where A(r) = r for r ∈ [0, ∞). +When A(r) = r2 (resp. ln(r)), the time-scale representation {UA [j] X(t)}t∈R,j∈Z is the +so-called scalogram (resp. logarithmic scalogram) of X in the field of signal processing +[1, Figure 8]. In the following, we consider two types of functions for A: A(r) = rν for +some ν ∈ (0, ∞) or A(r) = ln(r). +In practice, in order to make the features stable to actions of small diffeomorphisms, +such as deformations, to X, the transformation UA[j] is followed by a convolution with +a low-pass filter φJ [1, 31]: +SA +J [j]X(t) := +� +UA[j]X +� +⋆ φJ(t) = +� +R +UA[j]X(s)φJ(t − s)ds, J ∈ Z, +where +φJ(t) = 1 +2J φ +� t +2J +� + +5 +and φ is a real-valued function, usually chosen to be the father wavelet associated with +ψ [30]. We denote the Fourier transform of ψR by � +ψR , i.e., +� +ψR(λ) = +� +R +e−itλψR(t)dt. +Similarly, the Fourier transform of ψI is denoted as � +ψI. Because ψR and ψI are a Hilbert +pair (5), +� +ψI(λ) = −i sgn(λ)� +ψR(λ), +(7) +where +sgn(λ) = + + + +−1 +if λ < 0, +0 +if λ = 0, ++1 +if λ > 0. +For each j ∈ Z, the Fourier transform of the scaled wavelets +ψR,j(t) := 1 +2j ψR +� t +2j +� +and ψI,j(t) := 1 +2j ψI +� t +2j +� +are � +ψR(2j·) and � +ψI(2j·), respectively. +In terms of notation, � +ψR,j(·) = � +ψR(2j·) and +� +ψI,j(·) = � +ψI(2j·). +All assumptions needed in this paper about the analytic wavelet +ψ and the low-pass filter φJ are summarized as follows. +Assumption 1. For the real part ψR of the analytic wavelet ψ, we assume that � +ψR ∈ +L1(R) ∩ L2(R) and there exists a bounded and continuous function CψR : R → C with +CψR(0) ̸= 0 such that +� +ψR(λ) = CψR(λ)|λ|α, λ ∈ R, +for a certain α > 0. For the averaging function φ, we assume that it is real-valued and +�φ ∈ L1(R) ∩ L2(R). +2.2. Stationary Gaussian processes. Given the relative breadth of analytical tools +for Gaussian processes [28, 37, 38] and their ubiquity in applications [29], we consider +stationary Gaussian processes as a model for the input X of the analytic wavelet trans- +form. Let W be a complex-valued Gaussian random measure on R satisfying +W(∆1) = W(−∆1), +E[W(∆1)] = 0, +and +E +� +W(∆1)W(∆2) +� += Leb(∆1 ∩ ∆2) +for any ∆1, ∆2 ∈ B(R), where Leb is the Lebesgue measure on R and B(R) is the +Borel σ-algebra on R. Let (Ω, FW, P) be a probability space, where the σ-algebra FW +is generated by W. +If X is a mean-square continuous and stationary real Gaussian random process with +constant mean µX and covariance function RX, by the Bochner-Khinchin theorem [20, + +6 +G. R. LIU, Y. C. SHEU, AND H. T. WU +Chapter 4], there exists a unique nonnegative measure FX : B(R) → [0, ∞) such that +FX(∆) = FX(−∆) for any ∆ ∈ B(R) and +RX(t) = +� +R +eiλtFX(dλ), t ∈ R. +The measure FX is called the spectral measure of the covariance function RX. +Assumption 2. The spectral measure FX is absolutely continuous with respect to the +Lebesgue measure with a density function fX ∈ L1(R). It has one of the forms: (a) +fX(λ) = CX(λ) or (b) +fX(λ) = CX(λ) +|λ|1−β , +where β ∈ (0, 1) is the Hurst index of long-range dependence and CX : R → [0, ∞) is +bounded and continuous. +For convenience of presentation, we set 00 = 1, by which Assumption 2(a) can be +viewed as a limiting case of Assumption 2(b). Under Assumption 2, X can be expressed +as a Wiener integral as follows +X(t) = µX + +� +R +eitλ� +fX(λ)W(dλ), t ∈ R. +Because +� +R ψj(s)ds = 0, by the stochastic Fubini theorem [40, Theorem 2.1], +X ⋆ ψj(t) = +� +R +eitλ � +ψR(2jλ) +� +fX(λ)W(dλ) + i +� +R +eitλ� +ψI(2jλ) +� +fX(λ)W(dλ). +(8) +Both the real and imaginary parts of X ⋆ ψj(t) are normal random variables with mean +zero and variance +σ2 +j = +� +R +|� +ψR(2jλ)|2fX(λ)dλ = +� +R +|� +ψI(2jλ)|2fX(λ)dλ. +For k ∈ N ∪ {0}, denote Lk (u) to be the Laguerre polynomial of degree k with the +formula +Lk (u) = eu +k! +dk +duk +� +e−uuk� +, k ∈ N ∪ {0}. +We have the following first lemma. +Lemma 1. For x1, x2 ∈ R and function A : (0, ∞) → R satisfying +� ∞ +0 +� +A( +√ +2r) +�2 +e−rdr < ∞, +we have the expansion +A (|x1 + ix2|) = +� +m,n∈N∪{0} +Cm,n +Hm(x1) +√ +m! +Hn(x2) +√ +n! +, +(9) +where +Hm(x) = (−1)me +x2 +2 dm +dxme− x2 +2 , m = 0, 1, 2, . . ., + +7 +are the (probabilistic) Hermite polynomials, +Cm,n = +� hmhncA,(m+n)/2 +for m, n ∈ 2N ∪ {0}, +0 +otherwise, +(10) +hm = (−1) +m +2 +√ +m! +2 +m +2 � m +2 +� +! for m ∈ 2N ∪ {0}, +(11) +and +cA,(m+n)/2 = +� ∞ +0 +A( +√ +2u)L m+n +2 +(u) e−udu. +(12) +The proof of Lemma 1 is provided in A. The constants {cA,(m+n)/2}m,n∈2N∪{0} in Lemma +1 for practical cases are shown as follows. +• A(r) = rν, ν ∈ (0, ∞): +cA, ℓ +2 = 2 +ν +2 Γ(ν +2 + 1) +� ℓ +2 − ν +2 − 1 +ℓ +2 +� +, ℓ ∈ 2N. +(13) +For the binomial coefficient above, we note that +� ℓ +2 − ν +2 − 1 +ℓ +2 +� += +�ℓ +2 − ν +2 − 1 +� �ℓ +2 − ν +2 − 2 +� +· · · +� +1 − ν +2 +� � +−ν +2 +� �ℓ +2! +�−1 +. +Especially, when ν = 2, +cA,0 = 2, cA,1 = −2, +and cA, ℓ +2 = 0 for ℓ ∈ {4, 6, 8, . . .}. +• A(r) = ln(r): cA,0 = 1 +2 ln 2 − 1 +2γ, where γ is the Euler-Mascheroni constant, and +cA, ℓ +2 = −1 +ℓ +for ℓ ∈ 2N. +By (8) and Lemma 1, for the case A(r) = rν with ν > 0, +UA[j]X(t) =σν +j +� +m,n∈N∪{0} +Cm,n +√ +m!n! +Hm +� 1 +σj +� +R +eitλ � +ψR(2jλ) +� +fX(λ)W(dλ) +� +× Hn +� 1 +σj +� +R +eitλ� +ψI(2jλ) +� +fX(λ)W(dλ) +� +. +(14) +For the case A(r) = ln(r), +UA[j]X(t) = ln(σj) + +� +m,n∈N∪{0} +Cm,n +√ +m!n! +Hm +� 1 +σj +� +R +eitλ � +ψR(2jλ) +� +fX(λ)W(dλ) +� +× Hn +� 1 +σj +� +R +eitλ� +ψI(2jλ) +� +fX(λ)W(dλ) +� +. +(15) + +8 +G. R. LIU, Y. C. SHEU, AND H. T. WU +Let H = {f ∈ L2(R) | f(−λ) = f(λ) for all λ ∈ R} be a complex Hilbert space +with the inner product ⟨f, g⟩ = +� +R f(λ)g(λ)dλ. Given an integer m ≥ 1, we denote +the m-th tensor product of the Hilbert space H by H +⊗m. The m-th symmetric tensor +product of H is denoted by H +⊙m, which contains those functions f ∈ H +⊗m satisfying +f(λp(1), . . . , λp(m)) = f(λ1, . . . , λm) for any permutation (p(1), p(2), . . . , p(m)) of the set +{1, 2, . . . , m}. For any f ∈ H +⊗m, the m-fold Wiener-Itˆo integrals of f with respect to +the random measure W is defined by +Im(f) = +� +′ +Rm f(λ1, . . . , λm)W(dλ1) · · ·W(dλm), +where +� ′ means that the integral excludes the diagonal hyperplanes λk = ∓λk′ for +k, k +′ ∈ {1, . . . , m} and k ̸= k +′ [28]. Lemma 2 below, which comes from [28, Theorem 4.3 +and Proposition 5.1] and [37, Theorem 2.7.7 and Theorem 2.7.10], provides a significant +link between nonlinear functions of normal random variables and Wiener-Itˆo integrals. +Lemma 2 (Itˆo’s formula and Product Formula [28, 37]). Let f ∈ H be such that ∥f∥H = +1. Then, for any integer m ≥ 1, we have +Hm +�� +R +f(λ)W(dλ) +� += Im +� +f ⊗m� +. +For any m, n ≥ 1. If f ∈ H +⊙m and g ∈ H +⊙n, then +Im(f)In(g) = +m∧n +� +r=0 +r! +�m +r +��n +r +� +Im+n−2r (f ⊗r g) , +where f ⊗r g is the rth contraction of f and g defined as +f ⊗r g(λ1:m+n−2r) = +� +Rr f(τ1:r, λ1:m−r)g(−τ1:r, λm−r+1:m+n−2r)dτ1 · · · dτr +for r = 1, 2, . . . , m ∧ n. When r = 0, set f ⊗0 g = f ⊗ g. +Here, for any integers p1 < p2, we denote (λp1, λp1+1, . . . , λp2) by λp1:p2 to simplify the +lengthy expressions. By Lemma 2, UA[j]X in (14) and (15) can be further expressed +as triple summations of Wiener-Itˆo integrals. However, the obtained decomposition of +UA[j]X is not orthogonal due to the double sum over m and n in (14) and (15). On +the other hand, because of the Hilbert pair relation (7) between the real and imaginary +parts of the analytic wavelet, the weighted sum of products of sign functions in (16) will +naturally pop out. For obtaining an orthogonal Wiener chaos decomposition of UA[j]X +(Theorem 1 below), the following lemma plays a key role in merging non-orthogonal +terms in the triple summations obtained from (14), (15), and Lemma 2. The proof of +Lemma 3 is provided in B. + +9 +Lemma 3. For ℓ ∈ 2N, let P[ℓ] represent the set of permutations of {1, 2, . . . , ℓ}. For +{λ1, λ2, . . . , λℓ} ⊂ R \ {0}, define +B(ℓ, λ1:ℓ) = 1 +ℓ! +� +p∈P [ℓ] +� +m,n∈2N∪{0} +m+n=ℓ +��m +2 +� +! +�n +2 +� +! +�−1 +(−1) +n +2 +ℓ� +k=ℓ−n+1 +sgn(λp(k)). +(16) +The equality +B(ℓ, λ1:ℓ) = +� 2ℓ ℓ +2! (ℓ!)−1 +if N(λ1:ℓ) = ℓ/2, +0 +if N(λ1:ℓ) ̸= ℓ/2, +(17) +holds, where N(λ1:ℓ) is the number of negative elements in {λ1, λ2, . . . , λℓ}. +3. Main results +Theorem 1. Under Assumptions 1 and 2, the process UA[j]X defined in (6) can be +expressed as a series of Wiener-Itˆo integrals as follows. +(a) For A(r) = rν, where ν ∈ (0, ∞), +UA[j]X(t) = E +� +UA[j]X(t) +� ++ σν +j +� +ℓ∈2N +� +′ +Rℓ Q(ℓ) +t,j (λ1, . . . , λℓ)W(dλ1) · · ·W(dλℓ), +(18) +where E +� +UA[j]X(t) +� += σν +j cA,0, σj = ( +� +R |� +ψR(2jλ)|2fX(λ)dλ)1/2, +Q(ℓ) +t,j(λ1, . . . , λℓ) = cℓ +� +σ−ℓ +j +ℓ� +k=1 +eitλk � +ψR(2jλk) +� +fX(λk) +� +1{N(λ1:ℓ)=ℓ/2}, +cℓ = (−2) +ℓ +2 (ℓ +2!)(ℓ!)−1 cA, ℓ +2, +(19) +N(λ1:ℓ) is the number of negative elements in {λk}ℓ +k=1, and cA, ℓ +2 is defined in (12). +(b) The representation (18) can also be applied to the case A(r) = ln(r) with slight +notation modification: ν = 0 and E +� +UA[j]X(t) +� += cA,0 + ln(σj). +The proof of Theorem is provided in C. Corollary 1 below is an easy implication of +Theorem 1, whose proof is provided in D. +Definition 1. The Wasserstein metric is defined by +dW(Z1, Z2) = sup {|E [h(Z1)] − E [h(Z2)]| | h : R → R is Lipschitz and ∥h∥Lip ≤ 1 } +for any random variables Z1 and Z2. +Corollary 1. Given two stationary Gaussian processes X1 and X2 with spectral densities +fX1 and fX2, denote +σ2 +p,j = E +� +|W[j]Xp|2� +, p = 1, 2. + +10 +G. R. LIU, Y. C. SHEU, AND H. T. WU +Suppose that the wavelet ψ satisfies Assumption 1. For j ∈ Z, we have +dW(UA[j]X1, UA[j]X2) ≥ +� 2 +ν +2 Γ( ν +2 + 1) +��σν +1,j − σν +2,j +�� +if A(r) = rν with ν > 0; +|ln σ1,j − ln σ2,j| +if A(r) = ln(r). +Especially, if A(r) = ln(r) and +(20) +fXp(λ) = CXp(λ) +|λ|1−βp , λ ∈ R \ {0}, p = 1, 2, +for some Hurst parameters β1, β2 ∈ (0, 1) and CX1, CX2 ∈ Cb(R, [0, ∞)), then +2(ln 2)−1 lim +j→∞ +j−1dW(UA[j]X1, UA[j]X2) ≥ |β1 − β2| . +In view of that the Hurst parameters β1 and β2 in (20) give the strength of long-range +dependence of X1 and X2, Corollary 1 shows that if two stationary Gaussian processes +have remarkable difference on the strength of long-range dependence, this discrepancy +will also be reflected in the distribution distance between their logarithmic scalograms, +especially at large-scales. +By Theorem 1 and the stochastic Fubini theorem [40], for the case A(r) = rν, where +ν ∈ (0, ∞), we have +SA +J [j]X(t) = E +� +SA +J [j]X(t) +� ++ +� +ℓ∈2N +� +′ +Rℓ s(ℓ) +t,j(λ1, . . . , λℓ)W(dλ1) · · ·W(dλℓ), +(21) +where +s(ℓ) +t,j(λ1, . . . , λℓ) = σν +j Q(ℓ) +t,j(λ1, . . . , λℓ) � +φJ(λ1 + · · · + λℓ) +(22) +and +E +� +SA +J [j]X(t) +� += +� +R +E +� +UA[j]X(s) +� +φJ(t − s)ds = σν +j cA,0 �φ(0) +for all j, J ∈ Z and t ∈ R. The representation (21) can also be applied to the case A(r) = +ln(r) with slight notation modification: ν = 0 and E +� +SA +J [j]X(t) +� += (cA,0 + ln(σj)) �φ(0). +For any d ∈ N, j1, j2, . . . , jd ∈ Z, and t1, t2, . . . , td ∈ R, let F = (F1, . . . , Fd) with +Fm = 2 +J +2 � +SA +J [jm]X(2Jtm) − E +� +SA +J [jm]X(2Jtm) +�� +, m = 1, 2, . . . , d. +From (21) and (22), for each K ∈ 2N, rewrite F = F≤K+F>K with F≤K = (F1,≤K, . . . , Fd,≤K) +and F>K = (F1,>K, . . . , Fd,>K), where +(23) +Fm,≤K := 2 +J +2 +� +ℓ∈{2,4,...,K} +Iℓ +� +s(ℓ) +2Jtm,jm +� +and +(24) +Fm,>K := 2 +J +2 +� +ℓ∈{K+2,K+4,...} +Iℓ +� +s(ℓ) +2Jtm,jm +� +for m = 1, 2, . . . , d. + +11 +Let N (resp. N≤K) be a d-dimensional normal random vector with the same covariance +matrix as that of F (resp. +F≤K). +Below, we first explore the smooth Wasserstein +distance [5, 16] between F and N, and conclude this section with a corollary concerning +the Kolmogorov distance [37, Definition C.2.1] between F and N. +Definition 2. The smooth Wasserstein distance between the distribution of Rd-valued +random variables F and N is denoted and defined by +dH2(F, N) = sup +h∈H2 +��E [h (F)] − E [h (N)] +��. +Here, the class H2 of the test functions is defined as +H2 = +� +h ∈ C1(Rd) | ∥h∥∞ ≤ 1, ∥h +′∥∞ ≤ 1, h +′ is Lipschitz, and ∥h +′∥Lip ≤ 1 +� +, +where C1(Rd) is the space of continuously differentiable functions on Rd. +To explore the smooth Wasserstein distance between F and N, we bound dH2(F, N) +by the triangle inequality +|E [h (F)] − E [h (N)]| +(25) +≤ |E [h (F)] − E [h (F≤K)]| + |E [h (F≤K)] − E [h (N≤K)]| + |E [h (N≤K)] − E [h (N)]| +and control the right hand side term by term. +Lemma 4. For the case A(r) = rν with ν ∈ (0, ∞) \ 2N, under Assumptions 1 and 2 +with 2α + β ≥ 1, for any h ∈ H2, there exists a constant C ≥ 0 such that +(26) +|E [h (F)] − E [h (F≤K)]| ≤ dCK− ν +2 − 1 +4 +ε +and +(27) +|E [h (N≤K)] − E [h (N)]| ≤ dCK− ν +2 − 1 +4+ε +for a small ε > 0 and all J ∈ Z and K ∈ 2N. The inequalities (26) and (27) can also +be applied to the case A(r) = ln(r) with slight notation modification: ν = 0 and ε = 0. +The proof of Lemma 4 is provided in E. Lemma 4 says that even if the expansion of +F is comprised of infinite Wiener chaos, particularly when ν ∈ (0, ∞) \ 2N, the first and +third terms in (25) still converge to zero when K → ∞ uniformly with respect to J. +The constant C in Lemma 4 only depends on j1, . . . , jd, fX, ψ, φ, and A. Clearly, the +smaller ν is, the slower the convergence is. More details can be found in (E.12). +In the following, we plan to apply Stein’s method to analyze the second term in (25). +Lemma 5 below is a slight modification of [37, Theorem 6.1.2], in which the test function +h is twice differentiable. How to relax the twice differentiability assumption on the test +function h is described in F. + +12 +G. R. LIU, Y. C. SHEU, AND H. T. WU +Lemma 5 ([37]). For any integer d ≥ 2, let S = (S1, . . . , Sd) be a FW-measurable +random vector such that E[|Sm|4] + E[∥DSm∥4 +H] < ∞ and E[Sm] = 0 for m = 1, . . . , d. +Let [C(m, n)]1≤m,n≤d be a non-negative definite matrix in Rd×d, and let NC be a d- +dimensional normal random vector with mean zero and covariance matrix C. Then, for +any function h ∈ H2, we have +|E[h(S)] − E[h(NC)]| ≤ 1 +2∥h +′∥Lipρ, +(28) +where +∥h +′∥Lip = max +1≤m≤d +sup +x,y∈Rd +x̸=y +��� ∂h +∂xm(x) − +∂h +∂xm(y) +��� +∥x − y∥ +, +ρ = +� +� +� +� +d +� +m,n=1 +E +�� +C(m, n) − +� +DSn, −DL−1Sm +��2� +, +L−1 is the pseudo-inverse of the infinitesimal generator of the Ornstein-Uhlenbeck semi- +group, and D is the Malliavin derivative defined in J. +If Sm = Fm,≤K, by the Meyer’s inequality (see (J.2) and (J.3) in the appendix), the +condition E[|Sm|4] + E[∥DSm∥4 +H] < ∞ holds. On the other hand, because Fm,≤K is a +linear combination of Wiener chaos of order greater than or equal to 2, E[Fm,≤K] = 0. +Denote the covariance matrix of (F1,≤K, . . . , Fd,≤K) by CJ,K ∈ Rd×d; that is, +CJ,K(m, n) := E[Fm,≤KFn,≤K], +where 1 ≤ m, n ≤ d, which satisfies the requirement of non-negative definite for the +matrix C in Lemma 5. Because all conditions in Lemma 5 are satisfied, the inequality +(28) holds with S = F≤K and C = CJ,K. By making use of the explicit structure of the +Wiener chaos decomposition of F≤K derived from Theorem 1 (see also (21) and (23)), +we get the following results. +Proposition 1. Under Assumptions 1 and 2 with 2α + β ≥ 1, for any t1, ..., td ∈ R, +j1, ..., jd ∈ Z, and h ∈ H2, there exists a constant C such that +|E[h(F≤K)] − E[h(NCJ,K)]| ≤ C2− J +2 + + +� +ℓ∈{2,4,...,K} +|cℓ| +√ +ℓ!3 +ℓ +2 + + +2 +(29) +for all J ∈ Z and K ∈ 2N, where cℓ is defined in (19). +The proof of Proposition 1 is provided in G. The constant C in (29) only depends on +A, j1, . . . , jd, fX, ψ, and φ. More details can be found in (G.8). +Theorem 2. If Assumptions 1 and 2 hold and 2α + β ≥ 1, then for any d ∈ N, +t1, . . . , td ∈ R, and j1, . . . , jd ∈ Z, the sequence of J-dependent random vectors +F = 2J/2 � +SA +J [j1]X(2Jt1) − E +� +SA +J [j1]X(2Jt1) +� +, . . . , SA +J [jd]X(2Jtd) − E +� +SA +J [jd]X(2Jtd) +�� +, J ∈ Z, + +13 +satisfies +dH2 (F, N) ≤ + + + + + +O(2− J +2 ) +for A(r) = rν with ν ∈ 2N, +O(J− ν +2 − 1 +4+ε) +for A(r) = rν with ν ∈ (0, ∞) \ 2N, +O(J− 1 +4) +for A(r) = ln(r) +(30) +for any ε > 0 when J → ∞, where N is a zero-mean d-dimensional normal random +vector with the same covariance matrix as F. Furthermore, +(31) +lim +J→∞E +� +FTF +� += +� +κm,n +� +R +eiλ(tm−tn)|�φ(λ)|2dλ +� +1≤m,n≤d +, +where +κm,n = 1 +2π +� +R +Cov +� +UA[jm]X(τ), UA[jn]X(0) +� +dτ. +The proof of Theorem 2 is provided in H. Recall that the Kolmogorov distance between +the distribution of Rd-valued random variables F and N is denoted and defined by +dKol(F, N) = +sup +z1,...,zd∈R +��P (F ∈ (−∞, z1] × . . . × (−∞, zd]) +− P (N ∈ (−∞, z1] × . . . × (−∞, zd]) +��. +Proposition 2.6 in [18] allows us to extract Kolmogorov distance bounds between F and +its normal counterpart N from the smooth Wasserstein distance bounds in Theorem 2. +To keep the paper self-contained, the proof of Corollary 2 is provided in I. +Corollary 2. Let the assumptions and notation of Theorem 2 prevail. The J-dependent +random vectors F satisfies +dKol (F, N) ≤ + + + + + +O(2− J +6 ) +for A(r) = rν with ν ∈ 2N, +O(J− ν +6 − 1 +12 +ε) +for A(r) = rν with ν ∈ (0, ∞) \ 2N, +O(J− 1 +12) +for A(r) = ln(r) +for any ε > 0 when J → ∞, where N is a zero-mean d-dimensional normal random +vector with the same covariance matrix as the corresponding F. +4. Discussion and conclusions +We derived the Wiener chaos decomposition of the modulus of the analytic wavelet +transform and its variants of stationary Gaussian processes and proved a quantitative +central limit theorem for its moving average. Because the complex modulus performs +a square root on the scalogram, we observed that Wiener chaos decompositions of the +modulus wavelet transform and the scalogram have a significant difference. The former +consists of infinite Wiener chaos, while the latter consists of only finite Wiener chaos. +Such differences affect the convergence speed of the Gaussian approximation error of +their respective moving averages. + +14 +G. R. LIU, Y. C. SHEU, AND H. T. WU +The modulus wavelet transform, which may be further transformed by a nonlinear +function A, is a core component of the scattering transform [31], in which the composition +of the modulus wavelet transform coupled with the moving average +SA +J [j, j + δ]X(t) = +� +R +UA[j + δ]UA[j]X(s)φJ(t − s)ds, j ∈ Z, δ ∈ N+, +(32) +was proposed to extract more detailed features from X. In order to make sure that +SA +J [j, j + δ] is a non-expansive map [31, Proposition 2.5], the complex modulus, which +corresponds to the case A(r) = r, is used as the activation function. +As we have +observed in this work, UA[j]X in (32) is a non-Gaussian process, which consists of +infinite Wiener chaos for the non-expansive case A(r) = r. Therefore, to further analyze +the second-layer modulus wavelet transform of UA[j]X, developing new techniques to +handle the nonlinear interaction across different layers and analytic wavelet is necessary, +and we will report the results in our future work. 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Proof of Lemma 1 +Because +� +R +� +R +A +�� +x2 +1 + x2 +2 +� 1 +2πe− +x2 +1+x2 +2 +2 +dx1dx2 = +� ∞ +0 +A(r)e− r2 +2 rdr = +� ∞ +0 +A( +√ +2r)e−rdr < ∞ + +17 +and the set of normalized probabilist’s Hermite polynomials { +1 +√ +m!Hm}m∈N∪{0} forms an +orthonormal basis for the Gaussian Hilbert space L2 � +R, +1 +√ +2πe− x2 +2 dx +� +, the expansion (9) +holds with +Cm,n = +� +R +� +R +A +�� +y2 +1 + y2 +2 +� Hm(y1) +√ +m! +Hn(y2) +√ +n! +1 +2πe− +y2 +1+y2 +2 +2 +dy1dy2 +(A.1) +for m, n ∈ N∪{0}. Because A( +� +y2 +1 + y2 +2) is an even function of y1 and y2 and Hm(−y) = +(−1)mHm(y) for any y ∈ R and m ∈ N ∪ {0}, Cm,n = 0 if m or n is odd. Hence, we only +need to compute Cm,n for the cases m, n ∈ 2N∪{0}. By converting to polar coordinates, +(A.1) can be rewritten as +Cm,n = +� ∞ +0 +A(r) +�� 2π +0 +Hm(r cos θ) +√ +m! +Hn(r sin θ) +√ +n! +dθ +� 1 +2πe− r2 +2 rdr. +(A.2) +From [33], in which the physicists’ Hermite polynomials were used, +� 2π +0 +Hm(r cos θ) +√ +m! +Hn(r sin θ) +√ +n! +dθ = 2πhmhnL m+n +2 +�r2 +2 +� +, +(A.3) +where hm is defined in (11) and L m+n +2 +(z) is the Laguerre polynomial of degree m+n +2 . We +obtain (10) by substituting (A.3) into (A.2) as follows +Cm,n = 2πhmhn +� ∞ +0 +A(r)L m+n +2 +�r2 +2 +� 1 +2πe− r2 +2 rdr += hmhn +� ∞ +0 +A( +√ +2u)L m+n +2 +(u) e−udu. +Appendix B. Proof of Lemma 3 +For any n ∈ {0, 2, . . . , ℓ}, where ℓ ∈ 2N, and p ∈ P[ℓ], if λk ̸= 0 for k = 1, 2, . . . , ℓ, +then +ℓ� +k=ℓ−n+1 +sgn(λp(k)) = 1{N(λ1:ℓ,n,p)∈2Z} − 1{N(λ1:ℓ,n,p)/∈2Z}, +(B.1) +where N(λ1:ℓ, n, p) is the cardinality of the set {k | ℓ − n + 1 ≤ k ≤ ℓ, λp(k) < 0}. Let +N = N(λ1:ℓ) ≤ ℓ be the number of negative elements in {λk}ℓ +k=1. By (B.1) and the +probability mass function of hypergeometric random variables, +1 +ℓ! +� +p∈P [ℓ] +ℓ� +k=ℓ−n+1 +sgn(λp(k)) += 1 +ℓ! +� +p∈P [ℓ] +1{N(λ1:ℓ,n,p)∈2Z} − 1 +ℓ! +� +p∈P [ℓ] +1{N(λ1:ℓ,n,p)/∈2Z} = +�ℓ +n +�−1 +an, + +18 +G. R. LIU, Y. C. SHEU, AND H. T. WU +where +(B.2) +an = +n +� +q=0 +(−1)q +�N +q +��ℓ − N +n − q +� +. +Hence, (16) can be rewritten as +B(ℓ, λ1:ℓ) = +� +n∈{0,2,...,ℓ} +��ℓ +2 − n +2 +� +! +�n +2 +� +! +�−1 +(−1) +n +2 +�ℓ +n +�−1 +an = +� +n∈{0,2,...,ℓ} +wnan, +(B.3) +where +wn = 1 +ℓ! +n!(ℓ − n)! +(n/2)!(ℓ/2 − n/2)!(−1) +n +2 . +(B.4) +Case 1 (N = ℓ/2): In this case, for n ∈ {0, 2, . . . , ℓ}, +an = +n +� +q=0 +(−1)q +�ℓ/2 +q +�� ℓ/2 +n − q +� += (−1)n/2 +�ℓ/2 +n/2 +� +, +(B.5) +where the last equality follows from [44, Identity 81 in Page 61]. By substituting (B.5) +into (B.3), B(ℓ, λ1:ℓ) can be rewritten as follows +B(ℓ, λ1:ℓ) = +�ℓ +2! +� 1 +ℓ! +� +n∈{0,2,...,ℓ} +� n +n/2 +�� +ℓ − n +ℓ/2 − n/2 +� += +�ℓ +2! +� 1 +ℓ!2ℓ, +where the last equality follows from [46]. +Case 2 (N > ℓ/2): First of all, since the constant an in (B.2) is recognized to +be the coefficient of xn in the polynomial Pℓ,N(x) = (1 − x)N(1 + x)ℓ−N (in terms of +notation, an = [xn]Pℓ,N(x)), from (B.3), B(ℓ, λ1:ℓ) can be viewed as a weighted sum of +the coefficients of {x0, x2, . . . , xℓ} in Pℓ,N(x), i.e., +B(ℓ, λ1:ℓ) = +� +n∈{0,2,...,ℓ} +wnan = +� +n∈{0,2,...,ℓ} +wn[xn]Pℓ,N(x). +(B.6) +Because Pℓ,N(x) can be rewritten as +Pℓ,N(x) =(1 − x)N +�ℓ−N +� +k=0 +bk(1 − x)k +� += +ℓ−N +� +h=0 +bℓ−N−h(1 − x)ℓ−h +for some constants b0, b1, . . . , bℓ−N, (B.6) can be rewritten as +B(ℓ, λ1:ℓ) = +� +n∈{0,2,...,ℓ} +wn[xn] +�ℓ−N +� +h=0 +bℓ−N−h(1 − x)ℓ−h +� += +ℓ−N +� +h=0 +bℓ−N−hAh, +(B.7) + +19 +where +(B.8) +Ah := +� +n∈{0,2,...,ℓ} +wn[xn](1 − x)ℓ−h. +By (B.4), for each h ∈ {0, 1, . . . , ℓ − N}, +Ah = 1 +ℓ! +� +n∈{0,2,...,ℓ} +n≤ℓ−h +n!(ℓ − n)! +(n/2)!(ℓ/2 − n/2)!(−1) +n +2 +�ℓ − h +n +� += 1 +ℓ!(ℓ − h)! +� +n∈{0,2,...,ℓ} +n≤ℓ−h +(ℓ − n)! +(n/2)!(ℓ/2 − n/2)!(−1) +n +2 +1 +(ℓ − n − h)!. +(B.9) +Obviously, A0 = 0. For h ∈ {1, . . . , ℓ − N} ∩ 2N, from (B.9), +Ah = 1 +ℓ!(ℓ − h)! 2 +h +2 +� +n∈{0,2,...,ℓ} +n≤ℓ−h +(ℓ − n − 1)(ℓ − n − 3) · · ·(ℓ − n − h + 1) +(n/2)!(ℓ/2 − n/2 − h/2)! +(−1) +n +2 += 1 +ℓ! +(ℓ − h)! 2 +h +2 +(ℓ/2 − h/2)! +(ℓ−h)/2 +� +k=0 +(ℓ − 2k − 1)(ℓ − 2k − 3) · · · (ℓ − 2k − h + 1) +� +�� +� +h/2 terms +�(ℓ − h)/2 +k +� +(−1)k. +Under the condition h < ℓ +2, which is satisfied because N > ℓ +2, we have +(ℓ−h)/2 +� +k=0 +kq +�(ℓ − h)/2 +k +� +(−1)k = 0, +q = 0, 1, . . . , h +2. +Hence, we obtain Ah = 0 for h ∈ {1, . . . , ℓ − N} ∩ 2N. +For h ∈ {1, . . . , ℓ − N} \ 2N, from (B.9), +Ah = 1 +ℓ!(ℓ − h)! 2 +h+1 +2 +� +n∈{0,2,...,ℓ} +n≤ℓ−h +(ℓ − n − 1)(ℓ − n − 3) · · ·(ℓ − n − h + 2) +(n/2)!(ℓ/2 − n/2 − h/2 − 1/2)! +(−1) +n +2 += 1 +ℓ! +(ℓ − h)! 2 +h+1 +2 +(ℓ/2 − h/2 − 1/2)! +(ℓ−h−1)/2 +� +k=0 +(ℓ − 2k − 1)(ℓ − 2k − 3) · · ·(ℓ − 2k − h + 2) +� +�� +� +(h−1)/2 terms +�(ℓ − h − 1)/2 +k +� +(−1)k. +Under the condition h < ℓ +2, which is satisfied again because N > ℓ +2, we have +(ℓ−h−1)/2 +� +k=0 +kq +�(ℓ − h − 1)/2 +k +� +(−1)k = 0, +q = 0, 1, . . . , h − 1 +2 +. +Hence, we also have Ah = 0 for h ∈ {1, . . . , ℓ − N} \ 2N. +In summary, we have Ah = 0 for h ∈ {0, 1, . . ., ℓ−N}. Therefore, by (B.7), B(ℓ, λ1:ℓ) = +0 if N > ℓ/2. + +20 +G. R. LIU, Y. C. SHEU, AND H. T. WU +Case 3 (N < ℓ/2): For this case, we consider the following decomposition of Pℓ,N(x): +Pℓ,N(x) =(1 + x)ℓ−N +� N +� +k=0 +�bk(1 + x)k +� += +N +� +h=0 +�bN−h(1 + x)ℓ−h, +where �b0, . . . ,�bN ∈ R. Similar to (B.7), we have +B(ℓ, λ1:ℓ) = +N +� +h=0 +�bN−h �Ah, +where �Ah := +� +n∈{0,2,...,ℓ} +wn[xn](1 + x)ℓ−h. Because [xn](1 + x)ℓ−h = +�ℓ−h +n +� += [xn](1 − x)ℓ−h +for any nonnegative even integer n, �Ah = Ah = 0 for any nonnegative integer h < ℓ/2, +where Ah is defined in (B.8). Therefore, B(ℓ, λ1:ℓ) = 0 for the case N < ℓ/2. +Appendix C. Proof of Theorem 1 +We only prove the result for the case A(r) = rν, where ν > 0. The proof for the case +A(r) = ln(r) is similar, so we omit it. First of all, we denote +qR(λ) = � +ψR(2jλ) +� +fX(λ), qI(λ) = � +ψI(2jλ) +� +fX(λ), +qR,t(λ) = eitλqR(λ), and qI,t(λ) = eitλqI(λ). +By Itˆo’s formula in Lemma 2, for any +m, n ∈ N, +Hm +� 1 +σj +� +R +qR,t(λ)W(dλ) +� +Hn +� 1 +σj +� +R +qI,t(λ)W(dλ) +� += σ−(m+n) +j +Im +� +q⊗m +R,t +� +In +� +q⊗n +I,t +� +, +(C.1) +where q⊗m +R,t is the m-fold tensor product of qR,t, i.e., +q⊗m +R,t (λ1, . . . , λm) = +m +� +k=1 +qR,t(λk), λ1, . . . , λm ∈ R. +By default, I0 = 1. By substituting (C.1) into (14), we obtain +UA[j]X(t) = σν +j +� +m,n∈N∪{0} +Cm,n +√ +m!n! +σ−(m+n) +j +Im +� +q⊗m +R,t +� +In +� +q⊗n +I,t +� +. +(C.2) +By the product formula in Lemma 2 (see also [14, Proposition A.1]), +Im +� +q⊗m +R,t +� +In +� +q⊗n +I,t +� += +m∧n +� +r=0 +r! +�m +r +��n +r +� +Im+n−2r(q⊗m +R,t ⊗r q⊗n +I,t ), +(C.3) +where ⊗ℓ is the contraction operator defined in Lemma 2, or more precisely, +q⊗m +R,t ⊗r q⊗n +I,t (λ1, . . . , λm+n−2r) = +� +Rr q⊗m +R,t (λ1, . . . , λm−r,u1, u2, . . . , ur) +q⊗n +I,t (λm−r+1, . . . , λm+n−2r, −u1, − u2, . . . , −ur)du1 · · · dur + +21 +for every r ∈ {1, 2, . . . , m ∧ n}. For r = 0, +q⊗m +R,t ⊗0 q⊗n +I,t (λ1, . . . , λm+n) =q⊗m +R,t (λ1, . . . , λm)q⊗n +I,t (λm+1, . . . , λm+n). +Because (7) implies qI(λ) = −i sgn(2jλ)qR(λ), +� +R +qR(λ)qI(−λ)dλ = i +� +R +|� +ψR(2jλ)|2fX(λ) sgn(λ)dλ = 0. +Hence, for any m, n ∈ N, r ∈ {1, 2, . . . , m ∧ n}, and λ1, . . . , λm+n−2r ∈ R, +q⊗m +R,t ⊗r q⊗n +I,t (λ1, . . . , λm+n−2r) = 0. +It implies that (C.3) can be simplified as +(C.4) +Im +� +q⊗m +R,t +� +In +� +q⊗n +I,t +� += Im+n(q⊗m +R,t ⊗ q⊗n +I,t ). +By (10) and (C.4), the series (C.2) can be rewritten as +UA[j]X(t) =σν +j +� +m,n∈2N∪{0} +cA, m+n +2 +hmhn +√ +m!n! +σ−(m+n) +j +Im+n(q⊗m +R,t ⊗ q⊗n +I,t ) +=σν +j cA,0 + σν +j +� +ℓ∈2N +Iℓ +� +Q(ℓ) +t,j +� +, +(C.5) +where Q(ℓ) +t,j : Rℓ → C is defined as +Q(ℓ) +t,j = cA, ℓ +2σ−ℓ +j 2− ℓ +2(−1) +ℓ +2 +� +m,n∈2N∪{0} +m+n=ℓ +��m +2 +� +! +�n +2 +� +! +�−1 +q⊗m +R,t ⊗ q⊗n +I,t . +Because qI,t(·) = −i sgn(2j·)qR,t(·) = −i sgn(·)qR,t(·), we have +Q(ℓ) +t,j(λ1, . . . , λℓ) =cA, ℓ +2σ−ℓ +j 2− ℓ +2(−1) +ℓ +2q⊗ℓ +R,t(λ1, . . . , λℓ) +× +� +m,n∈2N∪{0} +m+n=ℓ +��m +2 +� +! +�n +2 +� +! +�−1 +(−1) +n +2 +ℓ� +k=ℓ−n+1 +sgn(λk) +(C.6) +for all (λ1, . . . , λℓ) ∈ Rℓ. By default, +ℓ� +k=ℓ+1 +sgn(λk) = 1. +Denote the symmetrization of Q(ℓ) +t,j with respect to (λ1, . . . , λℓ) by Q(ℓ) +t,j, which is defined +as +Q(ℓ) +t,j(λ1, . . . , λℓ) = 1 +ℓ! +� +p∈P [ℓ] +Q(ℓ) +t,j (λp(1), λp(2), . . . , λp(ℓ)), +where P[ℓ] represents the set of permutations of {1, 2, . . . , ℓ}. By the property +Iℓ +� +Q(ℓ) +t,j +� += Iℓ +� +Q(ℓ) +t,j +� +, + +22 +G. R. LIU, Y. C. SHEU, AND H. T. WU +the series representation (C.5) for UA[j]X(t) can be rewritten as +UA[j]X(t) = σν +j cA,0 + σν +j +� +ℓ∈2N +Iℓ +� +Q(ℓ) +t,j +� +. +(C.7) +From (C.6), +Q(ℓ) +t,j (λ1, λ2, . . . , λℓ) = cA, ℓ +2σ−ℓ +j 2− ℓ +2(−1) +ℓ +2q⊗ℓ +R,t(λ1:ℓ)B(ℓ, λ1:ℓ), +(C.8) +where λ1:ℓ = (λ1, . . . , λℓ) and +B(ℓ, λ1:ℓ) = 1 +ℓ! +� +p∈P [ℓ] +� +m,n∈2N∪{0} +m+n=ℓ +��m +2 +� +! +�n +2 +� +! +�−1 +(−1) +n +2 +ℓ� +k=ℓ−n+1 +sgn(λp(k)). +For any positive even integer ℓ and {λ1, λ2, . . . , λℓ} ⊂ R \ {0}, Lemma 3 shows that +B(ℓ, λ1:ℓ) = +� 2ℓ ℓ +2! (ℓ!)−1 +if N(λ1:ℓ) = ℓ/2, +0 +if N(λ1:ℓ) ̸= ℓ/2, +(C.9) +where N(λ1:ℓ) is the number of negative elements in {λk}ℓ +k=1. By substituting (C.9) into +(C.8), +Q(ℓ) +t,j(λ1, λ2, . . . , λℓ) = (−2) +ℓ +2 (ℓ +2!)(ℓ!)−1 cA, ℓ +2σ−ℓ +j q⊗ℓ +R,t(λ1:ℓ)1{N(λ1:ℓ)=ℓ/2}. +(C.10) +The proof of Theorem 1 is concluded by substituting (C.10) into (C.7) and noticing that +the Lebesque measure of {(λ1, λ2, . . . , λℓ) ∈ Rℓ | λk = 0 for some k} is zero. +Appendix D. Proof of Corollary 1 +In the following, we only give details for the case A(r) = ln(r). The proof for A(r) = rν +is similar and we omit it. Because h(x) = x is one of the test functions for the Wasserstein +metric, +dW +� +UA[j]X1, UA[j]X2 +� +≥ +��E +� +UA[j]X1 +� +− E +� +UA[j]X2 +��� = |ln σ1,j − ln σ2,j| , +(D.1) +where the last equality follows from Theorem 1. If the spectral densities of X1 and X2 +have the form (20) and the wavelet function ψR satisfies Assumption 1, then by the +dominated convergence theorem, +lim +j→∞2−βpj/2σp,j = +� +CXp(0) +� +R +|� +ψR(λ)|2|λ|βp−1dλ +� 1 +2 +, +that is, +lim +j→∞ − 1 +2(ln 2)βpj + ln σp,j = 1 +2 ln +� +CXp(0) +� +R +|� +ψR(λ)|2|λ|βp−1dλ +� +(D.2) +for p = 1, 2. By (D.1), we obtain +lim +j→∞ +j−1dW(UA[j]X1, UA[j]X2) ≥ lim +j→∞ +���� +ln σ1,j +j +− ln σ2,j +j +���� = 1 +2(ln 2) |β1 − β2| , +where the equality follows from (D.2). + +23 +Appendix E. Proof of Lemma 4 +(a) First of all, because ∥h +′∥∞ ≤ 1, +|E [h (F)] − E [h (F≤K)]| ≤ E +� +d +� +m=1 +|Fm,>K| +� +. +By the orthogonal property +E +� +Iℓ(s(ℓ) +2Jtm,jm)Iℓ′(s(ℓ +′) +2Jtm,jm) +� += +� +ℓ!∥s(ℓ) +2Jtm,jm∥2 +2 +if ℓ = ℓ +′, +0 +if ℓ ̸= ℓ +′, +and (24), we have +(E|Fm,>K|)2 ≤E +� +|Fm,>K|2� +=2J +� +ℓ∈{K+2,K+4,...} +ℓ! ∥s(ℓ) +2Jtm,jm∥2 +2. +(E.1) +For any t ∈ R and j ∈ Z, according to the definition of s(ℓ) +t,j in (22), +∥s(ℓ) +t,j∥2 +2 ≤σ2ν−2ℓ +j +c2 +ℓ +� +Rℓ +� ℓ� +k=1 +fX⋆ψR,j(λk) +� +|� +φJ(λ1 + · · · + λℓ)|2dλ1 · · · λℓ +=2−Jσ2ν−2ℓ +j +c2 +ℓ +� +R +f ⋆ℓ +X⋆ψR,j(2−Jη)|�φ(η)|2dη, +(E.2) +where +fX⋆ψR,j(λk) = +���� +ψR(2jλk) +��� +2 +fX(λk) +and f ⋆ℓ +X⋆ψR,j is the ℓ-fold convolution of fX⋆ψR,j with itself. +By the nonnegativity of +fX⋆ψR,j and σ2 +j = +� +R fX⋆ψR,j(λ)dλ, +f ⋆ℓ +X⋆ψR,j(η) = +� +R +f ⋆(ℓ−1) +X⋆ψR,j(η − ζ)fX⋆ψR,j(ζ)dζ ≤ ∥f ⋆(ℓ−1) +X⋆ψR,j∥∞σ2 +j. +It implies that for all positive integer ℓ, +∥f ⋆ℓ +X⋆ψR,j∥∞ ≤ ∥fX⋆ψR,j∥∞σ2(ℓ−1) +j +. +(E.3) +Note that ∥fX⋆ψR,j∥∞ < ∞ under the assumption 2α + β ≥ 1. By using (E.3) to bound +the integrand in (E.2), +∥s(ℓ) +t,j∥2 +2 ≤2−Jσ2ν−2 +j +c2 +ℓ∥fX⋆ψR,j∥∞∥�φ∥2 +2. +(E.4) +By applying (E.4) to (E.1), +(E|Fm,>K|)2 ≤ σ2ν−2 +jm +∥fX⋆ψR,jm∥∞∥�φ∥2 +2 +� +ℓ∈{K+2,K+4,...} +ℓ!c2 +ℓ. +(E.5) + +24 +G. R. LIU, Y. C. SHEU, AND H. T. WU +From the definition of cℓ in (19), +� +ℓ∈{K+2,K+4,...} +ℓ!c2 +ℓ = +� +ℓ∈{K+2,K+4,...} +2ℓ +�ℓ +2! +�2 +(ℓ!)−1c2 +A, ℓ +2 +(E.6) +For A(r) = rν with ν > 0, +cA, ℓ +2 = 2 +ν +2 Γ(ν +2 + 1) +� ℓ +2 − ν +2 − 1 +ℓ +2 +� +. +By L’Hospital’s rule, +ln |cA, ℓ +2+1| − ln |cA, ℓ +2| +ln( ℓ +2 + 1) − ln ℓ +2 += +ln | +� ℓ +2− ν +2 +ℓ +2 +1 +� +| − ln | +� ℓ +2− ν +2 −1 +ℓ +2 +� +| +ln( ℓ +2 + 1) − ln ℓ +2 +=ln( ℓ +2 − ν +2) − ln( ℓ +2 + 1) +ln( ℓ +2 + 1) − ln ℓ +2 +→ −ν +2 − 1 +(E.7) +as ℓ → ∞. From (E.7), we know that for any ε > 0, there exists a constant C1(ν, ε) > 0 +such that +(E.8) +|cA, ℓ +2| ≤ C1(ν, ε)ℓ− ν +2 −1+ε +for any integer ℓ ∈ 2N. For A(r) = ln(r), cA, ℓ +2 = −ℓ−1 for ℓ ∈ 2N, which can be viewed +as a special case of (E.8) with ν = 0 and ε = 0. On the other hand, by Stirling’s formula, +or more precisely +√ +2πn +�n +e +�n +e +1 +12n+1 < n! < +√ +2πn +�n +e +�n +e +1 +12n +(E.9) +for all n ∈ N, +(E.10) +2ℓ +�ℓ +2! +�2 +(ℓ!)−1 ≤ (2πℓ) +1 +2 +for any integer ℓ ∈ 2N. By applying the inequalities (E.8) and (E.10) to (E.6), there +exists a constant C2(ν, ε) > 0 such that +� +ℓ∈{K+2,K+4,...} +ℓ!c2 +ℓ ≤(C1(ν, ε))2 +� +ℓ∈{K+2,K+4,...} +(2πℓ) +1 +2ℓ−ν−2+2ε +≤C2(ν, ε)K−ν− 1 +2 +2ε +(E.11) +for all K ∈ 2N. By combining (E.5) and (E.11), we obtain +(E|Fm,>K|)2 ≤ σ2ν−2 +jm +∥fX⋆ψR,jm∥∞∥�φ∥2 +2C2(ν, ε)K−ν− 1 +2+2ε, +(E.12) +which leads to the inequality (26). +(b) By the Wiener chaos decomposition of F≤K in (23), for any m, n ∈ {1, . . . , d}, +E [Fm,≤K, Fn,≤K] = 2J +� +ℓ∈{2,4,...,K} +ℓ!⟨s(ℓ) +2Jtm,jm, s(ℓ) +2Jtn,jn⟩, + +25 +where +⟨s(ℓ) +2Jtm,jm, s(ℓ) +2Jtn,jn⟩ = +� +Rℓ s(ℓ) +2Jtm,jm(λ1, . . . , λℓ)s(ℓ) +2Jtn,jn(λ1, . . . , λℓ)dλ1 · · ·dλℓ. +Let +� +N(ℓ)� +ℓ∈2N = +�� +N(ℓ) +1 , . . . , N(ℓ) +d +�� +ℓ∈2N be a sequence of independent d-dimensional +normal random vectors with mean zero and E +� +N(ℓ) +m N(ℓ) +n +� += 2Jℓ!⟨s(ℓ) +2Jtm,jm, s(ℓ) +2Jtn,jn⟩ for +m, n ∈ {1, . . . , d}. Then, +N +d= +� +ℓ∈2N +N(ℓ) +and +N≤K +d= +� +ℓ∈{2,4,...,K} +N(ℓ), +where +d= means the equality is in the distribution sense. For any Lipschitz differentiable +function h : Rd → R with Lipschitz constant ∥h +′∥∞ ≤ 1, +|E [h (N≤K)] − E [h (N)]| ≤ +d +� +m=1 +E +������ +� +ℓ∈{K+2,K+4,...} +N(ℓ) +m +������ += +� +2 +π +d +� +m=1 + + +� +ℓ∈{K+2,K+4,...} +E|N(ℓ) +m |2 + + +1 +2 += +� +2 +π +d +� +m=1 + + +� +ℓ∈{K+2,K+4,...} +2Jℓ!∥s(ℓ) +2Jtm,jm∥2 +2 + + +1 +2 +. +(E.13) +The summation inside the square brackets in the last term is the same as the upper +bound of (E|Fm,>K|)2 in (E.1). Hence, we can directly apply (26) to the right hand side +of (E.13) to obtain +|E [h (N≤K)] − E [h (N)]| ≤ dCK− ν +2 − 1 +4+ε. +Appendix F. Proof of Lemma 5 +For any twice differentiable function h, [37, Theorem 6.1.2] shows that +|E[h(S)] − E[h(NC)]| ≤ 1 +2 +� +max +1≤m,n≤d sup +x∈Rd +��� +∂2h +∂xm∂xn +(x) +��� +� +ρ. +(F.1) +It is clear that +max +1≤m,n≤d sup +x∈Rd +��� +∂2h +∂xm∂xn +(x) +��� ≤ ∥h +′∥Lip +(F.2) +for any twice differentiable function h in H2, so we only need to show that (28) holds +for non-twice differentiable functions in H2. For such class of functions h, we define +hε(x) = +� +Rd +1 +(2πε) +d +2 e− ∥z∥2 +2ε h(x − z)dz, x ∈ Rd. + +26 +G. R. LIU, Y. C. SHEU, AND H. T. WU +For any ε > 0, hε is twice differentiable, so (F.1) and (F.2) imply that +|E[hε(S)] − E[hε(NC)]| ≤1 +2∥h +′ +ε∥Lipρ. +(F.3) +Because h is continuous and bounded, by the dominated convergence theorem, +lim +ε→0|E[hε(S)] − E[hε(NC)]| = |E[h(S)] − E[h(NC)]|. +(F.4) +On the other hand, because h is continuously differentiable and ∥ ∂h +∂xm∥∞ ≤ 1 for m ∈ +{1, 2, . . . , d}, +∂hε +∂xm +(x) = +� +Rd +1 +(2πε) +d +2 e− ∥z∥2 +2ε +∂h +∂xm +(x − z)dz. +Hence, for any x, y ∈ Rd with x ̸= y, +���� +∂hε +∂xm +(x) − ∂hε +∂xm +(y) +���� ≤ +� +Rd +1 +(2πε) +d +2 e− ∥z∥2 +2ε +���� +∂h +∂xm +(x − z) − ∂h +∂xm +(y − z) +���� dz, +≤ ∥ ∂h +∂xm +∥Lip∥x − y∥. +(F.5) +The inequality (F.5) implies that +lim sup +ε→0 +∥h +′ +ε∥Lip = max +1≤m≤dlim sup +ε→0 +sup +x,y∈Rd +x̸=y +��� ∂hε +∂xm(x) − ∂hε +∂xm(y) +��� +∥x − y∥ +≤ +max +m∈{1,2,...,d}∥ ∂h +∂xm +∥Lip. +(F.6) +From (F.3), (F.4), and (F.6), we know that (F.1) holds for any h ∈ H2. +Appendix G. Proof of Proposition 1 +The proof of (29) consists of three parts. First, we rewrite +E +�� +CJ,K(m, n) − +� +DFn,≤K, −DL−1Fm,≤K +��2� +by making use of the Wiener chaos decomposition of F≤K in (23) as follows. By the +orthogonal property of Wiener-Itˆo integrals, the covariance CJ,K(m, n) can be expressed +as +E [Fm,≤KFn,≤K] =2J +� +ℓ∈{2,4,...,K} +E +� +Iℓ +� +s(ℓ) +2Jtm,jm +� +Iℓ +� +s(ℓ) +2Jtn,jn +�� +=2J +� +ℓ∈{2,4,...,K} +ℓ! s(ℓ) +2Jtm,jm ⊗ℓ s(ℓ) +2Jtn,jn. +(G.1) +By Lemma 6 in J and (23) again, +DFn,≤K = 2 +J +2 +� +ℓ∈{2,4,...,K} +ℓIℓ−1 +� +s(ℓ) +2Jtn,jn +� +. +(G.2) + +27 +By the definition of L−1 in (J.4), +−DL−1Fm,≤K =2 +J +2 D + + +� +ℓ∈{2,4,...,K} +1 +ℓIℓ +� +s(ℓ) +2Jtm,jm +� + + +=2 +J +2 +� +ℓ∈{2,4,...,K} +Iℓ−1 +� +s(ℓ) +2Jtm,jm +� +. +(G.3) +By (G.2), (G.3), and the product formula in Lemma 2, +⟨DFn,≤K, −DL−1Fm,≤K⟩ = 2J� +� +ℓ∈{2,4,...,K} +ℓIℓ−1 +� +s(ℓ) +2Jtn,jn +� +, +� +ℓ∈{2,4,...,K} +Iℓ−1 +� +s(ℓ) +2Jtm,jm +� � += 2J +� +ℓ∈{2,4,...,K} +ℓ! s(ℓ) +2Jtm,jm ⊗ℓ s(ℓ) +2Jtn,jn ++ 2J +� +ℓ∈{2,4,...,K} +ℓ +ℓ−1 +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +�2 +I2ℓ−2r +� +s(ℓ) +2Jtm,jm ⊗r s(ℓ) +2Jtn,jn +� ++ 2J +� +ℓ,ℓ′∈{2,4,...,K} +ℓ̸=ℓ′ +ℓ +ℓ∧ℓ′ +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +��ℓ +′ − 1 +r − 1 +� +Iℓ+ℓ′−2r +� +s(ℓ) +2Jtm,jm ⊗r s(ℓ) +2Jtn,jn +� +. +(G.4) +By (G.1), (G.4), and the Minkowski inequality, +� +E +� +(CJ,K(m, n) − ⟨DFn,≤K, −DL−1Fm,≤K⟩)2� +≤2J +� +ℓ∈{2,4,...,K} +ℓ +ℓ−1 +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +�2� +E +����I2ℓ−2r +� +s(ℓ) +2Jtm,jm ⊗r s(ℓ) +2Jtn,jn +���� +2� ++2J +� +ℓ,ℓ′∈{2,4,...,K} +ℓ̸=ℓ′ +ℓ +ℓ∧ℓ′ +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +��ℓ +′ − 1 +r − 1 +�� +E +����Iℓ+ℓ′−2r +� +s(ℓ) +2Jtm,jm ⊗r s(ℓ′) +2Jtn,jn +���� +2� +. +By noticing that +E +����Iℓ+ℓ′−2r +� +s(ℓ) +2J tm,jm ⊗r s(ℓ +′) +2Jtn,jn +���� +2� +≤ (ℓ + ℓ +′ − 2r)! +���s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn +��� +2 +2 + +28 +G. R. LIU, Y. C. SHEU, AND H. T. WU +for all ℓ, ℓ +′ ∈ {2, 4, . . . , K} and r ∈ Λℓ,ℓ′, where Λℓ,ℓ = {1, 2, . . . , ℓ − 1} and Λℓ,ℓ′ = +{1, 2, . . . , ℓ ∧ ℓ +′} for the case ℓ ̸= ℓ +′, we obtain +� +E +�� +CJ,K(m, n) − +� +DFn,≤K, −DL−1Fm,≤K +��2� +≤2J +� +ℓ∈{2,4,...,K} +ℓ +ℓ−1 +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +�2� +(2ℓ − 2r)! +���s(ℓ) +2Jtm,jm ⊗r s(ℓ) +2Jtn,jn +��� +2 +(G.5) ++2J +� +ℓ,ℓ′∈{2,4,...,K} +ℓ̸=ℓ′ +ℓ +ℓ∧ℓ′ +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +��ℓ +′ − 1 +r − 1 +�� +(ℓ + ℓ +′ − 2r)! +���s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn +��� +2 . +Second, we calculate an upper bound for +���s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn +��� +2, where s(ℓ) +t,j is de- +fined in (22). To simplify the tedious notation, for any integers p1 < p2, we denote +(λp1, λp1+1, . . . , λp2) by λp1:p2 and λp1 + λp1+1 + . . . + λp2 by λ+ +p1:p2. From (22), +���s(ℓ) +2Jtm,jm(λ1:ℓ) +��� ≤σν−ℓ +jm |cℓ| +� ℓ� +k=1 +|� +ψR(2jmλk)| +� +fX(λk) +� +|� +φJ(λ+ +1:ℓ)|. +For any ℓ, ℓ +′ ∈ 2N, r ∈ Λℓ,ℓ′ and tm, tn ∈ R, +���s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn +� +λ1:ℓ+ℓ′−2r +���� +≤σν−ℓ +jm σν−ℓ +′ +jn +|cℓcℓ′| +�ℓ−r +� +k=1 +|� +ψR(2jmλk)| +� +fX(λk) +�  + +ℓ+ℓ +′−2r +� +k=ℓ−r+1 +|� +ψR(2jnλk)| +� +fX(λk) + + +× +� +Rr +� r� +k=1 +|� +ψR(2jmτk)� +ψR(2jnτk)|fX(τk) +� +|� +φJ(λ+ +1:ℓ−r + τ + +1:r)� +φJ(λ+ +ℓ−r+1:ℓ+ℓ′−2r − τ + +1:r)|dτ1 · · · dτr. +Hence, +∥s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn∥2 +2 +≤σ2ν−2ℓ +jm +σ2ν−2ℓ +′ +jn +c2 +ℓc2 +ℓ′ +� +Rℓ+ℓ′ +�ℓ−r +� +k=1 +|� +ψR(2jmλk)|2fX(λk) +�  + +ℓ+ℓ +′−2r +� +k=ℓ−r+1 +|� +ψR(2jnλk)|2fX(λk) + + +× +� r� +k=1 +|� +ψR(2jmτk)||� +ψR(2jnτk)|fX(τk) +� � r� +k=1 +|� +ψR(2jmηk)||� +ψR(2jnηk)|fX(ηk) +� +× |� +φJ(λ+ +1:ℓ−r + τ + +1:r)� +φJ(λ+ +ℓ−r+1:ℓ+ℓ′−2r − τ + +1:r)| +× |� +φJ(λ+ +1:ℓ−r + η+ +1:r)� +φJ(λ+ +ℓ−r+1:ℓ+ℓ′−2r − η+ +1:r)| dτ1 · · · dτr dη1 · · · dηr dλ1 · · · dλℓ+ℓ′−2r. + +29 +By considering the change of variables + + + + + + + + + + + + + + + + + +uk = τk, k = 1, . . . , r − 1 if r ≥ 2, +vk = ηk, k = 1, . . . , r − 1 if r ≥ 2, +wk = λk, k = 1, . . . , ℓ + ℓ +′ − 2r − 1, +x = 2J � +λ+ +1:ℓ−r + τ + +1:r +� +, +y = 2J � +λ+ +ℓ−r+1:ℓ+ℓ′−2r − τ + +1:r +� +, +z = 2J � +λ+ +1:ℓ−r + η+ +1:r +� +and noting that � +φJ(·) = �φ(2J·), +∥s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn∥2 +2 +(G.6) +≤2−3Jσ2ν−2ℓ +jm +σ2ν−2ℓ +′ +jn +c2 +ℓc2 +ℓ′ +� +Rℓ+ℓ′ +�ℓ−r +� +k=1 +|� +ψR(2jmwk)|2fXwk) +�  + +ℓ+ℓ +′−2r−1 +� +k=ℓ−r+1 +|� +ψR(2jnwk)|2fX(wk) + + +× +�r−1 +� +k=1 +|� +ψR(2jmuk)||� +ψR(2jnuk)|fX(uk) +� �r−1 +� +k=1 +|� +ψR(2jmvk)||� +ψR(2jnvk)|fX(vk) +� +× |� +ψR(2jmu∗)||� +ψR(2jnu∗)|fX(u∗)|� +ψR(2jmv∗)||� +ψR(2jnv∗)|fX(v∗)|� +ψR(2jnw∗)|2fX(w∗) +× |�φ(x)� +φJ(y)�φ(z)�φ(x + y − z)| du1 · · · dur−1dv1 · · · dr−1dw1 · · · dwℓ+ℓ′−2r−1dxdydz, +where u∗ = 2−Jx − u1 − · · · − ur−1 − w1 − · · · − wℓ−r, v∗ = 2−Jz − v1 − · · · − vr−1 − w1 − +· · · − wℓ−r, and w∗ = 2−Jx + 2−Jy − w1 − · · · − wℓ+ℓ′−2r−1. From (G.6), we obtain +∥s(ℓ) +2Jtm,jm ⊗r s(ℓ +′) +2Jtn,jn∥2 +2 ≤ C22−3Jc2 +ℓc2 +ℓ′, +(G.7) +where +C2 = ∥�φ∥∞∥�φ∥3 +1 max +1≤m,n≤d{σ2ν−2 +jm +σ2ν−4 +jn +M3, σ2ν−2 +jm +σ2ν−2 +jn +M2} +(G.8) +and +M = +max +1≤m,n≤d∥� +ψR(2jmu)� +ψR(2jnu)fX(u)∥∞ < ∞ +under the assumption 2α + β ≥ 1. +Third, we rewrite the summation (G.5) in a more compact form by using the inequality +(G.7). The summation (G.5) can be bounded as follows +� +E +�� +CJ,K(m, n) − +� +DFn,≤K, −DL−1Fm,≤K +��2� +≤ C 2− 1 +2 J [P1(K) + P2(K)] , +(G.9) +where +P1(K) = +� +ℓ∈{2,4,...,K} +ℓ +ℓ−1 +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +�2� +(2ℓ − 2r)! c2 +ℓ +(G.10) + +30 +G. R. LIU, Y. C. SHEU, AND H. T. WU +0 +50 +100 +150 +200 +250 +300 +2.8 +2.9 +3 +3.1 +3.2 +3.3 +3.4 +3.5 +3.6 +3.7 +3.8 +Figure G.1. Behavior of the ratio Θ1(ℓ + 1)/Θ1(ℓ) +and +P2(K) = +� +ℓ,ℓ′∈{2,4,...,K} +ℓ̸=ℓ′ +ℓ +ℓ∧ℓ′ +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +��ℓ +′ − 1 +r − 1 +�� +(ℓ + ℓ +′ − 2r)! |cℓcℓ′|. +(G.11) +Denote +Θ1(ℓ) = +1 +(ℓ − 1)! +ℓ−1 +� +r=1 +(r − 1)! +�ℓ − 1 +r − 1 +�2� +(2ℓ − 2r)! +for ℓ ∈ N with Θ1(1) = 0, which leads to +P1(K) = +� +ℓ∈{2,4,...,K} +c2 +ℓ ℓ! Θ1(ℓ) . +Observe that for every ℓ ∈ N, +Θ1(ℓ) = +ℓ−1 +� +r=1 +�ℓ − 1 +r − 1 +���2ℓ − 2r +ℓ − r +� += −1 + +ℓ−1 +� +k=0 +�ℓ − 1 +k +���2k +k +� +. +(G.12) +Because +�2k +k +� +≤ 4k for all k ∈ N ∪ {0}, (G.12) implies that +Θ1(ℓ) ≤ −1 + +ℓ−1 +� +k=0 +�ℓ − 1 +k +� +2k = −1 + 3ℓ−1, ℓ ∈ N. +(G.13) +See Figure G.1 for the behavior of the ratio Θ1(ℓ + 1)/Θ1(ℓ), which tends to 3 when +ℓ → ∞. By substituting the estimate (G.13) into (G.10), +P1(K) ≤ +� +ℓ∈{2,4,...,K} +c2 +ℓ ℓ! 3ℓ−1. +(G.14) + +31 +On the other hand, +P2(K) = +� +ℓ,ℓ +′∈{2,4,...,K} +ℓ̸=ℓ +′ +|cℓcℓ′| +√ +ℓ! +√ +ℓ +′!Θ2(ℓ, ℓ +′), +(G.15) +where +Θ2(ℓ, ℓ +′) =ℓ +ℓ∧ℓ +′ +� +r=1 +(ℓ − 1)!/ +√ +ℓ! +� +(ℓ − r)! +� +(r − 1)! +(ℓ +′ − 1)!/ +√ +ℓ +′! +� +(ℓ +′ − r)! +� +(r − 1)! +� +(ℓ + ℓ +′ − 2r)! +� +(ℓ − r)! +� +(ℓ +′ − r)! += +ℓ∧ℓ +′ +� +r=1 +r +ℓ +′ +√ +ℓ! +� +(ℓ − r)! +√ +r! +√ +ℓ +′! +� +(ℓ +′ − r)! +√ +r! +� +(ℓ + ℓ +′ − 2r)! +� +(ℓ − r)! +� +(ℓ +′ − r)! +. +(G.16) +Because +r +ℓ′ ≤ 1 and (ℓ + ℓ +′ − 2r)! ≤ +� +(2ℓ − 2r)! +� +(2ℓ +′ − 2r)! for all ℓ, ℓ +′ ∈ N and +r ∈ {1, 2, . . . , ℓ ∧ ℓ +′}, (G.16) implies that +Θ2(ℓ, ℓ +′) ≤ +ℓ∧ℓ +′ +� +r=1 +�� +ℓ +ℓ − r +��� +ℓ +′ +ℓ +′ − r +� ��2ℓ − 2r +ℓ − r +�� 1 +4 ��2ℓ +′ − 2r +ℓ +′ − r +�� 1 +4 +≤ + + + +ℓ∧ℓ +′ +� +r=1 +� +ℓ +ℓ − r +���2ℓ − 2r +ℓ − r +� + + +1 +2  + + +ℓ∧ℓ +′ +� +r=1 +� +ℓ +′ +ℓ +′ − r +���2ℓ +′ − 2r +ℓ +′ − r +� + + +1 +2 +≤ + + + +ℓ∧ℓ +′ +� +r=1 +� +ℓ +ℓ − r +� +2ℓ−r + + + +1 +2  + + +ℓ∧ℓ +′ +� +r=1 +� +ℓ +′ +ℓ +′ − r +� +2ℓ +′−r + + + +1 +2 +≤ 3 +ℓ +2 3 +ℓ +′ +2 , +(G.17) +where the second inequality follows from the Cauchy-Schwarz inequality and the third +inequality follows from the fact +�2ℓ−2r +ℓ−r +� +≤ 4ℓ−r. By substituting (G.17) into (G.15), +P2(K) ≤ +� +ℓ,ℓ +′∈{2,4,...,K} +ℓ̸=ℓ +′ +|cℓcℓ′| +√ +ℓ! +√ +ℓ +′! 3 +ℓ +2 3 +ℓ +′ +2 . +(G.18) +By combining (G.14) and (G.18), we get +P1(K) + P2(K) ≤ + + +� +ℓ∈{2,4,...,K} +|cℓ| +√ +ℓ! 3 +ℓ +2 + + +2 +. +(G.19) +By substituting (G.19) into (G.9), we obtain (29). +Appendix H. Proof of Theorem 2 +• For the case A(r) = rν with ν ∈ 2N, (13) shows that cA, ℓ +2 = 0 if ℓ > ν. It implies +that the decomposition (21) of SA +J [j]X is only comprised of finite Wiener chaos. +Therefore, the truncation step (24) for the decomposition of F is unnecessary +and we can obtain the result immediately from Proposition 1 with K = ν. + +32 +G. R. LIU, Y. C. SHEU, AND H. T. WU +• For the case A(r) = rν with ν ∈ (0, ∞) \ 2N, by (19) and (E.8), for any ε > 0, +there exists a constant C1(ν, ε) such that +|cℓ| ≤ C1(ν, ε)2 +ℓ +2 (ℓ +2!)(ℓ!)−1ℓ− ν +2 −1+ε +(H.1) +for all ℓ ∈ 2N. The inequality (H.1) also holds for A(r) = ln(r) with the notation +replacement: ν = 0 and ε = 0. By Stirling’s formula (E.9), +2 +ℓ +2 (ℓ +2!)(ℓ!)− 1 +2 ≤ 2ℓ +1 +4 +for all ℓ ∈ 2N. Hence, there exists a constant C2(ν, ε) such that +� +ℓ∈{2,4,...,K} +|cℓ| +√ +ℓ!3 +ℓ +2 ≤ C2(ν, ε) +K/2 +� +ℓ=1 +ℓ− ν +2 − 3 +4 +ε3ℓ. +For the summation above, for each fixed ν ∈ [0, ∞) \ 2N, there exists a threshold +Kν > 0 such that +K/2 +� +ℓ=1 +ℓ− ν +2 − 3 +4 +ε3ℓ ≤ 31+ K +2 (K/2)− ν +2 − 3 +4 +ε +for any even integer K > Kν. Therefore, for any ε > 0, there exists a constant +C3(ν, ε) > 0 such that + + +� +ℓ∈{2,4,...,K} +|cℓ| +√ +ℓ!3 +ℓ +2 + + +2 +≤ C3(ν, ε)3KK−ν− 3 +2 +ε +(H.2) +for any even integer K > Kν. By substituting (H.2) into (29), +|E[h(F≤K)] − E[h(NCJ,K)]| ≤ CC3(ν, ε)2− J +2 3KK−ν− 3 +2 +ε +(H.3) +for any even integer K > Kν. Finally, by applying the results of Lemma 4 and +(H.3) to (25), we obtain that +|E[h(F)] − E[h(N)]| ≤ C4(ν, ε) +� +K− ν +2 − 1 +4+ε + 2− J +2 3KK−ν− 3 +2 +ε� +for a certain constant C4(ν, ε) independent of J and K when K ≥ Kν. The +asymptotic behavior (30) is obtained by choosing K = 2⌊J +4 log3 2⌋. +Finally, because both sides of (31) can be computed by the Wiener chaos expansions +(18) and (21) of UA[j]X and SA +J [j]X, the verification of (31) is omitted. +Appendix I. Proof of Corollary 2 +The idea of proof is originated from the work [18], in which more complicated cases +were considered. However, we only need part of it. For the convenience of readers, we +will sketch its proof as follows. First of all, we define + +33 +-1 +0 +1 +z +z+ /2 +z+ +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) h and hz,δ +(b) h(−1,−1),2 +Figure I.1. Approximation of indicator functions +h(x) = + + + + + + + +1 +for x ≤ −1, +1 − 1 +2(1 + x)2 +for x ∈ (−1, 0], +1 +2(1 − x)2 +for x ∈ (0, 1], +0 +for x > 1, +and hz,δ(x) = h +� 2 +δ +� +x − +� +z + δ +2 +��� +for any z ∈ R and δ > 0. See Figure I.1(a) for the +graphs of functions h and hz,δ. For any d ∈ N and z = (z1, . . . , zd) ∈ Rd, let +hz,δ(x) = +d +� +k=1 +hzk,δ(xk), x = (x1, . . . , xd) ∈ Rd. +See Figure I.1(b) for the graph of function hz,δ with z = (−1, −1) and δ = 2. For any +fixed z ∈ Rd, the indicator function 1x≤z is bounded above by hz,δ(x), where x ≤ z +means that xk ≤ zk for all k = 1, . . . , d. Hence, +P (F ≤ z) − P (N ≤ z) ≤E [hz,δ(F)] − E [hz,δ(N)] + E [hz,δ(N)] − P (N ≤ z) +≤E [hz,δ(F)] − E [hz,δ(N)] + P(N ≤ z + δ) − P (N ≤ z) , +where z+δ means that z+δ11×d. Because ∥hz,δ∥∞ = 1, ∥h +′ +z,δ∥∞ = 2 +δ and ∥h +′ +z,δ∥Lip = ( 2 +δ)2, +we have +� +1 + +�2 +δ +�2�−1 +hz,δ ∈ H2 for any z ∈ Rd and δ > 0, which implies that +|E [hz,δ(F)] − E [hz,δ(N)]| ≤ +� +1 + +�2 +δ +�2� +dH2 (F, N) . +(I.1) +On the other hand, from Nazarov’s anti-concentration inequality [12, Lemma A.1], we +have +P(N ≤ z + δ) − P (N ≤ z) ≤ +δ +� +min +1≤m≤dE[F 2m] +�√ +2 ln d + 2 +� +(I.2) + +0.90.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +00.5 +0 +-3 +-2 +-1 +0 +C1 +1 +2 +0 +-2 +-1 +3 +234 +G. R. LIU, Y. C. SHEU, AND H. T. WU +for any z ∈ Rd and δ > 0. Hence, +P (F ≤ z) − P (N ≤ z) ≤ +� +1 + +�2 +δ +�2� +dH2 (F, N) + +δ +� +min +1≤m≤dE[F 2 +m] +�√ +2 ln d + 2 +� +. +(I.3) +By taking δ = 2 +�� +min +1≤m≤dE[F 2m]( +√ +2 ln d + 2)−1dH2 (F, N) +� 1 +3 +, (I.3) leads to the upper +bound +P (F ≤ z) − P (N ≤ z) ≤ 3 + + +√ +2 ln d + 2 +� +min +1≤m≤dE[F 2m] + + +2 +3 +(dH2 (F, N)) +1 +3 + dH2 (F, N) . +(I.4) +For finding a lower bound for P (F ≤ z) − P (N ≤ z), we first observe that +P (N ≤ z) − P (F ≤ z) ≤P (N ≤ z) − E [hz−δ,δ(N)] + E [hz−δ,δ(N)] − E [hz−δ,δ(F)] +≤P (N ≤ z) − P (N ≤ z − δ) + E [hz−δ,δ(N)] − E [hz−δ,δ(F)] +(I.5) +for any z ∈ Rd and δ > 0. By applying (I.1) and (I.2) to (I.5), we obtain a lower bound +for P (F ≤ z)−P (N ≤ z), which is the negative of the right hand side of (I.4). Therefore, +dKol (F, N) ≤ 3 + + +√ +2 ln d + 2 +� +min +1≤m≤dE[F 2m] + + +2 +3 +(dH2 (F, N)) +1 +3 + dH2 (F, N) . +Because Theorem 2 shows that lim +J→∞dH2 (F, N) = 0 and lim +J→∞E[F 2 +m] = κm,m∥�φ∥2 +2, we +obtain +dKol (F, N) ≤ O +� +(dH2 (F, N)) +1 +3 +� +(I.6) +when J → ∞. The statement of Corollary 2 follows by combining (30) and (I.6). +Appendix J. Elements of Malliavin Calculus +Let S denote the set of all random variables S of the form +s(W(f1), . . . , W(fn)), +where n ≥ 1, s : Rn → R is a C∞-function such that s and its partial derivatives have +at most polynomial growth, and fi ∈ H, i = 1, . . . , n. We denote by L2(Ω) the set of +FW-measurable random variables whose second moments exist. The space S is dense in +L2(Ω) [37, Lemma 2.3.1]. +Definition 3. For every integers n, p ≥ 1, the p-th Malliavin derivative of +S = s(W(f1), . . . , W(fn)) ∈ S + +35 +with respect to W is defined by +DpS = +n +� +i1,...,ip=1 +∂ps +∂xi1 · · · ∂xip +(W(f1), . . . , W(fn))fi1 ⊗ · · · ⊗ fip. +(J.1) +Because the sum in (J.1) runs over all partial derivatives, DpS belongs to L2(Ω, H +⊙p). +The Malliavin derivative of multiple Wiener-Itˆo integrals have the following property. +The proof can be found in [37, Proposition 2.7.4]. +Lemma 6 ([37]). For every integer p ≥ 1 and u ∈ H +⊗p, +DIp(u) = pIp−1(˜u), +where +˜u(λ1:p) = 1 +p! +� +π +u(λπ(1), . . . , λπ(p)), +is the canonical symmetrization of u and +Ip−1(˜u)(·) = +� +′ +Rp−1 ˜u(λ1, . . . , λp−1, ·)W(dλ1) · · ·W(dλp−1). +On the other hand, for all r ∈ N, p ∈ N and q ∈ [1, ∞], +∥Ip(f)∥Dr,q ≤ cr,p,q∥f∥H⊗p, f ∈ H +⊙p, +(J.2) +where cr,p,q > 0 is an universal constant and +∥Ip(f)∥Dr,q := +� +E[|Ip(f)|q] + E[∥DIp(f)∥q +H] + · · · + E[∥DrIp(f)∥q +H +⊗r] +�1/q +. +(J.3) +We note that (J.2) is a special case of the Meyer’s inequality [38, Proposition 1.5.7]. +Definition 4. For any F ∈ L2(Ω, FW, P) with the Wiener chaos expansion +F = E[F] + +∞ +� +p=1 +Ip (gp) , +where gp ∈ H +⊙p, the pseudo-inverse of the infinitesimal generator L of the Ornstein- +Uhlenbeck semigroup, denoted as L−1, is defined as +(J.4) +L−1F = − +∞ +� +p=1 +1 +p Ip (gp) . +The details about the semigroup of the Ornstein-Uhlenbeck semigroup and its infini- +tesimal generator can be found in [37, Section 2.8]. +(Gi-Ren Liu) Department of Mathematics, National Cheng-Kung University, Tainan, +Taiwan +(Yuan-Chung Sheu) Department of Applied Mathematics, National Yang Ming Chiao +Tung University, Hsinchu, Taiwan + +36 +G. R. LIU, Y. C. SHEU, AND H. T. WU +(Hau-Tieng Wu) Department of Mathematics and Department of Statistical Science, +Duke University, Durham, NC, USA + diff --git a/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/load_file.txt b/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43d35cb2441a93752e8b9fe1b5cff66694c00c5e --- /dev/null +++ b/NtAzT4oBgHgl3EQfk_1W/content/tmp_files/load_file.txt @@ -0,0 +1,1418 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf,len=1417 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='01540v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='PR] 4 Jan 2023 GAUSSIAN APPROXIMATION FOR THE MOVING AVERAGED MODULUS WAVELET TRANSFORM AND ITS VARIANTS GI-REN LIU, YUAN-CHUNG SHEU, AND HAU-TIENG WU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The moving average of the complex modulus of the analytic wavelet trans- form provides a robust time-scale representation for signals to small time shifts and deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In this work, we derive the Wiener chaos expansion of this represen- tation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The expansion allows us to obtain a lower bound for the Wasserstein dis- tance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance be- tween the distributions of a random vector derived from the time-scale representation and its normal counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It is worth mentioning that the expansion consists of infi- nite Wiener chaos and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases, and we provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Introduction The wavelet transform (WT) is one of the most useful tools in signal processing [15, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It performs a scale decomposition of an input signal X by convolving it with a family of functions {ψj}j∈Z, which are generated by dilating a mother wavelet ψ in the way ψj(t) = 2−jψ(2−jt), where t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The WT of X, denoted by {W[j]X(t)}t∈R,j∈Z, where W[j]X(t) := X ⋆ ψj(t), provides both the magnitude and phase information of X in the time-scale domain when the mother wavelet ψ is complex-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' This information has been applied to analyze the heart rate variability [39], detect the seizure through the Electroencephalography (EEG) signals [17], and prove the existence of intermittency in the local field potentials recorded from patients with Parkinson’s disease [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The WT also serves as a feature extractor in the field of machine learning [10, 24, 25, 45, 47], particularly when the invariance to small temporal shifts is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In light of the fact that a small temporal shift of X produces a phase shift on {W[j]X(t)}t∈R,j∈Z, usually the modulus of the wavelet coefficients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', {UA[j]X}j∈Z := {A(|W[j]X(t)|)}t∈R,j∈Z with A(r) = r) and its variants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', A(r) = r2 [6] and A(r) = ln(r) [7, 21]) are used [19, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In order to make the features stable to actions of small shifts 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Primary 60G60, 60H05, 62M15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Secondary 35K15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' analytic wavelet transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' complex modulus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Wiener-Itˆo decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Gaussian approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Stein’s method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Malliavin calculus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' smooth Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 1 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU and deformations to X, practitioners consider SA J [j], which comes from convolving the output of UA[j] with a low-pass filter φJ [31]: (1) SA J [j]X(t) := � UA[j]X � ⋆ φJ(t), where J is an integer used to control the width of the low-pass filter φJ through φJ(t) = 2−Jφ � 2−Jt � and φ is a real-valued function, usually chosen to be the father wavelet associated with ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In the work [31], {SA J [j]}j∈Z is called the first-order scatter- ing transform and its higher-order counterparts � UA[jn] · · ·UA[j2]UA[j1]X � ⋆ φJ, where n ∈ N and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , jn ∈ Z, are introduced to improve the mathematical understanding of convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In the presence of noise and interferences, the input signal X is usually viewed as a random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In [3, 4, 11, 32], some properties of random processes, such as stationarity and self-similarity, are characterized by corresponding properties of their wavelet transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Especially, for stationary random process X, which may be deformed by a stationarity-breaking operator, how to use the instantaneous wavelet spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', E � UA[j]X(t) � with A(r) = r2) to determine the second-order statistical properties of X was discussed in [22, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In [41], the authors considered a discrete-time analog of the transformation W[j]X Zj,t = �� k∈Z vi,j(γjt − k)X(k) � i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',d , (2) where d is a fixed positive integer corresponding to the number of WT coefficients we are interested in, {γj}j∈N is a divergent sequence of positive integers corresponding to the scale, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', γj = 2j, t ∈ Z corresponding to the time, vi,j : Z → R corresponding to the discretized wavelet satisfies � k∈Z v2 i,j(k) < ∞, and {X(k)}k∈Z is a sequence of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=') real-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' When j → ∞, they proved that the sequence of d-dimensional vectors n−1/2 j nj−1 � t=0 � Z2 j,t − E � Z2 j,t �� , (3) converges to a d-dimensional normal random vector in distribution sense under some conditions on vi,j and width of the averaging nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' random sequence {X(k)}k∈Z in (2) was extended to long-range dependent Gaussian sequences in [35] as well as sub- ordinated Gaussian sequences with long-term or short-term memory in [13, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Due to the difference in the strength of dependence of the sequence X, the limiting theorem for the wavelet coefficients in [13] was proved by the so-called large scale reduction principle, while the central limit theorem for (3) in [43, Theorem 1] was proved by showing that the sequence {Z2 j,t −E � Z2 j,t � }t is strong mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Especially, when the sequence X is gen- erated from a Hermite polynomial of a stationary Gaussian sequence with long memory, the authors of [14] showed that after suitable renormalization, (3) converges either to a Gaussian process or to a Rosenblatt process when j → ∞ depending on the order of 3 Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the discrete wavelet packet decomposition, which performs the discrete WT iteratively without taking account of any nonlinear activation functions, of stationary random processes, [2] proves that the outputs of decomposition converge in distribution to white Gaussian processes when the resolution level of decomposition increases by showing that the third and higher-order cumulants of the outputs converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In [9], the authors considered a transformation similar to (1) as follows 2j/2|X ⋆ ψ| ⋆ ψj(t), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (4) When X is a fractional Brownian motion, the authors of [9] applied a central limit theorem for sums of locally dependent random variables [8] to show that the marginal distribution of the random process defined in (4) converges to a complex normal distribu- tion when j → ∞ [9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The limit of (4) was also studied in our previous work [27] in the case when X is a stationary Gaussian process with long-range or short-range dependence and ψ is a real-valued wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In this work, we analyze the smooth Wasserstein distance and the Kolmogorov distance between the finite-dimensional distributions of the centralized SA J [j]X and its Gaussian counterpart, which is a normal random vector with the same covariance structure as SA J [j]X, in the case when X is a stationary Gaussian process and ψ is an analytic wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' According to our literature survey, there is no previous work on establishing the spectral representation of the modulus, or more general transform, of WT of Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It is indispensable for calculating the convolution UA[j]X ⋆ φJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The first contribution of this work is to provide spectral representations for UA[j]X and SA J [j]X for homogeneous or logarithmic functions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The representation is expressed as a linear combination of finite or infinite orthogonal Wiener chaos, depending on function A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The second contribution of this work is to provide a quantitative central limit theorem for SA J [j]X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' We note that the quantitative central limit theorem for random processes expressed as a linear combination of finite Wiener chaos has been well analyzed [36, 37, 38], but it is unclear how to deal with the case when there are infinite Wiener chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' To achieve these goals, we apply the Malliavin calculus and combinatorial techniques to analyze the complex modulus of the wavelet coefficients W[j]X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' This part has never been considered in existing works, including our previous work [26] in which the wavelet ψ is real-valued and A(r) = r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It is worth mentioning that the orthogonal expansion of SA J [j]X is an infinite sum of Wiener chaos, particularly for the practical cases A(r) = r and A(r) = ln(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In order to apply the general upper bound for the Gaussian approximation error in [36, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7] and [37, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2)] to the case of infinite sum of Wiener chaos, we simplify the complexity of the expression of the upper bound and provide a more concise expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' From the more concise expression for the upper bound (Proposition 1), we obtain upper bounds for the smooth Wasserstein distance and the Kolmogorov distance between the laws of the centralized SA J [j]X and its Gaussian counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The upper bounds depend on the window length of the moving average and the function A (Theorem 2 and Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In Section 2, we summarize necessary material for WT when the wavelet is analytic and present some preliminaries about the Wiener-Itˆo integrals and the Malliavin calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In Section 3, we state our main results, including Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proofs of our main results and some technical lemmas are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Wavelet transform with an analytic wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let ψ be a complex-valued function in L1(R) ∩ L2(R) defined through two real-valued mother wavelets ψR and ψI as follows ψ(t) = ψR(t) + iψI(t), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The function ψ is called an analytic wavelet if ψR and ψI meet the Hilbert pair require- ment (5) ψI(t) = 1 π � ∞ −∞ ψR(τ) t − τ dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Given a signal X : R → R, the family of analytic wavelets ψj(t) = 1 2j ψ � t 2j � , j ∈ Z, defines the WT of X [23] through the convolution of ψj with the input X: W[j]X(t) = X ⋆ ψj(t) = � R X(s)ψj(t − s)ds, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The magnitude information returned by the analytic wavelet transform UA [j] X :=A(|W[j]X(t)|), j ∈ Z, (6) describes the envelope of wavelet coefficients of X, where A(r) = r for r ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' When A(r) = r2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ln(r)), the time-scale representation {UA [j] X(t)}t∈R,j∈Z is the so-called scalogram (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' logarithmic scalogram) of X in the field of signal processing [1, Figure 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In the following, we consider two types of functions for A: A(r) = rν for some ν ∈ (0, ∞) or A(r) = ln(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In practice, in order to make the features stable to actions of small diffeomorphisms, such as deformations, to X, the transformation UA[j] is followed by a convolution with a low-pass filter φJ [1, 31]: SA J [j]X(t) := � UA[j]X � ⋆ φJ(t) = � R UA[j]X(s)φJ(t − s)ds, J ∈ Z, where φJ(t) = 1 2J φ � t 2J � 5 and φ is a real-valued function, usually chosen to be the father wavelet associated with ψ [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' We denote the Fourier transform of ψR by � ψR , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', � ψR(λ) = � R e−itλψR(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Similarly, the Fourier transform of ψI is denoted as � ψI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because ψR and ψI are a Hilbert pair (5), � ψI(λ) = −i sgn(λ)� ψR(λ), (7) where sgn(λ) = \uf8f1 \uf8f2 \uf8f3 −1 if λ < 0, 0 if λ = 0, +1 if λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For each j ∈ Z, the Fourier transform of the scaled wavelets ψR,j(t) := 1 2j ψR � t 2j � and ψI,j(t) := 1 2j ψI � t 2j � are � ψR(2j·) and � ψI(2j·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In terms of notation, � ψR,j(·) = � ψR(2j·) and � ψI,j(·) = � ψI(2j·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' All assumptions needed in this paper about the analytic wavelet ψ and the low-pass filter φJ are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the real part ψR of the analytic wavelet ψ, we assume that � ψR ∈ L1(R) ∩ L2(R) and there exists a bounded and continuous function CψR : R → C with CψR(0) ̸= 0 such that � ψR(λ) = CψR(λ)|λ|α, λ ∈ R, for a certain α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the averaging function φ, we assume that it is real-valued and �φ ∈ L1(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Stationary Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Given the relative breadth of analytical tools for Gaussian processes [28, 37, 38] and their ubiquity in applications [29], we consider stationary Gaussian processes as a model for the input X of the analytic wavelet trans- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let W be a complex-valued Gaussian random measure on R satisfying W(∆1) = W(−∆1), E[W(∆1)] = 0, and E � W(∆1)W(∆2) � = Leb(∆1 ∩ ∆2) for any ∆1, ∆2 ∈ B(R), where Leb is the Lebesgue measure on R and B(R) is the Borel σ-algebra on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let (Ω, FW, P) be a probability space, where the σ-algebra FW is generated by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' If X is a mean-square continuous and stationary real Gaussian random process with constant mean µX and covariance function RX, by the Bochner-Khinchin theorem [20, 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU Chapter 4], there exists a unique nonnegative measure FX : B(R) → [0, ∞) such that FX(∆) = FX(−∆) for any ∆ ∈ B(R) and RX(t) = � R eiλtFX(dλ), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The measure FX is called the spectral measure of the covariance function RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The spectral measure FX is absolutely continuous with respect to the Lebesgue measure with a density function fX ∈ L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It has one of the forms: (a) fX(λ) = CX(λ) or (b) fX(λ) = CX(λ) |λ|1−β , where β ∈ (0, 1) is the Hurst index of long-range dependence and CX : R → [0, ∞) is bounded and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For convenience of presentation, we set 00 = 1, by which Assumption 2(a) can be viewed as a limiting case of Assumption 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Under Assumption 2, X can be expressed as a Wiener integral as follows X(t) = µX + � R eitλ� fX(λ)W(dλ), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because � R ψj(s)ds = 0, by the stochastic Fubini theorem [40, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1], X ⋆ ψj(t) = � R eitλ � ψR(2jλ) � fX(λ)W(dλ) + i � R eitλ� ψI(2jλ) � fX(λ)W(dλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (8) Both the real and imaginary parts of X ⋆ ψj(t) are normal random variables with mean zero and variance σ2 j = � R |� ψR(2jλ)|2fX(λ)dλ = � R |� ψI(2jλ)|2fX(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For k ∈ N ∪ {0}, denote Lk (u) to be the Laguerre polynomial of degree k with the formula Lk (u) = eu k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' dk duk � e−uuk� , k ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' We have the following first lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For x1, x2 ∈ R and function A : (0, ∞) → R satisfying � ∞ 0 � A( √ 2r) �2 e−rdr < ∞, we have the expansion A (|x1 + ix2|) = � m,n∈N∪{0} Cm,n Hm(x1) √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hn(x2) √ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , (9) where Hm(x) = (−1)me x2 2 dm dxme− x2 2 , m = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', 7 are the (probabilistic) Hermite polynomials, Cm,n = � hmhncA,(m+n)/2 for m, n ∈ 2N ∪ {0}, 0 otherwise, (10) hm = (−1) m 2 √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2 m 2 � m 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' for m ∈ 2N ∪ {0}, (11) and cA,(m+n)/2 = � ∞ 0 A( √ 2u)L m+n 2 (u) e−udu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (12) The proof of Lemma 1 is provided in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The constants {cA,(m+n)/2}m,n∈2N∪{0} in Lemma 1 for practical cases are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' A(r) = rν, ν ∈ (0, ∞): cA, ℓ 2 = 2 ν 2 Γ(ν 2 + 1) � ℓ 2 − ν 2 − 1 ℓ 2 � , ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (13) For the binomial coefficient above, we note that � ℓ 2 − ν 2 − 1 ℓ 2 � = �ℓ 2 − ν 2 − 1 � �ℓ 2 − ν 2 − 2 � · · � 1 − ν 2 � � −ν 2 � �ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Especially, when ν = 2, cA,0 = 2, cA,1 = −2, and cA, ℓ 2 = 0 for ℓ ∈ {4, 6, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' A(r) = ln(r): cA,0 = 1 2 ln 2 − 1 2γ, where γ is the Euler-Mascheroni constant, and cA, ℓ 2 = −1 ℓ for ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By (8) and Lemma 1, for the case A(r) = rν with ν > 0, UA[j]X(t) =σν j � m,n∈N∪{0} Cm,n √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hm � 1 σj � R eitλ � ψR(2jλ) � fX(λ)W(dλ) � × Hn � 1 σj � R eitλ� ψI(2jλ) � fX(λ)W(dλ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (14) For the case A(r) = ln(r), UA[j]X(t) = ln(σj) + � m,n∈N∪{0} Cm,n √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hm � 1 σj � R eitλ � ψR(2jλ) � fX(λ)W(dλ) � × Hn � 1 σj � R eitλ� ψI(2jλ) � fX(λ)W(dλ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (15) 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU Let H = {f ∈ L2(R) | f(−λ) = f(λ) for all λ ∈ R} be a complex Hilbert space with the inner product ⟨f, g⟩ = � R f(λ)g(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Given an integer m ≥ 1, we denote the m-th tensor product of the Hilbert space H by H ⊗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The m-th symmetric tensor product of H is denoted by H ⊙m, which contains those functions f ∈ H ⊗m satisfying f(λp(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λp(m)) = f(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm) for any permutation (p(1), p(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , p(m)) of the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any f ∈ H ⊗m, the m-fold Wiener-Itˆo integrals of f with respect to the random measure W is defined by Im(f) = � ′ Rm f(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm)W(dλ1) · · ·W(dλm), where � ′ means that the integral excludes the diagonal hyperplanes λk = ∓λk′ for k, k ′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , m} and k ̸= k ′ [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 2 below, which comes from [28, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1] and [37, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10], provides a significant link between nonlinear functions of normal random variables and Wiener-Itˆo integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 2 (Itˆo’s formula and Product Formula [28, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let f ∈ H be such that ∥f∥H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Then, for any integer m ≥ 1, we have Hm �� R f(λ)W(dλ) � = Im � f ⊗m� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' If f ∈ H ⊙m and g ∈ H ⊙n, then Im(f)In(g) = m∧n � r=0 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �m r ��n r � Im+n−2r (f ⊗r g) , where f ⊗r g is the rth contraction of f and g defined as f ⊗r g(λ1:m+n−2r) = � Rr f(τ1:r, λ1:m−r)g(−τ1:r, λm−r+1:m+n−2r)dτ1 · · · dτr for r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , m ∧ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' When r = 0, set f ⊗0 g = f ⊗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Here, for any integers p1 < p2, we denote (λp1, λp1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λp2) by λp1:p2 to simplify the lengthy expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By Lemma 2, UA[j]X in (14) and (15) can be further expressed as triple summations of Wiener-Itˆo integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' However, the obtained decomposition of UA[j]X is not orthogonal due to the double sum over m and n in (14) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' On the other hand, because of the Hilbert pair relation (7) between the real and imaginary parts of the analytic wavelet, the weighted sum of products of sign functions in (16) will naturally pop out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For obtaining an orthogonal Wiener chaos decomposition of UA[j]X (Theorem 1 below), the following lemma plays a key role in merging non-orthogonal terms in the triple summations obtained from (14), (15), and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof of Lemma 3 is provided in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For ℓ ∈ 2N, let P[ℓ] represent the set of permutations of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ} ⊂ R \\ {0}, define B(ℓ, λ1:ℓ) = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] � m,n∈2N∪{0} m+n=ℓ ��m 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 (−1) n 2 ℓ� k=ℓ−n+1 sgn(λp(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (16) The equality B(ℓ, λ1:ℓ) = � 2ℓ ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1 if N(λ1:ℓ) = ℓ/2, 0 if N(λ1:ℓ) ̸= ℓ/2, (17) holds, where N(λ1:ℓ) is the number of negative elements in {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Main results Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Under Assumptions 1 and 2, the process UA[j]X defined in (6) can be expressed as a series of Wiener-Itˆo integrals as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (a) For A(r) = rν, where ν ∈ (0, ∞), UA[j]X(t) = E � UA[j]X(t) � + σν j � ℓ∈2N � ′ Rℓ Q(ℓ) t,j (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ)W(dλ1) · · ·W(dλℓ), (18) where E � UA[j]X(t) � = σν j cA,0, σj = ( � R |� ψR(2jλ)|2fX(λ)dλ)1/2, Q(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) = cℓ � σ−ℓ j ℓ� k=1 eitλk � ψR(2jλk) � fX(λk) � 1{N(λ1:ℓ)=ℓ/2}, cℓ = (−2) ℓ 2 (ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=')(ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1 cA, ℓ 2, (19) N(λ1:ℓ) is the number of negative elements in {λk}ℓ k=1, and cA, ℓ 2 is defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (b) The representation (18) can also be applied to the case A(r) = ln(r) with slight notation modification: ν = 0 and E � UA[j]X(t) � = cA,0 + ln(σj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof of Theorem is provided in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Corollary 1 below is an easy implication of Theorem 1, whose proof is provided in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The Wasserstein metric is defined by dW(Z1, Z2) = sup {|E [h(Z1)] − E [h(Z2)]| | h : R → R is Lipschitz and ∥h∥Lip ≤ 1 } for any random variables Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Given two stationary Gaussian processes X1 and X2 with spectral densities fX1 and fX2, denote σ2 p,j = E � |W[j]Xp|2� , p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU Suppose that the wavelet ψ satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For j ∈ Z, we have dW(UA[j]X1, UA[j]X2) ≥ � 2 ν 2 Γ( ν 2 + 1) ��σν 1,j − σν 2,j �� if A(r) = rν with ν > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' |ln σ1,j − ln σ2,j| if A(r) = ln(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Especially, if A(r) = ln(r) and (20) fXp(λ) = CXp(λ) |λ|1−βp , λ ∈ R \\ {0}, p = 1, 2, for some Hurst parameters β1, β2 ∈ (0, 1) and CX1, CX2 ∈ Cb(R, [0, ∞)), then 2(ln 2)−1 lim j→∞ j−1dW(UA[j]X1, UA[j]X2) ≥ |β1 − β2| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In view of that the Hurst parameters β1 and β2 in (20) give the strength of long-range dependence of X1 and X2, Corollary 1 shows that if two stationary Gaussian processes have remarkable difference on the strength of long-range dependence, this discrepancy will also be reflected in the distribution distance between their logarithmic scalograms, especially at large-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By Theorem 1 and the stochastic Fubini theorem [40], for the case A(r) = rν, where ν ∈ (0, ∞), we have SA J [j]X(t) = E � SA J [j]X(t) � + � ℓ∈2N � ′ Rℓ s(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ)W(dλ1) · · ·W(dλℓ), (21) where s(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) = σν j Q(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) � φJ(λ1 + · · · + λℓ) (22) and E � SA J [j]X(t) � = � R E � UA[j]X(s) � φJ(t − s)ds = σν j cA,0 �φ(0) for all j, J ∈ Z and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The representation (21) can also be applied to the case A(r) = ln(r) with slight notation modification: ν = 0 and E � SA J [j]X(t) � = (cA,0 + ln(σj)) �φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any d ∈ N, j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , jd ∈ Z, and t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , td ∈ R, let F = (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , Fd) with Fm = 2 J 2 � SA J [jm]X(2Jtm) − E � SA J [jm]X(2Jtm) �� , m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' From (21) and (22), for each K ∈ 2N, rewrite F = F≤K+F>K with F≤K = (F1,≤K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , Fd,≤K) and F>K = (F1,>K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , Fd,>K), where (23) Fm,≤K := 2 J 2 � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} Iℓ � s(ℓ) 2Jtm,jm � and (24) Fm,>K := 2 J 2 � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} Iℓ � s(ℓ) 2Jtm,jm � for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 11 Let N (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' N≤K) be a d-dimensional normal random vector with the same covariance matrix as that of F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' F≤K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Below, we first explore the smooth Wasserstein distance [5, 16] between F and N, and conclude this section with a corollary concerning the Kolmogorov distance [37, Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1] between F and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The smooth Wasserstein distance between the distribution of Rd-valued random variables F and N is denoted and defined by dH2(F, N) = sup h∈H2 ��E [h (F)] − E [h (N)] ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Here, the class H2 of the test functions is defined as H2 = � h ∈ C1(Rd) | ∥h∥∞ ≤ 1, ∥h ′∥∞ ≤ 1, h ′ is Lipschitz, and ∥h ′∥Lip ≤ 1 � , where C1(Rd) is the space of continuously differentiable functions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' To explore the smooth Wasserstein distance between F and N, we bound dH2(F, N) by the triangle inequality |E [h (F)] − E [h (N)]| (25) ≤ |E [h (F)] − E [h (F≤K)]| + |E [h (F≤K)] − E [h (N≤K)]| + |E [h (N≤K)] − E [h (N)]| and control the right hand side term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the case A(r) = rν with ν ∈ (0, ∞) \\ 2N, under Assumptions 1 and 2 with 2α + β ≥ 1, for any h ∈ H2, there exists a constant C ≥ 0 such that (26) |E [h (F)] − E [h (F≤K)]| ≤ dCK− ν 2 − 1 4 +ε and (27) |E [h (N≤K)] − E [h (N)]| ≤ dCK− ν 2 − 1 4+ε for a small ε > 0 and all J ∈ Z and K ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The inequalities (26) and (27) can also be applied to the case A(r) = ln(r) with slight notation modification: ν = 0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof of Lemma 4 is provided in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 4 says that even if the expansion of F is comprised of infinite Wiener chaos, particularly when ν ∈ (0, ∞) \\ 2N, the first and third terms in (25) still converge to zero when K → ∞ uniformly with respect to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The constant C in Lemma 4 only depends on j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , jd, fX, ψ, φ, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Clearly, the smaller ν is, the slower the convergence is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' More details can be found in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In the following, we plan to apply Stein’s method to analyze the second term in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 5 below is a slight modification of [37, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2], in which the test function h is twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' How to relax the twice differentiability assumption on the test function h is described in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU Lemma 5 ([37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any integer d ≥ 2, let S = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , Sd) be a FW-measurable random vector such that E[|Sm|4] + E[∥DSm∥4 H] < ∞ and E[Sm] = 0 for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let [C(m, n)]1≤m,n≤d be a non-negative definite matrix in Rd×d, and let NC be a d- dimensional normal random vector with mean zero and covariance matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Then, for any function h ∈ H2, we have |E[h(S)] − E[h(NC)]| ≤ 1 2∥h ′∥Lipρ, (28) where ∥h ′∥Lip = max 1≤m≤d sup x,y∈Rd x̸=y ��� ∂h ∂xm(x) − ∂h ∂xm(y) ��� ∥x − y∥ , ρ = � � � � d � m,n=1 E �� C(m, n) − � DSn, −DL−1Sm ��2� , L−1 is the pseudo-inverse of the infinitesimal generator of the Ornstein-Uhlenbeck semi- group, and D is the Malliavin derivative defined in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' If Sm = Fm,≤K, by the Meyer’s inequality (see (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) and (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) in the appendix), the condition E[|Sm|4] + E[∥DSm∥4 H] < ∞ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' On the other hand, because Fm,≤K is a linear combination of Wiener chaos of order greater than or equal to 2, E[Fm,≤K] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Denote the covariance matrix of (F1,≤K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , Fd,≤K) by CJ,K ∈ Rd×d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' that is, CJ,K(m, n) := E[Fm,≤KFn,≤K], where 1 ≤ m, n ≤ d, which satisfies the requirement of non-negative definite for the matrix C in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because all conditions in Lemma 5 are satisfied, the inequality (28) holds with S = F≤K and C = CJ,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By making use of the explicit structure of the Wiener chaos decomposition of F≤K derived from Theorem 1 (see also (21) and (23)), we get the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Under Assumptions 1 and 2 with 2α + β ≥ 1, for any t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', td ∈ R, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', jd ∈ Z, and h ∈ H2, there exists a constant C such that |E[h(F≤K)] − E[h(NCJ,K)]| ≤ C2− J 2 \uf8eb \uf8ed � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} |cℓ| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 ℓ 2 \uf8f6 \uf8f8 2 (29) for all J ∈ Z and K ∈ 2N, where cℓ is defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof of Proposition 1 is provided in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The constant C in (29) only depends on A, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , jd, fX, ψ, and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' More details can be found in (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' If Assumptions 1 and 2 hold and 2α + β ≥ 1, then for any d ∈ N, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , td ∈ R, and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , jd ∈ Z, the sequence of J-dependent random vectors F = 2J/2 � SA J [j1]X(2Jt1) − E � SA J [j1]X(2Jt1) � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , SA J [jd]X(2Jtd) − E � SA J [jd]X(2Jtd) �� , J ∈ Z, 13 satisfies dH2 (F, N) ≤ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 O(2− J 2 ) for A(r) = rν with ν ∈ 2N, O(J− ν 2 − 1 4+ε) for A(r) = rν with ν ∈ (0, ∞) \\ 2N, O(J− 1 4) for A(r) = ln(r) (30) for any ε > 0 when J → ∞, where N is a zero-mean d-dimensional normal random vector with the same covariance matrix as F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Furthermore, (31) lim J→∞E � FTF � = � κm,n � R eiλ(tm−tn)|�φ(λ)|2dλ � 1≤m,n≤d , where κm,n = 1 2π � R Cov � UA[jm]X(τ), UA[jn]X(0) � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof of Theorem 2 is provided in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Recall that the Kolmogorov distance between the distribution of Rd-valued random variables F and N is denoted and defined by dKol(F, N) = sup z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',zd∈R ��P (F ∈ (−∞, z1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' × (−∞, zd]) − P (N ∈ (−∞, z1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' × (−∞, zd]) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6 in [18] allows us to extract Kolmogorov distance bounds between F and its normal counterpart N from the smooth Wasserstein distance bounds in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' To keep the paper self-contained, the proof of Corollary 2 is provided in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let the assumptions and notation of Theorem 2 prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The J-dependent random vectors F satisfies dKol (F, N) ≤ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 O(2− J 6 ) for A(r) = rν with ν ∈ 2N, O(J− ν 6 − 1 12 +ε) for A(r) = rν with ν ∈ (0, ∞) \\ 2N, O(J− 1 12) for A(r) = ln(r) for any ε > 0 when J → ∞, where N is a zero-mean d-dimensional normal random vector with the same covariance matrix as the corresponding F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Discussion and conclusions We derived the Wiener chaos decomposition of the modulus of the analytic wavelet transform and its variants of stationary Gaussian processes and proved a quantitative central limit theorem for its moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because the complex modulus performs a square root on the scalogram, we observed that Wiener chaos decompositions of the modulus wavelet transform and the scalogram have a significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The former consists of infinite Wiener chaos, while the latter consists of only finite Wiener chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Such differences affect the convergence speed of the Gaussian approximation error of their respective moving averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU The modulus wavelet transform, which may be further transformed by a nonlinear function A, is a core component of the scattering transform [31], in which the composition of the modulus wavelet transform coupled with the moving average SA J [j, j + δ]X(t) = � R UA[j + δ]UA[j]X(s)φJ(t − s)ds, j ∈ Z, δ ∈ N+, (32) was proposed to extract more detailed features from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In order to make sure that SA J [j, j + δ] is a non-expansive map [31, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5], the complex modulus, which corresponds to the case A(r) = r, is used as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' As we have observed in this work, UA[j]X in (32) is a non-Gaussian process, which consists of infinite Wiener chaos for the non-expansive case A(r) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, to further analyze the second-layer modulus wavelet transform of UA[j]X, developing new techniques to handle the nonlinear interaction across different layers and analytic wavelet is necessary, and we will report the results in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' To sum up, our current work is not only interesting from the wavelet transform perspective but also paves a way toward a theoretical understanding of the scattering transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Acknowledgement This work benefited from support of the National Center for Theoretical Science (NCTS, Taiwan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Liu’s work was supported by the National Science and Technol- ogy Council under Grant 110-2628-M-006-003-MY3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendices Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Lemma 1 Because � R � R A �� x2 1 + x2 2 � 1 2πe− x2 1+x2 2 2 dx1dx2 = � ∞ 0 A(r)e− r2 2 rdr = � ∞ 0 A( √ 2r)e−rdr < ∞ 17 and the set of normalized probabilist’s Hermite polynomials { 1 √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='Hm}m∈N∪{0} forms an orthonormal basis for the Gaussian Hilbert space L2 � R, 1 √ 2πe− x2 2 dx � , the expansion (9) holds with Cm,n = � R � R A �� y2 1 + y2 2 � Hm(y1) √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hn(y2) √ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 1 2πe− y2 1+y2 2 2 dy1dy2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) for m, n ∈ N∪{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because A( � y2 1 + y2 2) is an even function of y1 and y2 and Hm(−y) = (−1)mHm(y) for any y ∈ R and m ∈ N ∪ {0}, Cm,n = 0 if m or n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, we only need to compute Cm,n for the cases m, n ∈ 2N∪{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By converting to polar coordinates, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) can be rewritten as Cm,n = � ∞ 0 A(r) �� 2π 0 Hm(r cos θ) √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hn(r sin θ) √ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' dθ � 1 2πe− r2 2 rdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) From [33], in which the physicists’ Hermite polynomials were used, � 2π 0 Hm(r cos θ) √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hn(r sin θ) √ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' dθ = 2πhmhnL m+n 2 �r2 2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) where hm is defined in (11) and L m+n 2 (z) is the Laguerre polynomial of degree m+n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' We obtain (10) by substituting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) as follows Cm,n = 2πhmhn � ∞ 0 A(r)L m+n 2 �r2 2 � 1 2πe− r2 2 rdr = hmhn � ∞ 0 A( √ 2u)L m+n 2 (u) e−udu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Lemma 3 For any n ∈ {0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ}, where ℓ ∈ 2N, and p ∈ P[ℓ], if λk ̸= 0 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ, then ℓ� k=ℓ−n+1 sgn(λp(k)) = 1{N(λ1:ℓ,n,p)∈2Z} − 1{N(λ1:ℓ,n,p)/∈2Z}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) where N(λ1:ℓ, n, p) is the cardinality of the set {k | ℓ − n + 1 ≤ k ≤ ℓ, λp(k) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let N = N(λ1:ℓ) ≤ ℓ be the number of negative elements in {λk}ℓ k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) and the probability mass function of hypergeometric random variables, 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] ℓ� k=ℓ−n+1 sgn(λp(k)) = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] 1{N(λ1:ℓ,n,p)∈2Z} − 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] 1{N(λ1:ℓ,n,p)/∈2Z} = �ℓ n �−1 an, 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU where (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) an = n � q=0 (−1)q �N q ��ℓ − N n − q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, (16) can be rewritten as B(ℓ, λ1:ℓ) = � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} ��ℓ 2 − n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 (−1) n 2 �ℓ n �−1 an = � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wnan, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) where wn = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ/2 − n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (−1) n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) Case 1 (N = ℓ/2): In this case, for n ∈ {0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ}, an = n � q=0 (−1)q �ℓ/2 q �� ℓ/2 n − q � = (−1)n/2 �ℓ/2 n/2 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) where the last equality follows from [44, Identity 81 in Page 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3), B(ℓ, λ1:ℓ) can be rewritten as follows B(ℓ, λ1:ℓ) = �ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} � n n/2 �� ℓ − n ℓ/2 − n/2 � = �ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2ℓ, where the last equality follows from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Case 2 (N > ℓ/2): First of all, since the constant an in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) is recognized to be the coefficient of xn in the polynomial Pℓ,N(x) = (1 − x)N(1 + x)ℓ−N (in terms of notation, an = [xn]Pℓ,N(x)), from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3), B(ℓ, λ1:ℓ) can be viewed as a weighted sum of the coefficients of {x0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , xℓ} in Pℓ,N(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', B(ℓ, λ1:ℓ) = � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wnan = � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wn[xn]Pℓ,N(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) Because Pℓ,N(x) can be rewritten as Pℓ,N(x) =(1 − x)N �ℓ−N � k=0 bk(1 − x)k � = ℓ−N � h=0 bℓ−N−h(1 − x)ℓ−h for some constants b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , bℓ−N, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) can be rewritten as B(ℓ, λ1:ℓ) = � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wn[xn] �ℓ−N � h=0 bℓ−N−h(1 − x)ℓ−h � = ℓ−N � h=0 bℓ−N−hAh, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7) 19 where (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) Ah := � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wn[xn](1 − x)ℓ−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4), for each h ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − N}, Ah = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} n≤ℓ−h n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ/2 − n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (−1) n 2 �ℓ − h n � = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} n≤ℓ−h (ℓ − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ/2 − n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (−1) n 2 1 (ℓ − n − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='. (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9) Obviously, A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − N} ∩ 2N, from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9), Ah = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2 h 2 � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} n≤ℓ−h (ℓ − n − 1)(ℓ − n − 3) · · ·(ℓ − n − h + 1) (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ/2 − n/2 − h/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (−1) n 2 = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2 h 2 (ℓ/2 − h/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ−h)/2 � k=0 (ℓ − 2k − 1)(ℓ − 2k − 3) · · · (ℓ − 2k − h + 1) � �� � h/2 terms �(ℓ − h)/2 k � (−1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Under the condition h < ℓ 2, which is satisfied because N > ℓ 2, we have (ℓ−h)/2 � k=0 kq �(ℓ − h)/2 k � (−1)k = 0, q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , h 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, we obtain Ah = 0 for h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − N} ∩ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − N} \\ 2N, from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9), Ah = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2 h+1 2 � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} n≤ℓ−h (ℓ − n − 1)(ℓ − n − 3) · · ·(ℓ − n − h + 2) (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ/2 − n/2 − h/2 − 1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (−1) n 2 = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ − h)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 2 h+1 2 (ℓ/2 − h/2 − 1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ−h−1)/2 � k=0 (ℓ − 2k − 1)(ℓ − 2k − 3) · · ·(ℓ − 2k − h + 2) � �� � (h−1)/2 terms �(ℓ − h − 1)/2 k � (−1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Under the condition h < ℓ 2, which is satisfied again because N > ℓ 2, we have (ℓ−h−1)/2 � k=0 kq �(ℓ − h − 1)/2 k � (−1)k = 0, q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , h − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, we also have Ah = 0 for h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − N} \\ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' In summary, we have Ah = 0 for h ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', ℓ−N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7), B(ℓ, λ1:ℓ) = 0 if N > ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 20 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU Case 3 (N < ℓ/2): For this case, we consider the following decomposition of Pℓ,N(x): Pℓ,N(x) =(1 + x)ℓ−N � N � k=0 �bk(1 + x)k � = N � h=0 �bN−h(1 + x)ℓ−h, where �b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ,�bN ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Similar to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7), we have B(ℓ, λ1:ℓ) = N � h=0 �bN−h �Ah, where �Ah := � n∈{0,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ℓ} wn[xn](1 + x)ℓ−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because [xn](1 + x)ℓ−h = �ℓ−h n � = [xn](1 − x)ℓ−h for any nonnegative even integer n, �Ah = Ah = 0 for any nonnegative integer h < ℓ/2, where Ah is defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, B(ℓ, λ1:ℓ) = 0 for the case N < ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Theorem 1 We only prove the result for the case A(r) = rν, where ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof for the case A(r) = ln(r) is similar, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' First of all, we denote qR(λ) = � ψR(2jλ) � fX(λ), qI(λ) = � ψI(2jλ) � fX(λ), qR,t(λ) = eitλqR(λ), and qI,t(λ) = eitλqI(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By Itˆo’s formula in Lemma 2, for any m, n ∈ N, Hm � 1 σj � R qR,t(λ)W(dλ) � Hn � 1 σj � R qI,t(λ)W(dλ) � = σ−(m+n) j Im � q⊗m R,t � In � q⊗n I,t � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) where q⊗m R,t is the m-fold tensor product of qR,t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=', q⊗m R,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm) = m � k=1 qR,t(λk), λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By default, I0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) into (14), we obtain UA[j]X(t) = σν j � m,n∈N∪{0} Cm,n √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' σ−(m+n) j Im � q⊗m R,t � In � q⊗n I,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) By the product formula in Lemma 2 (see also [14, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1]), Im � q⊗m R,t � In � q⊗n I,t � = m∧n � r=0 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �m r ��n r � Im+n−2r(q⊗m R,t ⊗r q⊗n I,t ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) where ⊗ℓ is the contraction operator defined in Lemma 2, or more precisely, q⊗m R,t ⊗r q⊗n I,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n−2r) = � Rr q⊗m R,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm−r,u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ur) q⊗n I,t (λm−r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n−2r, −u1, − u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , −ur)du1 · · · dur 21 for every r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , m ∧ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For r = 0, q⊗m R,t ⊗0 q⊗n I,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n) =q⊗m R,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm)q⊗n I,t (λm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because (7) implies qI(λ) = −i sgn(2jλ)qR(λ), � R qR(λ)qI(−λ)dλ = i � R |� ψR(2jλ)|2fX(λ) sgn(λ)dλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, for any m, n ∈ N, r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , m ∧ n}, and λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n−2r ∈ R, q⊗m R,t ⊗r q⊗n I,t (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λm+n−2r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It implies that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) can be simplified as (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) Im � q⊗m R,t � In � q⊗n I,t � = Im+n(q⊗m R,t ⊗ q⊗n I,t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By (10) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4), the series (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) can be rewritten as UA[j]X(t) =σν j � m,n∈2N∪{0} cA, m+n 2 hmhn √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' σ−(m+n) j Im+n(q⊗m R,t ⊗ q⊗n I,t ) =σν j cA,0 + σν j � ℓ∈2N Iℓ � Q(ℓ) t,j � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) where Q(ℓ) t,j : Rℓ → C is defined as Q(ℓ) t,j = cA, ℓ 2σ−ℓ j 2− ℓ 2(−1) ℓ 2 � m,n∈2N∪{0} m+n=ℓ ��m 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 q⊗m R,t ⊗ q⊗n I,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because qI,t(·) = −i sgn(2j·)qR,t(·) = −i sgn(·)qR,t(·), we have Q(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) =cA, ℓ 2σ−ℓ j 2− ℓ 2(−1) ℓ 2q⊗ℓ R,t(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) × � m,n∈2N∪{0} m+n=ℓ ��m 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 (−1) n 2 ℓ� k=ℓ−n+1 sgn(λk) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) for all (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) ∈ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By default, ℓ� k=ℓ+1 sgn(λk) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Denote the symmetrization of Q(ℓ) t,j with respect to (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) by Q(ℓ) t,j, which is defined as Q(ℓ) t,j(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] Q(ℓ) t,j (λp(1), λp(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λp(ℓ)), where P[ℓ] represents the set of permutations of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By the property Iℓ � Q(ℓ) t,j � = Iℓ � Q(ℓ) t,j � , 22 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU the series representation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) for UA[j]X(t) can be rewritten as UA[j]X(t) = σν j cA,0 + σν j � ℓ∈2N Iℓ � Q(ℓ) t,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7) From (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6), Q(ℓ) t,j (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) = cA, ℓ 2σ−ℓ j 2− ℓ 2(−1) ℓ 2q⊗ℓ R,t(λ1:ℓ)B(ℓ, λ1:ℓ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) where λ1:ℓ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) and B(ℓ, λ1:ℓ) = 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � p∈P [ℓ] � m,n∈2N∪{0} m+n=ℓ ��m 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �n 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �−1 (−1) n 2 ℓ� k=ℓ−n+1 sgn(λp(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any positive even integer ℓ and {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ} ⊂ R \\ {0}, Lemma 3 shows that B(ℓ, λ1:ℓ) = � 2ℓ ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1 if N(λ1:ℓ) = ℓ/2, 0 if N(λ1:ℓ) ̸= ℓ/2, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9) where N(λ1:ℓ) is the number of negative elements in {λk}ℓ k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8), Q(ℓ) t,j(λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) = (−2) ℓ 2 (ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=')(ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1 cA, ℓ 2σ−ℓ j q⊗ℓ R,t(λ1:ℓ)1{N(λ1:ℓ)=ℓ/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10) The proof of Theorem 1 is concluded by substituting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7) and noticing that the Lebesque measure of {(λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ) ∈ Rℓ | λk = 0 for some k} is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Corollary 1 In the following, we only give details for the case A(r) = ln(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof for A(r) = rν is similar and we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because h(x) = x is one of the test functions for the Wasserstein metric, dW � UA[j]X1, UA[j]X2 � ≥ ��E � UA[j]X1 � − E � UA[j]X2 ��� = |ln σ1,j − ln σ2,j| , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) where the last equality follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' If the spectral densities of X1 and X2 have the form (20) and the wavelet function ψR satisfies Assumption 1, then by the dominated convergence theorem, lim j→∞2−βpj/2σp,j = � CXp(0) � R |� ψR(λ)|2|λ|βp−1dλ � 1 2 , that is, lim j→∞ − 1 2(ln 2)βpj + ln σp,j = 1 2 ln � CXp(0) � R |� ψR(λ)|2|λ|βp−1dλ � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) for p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1), we obtain lim j→∞ j−1dW(UA[j]X1, UA[j]X2) ≥ lim j→∞ ���� ln σ1,j j − ln σ2,j j ���� = 1 2(ln 2) |β1 − β2| , where the equality follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 23 Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Lemma 4 (a) First of all, because ∥h ′∥∞ ≤ 1, |E [h (F)] − E [h (F≤K)]| ≤ E � d � m=1 |Fm,>K| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By the orthogonal property E � Iℓ(s(ℓ) 2Jtm,jm)Iℓ′(s(ℓ ′) 2Jtm,jm) � = � ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='∥s(ℓ) 2Jtm,jm∥2 2 if ℓ = ℓ ′, 0 if ℓ ̸= ℓ ′, and (24), we have (E|Fm,>K|)2 ≤E � |Fm,>K|2� =2J � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ∥s(ℓ) 2Jtm,jm∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) For any t ∈ R and j ∈ Z, according to the definition of s(ℓ) t,j in (22), ∥s(ℓ) t,j∥2 2 ≤σ2ν−2ℓ j c2 ℓ � Rℓ � ℓ� k=1 fX⋆ψR,j(λk) � |� φJ(λ1 + · · · + λℓ)|2dλ1 · · · λℓ =2−Jσ2ν−2ℓ j c2 ℓ � R f ⋆ℓ X⋆ψR,j(2−Jη)|�φ(η)|2dη, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) where fX⋆ψR,j(λk) = ���� ψR(2jλk) ��� 2 fX(λk) and f ⋆ℓ X⋆ψR,j is the ℓ-fold convolution of fX⋆ψR,j with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By the nonnegativity of fX⋆ψR,j and σ2 j = � R fX⋆ψR,j(λ)dλ, f ⋆ℓ X⋆ψR,j(η) = � R f ⋆(ℓ−1) X⋆ψR,j(η − ζ)fX⋆ψR,j(ζ)dζ ≤ ∥f ⋆(ℓ−1) X⋆ψR,j∥∞σ2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It implies that for all positive integer ℓ, ∥f ⋆ℓ X⋆ψR,j∥∞ ≤ ∥fX⋆ψR,j∥∞σ2(ℓ−1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) Note that ∥fX⋆ψR,j∥∞ < ∞ under the assumption 2α + β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By using (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) to bound the integrand in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2), ∥s(ℓ) t,j∥2 2 ≤2−Jσ2ν−2 j c2 ℓ∥fX⋆ψR,j∥∞∥�φ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) By applying (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1), (E|Fm,>K|)2 ≤ σ2ν−2 jm ∥fX⋆ψR,jm∥∞∥�φ∥2 2 � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='c2 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) 24 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU From the definition of cℓ in (19), � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='c2 ℓ = � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} 2ℓ �ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �2 (ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1c2 A, ℓ 2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) For A(r) = rν with ν > 0, cA, ℓ 2 = 2 ν 2 Γ(ν 2 + 1) � ℓ 2 − ν 2 − 1 ℓ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By L’Hospital’s rule, ln |cA, ℓ 2+1| − ln |cA, ℓ 2| ln( ℓ 2 + 1) − ln ℓ 2 = ln | � ℓ 2− ν 2 ℓ 2 +1 � | − ln | � ℓ 2− ν 2 −1 ℓ 2 � | ln( ℓ 2 + 1) − ln ℓ 2 =ln( ℓ 2 − ν 2) − ln( ℓ 2 + 1) ln( ℓ 2 + 1) − ln ℓ 2 → −ν 2 − 1 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7) as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' From (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7), we know that for any ε > 0, there exists a constant C1(ν, ε) > 0 such that (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) |cA, ℓ 2| ≤ C1(ν, ε)ℓ− ν 2 −1+ε for any integer ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For A(r) = ln(r), cA, ℓ 2 = −ℓ−1 for ℓ ∈ 2N, which can be viewed as a special case of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) with ν = 0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' On the other hand, by Stirling’s formula, or more precisely √ 2πn �n e �n e 1 12n+1 < n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' < √ 2πn �n e �n e 1 12n (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9) for all n ∈ N, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10) 2ℓ �ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �2 (ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1 ≤ (2πℓ) 1 2 for any integer ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By applying the inequalities (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10) to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6), there exists a constant C2(ν, ε) > 0 such that � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='c2 ℓ ≤(C1(ν, ε))2 � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} (2πℓ) 1 2ℓ−ν−2+2ε ≤C2(ν, ε)K−ν− 1 2 +2ε (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='11) for all K ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='11), we obtain (E|Fm,>K|)2 ≤ σ2ν−2 jm ∥fX⋆ψR,jm∥∞∥�φ∥2 2C2(ν, ε)K−ν− 1 2+2ε, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='12) which leads to the inequality (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (b) By the Wiener chaos decomposition of F≤K in (23), for any m, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d}, E [Fm,≤K, Fn,≤K] = 2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='⟨s(ℓ) 2Jtm,jm, s(ℓ) 2Jtn,jn⟩, 25 where ⟨s(ℓ) 2Jtm,jm, s(ℓ) 2Jtn,jn⟩ = � Rℓ s(ℓ) 2Jtm,jm(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ)s(ℓ) 2Jtn,jn(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λℓ)dλ1 · · ·dλℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Let � N(ℓ)� ℓ∈2N = �� N(ℓ) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , N(ℓ) d �� ℓ∈2N be a sequence of independent d-dimensional normal random vectors with mean zero and E � N(ℓ) m N(ℓ) n � = 2Jℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='⟨s(ℓ) 2Jtm,jm, s(ℓ) 2Jtn,jn⟩ for m, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Then, N d= � ℓ∈2N N(ℓ) and N≤K d= � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} N(ℓ), where d= means the equality is in the distribution sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any Lipschitz differentiable function h : Rd → R with Lipschitz constant ∥h ′∥∞ ≤ 1, |E [h (N≤K)] − E [h (N)]| ≤ d � m=1 E ������ � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} N(ℓ) m ������ = � 2 π d � m=1 \uf8ee \uf8f0 � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} E|N(ℓ) m |2 \uf8f9 \uf8fb 1 2 = � 2 π d � m=1 \uf8ee \uf8f0 � ℓ∈{K+2,K+4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='} 2Jℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='∥s(ℓ) 2Jtm,jm∥2 2 \uf8f9 \uf8fb 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='13) The summation inside the square brackets in the last term is the same as the upper bound of (E|Fm,>K|)2 in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, we can directly apply (26) to the right hand side of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='13) to obtain |E [h (N≤K)] − E [h (N)]| ≤ dCK− ν 2 − 1 4+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Lemma 5 For any twice differentiable function h, [37, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2] shows that |E[h(S)] − E[h(NC)]| ≤ 1 2 � max 1≤m,n≤d sup x∈Rd ��� ∂2h ∂xm∂xn (x) ��� � ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) It is clear that max 1≤m,n≤d sup x∈Rd ��� ∂2h ∂xm∂xn (x) ��� ≤ ∥h ′∥Lip (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) for any twice differentiable function h in H2, so we only need to show that (28) holds for non-twice differentiable functions in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For such class of functions h, we define hε(x) = � Rd 1 (2πε) d 2 e− ∥z∥2 2ε h(x − z)dz, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 26 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU For any ε > 0, hε is twice differentiable, so (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) imply that |E[hε(S)] − E[hε(NC)]| ≤1 2∥h ′ ε∥Lipρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) Because h is continuous and bounded, by the dominated convergence theorem, lim ε→0|E[hε(S)] − E[hε(NC)]| = |E[h(S)] − E[h(NC)]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) On the other hand, because h is continuously differentiable and ∥ ∂h ∂xm∥∞ ≤ 1 for m ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d}, ∂hε ∂xm (x) = � Rd 1 (2πε) d 2 e− ∥z∥2 2ε ∂h ∂xm (x − z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, for any x, y ∈ Rd with x ̸= y, ���� ∂hε ∂xm (x) − ∂hε ∂xm (y) ���� ≤ � Rd 1 (2πε) d 2 e− ∥z∥2 2ε ���� ∂h ∂xm (x − z) − ∂h ∂xm (y − z) ���� dz, ≤ ∥ ∂h ∂xm ∥Lip∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) The inequality (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) implies that lim sup ε→0 ∥h ′ ε∥Lip = max 1≤m≤dlim sup ε→0 sup x,y∈Rd x̸=y ��� ∂hε ∂xm(x) − ∂hε ∂xm(y) ��� ∥x − y∥ ≤ max m∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',d}∥ ∂h ∂xm ∥Lip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) From (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4), and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6), we know that (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) holds for any h ∈ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Proposition 1 The proof of (29) consists of three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' First, we rewrite E �� CJ,K(m, n) − � DFn,≤K, −DL−1Fm,≤K ��2� by making use of the Wiener chaos decomposition of F≤K in (23) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By the orthogonal property of Wiener-Itˆo integrals, the covariance CJ,K(m, n) can be expressed as E [Fm,≤KFn,≤K] =2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} E � Iℓ � s(ℓ) 2Jtm,jm � Iℓ � s(ℓ) 2Jtn,jn �� =2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' s(ℓ) 2Jtm,jm ⊗ℓ s(ℓ) 2Jtn,jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) By Lemma 6 in J and (23) again, DFn,≤K = 2 J 2 � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓIℓ−1 � s(ℓ) 2Jtn,jn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) 27 By the definition of L−1 in (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4), −DL−1Fm,≤K =2 J 2 D \uf8ee \uf8f0 � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} 1 ℓIℓ � s(ℓ) 2Jtm,jm � \uf8f9 \uf8fb =2 J 2 � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} Iℓ−1 � s(ℓ) 2Jtm,jm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) By (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3), and the product formula in Lemma 2, ⟨DFn,≤K, −DL−1Fm,≤K⟩ = 2J� � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓIℓ−1 � s(ℓ) 2Jtn,jn � , � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} Iℓ−1 � s(ℓ) 2Jtm,jm � � = 2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' s(ℓ) 2Jtm,jm ⊗ℓ s(ℓ) 2Jtn,jn + 2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ ℓ−1 � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 �2 I2ℓ−2r � s(ℓ) 2Jtm,jm ⊗r s(ℓ) 2Jtn,jn � + 2J � ℓ,ℓ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ′ ℓ ℓ∧ℓ′ � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 ��ℓ ′ − 1 r − 1 � Iℓ+ℓ′−2r � s(ℓ) 2Jtm,jm ⊗r s(ℓ) 2Jtn,jn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) By (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4), and the Minkowski inequality, � E � (CJ,K(m, n) − ⟨DFn,≤K, −DL−1Fm,≤K⟩)2� ≤2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ ℓ−1 � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 �2� E ����I2ℓ−2r � s(ℓ) 2Jtm,jm ⊗r s(ℓ) 2Jtn,jn ���� 2� +2J � ℓ,ℓ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ′ ℓ ℓ∧ℓ′ � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 ��ℓ ′ − 1 r − 1 �� E ����Iℓ+ℓ′−2r � s(ℓ) 2Jtm,jm ⊗r s(ℓ′) 2Jtn,jn ���� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By noticing that E ����Iℓ+ℓ′−2r � s(ℓ) 2J tm,jm ⊗r s(ℓ ′) 2Jtn,jn ���� 2� ≤ (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ���s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn ��� 2 2 28 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU for all ℓ, ℓ ′ ∈ {2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , K} and r ∈ Λℓ,ℓ′, where Λℓ,ℓ = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ − 1} and Λℓ,ℓ′ = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ ∧ ℓ ′} for the case ℓ ̸= ℓ ′, we obtain � E �� CJ,K(m, n) − � DFn,≤K, −DL−1Fm,≤K ��2� ≤2J � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ ℓ−1 � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 �2� (2ℓ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ���s(ℓ) 2Jtm,jm ⊗r s(ℓ) 2Jtn,jn ��� 2 (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) +2J � ℓ,ℓ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ′ ℓ ℓ∧ℓ′ � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 ��ℓ ′ − 1 r − 1 �� (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ���s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Second, we calculate an upper bound for ���s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn ��� 2, where s(ℓ) t,j is de- fined in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' To simplify the tedious notation, for any integers p1 < p2, we denote (λp1, λp1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λp2) by λp1:p2 and λp1 + λp1+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' + λp2 by λ+ p1:p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' From (22), ���s(ℓ) 2Jtm,jm(λ1:ℓ) ��� ≤σν−ℓ jm |cℓ| � ℓ� k=1 |� ψR(2jmλk)| � fX(λk) � |� φJ(λ+ 1:ℓ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any ℓ, ℓ ′ ∈ 2N, r ∈ Λℓ,ℓ′ and tm, tn ∈ R, ���s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn � λ1:ℓ+ℓ′−2r ���� ≤σν−ℓ jm σν−ℓ ′ jn |cℓcℓ′| �ℓ−r � k=1 |� ψR(2jmλk)| � fX(λk) � \uf8ee \uf8f0 ℓ+ℓ ′−2r � k=ℓ−r+1 |� ψR(2jnλk)| � fX(λk) \uf8f9 \uf8fb × � Rr � r� k=1 |� ψR(2jmτk)� ψR(2jnτk)|fX(τk) � |� φJ(λ+ 1:ℓ−r + τ + 1:r)� φJ(λ+ ℓ−r+1:ℓ+ℓ′−2r − τ + 1:r)|dτ1 · · · dτr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, ∥s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn∥2 2 ≤σ2ν−2ℓ jm σ2ν−2ℓ ′ jn c2 ℓc2 ℓ′ � Rℓ+ℓ′ �ℓ−r � k=1 |� ψR(2jmλk)|2fX(λk) � \uf8ee \uf8f0 ℓ+ℓ ′−2r � k=ℓ−r+1 |� ψR(2jnλk)|2fX(λk) \uf8f9 \uf8fb × � r� k=1 |� ψR(2jmτk)||� ψR(2jnτk)|fX(τk) � � r� k=1 |� ψR(2jmηk)||� ψR(2jnηk)|fX(ηk) � × |� φJ(λ+ 1:ℓ−r + τ + 1:r)� φJ(λ+ ℓ−r+1:ℓ+ℓ′−2r − τ + 1:r)| × |� φJ(λ+ 1:ℓ−r + η+ 1:r)� φJ(λ+ ℓ−r+1:ℓ+ℓ′−2r − η+ 1:r)| dτ1 · · · dτr dη1 · · · dηr dλ1 · · · dλℓ+ℓ′−2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 29 By considering the change of variables \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 uk = τk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , r − 1 if r ≥ 2, vk = ηk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , r − 1 if r ≥ 2, wk = λk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ + ℓ ′ − 2r − 1, x = 2J � λ+ 1:ℓ−r + τ + 1:r � , y = 2J � λ+ ℓ−r+1:ℓ+ℓ′−2r − τ + 1:r � , z = 2J � λ+ 1:ℓ−r + η+ 1:r � and noting that � φJ(·) = �φ(2J·), ∥s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn∥2 2 (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) ≤2−3Jσ2ν−2ℓ jm σ2ν−2ℓ ′ jn c2 ℓc2 ℓ′ � Rℓ+ℓ′ �ℓ−r � k=1 |� ψR(2jmwk)|2fXwk) � \uf8ee \uf8f0 ℓ+ℓ ′−2r−1 � k=ℓ−r+1 |� ψR(2jnwk)|2fX(wk) \uf8f9 \uf8fb × �r−1 � k=1 |� ψR(2jmuk)||� ψR(2jnuk)|fX(uk) � �r−1 � k=1 |� ψR(2jmvk)||� ψR(2jnvk)|fX(vk) � × |� ψR(2jmu∗)||� ψR(2jnu∗)|fX(u∗)|� ψR(2jmv∗)||� ψR(2jnv∗)|fX(v∗)|� ψR(2jnw∗)|2fX(w∗) × |�φ(x)� φJ(y)�φ(z)�φ(x + y − z)| du1 · · · dur−1dv1 · · · dr−1dw1 · · · dwℓ+ℓ′−2r−1dxdydz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' where u∗ = 2−Jx − u1 − · · · − ur−1 − w1 − · · · − wℓ−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' v∗ = 2−Jz − v1 − · · · − vr−1 − w1 − · · − wℓ−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' and w∗ = 2−Jx + 2−Jy − w1 − · · · − wℓ+ℓ′−2r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' From (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6), we obtain ∥s(ℓ) 2Jtm,jm ⊗r s(ℓ ′) 2Jtn,jn∥2 2 ≤ C22−3Jc2 ℓc2 ℓ′, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7) where C2 = ∥�φ∥∞∥�φ∥3 1 max 1≤m,n≤d{σ2ν−2 jm σ2ν−4 jn M3, σ2ν−2 jm σ2ν−2 jn M2} (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8) and M = max 1≤m,n≤d∥� ψR(2jmu)� ψR(2jnu)fX(u)∥∞ < ∞ under the assumption 2α + β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Third, we rewrite the summation (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) in a more compact form by using the inequality (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The summation (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) can be bounded as follows � E �� CJ,K(m, n) − � DFn,≤K, −DL−1Fm,≤K ��2� ≤ C 2− 1 2 J [P1(K) + P2(K)] , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9) where P1(K) = � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ ℓ−1 � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 �2� (2ℓ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' c2 ℓ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10) 30 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU 0 50 100 150 200 250 300 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8 Figure G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Behavior of the ratio Θ1(ℓ + 1)/Θ1(ℓ) and P2(K) = � ℓ,ℓ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ′ ℓ ℓ∧ℓ′ � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 ��ℓ ′ − 1 r − 1 �� (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' |cℓcℓ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='11) Denote Θ1(ℓ) = 1 (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ℓ−1 � r=1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' �ℓ − 1 r − 1 �2� (2ℓ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' for ℓ ∈ N with Θ1(1) = 0, which leads to P1(K) = � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} c2 ℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Θ1(ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Observe that for every ℓ ∈ N, Θ1(ℓ) = ℓ−1 � r=1 �ℓ − 1 r − 1 ���2ℓ − 2r ℓ − r � = −1 + ℓ−1 � k=0 �ℓ − 1 k ���2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='12) Because �2k k � ≤ 4k for all k ∈ N ∪ {0}, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='12) implies that Θ1(ℓ) ≤ −1 + ℓ−1 � k=0 �ℓ − 1 k � 2k = −1 + 3ℓ−1, ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='13) See Figure G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1 for the behavior of the ratio Θ1(ℓ + 1)/Θ1(ℓ), which tends to 3 when ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting the estimate (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='13) into (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='10), P1(K) ≤ � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} c2 ℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 3ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='14) 31 On the other hand, P2(K) = � ℓ,ℓ ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ ′ |cℓcℓ′| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' √ ℓ ′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='Θ2(ℓ, ℓ ′), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='15) where Θ2(ℓ, ℓ ′) =ℓ ℓ∧ℓ ′ � r=1 (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='/ √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (ℓ ′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='/ √ ℓ ′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ ′ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ ′ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' = ℓ∧ℓ ′ � r=1 r ℓ ′ √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' √ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' √ ℓ ′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ ′ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' √ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (ℓ ′ − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='16) Because r ℓ′ ≤ 1 and (ℓ + ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' ≤ � (2ℓ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � (2ℓ ′ − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' for all ℓ, ℓ ′ ∈ N and r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , ℓ ∧ ℓ ′}, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='16) implies that Θ2(ℓ, ℓ ′) ≤ ℓ∧ℓ ′ � r=1 �� ℓ ℓ − r ��� ℓ ′ ℓ ′ − r � ��2ℓ − 2r ℓ − r �� 1 4 ��2ℓ ′ − 2r ℓ ′ − r �� 1 4 ≤ \uf8f1 \uf8f2 \uf8f3 ℓ∧ℓ ′ � r=1 � ℓ ℓ − r ���2ℓ − 2r ℓ − r �\uf8fc \uf8fd \uf8fe 1 2 \uf8f1 \uf8f2 \uf8f3 ℓ∧ℓ ′ � r=1 � ℓ ′ ℓ ′ − r ���2ℓ ′ − 2r ℓ ′ − r �\uf8fc \uf8fd \uf8fe 1 2 ≤ \uf8f1 \uf8f2 \uf8f3 ℓ∧ℓ ′ � r=1 � ℓ ℓ − r � 2ℓ−r \uf8fc \uf8fd \uf8fe 1 2 \uf8f1 \uf8f2 \uf8f3 ℓ∧ℓ ′ � r=1 � ℓ ′ ℓ ′ − r � 2ℓ ′−r \uf8fc \uf8fd \uf8fe 1 2 ≤ 3 ℓ 2 3 ℓ ′ 2 , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='17) where the second inequality follows from the Cauchy-Schwarz inequality and the third inequality follows from the fact �2ℓ−2r ℓ−r � ≤ 4ℓ−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='17) into (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='15), P2(K) ≤ � ℓ,ℓ ′∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} ℓ̸=ℓ ′ |cℓcℓ′| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' √ ℓ ′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 3 ℓ 2 3 ℓ ′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='18) By combining (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='14) and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='18), we get P1(K) + P2(K) ≤ \uf8eb \uf8ed � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} |cℓ| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 3 ℓ 2 \uf8f6 \uf8f8 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='19) By substituting (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='19) into (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9), we obtain (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Theorem 2 For the case A(r) = rν with ν ∈ 2N, (13) shows that cA, ℓ 2 = 0 if ℓ > ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' It implies that the decomposition (21) of SA J [j]X is only comprised of finite Wiener chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, the truncation step (24) for the decomposition of F is unnecessary and we can obtain the result immediately from Proposition 1 with K = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' 32 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU For the case A(r) = rν with ν ∈ (0, ∞) \\ 2N, by (19) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8), for any ε > 0, there exists a constant C1(ν, ε) such that |cℓ| ≤ C1(ν, ε)2 ℓ 2 (ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=')(ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )−1ℓ− ν 2 −1+ε (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) for all ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The inequality (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) also holds for A(r) = ln(r) with the notation replacement: ν = 0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By Stirling’s formula (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='9), 2 ℓ 2 (ℓ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=')(ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' )− 1 2 ≤ 2ℓ 1 4 for all ℓ ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, there exists a constant C2(ν, ε) such that � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} |cℓ| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 ℓ 2 ≤ C2(ν, ε) K/2 � ℓ=1 ℓ− ν 2 − 3 4 +ε3ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the summation above, for each fixed ν ∈ [0, ∞) \\ 2N, there exists a threshold Kν > 0 such that K/2 � ℓ=1 ℓ− ν 2 − 3 4 +ε3ℓ ≤ 31+ K 2 (K/2)− ν 2 − 3 4 +ε for any even integer K > Kν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, for any ε > 0, there exists a constant C3(ν, ε) > 0 such that \uf8eb \uf8ed � ℓ∈{2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',K} |cℓ| √ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 ℓ 2 \uf8f6 \uf8f8 2 ≤ C3(ν, ε)3KK−ν− 3 2 +ε (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) for any even integer K > Kν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By substituting (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) into (29), |E[h(F≤K)] − E[h(NCJ,K)]| ≤ CC3(ν, ε)2− J 2 3KK−ν− 3 2 +ε (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) for any even integer K > Kν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Finally, by applying the results of Lemma 4 and (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) to (25), we obtain that |E[h(F)] − E[h(N)]| ≤ C4(ν, ε) � K− ν 2 − 1 4+ε + 2− J 2 3KK−ν− 3 2 +ε� for a certain constant C4(ν, ε) independent of J and K when K ≥ Kν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The asymptotic behavior (30) is obtained by choosing K = 2⌊J 4 log3 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Finally, because both sides of (31) can be computed by the Wiener chaos expansions (18) and (21) of UA[j]X and SA J [j]X, the verification of (31) is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Proof of Corollary 2 The idea of proof is originated from the work [18], in which more complicated cases were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' However, we only need part of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For the convenience of readers, we will sketch its proof as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' First of all, we define 33 1 0 1 z z+ /2 z+ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8 1 (a) h and hz,δ (b) h(−1,−1),2 Figure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Approximation of indicator functions h(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 for x ≤ −1, 1 − 1 2(1 + x)2 for x ∈ (−1, 0], 1 2(1 − x)2 for x ∈ (0, 1], 0 for x > 1, and hz,δ(x) = h � 2 δ � x − � z + δ 2 ��� for any z ∈ R and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' See Figure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1(a) for the graphs of functions h and hz,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any d ∈ N and z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , zd) ∈ Rd, let hz,δ(x) = d � k=1 hzk,δ(xk), x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , xd) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' See Figure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1(b) for the graph of function hz,δ with z = (−1, −1) and δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any fixed z ∈ Rd, the indicator function 1x≤z is bounded above by hz,δ(x), where x ≤ z means that xk ≤ zk for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, P (F ≤ z) − P (N ≤ z) ≤E [hz,δ(F)] − E [hz,δ(N)] + E [hz,δ(N)] − P (N ≤ z) ≤E [hz,δ(F)] − E [hz,δ(N)] + P(N ≤ z + δ) − P (N ≤ z) , where z+δ means that z+δ11×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because ∥hz,δ∥∞ = 1, ∥h ′ z,δ∥∞ = 2 δ and ∥h ′ z,δ∥Lip = ( 2 δ)2, we have � 1 + �2 δ �2�−1 hz,δ ∈ H2 for any z ∈ Rd and δ > 0, which implies that |E [hz,δ(F)] − E [hz,δ(N)]| ≤ � 1 + �2 δ �2� dH2 (F, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) On the other hand, from Nazarov’s anti-concentration inequality [12, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1], we have P(N ≤ z + δ) − P (N ≤ z) ≤ δ � min 1≤m≤dE[F 2m] �√ 2 ln d + 2 � (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5 0 3 2 1 0 C1 1 2 0 2 1 3 234 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU for any z ∈ Rd and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Hence, P (F ≤ z) − P (N ≤ z) ≤ � 1 + �2 δ �2� dH2 (F, N) + δ � min 1≤m≤dE[F 2 m] �√ 2 ln d + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) By taking δ = 2 �� min 1≤m≤dE[F 2m]( √ 2 ln d + 2)−1dH2 (F, N) � 1 3 , (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) leads to the upper bound P (F ≤ z) − P (N ≤ z) ≤ 3 \uf8eb \uf8ed √ 2 ln d + 2 � min 1≤m≤dE[F 2m] \uf8f6 \uf8f8 2 3 (dH2 (F, N)) 1 3 + dH2 (F, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) For finding a lower bound for P (F ≤ z) − P (N ≤ z), we first observe that P (N ≤ z) − P (F ≤ z) ≤P (N ≤ z) − E [hz−δ,δ(N)] + E [hz−δ,δ(N)] − E [hz−δ,δ(F)] ≤P (N ≤ z) − P (N ≤ z − δ) + E [hz−δ,δ(N)] − E [hz−δ,δ(F)] (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5) for any z ∈ Rd and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' By applying (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) to (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5), we obtain a lower bound for P (F ≤ z)−P (N ≤ z), which is the negative of the right hand side of (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Therefore, dKol (F, N) ≤ 3 \uf8eb \uf8ed √ 2 ln d + 2 � min 1≤m≤dE[F 2m] \uf8f6 \uf8f8 2 3 (dH2 (F, N)) 1 3 + dH2 (F, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Because Theorem 2 shows that lim J→∞dH2 (F, N) = 0 and lim J→∞E[F 2 m] = κm,m∥�φ∥2 2, we obtain dKol (F, N) ≤ O � (dH2 (F, N)) 1 3 � (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6) when J → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The statement of Corollary 2 follows by combining (30) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Elements of Malliavin Calculus Let S denote the set of all random variables S of the form s(W(f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , W(fn)), where n ≥ 1, s : Rn → R is a C∞-function such that s and its partial derivatives have at most polynomial growth, and fi ∈ H, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' We denote by L2(Ω) the set of FW-measurable random variables whose second moments exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The space S is dense in L2(Ω) [37, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For every integers n, p ≥ 1, the p-th Malliavin derivative of S = s(W(f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , W(fn)) ∈ S 35 with respect to W is defined by DpS = n � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=',ip=1 ∂ps ∂xi1 · · · ∂xip (W(f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , W(fn))fi1 ⊗ · · · ⊗ fip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) Because the sum in (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='1) runs over all partial derivatives, DpS belongs to L2(Ω, H ⊙p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The Malliavin derivative of multiple Wiener-Itˆo integrals have the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The proof can be found in [37, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Lemma 6 ([37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For every integer p ≥ 1 and u ∈ H ⊗p, DIp(u) = pIp−1(˜u), where ˜u(λ1:p) = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' � π u(λπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λπ(p)), is the canonical symmetrization of u and Ip−1(˜u)(·) = � ′ Rp−1 ˜u(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' , λp−1, ·)W(dλ1) · · ·W(dλp−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' On the other hand, for all r ∈ N, p ∈ N and q ∈ [1, ∞], ∥Ip(f)∥Dr,q ≤ cr,p,q∥f∥H⊗p, f ∈ H ⊙p, (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) where cr,p,q > 0 is an universal constant and ∥Ip(f)∥Dr,q := � E[|Ip(f)|q] + E[∥DIp(f)∥q H] + · · · + E[∥DrIp(f)∥q H ⊗r] �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='3) We note that (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='2) is a special case of the Meyer’s inequality [38, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' For any F ∈ L2(Ω, FW, P) with the Wiener chaos expansion F = E[F] + ∞ � p=1 Ip (gp) , where gp ∈ H ⊙p, the pseudo-inverse of the infinitesimal generator L of the Ornstein- Uhlenbeck semigroup, denoted as L−1, is defined as (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='4) L−1F = − ∞ � p=1 1 p Ip (gp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' The details about the semigroup of the Ornstein-Uhlenbeck semigroup and its infini- tesimal generator can be found in [37, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' (Gi-Ren Liu) Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan (Yuan-Chung Sheu) Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan 36 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' SHEU, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} +page_content=' WU (Hau-Tieng Wu) Department of Mathematics and Department of Statistical Science, Duke University, Durham, NC, USA' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfk_1W/content/2301.01540v1.pdf'} diff --git a/P9AyT4oBgHgl3EQf7frl/content/tmp_files/2301.00841v1.pdf.txt b/P9AyT4oBgHgl3EQf7frl/content/tmp_files/2301.00841v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c514730c8b458e247fb7824b594c0ae48718a7c --- /dev/null +++ b/P9AyT4oBgHgl3EQf7frl/content/tmp_files/2301.00841v1.pdf.txt @@ -0,0 +1,5189 @@ +Ranking Differential Privacy +Shirong Xu∗, Will Wei Sun†, and Guang Cheng‡, +Abstract +Rankings are widely collected in various real-life scenarios, leading to the leakage of +personal information such as users’ preferences on videos or news. To protect rankings, +existing works mainly develop privacy protection on a single ranking within a set of +ranking or pairwise comparisons of a ranking under the ϵ-differential privacy. This +paper proposes a novel notion called ϵ-ranking differential privacy for protecting ranks. +We establish the connection between the Mallows model (Mallows, 1957) and the +proposed ϵ-ranking differential privacy. This allows us to develop a multistage ranking +algorithm to generate synthetic rankings while satisfying the developed ϵ-ranking +differential privacy. Theoretical results regarding the utility of synthetic rankings in the +downstream tasks, including the inference attack and the personalized ranking tasks, +are established. For the inference attack, we quantify how ϵ affects the estimation +of the true ranking based on synthetic rankings. For the personalized ranking task, +we consider varying privacy preferences among users and quantify how their privacy +preferences affect the consistency in estimating the optimal ranking function. Extensive +numerical experiments are carried out to verify the theoretical results and demonstrate +the effectiveness of the proposed synthetic ranking algorithm. +Key Words: Differential Privacy, Learning Theory, Mallows Model, Ranking Data, Synthetic +Data +∗Department of Statistics, University of California, Los Angeles. Email: shirong@stat.ucla.edu +†Krannert School of Management, Purdue University. Email: sun244@purdue.edu. +‡Department of Statistics, University of California, Los Angeles. Email: guangcheng@ucla.edu. +1 +arXiv:2301.00841v1 [stat.ML] 2 Jan 2023 + +1 +Introduction +Ranking data commonly arises from various business scenarios, such as recommender systems +(Karatzoglou et al., 2013; Oliveira et al., 2020) and search engines (Dwork et al., 2001; Liu +et al., 2007). Generally, rankings are collected and utilized to learn users’ preferences to +items for providing appropriate recommendations in the future. A typical example is the +recommender system, which suggests new items for incoming users by pooling historical +information on users’ behaviors. Nevertheless, users’ rankings are usually highly sensitive +since they reveal their purchasing or political preferences (Yang et al., 2019; Lee, 2015). +Therefore, developing an effective mechanism to achieve privacy protection of ranking data +before being shared with an external party is of great need. +To ensure data privacy protection, it is a common practice to employ differential privacy +(DP; Dwork 2006) as a standard metric. Differential privacy formalizes privacy guarantee +in the mathematical language without imposing assumptions about data, possessing nice +properties such as immunity to post-processing and privacy composition for sanitized data +output by privacy-preserving mechanisms. Generally, differential privacy can be categorized +into two main classes, including central differential privacy (CDP; Dwork et al. 2006) and +local differential privacy (LDP; Wang et al. 2017). LDP perturbs data on the users’ side and +submits privatized data to servers, whereas CDP relies on a trusted data collector to protect +all data. LDP become popular in privacy protection and has now been applied in various +real scenarios, including Google Chrome browser (Erlingsson et al., 2014) and macOS (Tang +et al., 2017). +In the literature, various research efforts have been devoted to developing an effective +mechanism to protect ranking data under the differential privacy and establishing theoretical +results regarding statistical inference based on privatized ranking data. Depending on the +employment of either LDP or CDP, the privacy protection of ranking data can be divided +into two main classes. In the central model of DP, a trusted curator collects non-private +rankings from all users and carries out differentially private statistical inference (Hay et al., +2 + +2017; Lee, 2015; Shang et al., 2014; Lee, 2015; S´anchez et al., 2016; Busa-Fekete et al., +2021). Specifically, Lee (2015) developed efficient algorithms for eliciting the true ranking of +items under the central differential privacy and strategic manipulation. Shang et al. (2014) +proposed to utilize the Gaussian noise to contaminate the histogram of collected rankings for +rank aggregation. Hay et al. (2017) proposed several differentially private rank aggregation +algorithms using Laplace noise to protect single ranking from the released output. Li et al. +(2022) embedded the Laplace, the randomized response, and the exponential mechanisms +into Condorcet voting, developing a novel family of randomized voting rules for protecting +privacy of rankings. By contrast, in the local model of DP, ranking data are privatized via +some local mechanisms before they are submitted to a curator (Yang et al., 2019; Song et al., +2022; Yan et al., 2020; Alabi et al., 2022). Yan et al. (2020) employed the Laplace noise or +the randomized response mechanisms to randomly permute pairwise comparison preferences +of ranks. Song et al. (2022) proposed to collect differentially private rankings locally for +rank aggregation, which randomly permutes rankings via adding Gaussian noise to pairwise +comparisons between items. +In this paper, we develop a novel notion called ϵ-ranking differential privacy to protect +the position of any single item in a ranking, where a smaller ϵ leads to more stringent privacy +protection. Specifically, we first establish the definition of neighboring ranking that two +rankings are viewed as neighbors if they have identical partial orders between items when one +item is removed, and a synthetic ranking algorithm should have similar output distributions +for two neighboring rankings as input. The key motivation is that existing works mainly +focus on protecting pairwise comparisons between items’ ranks or a single ranking within +a set of rankings, formalizing privacy guarantees under the classical ϵ-differential privacy, +whereas few attempts have been made to directly protect the positions of items in a ranking. +A key disadvantage of the application of the classical ϵ-differential privacy to rankings is +that, for a multidimensional object like a ranking, the privacy protection usually relies on the +composition theorem for privacy accounting (Shang et al., 2014; Jeong et al., 2022), which +3 + +suffers from a loose privacy bound and complicated downstream privacy-utility analyses. In +contrast, the proposed ϵ-ranking DP uses a single privacy parameter ϵ to measure the degree +of privacy protection for a ranking instead of relying on the composition theorem (Kairouz +et al., 2015), avoiding an inaccurate computation of privacy budget. Under the developed +ϵ-ranking DP, we are capable of analyzing how privacy guarantee ϵ for each ranking ranking +affects the performance of downstream inference tasks based on privacy-preserving rankings. +For example, the proposed ϵ-ranking DP allows for personalized privacy preferences in the +personalized ranking task that users can control the privacy preferences over their rankings. +Additionally, classical data perturbation methods like the Laplace noise addition fails to +match with the ordinal nature of ranks, implicitly leading to a suboptimal privacy-utility +tradeoff for downstream tasks. We conducted extensive experiments to validate our theoretical +results and demonstrate the effectiveness of the synthetic ranking algorithm, showing that +the synthetic ranking algorithm produces privacy-preserving rankings with more utility for +downstream personalized learning task, which implicitly results from a better privacy-utility +tradeoff. +The contributions of this paper are summarized as follows. First, we develop an synthetic +ranking algorithm based on the multistage ranking algorithm (Fligner and Verducci, 1988; +Critchlow et al., 1991) to generate privacy-preserving rankings, which is proved to be +advantageous over the linear Laplace noise addition in retaining more information of ranking +regarding pairwise comparisons under the same privacy guarantee. The proposed algorithm +essentially generates synthetic rankings via the Mallows model (Mallows, 1957), establishing +a surprising connection between the Mallows model and the ϵ-ranking DP. Second, we analyze +the utility of the synthetic rankings by two downstream tasks, including the inference attack +of a ranking and the personalized ranking task. In the inference attack, we theoretically +quantify how ϵ affects the estimation of the central ranking based on generated synthetic +ranking, providing an optimal rate of ϵ adaptive to the number of synthetic rankings for not +correctly inferring the central ranking. For the personalized ranking task, we allow users to +4 + +have personalized privacy preferences and theoretically quantify the relation between users’ +privacy preferences and the regret in estimating the optimal ranking function. Particularly, +we derive the optimal order of privacy parameter ϵ adaptive to the number of users given +the consistency in estimating the optimal ranking function. Our theoretical results show +that when all users choose an adaptive privacy guarantee at the order O +�� +n−1 log1+ζ(n) +� +for any ζ > 0 the consistency in estimating the optimal ranking function is guaranteed. +Interestingly, our theoretical result is similar to that of Duchi and Ruan (2018) in providing +similar quantitative effect of ϵ on the convergence rate of estimation under the local models +of privacy. +The rest of the paper proceeds as follows. After introducing some necessary notations in +Section 1.1, Section 2 introduces the backgrounds of differential privacy, ranking data, and +the Mallows model. Section 3 formalizes the definition of the ϵ-ranking differential privacy +and develops a synthetic ranking algorithm satisfying ϵ-ranking DP. Section 4 establishes +theoretical results concerning the inference attack of the central ranking used to generate +synthetic rankings. Section 5 considers the situation that synthetic rankings are used for +downstream personalized ranking task, for which we establish theoretical results concerning +the consistency in estimating the optimal ranking function and quantify the corresponding the +privacy-utility tradeoff. Section 6 conducts extensive experiments to verify your theoretical +results. A brief summary is provided in Section 7 and all technical proofs are provided in the +Appendix. +1.1 +Notation +For a positive integer n, denote [n] = {1, ..., n} to be the n-set. For a set S, we let |S| denote +its cardinality. For two positive sequences {fn}∞ +n=1 and {gn}∞ +n=1, we denote that fn = O(gn) +if lim supn→∞ |fn|/gn < +∞. We let fn ≍ gn if fn = O(gn) and gn = O(fn). For a random +variable Xn and a sequence {an}∞ +n=1, we denote that Xn = op(an) is Xn/an converges to +zero in probability and Xn = Op(an) if Xn/an is stochastically bounded. For an integer K, +5 + +Υ(K) denotes the set of permutations of (1, 2, . . . , K). Let I(·) be the indicator function and +I(A) = 1 if A holds true and 0 otherwise. For a vector x, we let ∥x∥2 denote its l2-norm and +∥x∥∞ denote its l∞-norm. +2 +Preliminaries +This section introduces some basic concepts relating to differential privacy, ranking data, +and the Mallows model (Mallows, 1957; Fligner and Verducci, 1986) for ranking data, which +paves the way for us to propose a novel variant of the differential privacy for ranking data. +2.1 +Differential Privacy +Differential privacy has emerged as a rigorous framework for measuring the capacity of a +randomized mechanism in privacy protection, which is reflected by bounding the discrepancy +in the output distributions when any single record of the input changes. The most popular +definition of differential privacy is (ϵ, δ)-differential privacy. +Definition 1. +� +(ϵ, δ)-differential privacy +� +Let ϵ > 0 and δ ∈ [0, 1). Let S and S′ be two sets +of records of same length and define d : S × S → Z≥0 be the Hamming distance between two +sets. We say a randomized mechanism M : S → Z satisfies (ϵ, δ)-differential privacy with +respect to d if for any S, S′ ∈ S such that d(S, S′) = 1, we have +P(M(D) ∈ Z) ≤ eϵP(M(D′) ∈ Z) + δ, +for any Z ⊂ Z. If δ = 0, then M satisfies pure ϵ-differential privacy. +The intuition behind differential privacy is that inference on any single record in a dataset +is inaccurate in the sense that the output distribution stays less affected by the change of +any single record in the input dataset. +2.2 +Ranking Data and The Mallows Model +In ranking data, a user ranks a set of items according to a specific criterion. A typical example +of ranking is the preference ranking, which assigns ordinal ranks to items according to relative +6 + +preferences. Let Ω = {I1, . . . , Im} denote a set of m items and φΩ denote a ranking of items +in Ω. The ranking φΩ usually appears as an ordered list indicating the positions of items +in Ω under a specific metric, i.e., φΩ(Ii) = k means that the rank of item Ii is k. Without +loss of generality, we let the item with rank 1 refer to the most preferred item, and hence +items with higher ranks are less preferred in our setting. For example, a preference ranking +φΩ = (φΩ(I1), φΩ(I2), φΩ(I3)) = (3, 2, 1) indicates that item 3 is the most preferred item and +item 1 is the least preferred one. For ease of notation, φΩ will be abbreviated as φ in the +sequel when it causes no confusion. +Let Φ denote the random variable of the observed ranking φ, and Φ is a multi-variate +random variable taking values in Υ(|Ω|), where Υ(|Ω|) denotes the set of all permutations of +(1, . . . , |Ω|). Naturally, we suppose that P(Φ(Ii) > Φ(Ij)) > P(Φ(Ij) > Φ(Ii)) if and only if +the item Ij is better than the item Ii in quality. Here the randomness of P(Φ(Ii) > Φ(Ij)) +comes from the randomness of the observed ranking. This is a mild assumption that is +fulfilled for various ranking models, such as the random utility model (Walker and Ben-Akiva, +2002; Soufiani et al., 2014; Su, 2021) and the Mallows model (Mallows, 1957; Pearce and +Erosheva, 2022). +The Mallows model (Mallows, 1957) is a popular parametric model to model ranking +data. Specifically, the Mallows model with Kendall-τ distance (Fligner and Verducci, 1986; +Mandhani and Meila, 2009) generates synthetic rankings based on φ0. Specifically, a ranking φ +will be generated with a higher probability if φ is more aligned with φ0 in pairwise comparisons +of items’ ranks. The generation of rankings follows the distribution as +Pθ,φ0(φ) = +1 +Ψ(θ) exp +� +θT(φ, φ0) +� +, for any φ ∈ Υ(|Ω|), +where φ0 is the central ranking over the item set Ω, Ψ(θ) = � +φ∈Υ(|Ω|) exp +� +θT(φ, φ0) +� +, θ is +the dispersion parameter, and T(φ, φ0) = � +Ii,Ij∈Ω I +�� +φ(Ii) − φ(Ij) +�� +φ0(Ii) − φ0(Ij) +� +> 0 +� +can be viewed the number of concordant pairs, characterizing the difference between φ and +φ0. The Mallows model is widely used to model the distribution of preference rankings (D´esir +et al., 2016; Busa-Fekete et al., 2014), and φ0 usually refers to the true ranking of items. For +7 + +example, in recommender systems, φ0 refers to the true ranking of a set of items in quality, +and φ is the observed ranking by a specific user. +3 +Ranking Differential Privacy +Users’ ranking data are extremely sensitive in revealing their preferences and behaviors, +which are potentially interesting for marketing purposes (Jeckmans et al., 2013). Therefore, +it is of great importance to protect privacy of preference rankings before sharing them to an +external party. In this section, we propose the ϵ-ranking differential privacy. Subsequently, +we establish a connection between ϵ-ranking DP and the Mallows model, which can be +utilized to generate privacy-preserving rankings satisfying the proposed ϵ-ranking DP. The +connection is essentially derived from an inherent connection between the Mallows model +and the exponential mechanism of differential privacy (McSherry and Talwar, 2007; Lantz +et al., 2015). +Ranks of items indicate their positions among the set of items and implicitly reveal their +partial orders. Intuitively, in order to protect a ranking, the ranks of items should be noisy +in accordance with the nature of differential privacy. To this end, we propose a new variant +of neighboring set for rankings. +Definition 2. (Neighboring Ranking) Let φ and φ′ be two ordinal rankings on the same item +set Ω. We say φ and φ′ are neighboring rankings if there exists only one Ik ∈ Ω such that for +any Ii, Ij ∈ Ω \ {Ik} with i ̸= j +� +φ(Ii) − φ(Ij) +�� +φ′(Ii) − φ′(Ij) +� +> 0. +The main idea of neighboring ranking is that two ordinal rankings are consistent in all +partial orders of ranks of items except a specific item. Next, we propose our ϵ-ranking +differential privacy. +Definition 3. (ϵ-ranking Differential Privacy; ϵ-ranking DP) Let φ and φ′ be two neighboring +ordinal rankings on the item set Ω and M(φ) be a synthetic ranking algorithm producing +8 + +ordinal rankings for users. We say M satisfies ϵ-ranking differential privacy if +sup +�φ∈Υ(|Ω|) +��� log P(M(φ) = �φ) +P(M(φ′) = �φ) +��� ≤ ϵ, +where the randomness comes from the mechanism M. +The rationale of ϵ-ranking DP is that the output distributions of M(φ) and M(φ′) are +similar in the sense that inference on the ordinal rank of a single item based on output +synthetic rankings is statistically inaccurate. In other words, ϵ-ranking DP protects any +single rank within a ranking by ensuring that the change of any single rank in the ranking +produces little effect on the output distribution of rankings. The proposed ϵ-ranking DP uses +a single privacy parameter ϵ to measure the degree of privacy protection. +3.1 +Privacy-Preserving Rankings +In this section, we establish a connection between the Mallows model (Mallows, 1957; Fligner +and Verducci, 1986) and the proposed ϵ-ranking DP, which is derived from the fact that the +Mallows model belongs to the exponential family. Based on this connection, we propose +a synthetic ranking algorithm to generate privacy-preserving rankings, which is developed +based on the multistage ranking model (Fligner and Verducci, 1988; Critchlow et al., 1991). +Lemma 1. Let φ and φ′ be two neighboring rankings. The Mallows model satisfies that +�� log +� Pθ,φ(�φ) +Pθ,φ′(�φ) +��� ≤ ϵ for any �φ ∈ Υ(|Ω|) given that θ = 2−1ϵ|Ω|. +Lemma 1 demonstrates a connection between the Mallows model and the ϵ-ranking DP. +Specifically, under an appropriate choice of the dispersion parameter θ = 2−1ϵ|Ω|, using the +Mallows model to generate synthetic rankings satisfies the proposed ϵ-ranking DP. In other +words, protecting an observed ranking φ can be achieved by generating a counterpart ranking +�φ via the Mallows model with φ being the input ranking, where the privacy guarantee is +guarded under the developed ϵ-ranking DP. +To generate privacy-preserving rankings via the Mallows model, we propose an algorithm +based on the idea of the multistage ranking model (Fligner and Verducci, 1988; Critchlow et al., +9 + +Figure 1: The t-th iteration of the proposed synthetic ranking algorithm. +1991). The overall idea of the developed algorithm can be viewed as a shuffling mechanism. +To be more specific, the algorithm sequentially determines the ranks of items according to +their ranks in φ. As illustrated in Figure 1, the position of the item Ij in the synthetic +ranking �φ is determined by the sampling procedure with probabilities p(t) +i , i = 1, . . . , t − 1. +Here it should be noted that p(t) +i +> p(t) +j +for any i > j since position i is more aligned with the +position of item Ij in φ. The overall algorithm is summarized in Algorithm 1. +Algorithm 1 takes the ranking φ as an input and outputs a synthetic ranking �φ. In +Algorithm 1, the synthetic ranks of items are determined sequentially in a probabilistic +manner such that positions with more similar partial orders to the input ranking are more +likely to be chosen. Specifically, at the initialization step, the item with rank 1 in φ (denoted +as Ii) is chosen first, and its rank in �φ is set as 1 temporarily. Then the ranks of other items +in the synthetic ranking �φ are determined in an increasing order of their ranks of φ. At the +t-th iteration, the objective is to determine the relative position of the item with rank t to +the first t − 1 items whose relative positions are already specified and their item indexes are +stored in χ(t−1). Since positions are chosen in a probabilistic manner, we let V (t) +ϵ +denote the +random variable for the position taking values in ρ(t), and each element k ∈ ρ(t) corresponds +to a relative position. For each position k, τ(k, χ(t−1)) represents the number of concordant +10 + +§(Ii) = 1 +Φ(Ir) = 2 +(Iit-) = t - 1 +Item L +(t) +(t) +(t) +(t) +(t) +Probability: +Po +pi +P2 +Pt-2 +Pt-1 +Item L; +The ordinal rank of item L, is tAlgorithm 1: Privacy Preserving Ranking Algorithm +Input +: The ranking φ of the item set Ω and privacy guarantee ϵ +1 Initialization: Let φ−1 be the inverse function of ranking and choose the item Ii +satisfying φ−1(1) = Ii from Ω and set �φ(Ii) = 1 and χ(1) = {i}; +2 for t = 2, . . . , |Ω| do +3 +Select the item Ij from Ω such that the rank of Ij is t and define +ρ(t) = {0, . . . , t − 1}. +4 +Compute values +τ(k, χ(t−1)) = +� +l∈χ(t−1) +I +�� +t − φ(Il) +�� +k + 0.5 − �φ(Il) +� +> 0 +� +, k ∈ ρ(t). +5 +Sample a value k from ρ(t) and assign it to V (t) +ϵ +according to the probability as +p(t) +k = P(V (t) +ϵ += k) = +exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t−1)) +� +� +k∈ρ(t) exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t−1)) +�, k ∈ ρ(t). +6 +Define G(t)(V (t) +ϵ ) = {i : �φ(Ii) > V (t) +ϵ ++ 0.5, i ∈ χ(t−1)} and rearrange the ranking as +�φ(Ii) ← �φ(Ii) + 1, for i ∈ G(t)(V (t) +ϵ ), +�φ(Ij) ← V (t) +ϵ ++ 1. +7 +χ(t) ← χ(t−1) ∪ {j} +8 end +9 Output: Output synthetic ranking �φ. +pairs at position k, and a larger τ(k, χ(t−1)) indicates a higher probability p(t) +k +that position k +will be chosen. After V (t) +ϵ +is determined, the synthetic ranks of first t items will be rearranged +as demonstrated in step 6. The overall computational complexity of Algorithm 1 is O(|Ω|2) +Lemma 2. Let Aϵ be Algorithm 1. Given a ranking φ of the item set Ω = {I1, . . . , Im}, the +synthetic ranking Aϵ(φ) follows the following distribution, +P(Aϵ(φ) = �φ) = +exp +� +2−1ϵ|Ω|T(φ, �φ) +� +� +�φ∈Υ(|Ω|) exp +� +2−1ϵ|Ω|T(φ, �φ +�, for any �φ ∈ Υ(|Ω|), +where T(φ, �φ) = � +Ii,Ij∈Ω I +�� +φ(Ii) − φ(Ij) +���φ(Ii) − �φ(Ij) +� +> 0 +� +. +11 + +Lemma 2 shows that the synthetic ranking algorithm is essentially equivalent to sampling +from the Mallows model, and the basic idea is employing importance sampling to generate +synthetic ranking in that a ranking with larger value of T(φ, �φ) is generated with a larger +probability, providing a bridge to establish privacy parameter ϵ. +Theorem 1. Algorithm 1 possesses the following properties: +(1) Consistency in ranking optimality: For any Ii, Ij ∈ Ω with i ̸= j, it holds true that +� +P +� +Φ(Ii) > Φ(Ij) +� +− P +� +Φ(Ii) < Φ(Ij) +��� +P +��Φ(Ii) > �Φ(Ij) +� +− P +��Φ(Ii) < �Φ(Ij) +�� +> 0, +where �Φ = Aϵ(Φ) denotes the random synthetic ranking output by Algorithm 1 with Φ +being input ranking. +(2) Effect on ranking distribution: For any Ii, Ij ∈ Ω with Ii ̸= Ij, it holds true that +exp +� +(|Ω| − 1)−1ϵ +� +− 1 +exp +� +(|Ω| − 1)−1ϵ +� ++ 1 ≤ |2�ηij − 1| +|2ηij − 1| ≤ +exp +� 2|Ω|−3 +|Ω|−1 ϵ +� +− 1 +exp +� 2|Ω|−3 +|Ω|−1 ϵ +� ++ 1 +, +where ηij = P +� +Φ(Ii) > Φ(Ij) +� +and �ηij = P +��Φ(Ii) > �Φ(Ij) +� +. +(3) At the iteration t, the expectation and variance of V (t) +ϵ +are given as +E +� +V (t) +ϵ +� += (t − 1)qt +ϵ +qt +ϵ − 1 +− +qt +ϵ − qϵ +(qϵ − 1)(qt +ϵ − 1) +ϵ→+∞ +−−−−→ t − 1 +Var +� +V (t) +ϵ +� += (t − 1)2qt +ϵ +qt +ϵ − 1 +− 2E +� +V (t) +ϵ +� +qϵ − 1 ++ +qt +ϵ − qϵ +(qt +ϵ − 1)(qϵ − 1) − +� +E +� +V (t) +ϵ +��2 +ϵ→+∞ +−−−−→ 0, +where qϵ = exp +� +ϵ(|Ω| − 1)−1� +. +In Theorem 1, we present several properties of the proposed synthetic ranking algorithm. +To some extend, the properties in Theorem 1 establishes quantitative effect of the Mallows +model on the distribution of ranking. Specifically, property (1) shows that the partial order +between any two items’ ranks stays invariant at the population level after being processed +by Algorithm 1, which indicates that Ii is more likely to have a higher rank than Ij in a +synthetic ranking if this relation also holds for Φ. This property also implicitly permits the +12 + +invariance of the true ranking based on items’ quality values. Property (2) characterizes the +effect of the privacy parameter ϵ on the distribution of ranking. As ϵ decreases to zero, �ηij +gets closer to 1/2, which means that the partial order between Ii and Ij is indistinguishable. +Property (3) characterizes the relation between ϵ and V (t) +ϵ , where V (t) +ϵ +indicates the number of +correct partial relations retained in the synthetic ranking in the t-th iteration. As ϵ increases +to infinity, the distribution of V (t) +ϵ +converges to the constant t− 1 as expected. This is natural +since ϵ = ∞ refers to the non-private case as in the definition of differential privacy, and in +this case the output ranking �φ is identical to the input ranking φ. +3.2 +Comparison to Laplace Noise +In this section, we demonstrate the effectiveness of the synthetic ranking algorithm in +comparison with the Laplace mechanism (Dwork, 2006) under the developed ϵ-ranking +differential privacy. In the domain of differential privacy, the Laplace mechanism has emerged +as a popular technique to ensure privacy protection for numeric data due to its computational +simplicity. For releasing a ranking φ of an item set Ω = {I1, . . . , Im} in a secure manner, the +Laplace mechanism adds element-wise noises to ranks of items, which is given as +Mlap +λ (φ) = +� +φ(Ii) + ξi +� +i∈[m], +where ξi are i.i.d. random samples drawn from the Laplace distribution with mean 0 and +scale λ. +Lemma 3. Let φ and φ′ be two neighboring rankings on the item set Ω = {I1, . . . , Im}. +Given that λ = 2(m − 1)ϵ−1, it holds that +��� log P(Mlap +λ (φ) = r) +P(Mlap +λ (φ′) = r) +��� ≤ ϵ, +for any r ∈ Rm. +Lemma 3 shows that adding Laplace noise to ranks of items also achieves the proposed +ϵ-ranking differential privacy with a properly chosen scale λ = 2(m − 1)ϵ−1. +13 + +Lemma 4. Let Aϵ and Mlap +2(m−1)ϵ−1 denote Algorithm 1 and the Laplace mechanism satisfying +ϵ-ranking differential privacy, respectively. For any ranking φ of length m, it holds true that +for any ϵ > 0 +E +� +T +� +φ, Aϵ(φ) +�� +> E +� +T +� +φ, Mlap +2(m−1)ϵ−1(φ) +�� +, +where T(φ, �φ) = � +Ii,Ij∈Ω I +�� +φ(Ii) − φ(Ij) +���φ(Ii) − �φ(Ij) +� +> 0 +� +. +For two rankings φ and �φ, T(φ, �φ) calculates the number of concordant pairs between +φ and �φ. Clearly, T(φ, �φ) attains the maximum value when φ and �φ have identical partial +orders between items. In Lemma 4, we use the expected number of concordant pairs to +measure the utility of the proposed synthetic ranking algorithm and the Laplace mechanism +satisfying the same ϵ-ranking DP, showing that even though the Laplace mechanism can +achieve the same privacy guarantee, it is is less effective than our algorithm in preserving +relative positions of items in the raw ranking. This is because the framework of differential +privacy imposes privacy guarantee on top of the worst case. +(a) m=4 +(b) m=5 +(c) m=6 +Figure 2: These figures present the averaged T +� +φ, Aϵ(φ) +� +(red) and T +� +φ, Mlap +2(m−1)ϵ−1(φ) +� +(blue) in 20,000 replications with ϵ ∈ [0.1, 30], where φ is a ranking on the item set Ω = +{I1, . . . , Im} with φ(Ii) = i, i = 1, . . . , m. +To further validate theoretical results in Lemma 4, we carry out a simple experiment to +illustrate the improvement of the proposed synthetic ranking algorithm relative to the Laplace +mechanism. As can be seen in Figure 2, the proposed algorithm produces synthetic rankings +preserving more partial orders among items in all cases. Particularly, the improvement +14 + +1.0 +Expected number of concordant pairs +0.9 +0.8 +0.7 +0.6 +Synthetic Ranking Algorithm +Laplace Noise +0.5 +5 +10 +15 +20 +25 +30 +w1.0 +Expected number of concordant pairs +0.9 +0.8 +0.7 +0.6 +Synthetic Ranking Algorithm +0.5 +LaplaceNoise +0 +5 +10 +15 +20 +25 +301.0 +Expected number of concordant pairs +0.9 +0.8 +0.7 +0.6 +Synthetic Ranking Algorithm +0.5 +LaplaceNoise +0 +5 +10 +15 +20 +25 +30becomes more significant as the size of ranking m increases, showing that the proposed +synthetic ranking algorithm is highly competitive when applied to a large ranking size. +4 +Inference Attack of Ranking +A critical challenge in privacy protection is against inference attacks (Williams and McSherry, +2010; Sun et al., 2018; Wu et al., 2022), which aims to recover some sensitive information +contained in the true dataset based on the released data. For ranking data, obtaining rankings +of items is usually of interest to attackers. For example, preference rankings revealing users’ +preferences can be used for advertisement purpose (Mayer and Mitchell, 2012; Chen and +Stallaert, 2014). +In this section, we intend to study the effectiveness of the proposed synthetic ranking +algorithm in resisting inference attacks based on synthetic rankings. Specifically, as illustrated +in Figure 3, we assume that attackers can interact with the synthetic ranking algorithm +frequently, requesting multiple synthetic rankings of the same central ranking φ0. In addition, +attackers are assumed to have full knowledge of the Mallows model and privacy parameter ϵ +and intend to estimate the observed ranking. +Figure 3: The framework of inference attack of rankings. +Let S = {�φi}N +i=1 denote a set of synthetic rankings generated by Algorithm 1 with the +input ranking φ0 on the item set Ω. We first consider a simple case that privacy parameter +15 + +d. +A(Φo) +Attacker: +Φmle +中oϵ is known. Hence, it remains to estimate the observed ranking φ0 for attackers. The +log-likelihood function of S can be written as +L(φ) = N log +� +� +�φ∈Υ(|Ω|) +exp +� +2−1ϵ|Ω|T(φ, �φ +��� ++ ϵ|Ω| +2 +N +� +i=1 +T(φ, �φi). +Let �φmle denote the maximum likelihood estimation (MLE) of φ0, which is defined as +�φmle = argmin +φ∈Υ(|Ω|) +L(φ). +(1) +The consistency of �φmle to φ0 for any fixed ϵ is guaranteed by general results in Choirat +and Seri (2012), which proves the consistency of M-estimators in discrete parameter models. +In practice, the estimator �φmle is difficult to obtain, since the feasible region of �φmle is a +discrete space. Therefore, the problem of finding �φmle is known to be NP-hard (Meil˘a and +Bao, 2010; Young, 1986). This explicitly demonstrates the merits of the proposed ranking +algorithm in avoiding estimation of the observed ranking φ0 from a computational perspective. +Specifically, it is computationally heavy for attackers to obtain the correct ranking via MLE, +even though full knowledge of the synthetic ranking algorithm and the privacy parameter ϵ +are provided. +Theorem 2. Let φ0 denote the observed ranking and S = {�φi}N +i=1 denote a set of synthetic +rankings generated by Aϵ(φ0). For any ϵ > 0 and Ω, there exists a positive constant C0 such +that that +PS +� +�φmle ̸= φ0 +� +≥ C0 +exp(ϵ(|Ω| − 1)−1) + 1 +exp(ϵ(|Ω| − 1)−1) − 1 +� +2 +πN exp +� +− N +� +exp(ϵ(|Ω| − 1)−1) − 1 +�2� +. +Furthermore, PS +��φmle ̸= φ0 +� +is bounded away from 0 if ϵ = O +� +(|Ω| − 1) +√ +N −1� +. +Theorem 2 provides a lower bound for the estimation error of �φmle, quantifying the effect +of privacy parameter ϵ on the convergence of �φmle. The implication of Theorem 2 is two-fold. +First, the lower bound enlarges when the length of ranking |Ω| increases, showing that the +length of ranking amplifies the difficulty of the estimation of φ0. Additionally, setting ϵ to be +16 + +adaptive to the number of synthetic rankings at the order ϵ = O +� +(|Ω| − 1) +√ +N −1� +leads to +the inconsistency of �φmle, where attackers estimate φ0 inaccurately with a fixed probability, +regardless of the number of synthetic rankings requested. +5 +Ranking Differential Privacy in Personalized Rank- +ing +This section is devoted to exploring the application of ϵ-ranking DP to personalized ranking +and establishing theoretical results regarding the quantitative relation between ϵ-ranking DP +and the consistency of the estimated ranking function based on synthetic rankings. +5.1 +Personalized Ranking +A typical dataset in ranking problems consists of a set of triples (xu, yi, φu(Ii)), where +xu ∈ Rp denotes the p-dimensional feature vector of user or query u, yi ∈ Rq denotes the +q-dimensional feature vector of item i, and φu(Ii) represents the preference or relevance rank +from user u to item i. We suppose that the item set is fixed, and the objective of personalized +ranking problem (Balakrishnan and Chopra, 2012) is to select most preferred or relevant +items for a new user according to his/her preference or relevance. +To estimate the ranking function, it is a common practice to employ the pairwise ranking +loss (Rendle et al., 2012; Balakrishnan and Chopra, 2012). +Rn(f) = +1 +nm(m − 1) +n +� +u=1 +� +i̸=j +I +� +φu(Ii) > φu(Ij) +� +I +� +f(xu, yi) ≤ f(xu, yj) +� +. +(2) +As can be seen in (2), error occurs when f disagrees with two observed ranks in relative +orders, and hence minimizing (2) stimulates f to be consistent with observed ranks in ranking +as much as possible. +Let φu denote a realization of Φu and ηuij = P +� +Φu(Ii) > Φu(Ij) +� +denote the probability +that user u gives higher rank on Ii than Ij. The ranking risk can be written as +R(f) = +1 +m(m − 1) +� +i φu(Ij) +� +υ +� +f(xu, yi) − f(xu, yj) +� +, +(4) +where υ denote a surrogate loss function, such as logistic loss υ(x) = log(1+exp(−x)) (Rendle +et al., 2012; Zhu and Hastie, 2005), exponential loss υ(x) = exp(−x) (Schapire, 2003), and +ψ-loss υ(x) = min(1, (1 − x)+) (Shen et al., 2003; Dai et al., 2021). It should be noted that υ +cannot be hinge loss since the optimal ranking function under the hinge loss is not attainable +in general (Gao and Zhou, 2015; Dai et al., 2021). +5.2 +Differentially Private Personalized Ranking +To protect the observed rankings of users {φu}n +u=1, the proposed synthetic ranking algorithm +can be utilized to generate synthetic rankings {�φu}n +u=1 for the subsequent personalized +18 + +learning. Let �φu = Aϵu(φu) denote the synthetic ranking generated by Algorithm 1 satisfying +the ϵu-ranking DP for user u, where ϵu denotes the personalized privacy preference of user u. +Then, the differentially private pair-wise ranking task can be organized as +�Rn,υ(f) = +1 +nm(m − 1) +n +� +u=1 +� +i̸=j +I +� +�φu(Ii) > �φu(Ij) +� +υ +� +f(xu, yi) − f(xu, yj) +� +. +(5) +Let F denote the class of ranking functions of interest. The estimated ranking function �f is +obtained as +�f = argmin +f∈F +�Rn,υ(f) + λnJ(f), +(6) +where λn is a tuning parameter vanishing with n and J(·) is a regularization term. +Denote by �Φu = Aϵu +� +Φu) the random variable of synthetic ranking of user u. +Let +�ηuij = P +��Φu(Ii) > �Φu(Ij) +� +denote the probability that the user u gives a higher rank to Ii +than Ij in the synthetic ranking �φu. Correspondingly, the ranking risk of synthetic rankings +can be written as +�R(f) = +1 +m(m − 1) +� +i 0, it holds that +� +f ∗(xu, yi) − f ∗(xu, yj) +�� �f ∗(xu, yi) − �f ∗(xu, yj) +� +> 0, +for any user u and i, j ∈ [m]. Particularly, when ϵu = ϵ for any u, it holds true that +� +R(f) − R∗�exp((|Ω| − 1)−1ϵ) − 1 +exp((|Ω| − 1)−1ϵ) + 1 ≤ �R(f) − �R∗ ≤ +� +R(f) − R∗�exp +� 2|Ω|−3 +|Ω|−1 ϵ +� +− 1 +exp +� 2|Ω|−3 +|Ω|−1 ϵ +� ++ 1 +. +(7) +Lemma 6 shows that �f ∗ is consistent with f ∗ in ranking items under any privacy guarantee +ϵ, demonstrating there is no bias in optimality of using synthetic rankings for the personalized +learning. Furthermore, (7) establishes the quantitative relation between the excess risks +19 + +under the raw ranking and the synthetic ranking distributions, from which we can derive the +convergence rate of R( �f) − R∗ from that of �R( �f) − �R∗. By the upper bound in (7), we can +see that �R(f) − �R∗ tends to 0 for any f if ϵ goes to 0. This is as expected since ϵ = 0 refers +to the most private case, where synthetic ranks of items are randomly determined, and all +ranking functions degenerate simultaneously. +5.3 +Consistency in Differentially Private Personalized Ranking +In this section, we establish theoretical results regarding to the asymptotic behavior of �f, +quantifying how ϵ-ranking DP affects the convergence rate of R( �f) − R∗. We first define the +ranking υ-risk with respect to synthetic rankings as +�Rυ(f) = +1 +m(m − 1) +� +i̸=j +E +� +I +��Φu(Ii) − �Φu(Ii) > 0 +� +υ +� +f(xu, yi) − f(xu, yj) +�� +, +where the expectation is taken with respect to the randomness from users and the synthetic +ranking algorithm. We denote by �f ∗ +υ = argminf �Rυ(f) the optimal ranking funtion under +υ-loss. It is important to note that �f ∗ is not unique, and �f ∗ +υ can also be the optimal ranking +function in minimizing �R(f) when υ(·) is properly chosen, such as logistic loss (Gao and +Zhou, 2015), exponential loss (Gao and Zhou, 2015), and ψ-loss (Dai et al., 2021). +In this paper, we only consider those loss functions such that �R( �f ∗ +υ) = �R( �f ∗), and �f ∗ +will be referred to as �f ∗ +υ in the sequel. We denote e( �f, �f ∗) = �R( �f) − �R( �f ∗) and eυ( �f, �f ∗) = +�Rυ( �f) − �Rυ( �f ∗) as its excess risk under 0-1 loss and υ-loss, respectively. To quantify how +ϵ-ranking DP affects the excess risk, we first derive the convergence rate of e( �f, �f ∗) from that +eυ( �f, �f ∗), which combined with (7) characterizes the asymptotic behavior of R( �f) − R∗. The +derived convergence behavior of e( �f, �f ∗) from eυ( �f, �f ∗) is known to depend on the surrogate +loss υ (Zhang, 2004; Gao and Zhou, 2015). Before introducing the main theory, we first list +some assumptions. +Assumption 1. Let Fδ = {f ∈ F : eυ(f, �f ∗) ≤ δ} denote a subset of F such that the excess +υ-risk of any f ∈ Fδ is smaller than δ. Assume that there exist a constant α > 0 and a +sufficiently small constant δ > 0 such that supf∈Fδ e(f, �f ∗) ≤ δα. +20 + +Assumption 1 establishes a conversion relationship between eυ(f, �f ∗) and e(f, �f ∗), which +influences the asymptotic behavior of e(f, �f ∗). Specifically, α depends on the choice of loss +function, and particularly α = 1 for ψ-loss (Dai et al., 2021) and α = 1/2 for exponential +loss and logistic loss (Zhang, 2004; Bartlett et al., 2006; Gao and Zhou, 2015). Additionally, +Assumption 1 indicates that the convergence rate of e( �f, �f ∗) can be derived from that of +eυ( �f, �f ∗). Let �f ∗ +F = argminf∈F �Rυ(f) be the best ranking function in F in approximating �f ∗. +The excess risk eυ( �f, �f ∗) admits the decomposition as +eυ( �f, �f ∗) = eυ( �f, �f ∗ +F) + e( �f ∗ +F, �f ∗) = �Rυ( �f) − �Rυ( �f ∗ +F) + �Rυ( �f ∗ +F) − �Rυ( �f ∗), +where eυ( �f, �f ∗ +F) and eυ( �f ∗ +F, �f ∗) are usually referred to as estimation error and approximation +error, respectively. +In this paper, we mainly focus on the estimation error in the consistency of �f. Therefore, +in Assumption 2, we assume that the class of ranking functions is correctly specified and the +approximation error is ignorable. +Assumption 2. The class of ranking functions F is properly chosen such that eυ( �f ∗ +F, �f ∗) = 0. +Assumption 3. We assume that each user u has personal privacy preference ϵu and Ψu = +exp +� +(|Ω|−1)−1ϵu +� ++1 +exp +� +(|Ω|−1)−1ϵu +� +−1 is a sub-Gaussian random variable. +Assumption 3 considers varying privacy preference among users, which generally holds +true in real-life applications (Watson et al., 2015). Additionally, Assumption 3 imposes a +constraint on the distribution of users’ privacy preference that Ψu is sub-Gaussian distribution, +which implies that fewer users pursue stronger privacy protection on their rankings. +Assumption 4. For any user u, we assume that there exists some constants C2 > 0 and +0 < γ ≤ +∞ such that P +� +|2ηuij − 1| ≤ β +� +≤ C2βγ for 0 ≤ β ≤ 1. +Assumption 4 is known as the low-noise assumption (Bartlett et al., 2006; Shen et al., +2003) characterizing the behavior of 2ηuij − 1 around the decision boundary, which affects the +asymptotic behavior of �f. Particularly, γ = +∞ implies that the partial orders of preferences +are deterministic, resulting in optimal convergence rate of e( �f, �f ∗). +21 + +Lemma 7. Under Assumption 4, it holds that for any user u, P +� +|2�ηuij − 1| ≤ β +� +≤ C2Ψγ +uβγ, +where Ψu = +exp +� +(|Ω|−1)−1ϵu +� ++1 +exp +� +(|Ω|−1)−1ϵu +� +−1. +Assumption 5. Let Θ denote all parameters of functions in F. We assume that ∥Θ∥∞ ≤ CF +and there exists some positive constants C1 such that for any f1, f2 ∈ F +� +f1(xu, yi) − f2(xu, yi) +�2 ≤ C1 +� +∥xu∥2 +2 + ∥yi∥2 +2 +� +∥Θf1 − Θf2∥2 +∞, +where Θf denote the parameters of f. +Assumption 5 is a mild assumption, ensuring the smoothness of functions in F with +respect to input features and parameters, establishing the connection between the metric +entropy of F and the associated parameter space. +Theorem 3. Under Assumptions 1-5, it holds that for any minimizer �f defined in (6), there +exist some constants C3, C4 > 0 such that +P +� +e( �f, f ∗) ≥ δα +n +� +≤ 8C3 exp +� +− C4nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� +, +where E(Ψγ +u) = O(δn), E +1 +γ+2(Ψγ +u) +� +|Θ|n−1 log(n/|Θ|) +� γ+1 +γ+2 = O(δn), and |Θ| is the number of +parameters of f ∈ F and λnJ0 ≍ δn with J0 = max{J(f ∗ +F), 1}. +Theorem 3 quantifies the asymptotic behavior of e( �f, f ∗), as well as its relation to the +privacy guarantee. Specifically, the convergence rate δα +n is governed by γ, α, averaged privacy +preference E(Ψγ +u), and the complexity of F. Particularly, when α = 1 and γ = ϵ = ∞, the best +convergence rate is obtained as Op +� +|Θ|n−1 log(n/|Θ|) +� +, which matches the existing theoretical +results in Dai et al. (2021). Furthermore, it is interesting to note that, when E−1(Ψγ +u) = o(1) +, the rate δn gets slower compared with that of non-private case. For example, when ϵu = ϵ +for all users, E−1(Ψγ +u) is of the order ϵγ with ϵ = o(1), indicating that the convergence to the +optimal ranking function is slower when all users pursue more protection of their rankings +corresponding to a smaller ϵ. +22 + +Corollary 1. Under assumptions of Theorem 3, we have +R( �f) − R(f ∗) = Op +� +E(Ψu)E +α +γ+2(Ψγ +u) +� +|Θ|n−1 log(n/|Θ|) +� α(γ+1) +γ+2 � +. +Particularly, when ϵu = ϵ = o(1) for any user u and υ(·) is chosen such that α = 1, we have +R( �f) − R(f ∗) = Op +�� +|Θ|(|Ω| − 1)2n−1ϵ−2 log(n/|Θ|) +� γ+1 +γ+2� +. +Furthermore, we have R( �f) − R(f ∗) = op(1) if |Ω| +� +|Θ|n−1 log1+ζ(n/|Θ|) = O(ϵ) for some +ζ > 0. +Corollary 1 presents the convergence rate of excess risk of differentially private ranking +function �f, quantifying the effect of synthetic ranking algorithm. First, if all users choose a +constant privacy, the convergence rate becomes slower by a multiplicative constant compared +with the non-private case. Corollary 1 also sheds light on the best rate of privacy guarantee +achievable for all users under the consistency of pairwise learning. Specifically, when the +privacy guarantee ϵ of all users satisfies |Ω| +� +|Θ|n−1 log1+ζ(n/|Θ|) = O(ϵ) for some ζ > 0 for +any ζ > 0, pairwise learning can still estimate the optimal ranking function f ∗ well with +appropriately chosen parameter space F and loss function υ(·). +6 +Experiment +In this section, we conduct a series of numerical experiments on simulated datasets to validate +our theoretical results and demonstrate the effectiveness of the proposed algorithm. +6.1 +Privacy Guarantee +In the first simulation, we aim to provide empirical validations of the privacy guarantee of +the proposed synthetic ranking algorithm as stated in Lemma 1. To this end, we consider +a ranking σ with size m such that φ(Ii) = i for i = 1, . . . , m. Let S denote the set of all +possible neighboring rankings of φ and C denote the set of all possible permutations of φ. +For each φ′ ∈ S, we implement the proposed algorithm with privacy guarantee ϵ on φ and φ′ +23 + +in N times and then estimate the privacy guarantee by +�ϵ = max +φ′∈S max +�φ∈C +��� log +� �N +i=1 I(Aϵ(φ) = �φ) +�N +i=1 I(Aϵ(φ′) = �φ) +����, +(8) +where Aϵ(φ) denote the output ranking of the algorithm in the i-th replication. We consider +cases that (m, ϵ) ∈ {3, 4, 5} × {0.5 + 0.25 ∗ i, i = 0, . . . , 6} with N = 107 and the results are +reported in Figure 4. +(a) m=3 +(b) m=4 +(c) m=5 +Figure 4: Estimated privacy guarantee �ϵ v.s. the pre-specified privact guarantee +As can be seen in Figure 4, the estimated privacy guarantee �ϵ perfectly matches the +pre-specified privacy guarantee ϵ, which is consistent with our theoretical results established +in Lemma 1. +6.2 +Inference Attack +This simulation intends to verify theoretical results in Theorem 2 that an adaptive scheme of +ϵ regarding the number of synthetic rankings results in the deterioration of the estimation of +ranking. To this end, the simulation setting is organized as follows. First, for a ranking φ0 of +size m, we generate a set of synthetic rankings Sm = {�φi}N +i=1. Second, the estimation of φ0 +is implemented via maximum likelihood estimation based on Sm as in (1). We repeat the +above steps in R replications. Let �φ(i) +mle denote the maximum likelihood estimator in the i-th +replication. Then we estimate the probability of the inconsistency of MLE as +�P(�φmle ̸= φ0) = 1 +R +R +� +i=1 +I(�φ(i) +mle ̸= φ0). +24 + +1.75 +1.50 + �φu(Ij) +� +υ(f(xu, yi) − f(xu, yj)), +(9) +�flap = argmin +f∈F +1 +n +n +� +i=1 +� +i̸=j +I(�φlap +u +� +Ii) > �φlap +u (Ij) +� +υ(f(xu, yi) − f(xu, yj)), +(10) +where F is the class of two-tower models (Wang et al., 2021; Yang et al., 2020), which +employs two parallel neural networks to learn representations of users’ and items’ features +and computes preference scores by dot product of their representations. +Furthermore, to compare the performance of �flap and �f, we generate a set of new users +with their features following same setting as above. Specifically, we let {x′ +u}N +u=1 denote the +set of features of new users and generate their preferences as r′ +ui = αTx′ +u + βTyi. For a +ranking function f, we employ the following metric to evaluate its performance, +Lpair(f) = +1 +Nm(m − 1) +N +� +u=1 +� +i̸=j +I +� +r′ +ui > r′ +uj +� +I +� +f(x′ +u, i) > f(x′ +u, j) +� +. +(11) +For the hyper-parameter selection, we set two neural networks in the two-tower model to +be a 3-layer multilayer perceptron with 10 hidden units in each layer. We set N = 1, 000 and +27 + +consider cases that (m, n, ϵ) ∈ {15, 30} × {200, 250, 300, 350} × {1, 2, 3, 4}. In each case, 100 +users will be used for validation and the early-stopping method monitoring the validation +error is employed to tune the parameters of neural networks. We repeat each case in 50 runs +and report the averaged pairwise testing accuracies in Figure 7. +(a) ϵ = 1 +(b) ϵ = 2 +(c) ϵ = 3 +(d) ϵ = 4 +Figure 7: The averaged pairwise accuracies in 50 replications with different privacy guarantees +ϵ = 1, 2, 3, 4 and lengths of rankings m = 15, 30. +As seen in Figure 7, the averaged testing accuracies of the proposed method are significantly +higher than those of the Laplace noise in all scenarios, showing that the proposed synthetic +ranking algorithm outperforms the Laplace noise in generating more utility-preserving rankings +for the pairwise learning task. +28 + +0.775 +0.750 +0.725 +0.700 +0.675 +0.650 +0.625 +Synthetic Ranking Algorithm:m=15 +Laplace Noise:m=15 +0.600 +Synthetic RankingAlgorithm:m=30 +Laplace Noise: m=30 +200 +220 +240 +260 +280 +300 +320 +340 +Number of Users0.875 +0.850 +0.825 +Lpair( +0.800 +0.775 +0.750 +0.725 +Synthetic Ranking Algorithm:m=15 +Laplace Noise:m=15 +Synthetic Ranking Algorithm: m=30 +0.700 +Laplace Noise: m=30 +200 +220 +240 +260 +280 +300 +320 +340 +Number of Users0.90 +0.88 +I Accuracy: +0.86 +0.84 +Testing +0.82 +Synthetic Ranking Algorithm:m=15 +Laplace Noise:m=15 +0.80 +Synthetic Ranking Algorithm:m=30 +Laplace Noise: m=30 +200 +220 +240 +260 +280 +300 +320 +340 +Number of Users0.92 +Testing Accuracy: Lpair(i +0.90 +0.88 +0.86 +Synthetic Ranking Algorithm:m=15 +0.84 +Laplace Noise:m=15 +Synthetic Ranking Algorithm: m=30 +0.82 +Laplace Noise: m=30 +200 +220 +240 +260 +280 +300 +320 +340 +Numberof Users6.4 +Real Application +In this section, we evaluate the performance of the developed synthetic ranking algorithm +over the Sushi dataset, aiming to show that our synthetic ranking algorithm maintains more +information for downstream learning tasks in achieving a better generalization performance +compared with adding Laplace noise under the same privacy guarantee. +The Sushi dataset is publicly available at https://www.kamishima.net/sushi/, which +consists of preference rankings of 5,000 users over 10 kinds of sushis collected via a ques- +tionnaire survey, including “shrimp”, “sea eel”, “tuna”, “squid”, “sea urchin”, “salmon +roe”, “tamago”, “fatty tuna”, “tuna roll”, and “cucumber roll”. For this application, we +only consider the preference ranks of “tuna”, “salmon roe”, “tamago”, “fatty tuna”, and +“cucumber roll”, whose preference ranks are more consistent among users compared with +other sushis. +We compare the utility of rankings generated by the synthetic ranking algorithm and +the counterpart which adds Laplace noise via the pairwise learning task as in Section 6.3. +To be more specific, we first split the dataset into a training dataset of 3,000 users and a +testing dataset of 2,000 users. Second, the rankings in the training dataset are permuted +by the synthetic ranking algorithm and the Laplace noise satisfying the same ϵ-ranking DP. +Then, we estimate the ranking functions �f and �flap as in (9) and (10), respectively, where +the ranking function to be f(xu, yi) = αxu + βyi. We evaluate the performance of �f and +�flap by the pairwise test accuracy defined in (11) on the users in the testing dataset. +Figure 8 presents similar results as in Section 6.3 that the proposed method produces +more utility-preserving synthetic rankings for the pairwise learning task compared with the +Laplace noise under the same privacy guarantee. +7 +Summary +In this paper, we propose a new privacy metric called ϵ-ranking differential privacy for ranking +data. The basic idea is to generate a synthetic ranking in placement of the real ranking +29 + +Figure 8: The averaged relative utility in 50 replications with different privacy guarantees +ϵ being evenly spaced values ranging from 0.06 to 0.5. L∗ +pair denotes the averaged pairwise +accuracy of the ranking function trained on non-privatized training dataset in 50 replications. +for sharing while resisting inference on a specific rank within the ranking list. Based on +this notion, we further develop a synthetic ranking algorithm grounded in the multistage +ranking model to synthetic rankings satisfying the developed ϵ-ranking differential privacy, +which inherently permutes ordinal ranks of items in a probabilistic manner. Furthermore, we +analyze the utility of synthetic rankings yielded by the proposed algorithm in the downstream +inference attack and personalized ranking tasks, for which we establish statistical consistency +characterizing the corresponding utility-privacy tradeoff. It is worth noting that, unlike +existing methods employing noise addition approaches to produce differentially private +rankings, the proposed method implicitly perturbs rankings nonlinearly while accommodating +the requirement of strict privacy protection. +30 + +1.00 +Synthetic Ranking Algorithm +Laplace Noise +0.98 +0.96 +0.94 +0.92 +0.90 +0.88 +0.86 +0.84 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +wReferences +Alabi, D., Ghazi, B., Kumar, R., and Manurangsi, P. 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Kernel logistic regression and the import vector machine. +Journal of Computational and Graphical Statistics, 14(1):185–205. +36 + +Supplementary Materials +“Ranking Differential Privacy” +Shirong Xu, Will Wei Sun, and Guang Cheng +Proof of Lemma 1: Let φ and φ′ be two neighboring ranking differing such that +� +φ(Ii) − φ(Ij) +�� +φ′(Ii) − φ′(Ij) +� +> 0, +for any Ii, Ij ∈ Ω \ {Ik}. Therefore, for any synthetic ranking �φ and Ii, Ij ∈ Ω \ {Ik}, it +holds that +I +� +(φ(Ii) − φ(Ij))(�φ(Ii) − �φ(Ij)) > 0 +� +− I +� +(φ′(Ii) − φ′(Ij))(�φ(Ii) − �φ(Ij)) > 0 +� += 0. +Let MΩ = {(Ii, Ij) : Ii, Ij ∈ Ω, Ii ̸= Ij}. For any synthetic ranking �φ, we get +���T(φ, �φ) − T(φ′, �φ) +��� = +��� +1 +|MΩ| +� +(i,j)∈MΩ +I +� +(φ(Ii) − φ(Ij))(�φ(Ii) − �φ(Ij)) > 0 +� +− +1 +|MΩ| +� +(i,j)∈MΩ +I +� +(φ′(Ii) − φ′(Ij))(�φ(Ii) − �φ(Ij)) > 0 +���� += +��� +2 +|MΩ| +� +Ij∈Ω\{Ik} +I +� +(φ(Ik) − φ(Ij))(�φ(Ik) − �φ(Ij)) > 0 +� +− +2 +|MΩ| +� +Ij∈Ω\{Ik} +I +� +(φ′(Ik) − φ′(Ij))(�φ(Ik) − �φ(Ij)) > 0 +���� +≤ 2 +|Ω|. +Therefore, for any neighboring rankings φ and φ′, it holds that +max +�φ +���T(φ, �φ) − T(φ′, �φ) +��� ≤ 2 +|Ω|. +The desired result immediately follows by setting θ = 2−1|Ω|ϵ. +Proof of Lemma 2: Let A denote algorithm 1 and �φ = A(φ) denote the output synthetic +ranking. +Let Ω = {I1, I2, . . . , IL} denote the item set. +Suppose that �φ satisfies that +37 + +�φ(Ii) = sIi for i ∈ [L] denote the output synthetic ranking, where s = (sIi)i∈[L] is a +permutation of (i)i∈[L]. +According to the Algorithm 1, at the iteration t ∈ {2, . . . , L}, the selected item Ij has +rank t among the items, that is Ij = φ−1(t). Notice that all selected items in the previous +items all have smaller ordinal ranks. Therefore it is straightforward to see that +P(V (t) +ϵ += k) ∝ exp +� +ϵ(L − 1)−1k +� +, +for k = 0, . . . , t − 1. With this, we further have +P(V (t) +ϵ += k) = +exp +� +ϵk(L − 1)−1� +�t−1 +k=0 exp +� +ϵk(L − 1)−1�, k = 0, . . . , t − 1. +Define Ht(s) = {i : φ(Ii) < t, sIi < sφ−1(t), i ∈ [L]} for t = 2, . . . , L. It is easy to verify +that |Ht(s)| = k implies that V (t) = k. Notice that |H1(s)| = 0, it then follows that +P +� +A(φ) = �φ +� += +L +� +t=2 +exp(ϵ(L − 1)−1|Ht(s)|) +�t−1 +k=0 exp(ϵ(L − 1)−1k) +∝ exp +� +ϵ(L − 1)−1 +L +� +t=1 +|Ht(s)| +� +. +(12) +Next, it remains to prove that �L +t=1 |Ht(s)| = 2−1|Ω|(|Ω| − 1)T(φ, �φ). By the definition of +Ht(s), we have +|Ht(s)| = +� +i:φ(Ii) 0 +� += +� +i:φ(Ii) 0 +� +, +where Ij = φ−1(t). Summing over t from 1 to L yields that +L +� +t=1 +|Ht(s)| = +L +� +t=1 +� +i:φ(Ii) 0 +� += +L +� +j=1 +� +{i:φ(Ii)<φ(Ij)} +I +� +(φ(Ii) − φ(Ij))(�φ(Ii) − �φ(Ij)) > 0 +� +=1 +2 +L +� +j=1 +� +i̸=j +I +� +(φ(Ii) − φ(Ij))(�φ(Ii) − �φ(Ij)) > 0 +� +=1 +2|Ω|(|Ω| − 1)T(φ, �φ). +(13) +38 + +Plugging (13) into (12), it follows that +P +� +A(φ) = �φ +� +∝ exp(2−1ϵ|Ω|T(φ, �φ)). +This completes the proof. +Proof of Theorem 1: First, we prove the property (1). Let Ω = {I1, . . . , IL} be the item +set. We consider the case that P +� +Φ(Ii) > Φ(Ij) +� +> P +� +Φ(Ii) < Φ(Ij) +� +, which implies that +2ηij − 1 > 0. Furthermore, For any Ii, Ij ∈ Ω, we define two sets of rankings as +Y g +ij = +� +φ : +� +φ(Ii) +� +Ii∈Ω ∈ Υ(L), φ(Ii) > φ(Ij) +� +, +Y l +ij = +� +φ : +� +φ(Ii) +� +Ii∈Ω ∈ Υ(L), φ(Ii) < φ(Ij) +� +, +where Y g +ij and Y l +ij denote the ranking sets whose elements satisfying φ(Ii) > φ(Ij) and +φ(Ij) > φ(Ii), respectively. +Let �Φ = A(Φ) denote the random variable of synthetic ranking. Next, we intend to prove +P +��Φ(Ii) > �Φ(Ij) +� +> P +��Φ(Ii) < �Φ(Ij) +� +, which are given by +P +��Φ(Ii) > �Φ(Ij) +� += P +��Φ ∈ Y g +ij +� += +� +�φ∈Y g +ij +P +� +A(Φ) = �φ +� +, +P +��Φ(Ii) < �Φ(Ij) +� += P +��Φ ∈ Y l +ij +� += +� +�φ∈Y l +ij +P +� +A(Φ) = �φ +� +, +respectively. +Consider two synthetic rankings �φ1 ∈ Y g +ij and �φ2 ∈ Y l +ij satisfying that �φ1(Ii) = �φ2(Ij) = k1, +�φ1(Ij) = �φ2(Ii) = k2 with k1 > k2, and �φ1(Il) = �φ2(Il) for l ∈ [L] \ {i, j}. Notice that �φ1 +and �φ2 are a couple of ordinal rankings by the one-to-one correspondence between �φ1 ∈ Y g +ij +and �φ2 ∈ Y l +ij, therefore it is easy to see that |Y l +ij| = |Y g +ij|. By Lemma 2, we have +P +� +A(φ) = �φ1 +� +P +� +A(φ) = �φ2 +� = exp +� +2−1ϵL +� +T(φ, �φ1) − T(φ, �φ2) +�� +, +Therefore, it suffices to compare the values of quality function T to obtain the relative order +of P +� +A(φ +� += �φ1 +� +and P +� +A(φ) = �φ2 +� +. +39 + +For any ranking φ, define S(φ, k2, k1) = {i : k2 < φ(Ii) < k1, i ∈ [L]} as the subset of Ω +with ordinal ranks smaller than k1 and greater than k2. By the definitions of �φ1 and �φ2, it is +straightforward to verify that +S(�φ1, k2, k1) = S(�φ2, k2, k1). +Further, we denote that ∆ij(φ, �φ1) = I +� +(φ(Ii)−φ(Ij))(�φ1(Ii)−�φ1(Ij)) > 0 +� +. By the definition +of T, we get +T(φ, �φ1) − T(φ, �φ2) = +1 +L(L − 1) +� +Ii,Ij∈Ω +� +∆ij(φ, �φ1) − ∆ij(φ, �φ2) +� += +2 +L(L − 1) +� +l∈S(�φ1,k2,k1) +� +∆il(φ, �φ1) + ∆jl(φ, �φ1) − ∆jl(φ, �φ2) − ∆il(φ, �φ2) +� ++ +2 +L(L − 1) +� +∆ij(φ, �φ1) − ∆ij(φ, �φ2) +� +. +(14) +Next, we consider two cases that φ ∈ Y g +ij and φ ∈ Y l +ij. Notice that φ ∈ Y g +ij implies φ(Ii) > φ(Ij) +and ∆ij(φ, �φ1) − ∆ij(φ, �φ2) = 1. For any l ∈ S(�φ1, k2, k1), it can be verified that +∆il(φ, �φ1) + ∆jl(φ, �φ1) − ∆jl(φ, �φ2) − ∆il(φ, �φ2) = +� +2, φ(Ij) < φ(Il) < φ(Ii), +0, otherwise. +Therefore, if φ ∈ Y g +ij, (14) can be written as +T(φ, �φ1) − T(φ, �φ2) = 4|S(�φ1, k2, k1) ∩ S(φ, φ(Ij), φ(Ii))| + 2 +L(L − 1) +. +Hence, it follows that +2 +L(L − 1) ≤ T(φ, �φ1) − T(φ, �φ2) ≤ 4(L − 2) + 2 +L(L − 1) +. +Consequently, for any φ ∈ Y g +ij, it holds that +P +� +A(φ) ∈ Y g +ij +� +P +� +A(φ) ∈ Y l +ij +� = +� +�φ1∈Y g +ij P +� +A(φ) = �φ1 +� +� +�φ2∈Y l +ij P +� +A(φ) = �φ2 +� ≥ exp +� +(L − 1)−1ϵ +� +, +P +� +A(φ) ∈ Y g +ij +� +P +� +A(φ) ∈ Y l +ij +� = +� +�φ1∈Y g +ij P +� +A(φ) = �φ1 +� +� +�φ2∈Y l +ij P +� +A(φ) = �φ2 +� ≤ exp +�2L − 3 +L − 1 ϵ +� +. +40 + +Applying a similar argument for the case φ ∈ Y l +ij, we have +exp +� +(L − 1)−1ϵ +� +≤ P +� +A(φ) ∈ Y l +ij +� +P +� +A(φ) ∈ Y g +ij +� ≤ exp +�2L − 3 +L − 1 ϵ +� +. +Notice that Y g +ij and Y l +ij are mutually exclusive and Υ = Y g +ij ∪ Y l +ij, we obtain that +P +��Φ(Ii) > �Φ(Ij) +� +− P +��Φ(Ij) > �Φ(Ii) +� += +� +P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +�� +P +� +Φ ∈ Y g +ij +� ++ +� +P +��Φ ∈ Y g +ij|Φ ∈ Y l +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y l +ij +�� +P +� +Φ ∈ Y l +ij +� +. +(15) +By the symmetry between Y l +ij and Y g +ij, it follows that +P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +� +=P +��Φ ∈ Y l +ij|Φ ∈ Y l +ij +� +− P +��Φ ∈ Y g +ij|Φ ∈ Y l +ij +� +. +(16) +Plugging (16) into (15) yields that +P +��Φ(Ii) > �Φ(Ij) +� +− P +��Φ(Ij) > �Φ(Ii) +� += +� +P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +��� +P +� +Φ ∈ Y g +ij +� +− P +� +Φ ∈ Y l +ij +�� += +� +P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +��� +P +� +Φ(Ii) > Φ(Ij) +� +− P +� +Φ(Ij) > Φ(Ii) +�� += +� +P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +�� +(2ηij − 1) > 0. +(17) +By the fact that P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +− P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +� +> 0, it follows that +� +P +� +Φ(Ii) > Φ(Ij) +� +− P +� +Φ(Ii) < Φ(Ij) +��� +P +��Φ(Ii) > �Φ(Ij) +� +− P +��Φ(Ii) < �Φ(Ij) +�� +> 0, +This completes the proof of property (1). +Next, we turn to prove property (2). For ease of notation, we let A1 = P +��Φ ∈ Y g +ij|Φ ∈ Y g +ij +� +and A2 = P +��Φ ∈ Y l +ij|Φ ∈ Y g +ij +� +. By the fact that A1 +A2 = 1 and exp +� +(L−1)−1ϵ +� +≤ A1/A2 ≤ +exp( 2L−3 +L−1 ϵ), we have +A2 ≤ exp +� +− (L − 1)−1ϵ +� +A1 and A1 ≤ exp +�2L − 3 +L − 1 ϵ +� +A2. +41 + +These combined with the fact that A1 + A2 = 1 imply that +exp +� +(L − 1)−1ϵ +� +− 1 +exp +� +(L − 1)−1ϵ +� ++ 1 ≤ A1 − A2 ≤ +exp +� 2L−3 +L−1 ϵ +� +− 1 +exp +� 2L−3 +L−1 ϵ +� ++ 1. +Combined with (17), we get +exp +� +(L − 1)−1ϵ +� +− 1 +exp +� +(L − 1)−1ϵ +� ++ 1 ≤ |2�ηij − 1| +|2ηij − 1| ≤ +exp +� 2L−3 +L−1 ϵ +� +− 1 +exp +� 2L−3 +L−1 ϵ +� ++ 1, +where ηij = P +� +Φ(Ii) > Φ(Ij) +� +and �ηij = P +��Φ(Ii) > �Φ(Ij) +� +. This completes the proof of +property (2). +Next, we prove the property (3). In the t-th iteration, V (t) +ϵ +follows the distribution +P(V (t) +ϵ += k) = +exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t−1)) +� +� +k∈ρ(t) exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t−1)) +�, k = 0, . . . , t − 1. +Notice that items are selected sequentially by their ordinal ranks, hence t > σ(l) for l ∈ χ(t) +and V (t) +ϵ += k implies that τ(k, χ(t)) = k. Therefore, we have +E +� +V (t) +ϵ +� += +t−1 +� +k=0 +P(V (t) +ϵ += k)k = +t−1 +� +k=0 +k exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t)) +� +� +k∈ρ(t) exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t)) +�. +For ease of notation, we denote that Q = � +k∈ρ(t) exp +� +ϵ(|Ω| − 1)−1τ(k, χ(t)) +� +. We have +E +� +V (t) +ϵ +� += Q−1 +t−1 +� +k=1 +k exp +� +ϵ(|Ω| − 1)−1k +� +. +Straightforward algebra shows that +E +� +V (t) +ϵ +� += +Q−1(t − 1) exp +� +ϵ(|Ω| − 1)−1t +� +− Q−1 �t−1 +k=1 exp +� +ϵ(|Ω| − 1)−1k +� +exp +� +ϵ(|Ω| − 1)−1� +− 1 +=(t − 1)qt +ϵ +qt +ϵ − 1 +− +qt +ϵ − qϵ +(qϵ − 1)(qt +ϵ − 1), +where qϵ = exp +� +ϵ(|Ω| − 1)−1� +. As ϵ goes to infinity, it can be verified that E +� +V (t) +ϵ +� +converges +to t − 1. +Note that Var(V (t) +ϵ ) = E(V (t) +ϵ )2 − +� +E(V (t) +ϵ ) +�2, it remains to compute E(V (t) +ϵ )2. +E(V (t) +ϵ )2 = Q−1 +t−1 +� +k=1 +k2 exp +� +ϵ(|Ω| − 1)−1k +� +. +42 + +Applying a similar argument, we get +E(V (t) +ϵ )2 = +Q−1(t − 1)2 exp +� +ϵ(|Ω| − 1)−1t +� +exp +� +ϵ(|Ω| − 1)−1 +� +− 1 +− +Q−1 �t−1 +k=1(2k − 1) exp +� +ϵ(|Ω| − 1)−1k +� +exp +� +ϵ(|Ω| − 1)−1 +� +− 1 += +(t − 1)2 exp +� +ϵ(|Ω| − 1)−1t +� +exp +� +ϵ(|Ω| − 1)−1t +� +− 1 +− +Q−1 �t−1 +k=1(2k − 1) exp +� +ϵ(|Ω| − 1)−1k +� +exp +� +ϵ(|Ω| − 1)−1 +� +− 1 += +(t − 1)2 exp +� +ϵ(|Ω| − 1)−1t +� +exp +� +ϵ(|Ω| − 1)−1t +� +− 1 +− +2E(V (t) +ϵ ) +exp +� +ϵ(|Ω| − 1)−1 +� +− 1 ++ +exp +� +ϵ(|Ω| − 1)−1t +� +− exp +� +ϵ(|Ω| − 1)−1� +exp +� +ϵ(|Ω| − 1)−1t +� +− 1 +1 +exp +� +ϵ(|Ω| − 1)−1 +� +− 1 +. +It is easy to verify that limϵ→+∞ E(V (t) +ϵ )2 = (t − 1)2. This completes the whole proof. +Proof of Theorem 2: Notice that the minimizer of L(φ) lies in a discrete space. Therefore, +it is impossible to derive an analytic form of �φ. Without loss of generality, we suppose the +true ranking φ satisfies φ(Ii) = i for i = 1, . . . , L. +The proof we present here resembles that of Theorem 3.2 in Tang (2019). Let Q1 = +{φ : φ(I1) = 2, φ(I2) = 1} denote the set of rankings that item I1 and I2 exchange their +ranks and Q2 = {φ : φ(I1) = 1, φ(I2) = 2} denote the set of rankings that item I1 and +I2 have correct ranks. Here it should be noted that the true ranking is an element of Q2. +For any φ1 ∈ Q1, there exists an φ2 ∈ Q2 such that φ1(Il) = φ2(Il) for l ∈ {3, . . . , L}. Let +Πij(S) = �N +l=1 I +��φl(Ii) > �φl(Ij) +� +denote the frequency that the rank of item Ii is larger than +that of item Ij. If Π12(S) > N/2, it is easy to verify that +L(φ1) > L(φ2). +Therefore, it holds that +L(φ) ≤ max +φ2∈Q2 L(φ) < max +φ1∈Q1 L(φ). +Hence, Π12(S) > N/2 implies that �φ ̸= φ. +PS +� +�φmle ̸= φ +� +≥ PS +� +Π12(S) > N/2 +� +. +43 + +Recall that, in Algorithm 1, the ranks of I1 and I2 are determined by the probability +P +��φl(I2) < �φl(I1) +� += +1 +1 + exp(ϵ(|Ω| − 1)−1), for l = 1, . . . , N. +For ease of notation, we denote pϵ,Ω = +exp(ϵ(|Ω|−1)−1) +1+exp(ϵ(|Ω|−1)−1). It then follows that +PS +� +Π12(S) > N/2 +� +≥PS +� +Π21(S) ≤ ⌊N/2⌋ +� += +⌊N/2⌋ +� +k=0 +�N +k +� +pk +ϵ,Ω(1 − pϵ,Ω)N−k. +By Theorem 2 of Arratia and Gordon (1989), there exists some positive constants C0 +such that By approximating the binomial coefficient with Stirling’s formula, we get +PS +� +Π12(S) > N/2 +� +≥ C0 +exp(ϵ(|Ω| − 1)−1) + 1 +exp(ϵ(|Ω| − 1)−1) − 1 +� +2 +πN exp +� +− NDKL(1/2∥pϵ,Ω) +� +, +(18) +where DKL(1/2∥pϵ,Ω) denotes the KL-divergence between two Bernoulli random variables +with parameters 1/2 and pϵ,Ω, respectively. Applying the reverse Pinsker inequality (Sason, +2015) to the right-hand side of (18), we further have +PS +� +Π12(S) > N/2 +� +≥C0 +exp(ϵ(|Ω| − 1)−1) + 1 +exp(ϵ(|Ω| − 1)−1) − 1 +� +2 +πN exp +� +− 2N +� +exp(ϵ(|Ω| − 1)−1) − 1 +�2 +1 + exp(ϵ(|Ω| − 1)−1) +� +≥C0 +exp(ϵ(|Ω| − 1)−1) + 1 +exp(ϵ(|Ω| − 1)−1) − 1 +� +2 +πN exp +� +− 2N +� +exp(ϵ(|Ω| − 1)−1) − 1 +�2 +2 +� +≥C0 +exp(ϵ(|Ω| − 1)−1) + 1 +exp(ϵ(|Ω| − 1)−1) − 1 +� +2 +πN exp +� +− N +� +exp(ϵ(|Ω| − 1)−1) − 1 +�2� +. +Denote that C(ϵ) = exp(ϵ(|Ω| − 1)−1) − 1, the lower bound can be re-written as +PS +� +Π12(S) > N/2 +� +≥ C0 +� +8 +C2(ϵ)πN exp +� +− NC2(ϵ) +� +. +(19) +Further, by setting C(ϵ) = O +� +N −1/2� +, the right hand side of (19) is bounded away from 0 for +any N ≥ 1. The desired result immediately follows by seeing that C(ϵ) = O +� +N −1/2� +implies +ϵ = O +� +(|Ω| − 1)N −1/2� +. +Proof of Lemma 3: Without loss of generality, we suppose the raw ranking φ is +φ(Ii) = i, i ∈ [m]. +44 + +Let �φlap = Mlap +λ (φ) and �φ′ +lap = Mlap +λ (φ′) be outputs of the Laplace mechanism. For any +r ∈ Rm, the joint density functions of �φlap = r and �φ′ +lap = r can be written as +P(�φlap = r) = +1 +2mλm exp +� +− +�m +i=1 |ri − φ(Ii)| +λ +� +, +P(�φ′ +lap = r) = +1 +2mλm exp +� +− +�m +i=1 |ri − φ′(Ii)| +λ +� +. +By the definition of neighboring ranking, we can easily verify that +��� +m +� +i=1 +|ri − φ(Ii)| − +m +� +i=1 +|ri − φ′(Ii)| +��� ≤ 2(m − 1), +where the equality holds when φ′ satisfies φ′(Ii) = i + 1 for i = 1, . . . , m − 1 and φ′(I1) = m +and ri ≥ m for i = 1, . . . , m. Therefore, by setting λ = 2(m − 1)/ϵ, we have +��� log P(�φlap = r) +P(�φ′ +lap = r) +��� ≤ ϵ. +This completes the proof. +Proof of Lemma 4: Without loss of generality, we suppose that raw ranking φ satisfies +that φ(Ii) = i for i = 1, . . . , m. +Denote that �φlap = Mlap +2(m−1)ϵ−1(φ). For any i > j, we define a new random variable as +Zij = �φlap(Ii) − �φlap(Ij). By Corollary 2 of Nadarajah (2007), the cumulative distribution +function of Zij takes the form as +FZij(z) = +� +� +� +1 +2 exp +� +ϵ(z−i+j) +2(m−1) +� +− (z−i+j)ϵ +8(m−1) exp +� +ϵ(z−i+j) +2(m−1) +� +, z < i − j, +1 − 1 +2 exp +� +− ϵ(z−i+j) +2(m−1) +� ++ (z−i+j)ϵ +8(m−1) exp +� +− ϵ(z−i+j) +2(m−1) +� +z > i − j. +Therefore, we have +P +� +�φlap(Ii) > �φlap(Ij) +� += P +� +Zij > 0 +� += 1 − FZij(0) +=1 − 1 +2 exp +� +− ϵ(i − j) +2(m − 1) +� +− (i − j)ϵ +8(m − 1) exp +� +− ϵ(i − j) +2(m − 1) +� +. +Let U t−1 +ϵ += E +� �t−1 +j=1 I(�φlap(It) > �φlap(Ij)) +� +denote the expected number of correct partial +45 + +orders of item t and those items with lower ranks. It then follows that for any t ≥ 2, +E +� +U t−1 +ϵ +� += +t−1 +� +j=1 +P +� +�φlap(It) > �φlap(Ij) +� +=t − 1 − exp(−tλϵ) +2 +t−1 +� +j=1 +exp +� +jλϵ +� +− +t−1 +� +j=1 +jλϵ +4 exp +� +− jλϵ +� +=t − 1 − 1 − exp(−(t − 1)λϵ) +2(exp(λϵ) − 1) ++ λϵ(t − 1) exp(−(t − 1)λϵ) +4(exp(λϵ) − 1) +− λϵ +exp(−λϵ) − exp(−tλϵ) +4(1 − exp(−λϵ))2 +=t − 1 − 1 − p−(t−1) +ϵ +2(pϵ − 1) + λϵ(t − 1)p−(t−1) +ϵ +4(pϵ − 1) +− λϵ +pϵ − p−(t−2) +ϵ +4(pϵ − 1)2 +=t − 1 − +pt−1 +ϵ +− 1 +2(pt +ϵ − pt−1 +ϵ +) + +λϵ(t − 1) +4(pt +ϵ − pt−1 +ϵ +) − λϵ +pϵ − p−(t−2) +ϵ +4(pϵ − 1)2 +where pϵ = exp(λϵ) and λϵ = ϵ2−1(m − 1)−1. By property (3) of Theorem 1, we have +E +� +V t−1 +ϵ +� += t − 1 + +t +qt +ϵ − 1 − +1 +qϵ − 1, +where qϵ = p2 +ϵ. +Next, we turn to prove E +� +V t−1 +ϵ +� +− E +� +U t−1 +ϵ +� +> 0 for any 2 ≤ t ≤ M and ϵ > 0. +E +� +V t−1 +ϵ +� +− E +� +U t−1 +ϵ +� += +t +qt +ϵ − 1 − +1 +qϵ − 1 + +pt−1 +ϵ +− 1 +2(pt +ϵ − pt−1 +ϵ +) − +λϵ(t − 1) +4(pt +ϵ − pt−1 +ϵ +) + λϵ +pϵ − p−(t−2) +ϵ +4(pϵ − 1)2 . +Notice that +− λϵ(t − 1) +4(pt +ϵ − pt−1 +ϵ +) + λϵ +pϵ − p−(t−2) +ϵ +4(pϵ − 1)2 = λϵ(pt +ϵ − tpϵ + t − 1) +4(pt +ϵ − pt−1 +ϵ +)(pϵ − 1) ≥ 0, +for any pϵ ≥ 1. Then, when pϵ ≥ 2, we have +E +� +V t−1 +ϵ +� +− E +� +U t−1 +ϵ +� +≥ +t +qt +ϵ − 1 + pt−1 +ϵ +(pϵ − 1) − (pϵ + 1) +2(pϵ + 1)(pt +ϵ − pt−1 +ϵ +) +> 0. +Next, we consider the case that pϵ < 2. By the fact that 3/2λϵ ≥ pϵ − 1 when pϵ < 2, we get +λϵ(pt +ϵ − tpϵ + t − 1) +4(pt +ϵ − pt−1 +ϵ +)(pϵ − 1) ≥ (pt +ϵ − tpϵ + t − 1) +6(pt +ϵ − pt−1 +ϵ +) +. +Next, we analyze the ratio +Ra(pϵ, t) = +1 +p2ϵ−1 − +t +p2t +ϵ −1 +pt−1 +ϵ +−1 +2(ptϵ−pt−1 +ϵ +) + (ptϵ−tpϵ+t−1) +6(ptϵ−pt−1 +ϵ +) += +6(p2t +ϵ − tp2 +ϵ + t − 1)(pt +ϵ − pt−1 +ϵ +) +(pt +ϵ + 3pt−1 +ϵ +− tpϵ + t − 4)(p2 +ϵ − 1)(p2t +ϵ − 1) += +6(p2t +ϵ − tp2 +ϵ + t − 1)pt−1 +ϵ +(pt +ϵ + 3pt−1 +ϵ +− tpϵ + t − 4)(pϵ + 1)(p2t +ϵ − 1) += +6pt−1 +ϵ +(1 + pϵ)(pt +ϵ + 1) +(p2t +ϵ − tp2 +ϵ + t − 1) +p2t +ϵ + 3p2t−1 +ϵ +− tpt+1 +ϵ ++ (t − 5)pt +ϵ − 3pt−1 +ϵ ++ tpϵ − t + 4. +(20) +46 + +Using L’Hospital’s rule, we have limpϵ→1 Ra(pϵ, t) = 1 for any t ≥ 2. Furthermore, it can +easily verified that the numerator and denominator of (20) are both positive and increasing on +pϵ ∈ [1, 2] for any t ≥ 2 and the denominator is larger than the numerator for any pϵ ∈ [1, 2] +and t ≥ 2. Therefore, it follows that Ra(pϵ, t) ≤ 1, which implies that +E +� +V t−1 +ϵ +� +− E +� +U t−1 +ϵ +� +≥ 0, +where the equality holds if and only if pϵ = 1 indicating ϵ = 0. +To sum up, we get +E +� +T +� +φ, Aϵ(φ) +�� += 2 +m +� +t=2 +E[V t−1 +ϵ +] > 2 +m +� +t=2 +E[U t−1 +ϵ +] = E +� +T +� +φ, Mlap +2(m−1)ϵ−1(φ) +�� +, +for any ϵ > 0. This completes the proof. +Proof of Lemma 6: By property (1) of Theorem 1, we get +� +P +� +Φu(Ii) > Φ(Ij) +� +− P +� +Φu(Ii) < Φ(Ij) +��� +P +��Φu(Ii) > �Φu(Ij) +� +− P +��Φu(Ii) < �Φu(Ij) +�� +> 0. +This combined with Lemma 5 yields that +� +f ∗(xu, yi) − f ∗(xu, yj) +�� +�f ∗(xu, yi) − �f ∗(xu, yj) +� +> 0, +(21) +for any u ∈ [n] and i, j ∈ [m]. By the definitions of Euij(f) and �Euij(f), we have +Euij(f) − E∗ +uij = |2ηuij − 1|I +�� +f(xu, yi) − f(xu, yj) +�� +f ∗(xu, yi) − f ∗(xu, yj) +� +> 0 +� +, +�Euij(f) − �E∗ +uij = |2�ηuij − 1|I +�� +f(xu, yi) − f(xu, yj) +�� �f ∗(xu, yi) − �f ∗(xu, yj) +� +> 0 +� +. +The desired result immediately follows from property (2) of Theorem 1. +Proof of Lemma 7: By the property (1) in Theorem 1, we have +exp +� +(|Ωu| − 1)−1ϵu +� +− 1 +exp +� +(|Ωu| − 1)−1ϵu +� ++ 1 ≤ |2�ηuij − 1| +|2ηuij − 1| ≤ exp +� +ϵu +� +− 1 +exp +� +ϵu +� ++ 1. +Let Ψu = +exp +� +(|Ωu|−1)−1ϵu +� ++1 +exp +� +(|Ωu|−1)−1ϵu +� +−1, then it is straightforward to see that |2�ηuij − 1| ≤ β implies +|2ηuij − 1| ≤ Ψuβ. Following from Assumption C yields that +P +� +|2�ηuij − 1| ≤ β +� +≤ P +� +|2ηuij − 1| ≤ Ψuβ +� +≤ C1(Ψu)γβγ. +This completes the proof. +47 + +Lemma 8. Under Assumption 4, there exists some positive constants C4 > 0 such that +Var +� +Gf(u) − Gf∗(u) +� +≤ 2C4E +1 +1+γ � +Ψγ +u +�� +eυ(f, f ∗) +�γ/(γ+1). +Proof of Lemma 8 : By the law of total variance, we have +Var +� +Gf(u) − Gf∗(u) +� += E +� +Varu +� +Gf(u) − Gf∗(u) +�� ++ Var +� +Eu +� +Gf(u) − Gf∗(u) +�� +, +where Varu +� +Gf(u) − Gf∗(u) +� +is the conditional variance of Gf(u) − Gf∗(u) with fixed user u +and Eu +� +Gf(u) − Gf∗(u) +� +is the taken with respect to synthetic rankings conditional on user +u. +By assuming items are uniformly generated with fixed size, Varu +� +Gf(u) − Gf∗(u) +� +can be +upper bounded as +Varu +� +Gf(u) − Gf∗(u) +� += +1 +m2(m − 1)2Var +� � +i̸=j +I(�Φu(Ii) > �Φu(Ij)))υ +� +f(xu, yi) − f(f(xu, yj) +�� +≤ +1 +m(m − 1)2 +� +i̸=j +Var +� +I(�Φu(Ii) > �Φu(Ij)))υ +� +f(xu, yi) − f(xi, yj) +�� +, +where υ(f(xu, yi) − f(xu, yj)) = υ(f(xu, yi) − f(xu, yj)) − υ(f ∗(xi, yi) − f ∗(xi, yj)) and the +inequality follows from the fact that partial orders are partially correlated. +For any i, j ∈ [m], we have +Var +� +I(�Φu(Ii) > �Φu(Ij))υ +� +f(xu, yi) − f(xu, yj) +�� +≤P +��Φu(Ii) > �Φu(Ij) +� +υ2� +f(xu, yi) − f(xu, yj) +� ++ P +��Φu(Ij) > �Φu(Ii) +� +υ2� +f(xu, yj) − f(xu, yi) +� +=�ηuijυ2� +f(xu, yi) − f(xu, yj) +� ++ (1 − �ηuij)υ2� +f(xu, yj) − f(xu, yi) +� +≤K2 +υ +� +f(xu, yi) − f(xi, yj) − f ∗(xu, yi) + f ∗(xu, yj) +�2 +, +where the last inequality follows from the Lipschitz continuity of υ. For ease of notation, we +let gf(u, i, j) = f(xu, yi) − f(xu, yj). By the fact that f ∗ is the optimal minimizer, then for +48 + +each user u, Eu +� +Gf(u) − Gf∗(u) +� +can be lower-bounded as +Eu +� +Gf(u) − Gf∗(u) +� +(22) += +1 +m(m − 1) +� +i 0 and the first inequality follows from the fact that f ∗ is +the optimal minimizer and the inequality holds when �ηuij = 1/2 almost surely or f = f ∗. +Combining (22) and Lemma 7, it follows that +Eu +� +Gf(u) − Gf∗(u) +� +≥T1βEu +� +(gf(u, i, j) − gf∗(u, i, j))2� +− C2Ψγ +uβγ+1 +=T1( T2 +2C2 +) +γ+1 +γ E +γ+1 +γ +u +� +(gf(u, i, j) − gf∗(u, i, j))2� +Ψ−1 +u , +where the last equality follows by taking β = (2−1T1C−1 +2 ) +1 +γ Ψ−1 +u E +1 +γ � +(gf(u, i, j) − gf∗(u, i, j))2� +. +Therefore, it holds that for some positive constants C4 depending on γ, +Varu +� +Gf(u) − Gf∗(u) +� +≤ +C4 +|Ω| − 1Ψ +γ +γ+1 +u +E +γ +γ+1 +u +� +Gf(u) − Gf∗(u) +� +. +Taking the expectation of both sides yields that +E +� +Varu +� +Gf(u) − Gf∗(u) +�� +≤C4E +� +(|Ω| − 1)−1Ψ +γ +γ+1 +u +E +γ +γ+1 +u +� +Gf(u) − Gf∗(u) +�� +≤ +C4 +|Ω| − 1E +1 +1+γ � +Ψγ +u +�� +eυ(f, f ∗) +� +γ +γ+1, +(24) +where the last inequality follows from the H¨older’s inequality. +Next, we proceed to establish the relation between Var +� +Eu +� +Gf(u)−Gf∗(u) +�� +and eυ(f, f ∗). +Var +� +Eu +� +Gf(u) − Gf∗(u) +�� += E +� +E2 +u +� +Gf(u) − Gf∗(u) +�� +− e2 +υ(f, f ∗) +=E +� +1 +m(m − 1) +� +i 0 such that +E[Rn(Hij)] ≤ A1n−1/2E +� � 2σn(Hij) +0 +� +log N(Hij, v, ∥ · ∥U)dv +� +, +(27) +where ∥f∥U = +� +n−1 �n +u=1 DG2 +f(u) and σ2 +n(Hij) = supf∈Hij n−1 �n +u=1 DG2 +f(u). Next, we +50 + +derive an upper bound for log N(Hij, v, ∥ · ∥U). For any f1, f2 ∈ H, +∥f1 − f2∥2 +U =1 +n +n +� +u=1 +� +DGf1(u) − DGf2(u) +�2 += 1 +n +n +� +u=1 +� +Gf1(u) − Gf2(u) +�2 +≤1 +n +n +� +u=1 +1 +m(m − 1) +� +i̸=j +K2 +υ +� +gf1(u, i, j) − gf2(u, i, j) +�2 +≤1 +n +n +� +u=1 +1 +m(m − 1) +� +i̸=j +K2 +υ +� +2 +� +f1(xu, yi) − f2(xu, yi) +�2 + 2 +� +f1(xu, yj) − f2(xu, yj) +�2� +≤1 +n +n +� +u=1 +1 +m(m − 1) +� +i̸=j +K2 +υ +� +f1(xu, yi) − f2(xu, yi) +�2 +≤C1 +n +n +� +u=1 +� +∥xu∥2 +2 + 1 +m +� +i∈[m] +∥yi∥2 +2 +� +∥Θf1 − Θf2∥2 +∞, +where the first inequality follows from the fact that φ(·) is a Kυ-Lipschitz function and the +last inequality follows from Assumption 3. +For ease of notation, we denote that F 2(u) = ∥xu∥2 +2 + 1 +m +� +i∈[m] ∥yi∥2 +2. Notice that f ∈ Hij +implies that ∥Θf∥∞ ≤ 2jJ0. Let BΘ(2jJ0) denote the cube centered at the origin and of +length 2j+1J0 in R|Θ|. It then can be verified that +N(Hij, v, ∥ · ∥U) ≤ N +� +BΘ(2jJ0), +v +√C1∥F∥, ∥ · ∥ +� +, +where ∥F∥ = +� +n−1 �n +u=1 F 2(u). It then follows that +log N(Hij, v, ∥ · ∥U) ≤ |Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +, +(28) +Next, combining (27) and (28) yields that +E[Rn(Hij)] ≤ A1n−1/2E +� � 2σn(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +dv +� +=A1n−1/2E +� � 2σn(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +dvI +� +∥F∥2 > 2EF 2(u) +�� ++A1n−1/2E +� � 2σn(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +dvI +� +∥F∥2 ≤ 2EF 2(u) +�� +=V1 + V2. +51 + +Next, it suffices to bound V1 and V2 separately. For V1, we first note that +� 2σn(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +dv += +� +C12j+1J0∥F∥ +� 2 +σn(Hij) +√ +C12j+1J0∥F ∥ +0 +� +|Θ| log +� +max +�1 +v, 1 +�� +dϵ +≤ +� +C12j+1J0∥F∥ +� 1 +0 +� +|Θ| log +� +ϵ−1� +dϵ +≤ +� +C12j+1J0∥F∥ +� +|Θ| +� ∞ +1 +u−2� +log(u)du +≤A22j+1J0∥F∥ +� +|Θ|, +where A2 = √C1 +� ∞ +0 u−3/2du and the last inequality follows from the fact that +� +log(u) ≤ +√u − 1 for u ≥ 1. With this, V1 can be bounded as +V1 ≤A1n−1/2E +� +A22j+1J0∥F∥ +� +|Θ|I +� +∥F∥2 > 2EF 2(u) +�� +≤A1A22j+1J0n−1/2E +� +∥F∥ +� +|Θ|I +� +∥F∥2 > 2EF 2(u) +�� +≤A1A22j+1J0n−1/2� +|Θ|EF 2(u)P +� +∥F∥2 > 2EF 2(u) +� +≤A1A22j0+1J0n−1/2� +|Θ|EF 2(u)P +� +∥F∥2 > 2EF 2(u) +� +(29) +where the third inequality follows from H¨older’s inequality. By Bernstein’s inequality, the +right-hand side of (29) can be bounded as +P +� +∥F∥2 > 2EF 2(u) +� +≤ exp +� +− +1/2n2(EF 2(u))2 +nA4 + 1/3A3EF 2(u) +� +, +(30) +where A3 ≥ F(u) for any u and A4 ≥ 1 +n +�n +u=1 E[F 4(u)], which holds true by the fact that +X and Y are both compact spaces. Combining (29) and (30) shows that there exists some +positive constants A5 > 0 such that +V1 ≤ A5n−1. +To bound V2, we first note that for u ≥ 1 +du−2� +log(max{u, 1}) +du += +1 +u3� +log(u) +(−2 log(u) + 1). +(31) +52 + +Setting (31) to 0 yields that u = exp(1/2), hence u−2 log(max{u, 1}) is increasing for +1 ≤ u ≤ exp(1/2) and decreasing for u > exp(1/2). It follows that for any A > 0, +� +∞ +A +u−2� +log(max{u, 1})du = +� exp(1/2) +A +u−2� +log(max{u, 1})du + +� +∞ +exp(1/2) +u−2� +log(u)du +≤2−1/2A−1 − 2−1/2 exp(−1/2) + 2−1/2 exp(−1/2) + +� +∞ +exp(1/2) +u−2 +� +log(u) +du +≤2−1/2A−1 + A−a +� +∞ +exp(1/2) +u−2+a +� +log(u) +du, +where 0 ≤ a < 1. It can be verified that +� +∞ +exp(1/2) +u−2+a +√ +log(u)du is finite when 0 ≤ a < 1. Hence, +there exists constant A6 such that for any A > 0 +� +∞ +A +u−2� +log(max{u, 1})du ≤ A6A−1. +(32) +For V2, by the concavity of +� +log(x), it holds that +V2 =A1n−1/2E +� � 2σn(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0∥F∥ +v +, 1 +�� +dvI +� +∥F∥2 ≤ 2EF 2(u) +�� +≤A1n−1/2 +� 2√ +Eσ2n(Hij) +0 +� +|Θ| log +� +max +�√C12j+1J0 +� +2E[F 2(u)] +v +, 1 +�� +dv +=A1 +� +C1n−1/22j+1J0 +� +2|Θ|E[F 2(u)] +� +√ +Eσ2n(Hij) +√ +C12jJ0 +√ +2E[F 2(u)] +0 +� +log +� +max +�1 +v, 1 +�� +dv +≤A1 +� +C1n−1/22j+1J0 +� +2|Θ|E[F 2(u)] +� +∞ +√ +C12j+1J0 +√ +E[F 2(u)] +2√ +Eσ2n(Hij) +u−2� +log(max{u, 1})du +≤2A1A6 +� +C1n−1/2� +|Θ|Eσ2 +n(Hij), +(33) +where the last inequality follows from (32) +As proved in Theorem 2, we have +Eσ2 +n(Hij) ≤ 8CυE +� +R(Hij) +� ++ C5E +1 +1+γ � +Ψγ +u +� +M +γ +γ+1(i, j), +(34) +Plugging (34) into (33) yields that for some positive constants A7 > 0 +V2 ≤ A7 +� +n−1|Θ| +�� +8CυE +� +R(Hij) +� ++ C5E +1 +2(1+γ)� +Ψγ +u +� +(2iδn) +γ +2(γ+1) +� +. +53 + +Consequently, we get +E[Rn(Hij)] ≤ A6n−1 + A7 +� +n−1|Θ| +�� +8CυE +� +R(Hij) +� ++ C5E +1 +2(1+γ)� +Ψγ +u +� +(2iδn) +γ +2(γ+1) +� +. +(35) +Solving (35) yields that E[Rn(Hij)] ≤ A8n−1/2|Θ|−1/2(2iδij) +γ +2(γ+1)E +1 +2(1+γ)� +Ψγ +u +� ++ A9n−1|Θ| for +some constants A8, A9 > 0. +Provided that +� +n−1|Θ| log +� +|Θ|−1n) +� γ+1 +γ+2E +1 +γ+2� +Ψγ +u +� += O(δn), it follows that +E[Rn(Hij)] +M(i, j) +≤ A8n−1/2|Θ|1/2E +1 +2(1+γ)� +Ψγ +u +� +(2iδn) +γ+2 +2(γ+1) ++ A9n−1|Θ| +2iδn +≤ 1/4, +where the last inequality holds when n goes to infinity, which then implies E +� +supf∈Hij Un(f) +� +≤ +1/2M(i, j), and this completes the proof. +Proof of Theorem 3: By Assumption 2, it holds that for δn > 0, +P +� +e( �f, f ∗) ≥ δα +n +� +≤ P +� +eυ( �f, f ∗) ≥ δn +� +. +Next, we proceed to bound P +� +eφ( �f, f ∗ +φ) ≥ δn +� +. By the definition of �f, we have +P +� +eυ( �f, f ∗) ≥ δn +� +≤ P +� +sup +f∈H +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� +≡ I, +(36) +where J0 = max{J(f ∗ +F), 1}. Let i0 denote the integer such that 2i0δn ≥ supf eυ(f, f ∗) and j0 +denote the integer such that for any f ∈ F satisfies J(f) ≥ 2j0J0 must have ∥Θf∥min ≥ CF, +where ∥ · ∥min denotes the minimum absolute value of Θf. +For 0 ≤ i ≤ i0 and 0 ≤ j ≤ j0, we define +Hij = +� +f ∈ F : 2i−1δn < eυ(f, f ∗) ≤ 2iδn, 2j−1J0 < J(f) ≤ 2jJ0 +� +, 1 ≤ i ≤ i0, 1 ≤ j ≤ j0 +Hi0 = +� +f ∈ F : 2i−1δn < eυ(f, f ∗) ≤ 2iδn, J(f) ≤ J0 +� +, 1 ≤ i ≤ i0. +It is easy to verify that H can be represented as H = ∪i0 +i=1 ∪j0 +j=0 Hij. With this, (36) can be +54 + +upper bounded as +I =P +� +sup +f∈∪i0 +i=1∪j0 +j=0Hij +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� +≤ +i0 +� +i=0 +j0 +� +j=0 +P +� +sup +f∈Hij +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� += +i0 +� +i=1 +j0 +� +j=1 +P +� +sup +f∈Hij +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� ++ +i0 +� +i=1 +P +� +sup +f∈Hi0 +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� +≡ I1 + I2. +Therefore, it suffices to bound I1 and I2 separately. By Assumption 2, for any 1 ≤ i ≤ i0 and +j ≥ 0, we get +P +� +sup +f∈Hij +�Rn,υ(f ∗ +F) + λnJ0 − �Rn,υ(f) − λnJ(f) ≥ 0 +� +≤P +� +sup +f∈Hij +� �Rn,υ(f ∗ +F) − �Rn,υ(f) − �Rυ(f ∗ +F) + �Rυ(f) +� +≥ λn inf +f∈Hij(J(f) − J0) + inf +f∈Hij +�Rφ(f) − �Rυ(f ∗ +F) +� +≤P +� +sup +f∈Hij +� �Rn,υ(f ∗ +F) − �Rn,υ(f) − �Rυ(f ∗ +F) + �Rυ(f) +� +≥ λn(2j−1 − 1)J0 + 2i−1δn +� +≤P +� +sup +f∈Hij +� �Rn,υ(f ∗ +F) − �Rn,υ(f) − �Rυ(f ∗ +F) + �Rυ(f) +� +≥ λn(2j−1 − 1)J0 + 2i−1δn +� +≤P +� +sup +f∈Hij +� �Rn,υ(f ∗ +F) − �Rn,υ(f) − �Rυ(f ∗ +F) + �Rυ(f) +� +≥ M(i, j) +� +, +(37) +where M(i, j) = λn(2j−1 − 1)J0 + 2i−1δn for 1 ≤ i ≤ i0 and j ≥ 1 and M(i, 0) = 2i0δn for +i ≥ 1. Further, we define +Gf(u) = +1 +m(m − 1) +� +i̸=j +I +� +�φu(Ii) > �φu(Ij) +� +υ(gf(u, i, j)). +Then the right-hand side of (37) can be written as +P +� +sup +f∈Hij +� �Rn,υ(f ∗ +F) − �Rn,υ(f) − �Rυ(f ∗ +F) + �Rυ(f) +� +≥ M(i, j) +� +=P +� +sup +f∈Hij +�1 +n +n +� +u=1 +� +Gf∗ +F(u) − Gf(u) +� +− E +� +Gf∗ +F(u) − Gf(u) +�� +≥ M(i, j) +� +=P +� +sup +f∈Hij +�1 +n +n +� +u=1 +� +DGf(u) +� +− E +� +DGf(u) +�� +≥ M(i, j) +� +, +(38) +55 + +where DGf(u) = Gf∗ +F(u) − Gf(u). Let Un(f) = n−1 �n +u=1 +� +DGf(u) − E +� +DGf(u) +�� +, then (38) +can be re-written as +P +� +sup +f∈Hij +�1 +n +n +� +u=1 +� +DGf(u) +� +− E +� +DGf(u) +�� +≥ M(i, j) +� +=P +� +sup +f∈Hij +Un(f) − E +� +sup +f∈Hij +Un(f) +� +≥ M(i, j) − E +� +sup +f∈Hij +Un(f) +�� +≤P +� +sup +f∈Hij +Un(f) − E +� +sup +f∈Hij +Un(f) +� +≥ 1/2M(i, j) +� +, +(39) +where the last inequality follows from Lemma 9. With this, we have +I1 ≤ +i0 +� +i=1 +j0 +� +j=1 +P +� +sup +f∈Hij +Un(f) − E +� +sup +f∈Hij +Un(f) +� +≥ 1/2M(i, j) +� +, +I2 ≤ +i0 +� +i=1 +P +� +sup +f∈Hi0 +Un(f) − E +� +sup +f∈Hi0 +Un(f) +� +≥ 1/2M(i, j) +� +. +Clearly, it suffices to bound P +� +supf∈Hij Un(f) − E +� +supf∈Hij Un(f) +� +≥ 1/2M(i, j) +� +for any +1 ≤ i ≤ i0 and 0 ≤ j ≤ j0, to which we apply the Talagrand’s inequality in the following step. +Denote that E[σ2 +n(Hij)] = E[supf∈Hij n−1 �n +i=1 DG2 +f(u)]. Then, we establish the relation +between E[σ2 +n(Hij)] and M(i, j). By a symmetrization argument, we have +E[σ2 +n(Hij)] ≤E +� +sup +f∈Hij +1 +n +n +� +u=1 +DG2 +f(u) − E +� +DG2 +f(u) +�� ++ sup +f∈Hij +1 +n +n +� +u=1 +E +� +DG2 +f(u) +� +≤8CυE +� +R(Hij) +� ++ sup +f∈Hij +1 +n +n +� +u=1 +E +�� +Gf∗ +F(u) − Gf(u) +�2� +. +(40) +The right-hand side of (40) can be bounded by +sup +f∈Hij +1 +n +n +� +u=1 +E +�� +Gf∗ +F(u) − Gf(u) +�2� +≤2 +n +n +� +u=1 +E +�� +Gf∗ +F(u) − Gf∗(u) +�2� ++ sup +f∈Hij +2 +n +n +� +u=1 +E +�� +Gf∗(u) − Gf(u) +�2� +≤2 +n +n +� +u=1 +Var +� +Gf∗ +F(u) − Gf∗(u) +� ++ sup +f∈Hij +2 +n +n +� +u=1 +Var +� +Gf∗(u) − Gu(f) +� ++ 2e2 +υ(f ∗ +F, f ∗) + 2 sup +f∈Hij +e2 +υ(f, f ∗ +F) +≤4C4E +1 +1+γ � +Ψγ +u +� +sup +f∈Hij +� +eυ(f, f ∗) +�γ/(γ+1) + sup +f∈Hij +e2 +υ(f, f ∗ +F). +56 + +Suppose that E +� +Ψγ +u +� += O(δn). Since γ/(γ + 1) < 2, there exists some constants C5 such that +sup +f∈Hij +1 +n +n +� +u=1 +E +�� +Gf∗ +F(u) − Gf(u) +�2� +≤ C5E +1 +1+γ � +Ψγ +u +� +M +γ +γ+1(i, j). +(41) +Combined with Lemma 9, it holds that +E +� +σ2 +n(Hij) +� +≤ 2CυM(i, j) + C5E +1 +1+γ � +Ψγ +u +� +M +γ +γ+1(i, j), +where Cυ such that Gf(u) ≤ Cυ for any f ∈ F. Since M(i, j) is asymptotically smaller then +1, there exists some constants T4 such that +Eσ2 +n(Hij) ≤ T4E +1 +1+γ � +Ψγ +u +� +M +γ +γ+1(i, j), +(42) +for 1 ≤ i ≤ i0 and 0 ≤ j ≤ j0. Then, plugging (42) into (39) yields that for some constants +T5 > 0 +P +� +sup +f∈Hij +Un(f) − E +� +sup +f∈Hij +Un(f) +� +≥ 1/2M(i, j) +� +≤T1 exp +� +− nM(i, j) +2BT1 +log +� +1 + +B +� +M(i, j) +� +T4E +1 +1+γ (Ψγ +u)M γ/(γ+1)(i, j) +�� +≤T1 exp +� +− +T5 +2T1T4 +nE− +1 +1+γ (Ψγ +u) +� +M(i, j) +� γ+2 +γ+1� +, +where T5 satisfies that log(1 + x) ≥ T5x for x ∈ [0, (T4E +1 +1+γ (Ψγ +u))−1B +� +M(i, j) +�1/(1+γ)], which +holds true by the boundedness of M(i, j). +57 + +We let T6 = +T5 +2T1T4, then I3 can be bounded as +I1 = +i0 +� +i=1 +j0 +� +j=1 +P +� +sup +f∈Hij +Un(f) − E +� +sup +f∈Hij +Un(f) +� +≥ 1/2M(i, j) +� +≤ +i0 +� +i=1 +j0 +� +j=1 +T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u) +� +δ +γ+2 +γ+1 +n +(2i−1) +γ+2 +γ+1 − (λnJ0) +γ+2 +γ+1(2j−1 − 1) +γ+2 +γ+1 +�� +≤ +∞ +� +i=2 +∞ +� +j=1 +T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u) +� +δ +γ+2 +γ+1 +n +i − (λnJ0) +γ+2 +γ+1(j − 1) +��� ++ +∞ +� +j=1 +T1 exp +�� +− T6nE− +1 +1+γ (Ψγ +u) +� +δ +γ+2 +γ+1 +n +− (λnJ0) +γ+2 +γ+1(j − 1) +�� +≤T1 +exp(−2T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +) +1 − exp(−T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +)) +1 +1 − exp(−T6nE− +1 +1+γ (Ψγ +u))(λnJ0) +γ+2 +γ+1) ++ T1 exp(−T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +) +1 +1 − exp(−T6nE− +1 +1+γ (Ψγ +u))(λnJ0) +γ+2 +γ+1) +≤6T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� +, +where the second last inequality follows from Cauchy-Schwartz inequality and the last +inequality holds true when n goes to infinity. +Similarly, for I2, we have +I4 = +i0 +� +i=1 +P +� +sup +f∈Hi0 +Un(f) − E +� +sup +f∈Hi0 +Un(f) +� +≥ 1/2M(i, j) +� +≤ +i0 +� +i=1 +T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +(2i−1) +γ+2 +γ+1 + T6nE− +1 +1+γ (Ψγ +u)(λnJ0) +γ+2 +γ+1 +� +≤ +i0 +� +i=1 +T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +(2i−1 − 1/2) +γ+2 +γ+1 +� +≤T1 +exp +� +− 2−1T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� +1 − exp +� +− T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� ≤ 2T1 exp +� +− T6nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� +. +Finally, by setting C2 = T1 and C3 = T6 we get +I1 + I2 ≤ 8C2 exp +� +− C3nE− +1 +1+γ (Ψγ +u)δ +γ+2 +γ+1 +n +� +. +This completes the proof. +58 + +Proof of Corollary 1: First, by the property (2) in Theorem 3, we have +|2ηuij − 1| +|2�ηuij − 1| ≤ exp +� +(|Ω| − 1)−1ϵu +� ++ 1 +exp +� +(|Ω| − 1)−1ϵu +� +− 1 = Ψu. +By the definition of excess risk, we have +�R(f) − �R(f ∗) = +1 +m(m − 1)Eu +� � +i25.0 +25.44 +Strehl ratio over NIRCam FOV at 2.0 µm +>0.8 +0.84 +Strehl ratio over MIRI FOV at 5.6 µm +>0.8 +0.92 +Encircled Energy Stability over 24 hoursa +<2% +0.2%c +Encircled Energy Stability over 14 hoursb +<3% +0.53%c +Vignetting +None +None +a Aperture radius of 80 mas at a wavelength of 2µm from the mean over a +24 hour period, evaluated for a worst-case 10◦pitch maneuver. +b Aperture radius of 80 mas at a wavelength of 2µm from the mean over a +14 day period, evaluated for a worst-case hot-to-cold pitch maneuver. +c Typical performance values. Occasional larger instabilities are observed +due to tilt events, discussed in Section 6.2.1. +help answer some of the biggest questions facing humanity, +such as how the universe works and how we got here. +2. Telescope Requirements +The telescope architecture was designed to meet just a few +driving optical requirements derived by early science trade +studies, as presented in Table 1 alongside the on-orbit perfor- +mance values. A large, primary mirror aperture was required +to have a total unobscured collecting area greater than 25 m2 +that operated over the spectral range 0.6 µm to 27 µm. The +optical area x transmission product was the metric used to +specify the performance of the coatings, contamination, and +micrometeoroid damage. For this large aperture, the tele- +scope image quality was specified using the Strehl ratio met- +ric at 2 µm2 and 5.6 µm, with a diffraction-limited image +quality value of 0.8 over the NIRCam and MIRI fields of +view, respectively. Finally, a set of encircled energy stability +requirements were defined at time intervals of 24 hours and +14 days, specifying less than 2.3 % and 3.0 % RMS varia- +tions of the energy within a 80 mas radius aperture at 2µm +wavelength following a worst case hot-to-cold slew within +the field of regard (FOR). +The radiometric sensitivity for the observatory required the +observatory to have low backgrounds that enabled the detec- +tion of faint sources. The largest near-infrared background +component was to be from zodiacal light. In the longer wave- +lengths of the mid-infrared channel, the requirements permit- +ted the background to be dominated by thermal self-emission +from the observatory rather than the natural sky, but with that +2 A key requirements downscope was dropping the formal requirement for +1 µm performance, to reduce complexity in the integration and test (I&T) +program, with the expectation that an optical system meeting or exceeding +requirements at 2 µm would necessarily also yield very good performance at +1µm. This has proven the case in flight. See Section 7.4 for lesson learned. +thermal emission tightly constrained to yield the required un- +precedented mid-infrared sensitivity. The near-infrared back- +ground was limited by the observatory design and modeling, +careful control of particulate levels, and the use of low reflec- +tivity baffle materials. The low-background, mid-infrared en- +vironment is enabled by passively cooling the telescope and +science instruments to cryogenic operating temperatures for +the life of the mission. The observatory operates in a Lis- +sajous orbit about the Earth-Sun L2 point that permits a sun- +shield to continuously keep the telescope and science instru- +ment protected from the sun. (Rigby PASP Backgrounds) +provides a detailed discussion of the backgrounds and the +on-sky measurements. +All of the performance requirements were defined to be +met at 5.5 yr after launch, referred to as “End of Life” (EOL), +as that was the minimum required lifetime of the mission3. +The optical performance properties are expected to degrade +over time due exposure within the space environment (e.g., +Section 6.2.2). The optical performance requirement predic- +tions accounted for these degradations and used the worst- +case prediction for each input category. For example, the +encircled energy stability requirement used the worst-case +wavefront error, thermal distortion, and image motion all +happening at the same time, which is a condition that is +known to be statistically unlikely to occur. The worst case +predictions also included model uncertainty factors (MUFs) +to manage uncertainties and provide margin at the system +level. +Therefore, the performance measured during com- +missioning, the so-called “Beginning of Life” (BOL) optical +properties, should not be directly compared to the require- +ment EOL values. +In addition to the driving performance requirements, the +telescope had challenging design constraints in order for +the observatory to meet its system-level requirements. The +telescope total mass was specified to be less than 2460 kg +and the measured mass at launch was 2339kg. +The cen- +ter of mass lateral location was specified to fit within a +200mm×200mm envelope and the Observatory fit within +that envelope with 37.9 mm margin to the nearest boundary. +The stowed telescope volume was set to be within an enve- +lope 3.985 m × 4.114 m × 6.942 m (V1,V2,V3). Both +the mass and stowed telescope volume requirements were +driven by the capabilities of the Ariane 5 launch vehicle. +The deployed telescope volume expanded considerably to +9.386 m × 6.100 m × 7.971 m, which needed to be shaded +by the sunshield while pointing within the designated field of +regard. Finally, the power consumption of the telescope was +3 The only consumable onboard JWST is the propellant, which is pre- +dicted to have a lifetime of greater than 20 years (Menzel PASP Observa- +tory). + +4 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 1: The telescope and science instruments are sepa- +rated from the spacecraft bus and shielded from direct sun- +light by a 5-layer sunshield. The telescope’s primary and sec- +ondary mirrors are open to the celestial sky. When stowed +(below), the observatory volume is significantly reduced to +fit within the Ariane 5 fairing and pinned mechanisms enable +the structure to withstand the launch environment. The V co- +ordinate system origin is at the vertex of the primary mirror +surface, along the boresight. +set to be less than 50 W (measured 34 W) such that the total +power budget for the observatory was maintained. +3. Telescope Architecture +The observatory architecture consists of three major sys- +tems: the telescope and science instruments, the 5-layer sun- +shield, and the spacecraft bus (Nella et al. 2004; Lightsey +et al. 2012). Renderings of the observatory in its operational +and stowed configurations are shown in Figure 1. The tele- +scope and the science instruments are passively cooled by the +sunshield and thermally isolated from the spacecraft bus and +solar array that are on the warm, sun-facing side of the obser- +vatory. The telescope’s primary mirror (PM) and secondary +mirror (SM) are open to the celestial sky, but the tertiary and +fine steering mirror are enclosed within the aft optics system +(AOS). The telescope Cassegrain focus is formed near the +entrance to the AOS where an aperture mask is located for +stray light suppression. There are four science instruments +(SIs): the NIRCam (Rieke et al. 2005), NIRSpec (Jakobsen +et al. 2022), MIRI (Rieke et al. 2015; Wright et al. 2015), +and FGS/NIRISS (Doyon et al. 2012). All of the science in- +struments are mounted within the same structure, called the +integrated science instrument module (ISIM), which is blan- +keted from the celestial sky. The instrument drive electronics +and radiators are mounted on the exterior of the ISIM. Fig- +ure 2 shows an exploded view of the telescope components +along with the science instruments, thermal management sys- +tem and mechanical supports. +The telescope field of regard is restricted to pointings that +shield the telescope and science instruments from direct so- +lar illumination. The telescope boresight pitch constraints are +between pitch angles of 85◦ and 130◦ (0◦ pointed towards the +sun), roll about the telescope boresight is constrained to ±5◦, +and yaw is unconstrained to a full 360◦ around the sunline. +The observatory is in an orbit around the Earth-Sun L2 La- +grange libration point, such that it orbits the sun along with +the Earth over one year. While the Earth and L2 Lagrange +point sweep along the celestial sphere, the field of regard on- +sky visibility changes. At any instantaneous moment, 40% +of the sky is visible to the telescope, while over the course +of the year the full sky is observable. The total visibility pe- +riod over the course of a year increases with ecliptic latitude, +ranging from ∼100 days in the ecliptic plane to continuous +visibility in 5-degree-radius cones at the ecliptic poles. +The JWST telescope was designed to be stowed in or- +der to fit within the Ariane 5 fairing for launch. Following +launch, the observatory needed to be deployed into the oper- +ational configuration. The 18 primary mirror segments are +arranged in a close-packed configuration with twelve seg- +ments in the center section and three segments on each of +two wings which were folded back for launch. No segment +is located at the nominal center position, as it is filled with +the aft optical structure that holds the telescope tertiary and +fine steering mirrors. The secondary mirror was also stowed +such that it was face down above the center section, shielding +its surface from particulate redistribution during launch. The + +Front +V3 +6.100 m +V1 +V2 +Optical +Telescope +6.600 m +Element (OTE) +14.625 m +V3 +Side +Integrated Science +Instrument Module (slM) +Sunshield +Spacecraft Bus +21.197 m +Stowed Side +V3 +10.661 m +4.472 m +JWST_005THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +5 +Figure 2: This exploded view shows telescope components, including the telescope mirrors, the optomechanical structures, +control electronics, and the thermal management system. The integrated science instrument module includes the four science +instruments. +telescope deployments included separating from the space- +craft bus using a deployable tower assembly, driving the sec- +ondary mirror into position, rotating and latching the primary +mirror wings into position, and finally deploying the primary +mirror segment assemblies and the secondary mirror from +their launch locks. +The telescope alignment used image- +based sensing and control with the primary mirror segments +and secondary mirror. +3.1. Design Implementation +The optical design of the telescope is a three-mirror anas- +tigmat (TMA) that corrects spherical aberration, coma, astig- +matism, and field curvature (see Korsch (1972)). In addition +to the primary, secondary, and tertiary mirrors of the TMA, +the telescope also includes a fine steering mirror (FSM) +which is actively controlled to stabilize the line-of-sight opti- +cal path into the science instruments. A raytrace through the +telescope with annotations for the optical surface is shown in +Figure 3. Each of the 18 primary mirror segments are 1.32 m +(flat-to-flat) separated by 7 mm gaps (see Figure 4). The PM +circumscribed diameter is 6.64 m (segment flat-to-flat diam- +eter is 6.575 m, entrance pupil diameter is 6.6 m) with an +area equivalent to a 5.7 m circular filled aperture. The tele- +scope is f/20 with an effective focal length of 131.4 m. The +field is about ∼ 18′ × 9′ at the telescope focal plane. Each of +the four science instruments has one or more pick-off mirrors +near the focal plane of the telescope such that they observe +offset parts of the sky. The telescope optical parameters are +reported in Table 2. +Figure 3: The JWST telescope is a three mirror anastigmat +that has a primary mirror, secondary mirror, tertiary mirror, +and a fine steering mirror. Each of the four science instru- +ments has one or more pickoff mirrors near the telescope’s +focal surface such that they see different fields of view. +Each of the 18 primary mirror segments is controlled in the +six mechanical degrees of freedom (DoF) using a hexapod +with six actuators. The segments are semi-rigid with a radius +of curvature actuator at the center of each. During the align- +ment process, the radius of curvature for each primary mirror + +Aft Optics +Telescope +Secondary Mirror Support +Subsystem (AOS) +Electronic +Structure (SMsS) +Boxes +ISIM Radiators +ISIM +Science +Secondary Mirror +Instruments +Assembly (sMA) +ISIM +Harness +Primary Mirror Segment +Radiator +Assemblies (PMsA) +ISIM +Electronics +Backplane Support +Structure +Thermal Management +Components +Deployment Tower +Assembly (DTA) +JWST_012OTE!ISIM +V3 (anti-spacecraft) +(V1, V3 +V2 +origin) +V1 +Cassegrain +Tertiary +focus +mirror +Fine- +steering +Focal +Secondary +mirror +surface +Mirror +Primary→ +Mirror +Sunshield +Sunlight +JWST_0016 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 4: The primary mirror dimensions and the tiling con- +figuration with 18 hexagonal-shaped segments to form a tri- +contagon. The A-, B-, and C-segment prescriptions are sep- +arate and take advantage of the six-fold symmetry. +segment is matched and the primary and secondary mirrors +are optimally aligned. The gaps between segments, nomi- +nally 7 mm, were tracked during the deployment and align- +ments by bookkeeping the actuator resolver counts and in- +dependently monitoring the positions using linearly variable +differential transformers (LVDT) electromechanical sensors. +All of the telescope optics are made from Beryllium based +primarily on its low coefficient of thermal expansion (CTE) +over the mirror’s operating temperature range of ∼35-55 K. +Segments closer to the warm spacecraft core region have cor- +respondingly higher temperatures. +Mirror moves are controlled through the actuator drive +unit (ADU) that is housed on an electronics panel within the +spacecraft bus. The ADU provides state-of-health telemetry +for the telescope hardware, controls the 132 mirror actuators, +and polls telemetry like motor revolutions and strain gauges +on the telescope. Those signals pass through cryogenic elec- +tronics boxes: first the cold multiplexer units (CMUs) located +on the telescope backplane, and a cold junction box (CJB) +before returning to the ADU. In a separate mode, the ADU +controls the fine steering mirror linear voice coil motors and +provides telemetry for its x,y position, current, and tempera- +tures. +Passive stability is achieved through mechanical ∼1 Hz +isolators between the telescope and the spacecraft bus and +a thermally stable backplane support for the telescope op- +tics made from a lightweight composite. The reaction wheel +assemblies (RWA) and cryocooler compressor assemblies +housed within the spacecraft include vibration attenuators +within their subsystems. +The composite truss structure is +comprised of a center section that supports 12 segments, and +two wing sections, each supporting 3 segments. There is a +separate backplane support frame that carries the load from +the telescope structure and the ISIM. The secondary mirror is +Table 2: Telescope Optical Parameters +Mirror +RoC +Surface +Conic +V1 +V2 +V3 +Phys. Size +(mm) +(mm) +(mm) +(mm) +(mm) +Primary +15879.7 +concave +-0.9967 +0 +0 +0 +6605.2 +Secondary +1778.9 +convex +-1.6598 +7169.0 +0 +0 +738 +Tertiary +3016.2 +concave +-0.6595 +-796.3 +0 +-0.19 +728×517 +Fine Steering +flat +1047.8 +0 +-2.36 +172.5 +NOTE—The primary-to-secondary effective focal length (EFL) is 59400 mm., and the three- +mirror telescope EFL is 131400 mm. +mounted to a tripod that consists of four composite tubes and +connects to the primary center section in a four-bar linkage +(see Figure 2). +Active pointing stability is achieved using a control loop +that senses line of sight pointing with a fine guidance sensor +(FGS, ∼16 Hz) and corrects pointing errors using the FSM. +This controls line of sight pointing drifts with a 0.78 Hz con- +trol loop bandwidth. In addition, as the fine guidance sen- +sor can only provide pointing measurements about two axes +(image tip and tilt), the coarse pointing roll controller uses +star tracker and inertial reference unit (IRU) measurements +to stabilize the roll (clocking) about the boresight. +How- +ever, thermal distortions may cause drifts between the star- +tracker-based spacecraft pointing frame and the line of sight +frame, which cannot be measured and therefore would re- +main uncorrected. This could create a rotational drift about +the guide star, which manifests as image smear elsewhere in +the science instrument field of view (see Section 5.4.4). Ad- +ditionally, higher frequency image motion, due to reaction +wheel and cryocooler disturbances, results in a Gaussian im- +age blur. The image motion has a negligible effect on the +encircled energy, but may degrade the image quality defined +by the Strehl ratio requirement. +The telescope open architecture for passive cooling makes +it susceptible to stray light from the celestial sky and the ob- +servatory emission itself. There are successive layers of stray +light protection used to shield the science instrument focal +planes. The sunshield protects the optical path from direct +sunlight and enables passive cooling of the telescope and sci- +ence instruments that reduces the self emission. The spreader +bars that hold the sunshield layers in position have baffles on +their caps, called epaulettes, that shield the optical path from +the warm spreader bars. +There is an oversized, internal pupil stop near the FSM +to transfer the maximum PM collected energy and provide +a well defined pupil for wavefront sensing. A lightweight +structure, called the ‘frill’ (see Figure 1 top), extends from +the perimeter of the PM and substantially fills in the gap be- +tween the oversized pupil stop and the image of the PM at the + +1.32 meters +flat to flat +B1 +C6 +C1 +B6 +A1 +B2 +A6 +A2 +C5 +C2 +A5 +A3 +B5 +A4 +B3 +C4 +C3 +6.6 meters +B4 +flat to flat +JWST_002THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +7 +stop. By filling in this gap, the frill blocks celestial light from +behind and around the PM (‘truant’ stray light path) from be- +coming a source of background stray light. A similar baffle, +called the bib, extends below the frill and blocks the direct +path to the warm spacecraft core area. +The AOS enclosure blocks stray light around the tertiary +and FSM mirrors. The AOS entrance aperture is near the +Cassegrain focus of the telescope and provides an aperture +stop for stray light suppression. A baffle extending above the +FSM combined with careful telescope alignment, prevents +light from the sky passing through the AOS entrance aperture +and striking any instrument optics directly. This stray light +path, called the rogue path, was identified early in the de- +sign phase and precautions were taken to keep light from this +path from propagating directly through the SI optical trains +to their detectors. +In flight, however, scatter paths off SI +mechanical structure were found to produce unwanted back- +grounds from sources in this small region of sky (see Sec- +tion 5.3 in Rigby PASP Science, Section 7.10 lesson learned). +Fortunately, the rogue path stay light can be largely managed +through observation scheduling. +The integrated science instrument module is enclosed and +blanketed for stray light and thermal purposes. Each of the +science instruments also has an enclosure and the optical +paths include internal baffles. +4. Telescope Development +The telescope development was a long and complex pro- +cess that took place between 2002 and 2022. In this section, +we provide an overview of the telescope development phase +with discussions of the systems engineering, the new tech- +nologies developed, and the integration and testing needed to +verify the design and workmanship. +Prime contractor Northrop Grumman was responsible for +the telescope design and built a team that included optics lead +Ball Aerospace, composite lead ATK (later to be bought by +Northrop Grumman), and the L3Harris Corporation (orig- +inally part of Kodak, then ITT Exelis) for their large op- +tics integration heritage. The lead government organization +was NASA’s Goddard Space Flight Center, which took full +responsibility for the telescope and provided key facilities +where the mirrors and science instruments were integrated +with the backplane structure and tested at ambient temper- +atures for vibration and acoustics. The Jet Propulsion Lab- +oratory and NASA’s Marshall Space Flight Center helped +with technology development and mirror testing, while cryo- +genic tests took place at NASA’s Johnson Space Flight Cen- +ter. The Space Telescope Science Institute provided lead- +ership in developing telescope operations plans and helped +lead the commissioning efforts. Throughout the development +phase, groups of scientists and engineers worked collabora- +tively across these organizations to jointly design, integrate, +test, and commission JWST. +Further, the telescope development team sought input from +external expertise for technical decision making. The project +formed an independent Product Integrity Team (PIT) to pro- +vide technical engineering advice on the telescope. +The +PIT was led by Professor Duncan Moore of University of +Rochester and Professor James Wyant of University of Ari- +zona and included space- and ground-telescope experts from +Hubble, the Spitzer Space Telescope (Spitzer), Keck and +many others. Throughout the development phase, the PIT +independently reviewed all aspects of the optics program, +especially the integration and test campaign with its direct +ties to the verification plans. The early test concepts ma- +tured and improved significantly over time with input from +the optical PIT (e.g., Feinberg et al. (2006)). In cases where +performance predictions threatened requirements, the JWST +science requirements advisory board was convened, with rep- +resentation from the JWST Science Working Group, to eval- +uate the science impacts. +The telescope design process began with the mirror sub- +strate and moved outwards to the backplane and ultimately +to the whole telescope and observatory design. New tech- +nologies were developed in parallel. Sub-system develop- +ment schedules were managed such that the telescope would +meet the larger observatory schedules. Early engineering de- +sign units (EDUs) of the primary mirror segment and sec- +ondary mirror were important to prove out the manufactur- +ing process for these critical-path components. The primary +mirror EDU demonstrated technology readiness, TRL-6, by +carrying out successful acoustic and vibration tests that were +not completed in the earlier technology development period. +An engineering design unit partial version of the OTE center +section, called Pathfinder, allowed verification of assembly, +handling, and testing techniques. +4.1. Systems Engineering for the JWST Telescope +The JWST telescope development relied on systems en- +gineering principles, tools, and practices as described in +Menzel PASP Observatory. +The systems engineering ap- +proach for JWST is presented thoroughly in Lightsey & +Arenberg (2018). The telescope performance requirements +relied on detailed budgets for optical performance (Lightsey +et al. 2010), alignments (Glassman et al. 2016), and actuator +ranges (Barto et al. 2012). The budgets were used through- +out the mission development: to make design decisions and +carry out concept trades, to design the integration and test +program, to verify the requirements before launch, for oper- +ations planning, and to assess the on-orbit performance. +The optical performance requirements were managed +through the wavefront error budget that was based on the +image quality metrics of Strehl ratio and encircled energy + +8 +MCELWAIN, FEINBERG, & PERRIN ET AL. +(see Section 1). The top-level image quality requirements +put direct constraints on the system RMS wavefront error +(WFE). Specifically, the driving requirement of a Strehl ratio +of 0.8 at 2 µm wavelength required a system-wide RMS WFE +< 150 nm. A portion of this WFE budget was then allocated +to each subsystem. This top-level system-level performance +was used to make allocations to each of the sub-systems +(e.g., telescope, integrating structure, science instruments). +The driving encircled energy stability requirement was <3% +in a 80 mas radius aperture at 2µm wavelength over 14 days +following a worst case hot-to-cold slew. The encircled en- +ergy stability was budgeted by assigning wavefront error al- +locations in different spatial frequency composition regimes +based on capability of the active wavefront control. The low- +frequency modes were allocated based on the active global- +alignment aberration control, while the mid-frequency modes +were allocated based on the active primary mirror figure con- +trol. +The high-frequency modes were based on the static +high spatial frequency errors that were controlled during the +fabrication process. Each sub-system’s WFE allocation was +further divided into WFE static residuals, WFE stability, and +image motion. As a system, each allocation could either be +tested (e.g, mirror static high-frequency errors) or assessed +via integrated modeling (e.g., OTE stability). +JWST’s performance verification could not follow the tra- +ditional NASA paradigm to “test as you fly” (TAYF). The +test facilities were not available for end-to-end testing of an +observatory of this size operating at cryogenic temperatures. +Rather, the integration and test program developed for JWST +relied on incremental performance verification with testing +at the sub-system level and, to the extent possible, testing +at higher levels of assembly. The test conditions across the +observatory were dramatically different. For example, the +telescope and science instruments were cryovacuum tested +at their operational temperatures of ∼40 K in Chamber A at +NASA’s Johnson Space Center (see Section 4.3.4) whereas +the sunshield and spacecraft bus were cryovacuum tested at +temperatures ranging between 150–330 K in a more conven- +tional thermal-vacuum environment at Northrop Grumman +(McElwain et al. 2018). This piecewise verification relied +on subsystem test data to anchor the high-fidelity integrated +models used to make performance predictions for many of +the system-level requirements. +4.1.1. Integrated Modeling +Highly complex integrated modeling was a key enabling +capacity throughout all stages of development. The optical +performance estimates made use of ground test data, inte- +grated models, and simulations including the uncertainties +in wavefront sensing and control to verify the pre-launch +requirements (Figure 5). +Component design and ground +test performance was used to provide predictions for optical +alignments, component-level wavefront error, and ground-to- +flight effects. The dynamic components of the error budget +used test data as inputs to an extensive structural-thermal- +optical (STOP) integrated modeling process that predicted +wavefront stability (Knight et al. 2012) and line-of-sight im- +age motion (Johnston et al. 2004). The integrated modeling +for telescope performance made use of models of the struc- +ture, deployed thermal, thermal distortion, optical perfor- +mance, dynamics and attitude control, and stray light. Each +model was validated upon test data and conservative model +uncertainty factors were applied to bound the worst case per- +formance. Image motion predictions made use of exported +vibrations and a model of the deployed dynamics of the ob- +servatory. The telescope thermal distortion and pointing sta- +bility following a worst-case slew used a thermal model that +balanced the steady state at the hot attitude (pitched towards +the sun) and the cold attitude (pitched away from the sun) +with a worst case roll. The small temperature changes, less +than 15 mK, from those thermal extremes was predicted for +each of the thermal nodes on the telescope and used to de- +termine the mechanical displacements on the structure. The +repository of test data and integrated modeling results were +used as inputs to the Integrated Telescope Model (ITM) sim- +ulator developed by Ball Aerospace in order to predict the +optical performance, simulate data products for the develop- +ment of analysis tools, and to rehearse the telescope align- +ment process (Knight et al. 2012). +The same optical models were also used to inform pre- +flight modeling of point spread functions (in particular using +the software package WebbPSF, Perrin et al. 2014), which +were used extensively in science planning, in development +of proposal planning tools such as the exposure time calcu- +lator, and in development of data analyses pipelines. The +core Fourier optical simulations of PSFs were augmented +over time to become part of comprehensive high-fidelity data +simulators, such as MIRAGE for NIRCam, NIRISS, and +FGS data (Chambers et al. 2019; Hilbert et al. 2022) and +MIRISim for MIRI (Klaassen et al. 2021). These, along with +ITM, became critical enabling tools for the long campaign +of preflight rehearsals of the WFSC alignment process (Sec- +tion 4.2.3). +4.2. New Technologies Needed +Early in the mission development, three new telescope +technologies were identified that needed to be developed ex- +plicitly for JWST. The project made significant early invest- +ments in these enabling technologies to ensure they would +be at a technology readiness level (TRL) of 6 (TRL-6 = +fully functional model or prototype, demonstrated in a rel- +evant environment) or higher prior to the mission’s prelimi- +nary design review (PDR). The telescope technology matu- +ration program included primary mirror segment assemblies, + +THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +9 +Figure 5: JWST’s optical performance estimates were made using a combination of test data and integrated models. For require- +ment verifications, there were conservative model uncertainty factors applied and the end of life performance during worst case +conditions were assumed. +large precision cryogenic structures, and wavefront sensing +and control. In addition, new metrology capabilities were +needed to verify the performance, and a series of new inter- +ferometric techniques were developed to test the stability of +the composite structure and verify the optical performance +of the telescope at ambient and cryogenic temperatures (Saif +et al. 2021). The following discussion provides highlights +from the technology development program that was com- +pleted in 2006. +4.2.1. Primary Mirror Segment Assemblies +Low areal density mirrors were recognized as a key tech- +nology gap to enabling a ∼25 m2 aperture space telescope. +The areal density of the Hubble primary mirror is 240 kg +m−2, while JWST’s objective was < 26.5 kg m−2, which +was achieved. A mirror technology development program +was convened to evaluate and advance mirror technologies +through multiple programs, including the subscale Beryl- +lium mirror demonstrator (SBMD, Reed et al. 2001) and the +advanced mirror system demonstrator (AMSD, Stahl et al. +2004). The AMSD program evaluated ULE and Beryllium +with a wide variety of parameters such as the optical perfor- +mances achieved, control authority, mounting, and fabrica- +tion schedule. While ULE was deemed to have programmatic +advantages, it was found to have an astigmatism as it cooled +that was non-deterministic and would have added uncertainty +to the development. The decision to select O30 Beryllium, +a more isotropic form of Beryllium not previously used in +space telescopes, was made following the recommendation +from the Mirror Recommendation Board. Beryllium was se- +lected largely due to its small coefficient of thermal expan- +sion (CTE) within the telescope’s operational temperature +range, making it particularly advantageous during the cryo- +polishing fabrication process and achieving the telescope’s +passive stability objectives by not requiring active thermal +control. Beryllium is also light weight, advantageous given +the very tight mass constraints for the telescope (Feinberg +et al. 2012). Beryllium mirrors have flight heritage from pre- +vious space missions, including the Spitzer, the Infrared As- +tronomical Satellite (IRAS), and the Cosmic Background Ex- +plorer (COBE). +Active control of primary mirror segment position was +achieved using actuators mounted in a hexapod arrangement, +plus a center actuator for active control of radius of curva- +ture (see Figure 6). Specialized actuator mechanisms were +developed specifically for JWST in order to enable the active +positioning of the large segmented mirrors and to support the +mirrors during ground test and the launch environment. Each +mechanism makes use of a fine stage flexure and coarse drive +coupling to control the linear displacement (Warden 2006). +The actuators themselves have remarkable performance pa- +rameters including a fine step size of 7.7 nm resolution, with +2 nm of fine repeatability, over a 10 µm fine range. A coarse +drive coupling in the same mechanism provides a 58 nm step +size over a full 21 mm. Further, unlike ground telescope +active and adaptive optics, which often use electrostatic or +piezoelectric actuators, JWST’s actuators operate mechani- +cally via a gear train and flexures; the mechanical gear trains +hold position stiffly and precisely even when the actuator is +entirely unpowered, which is necessary to avoid undesired +waste heat into the cryogenic telescope. +The primary mirror segment development process required +new facilitization and made use of economies of scale to fab- +ricate the multiple segments in parallel. The mirror blanks +were made from O30 Beryllium through a hot isostatic press- +ing process by Brush Wellman. The blanks were then light + +StaticErrors +WFSC +Algorithms +Optical Alignments +Simulations +WFSC +Component WFE +Realizations +Processing +(PM, SM, TM, FSM, SIs) +with +ITM / MIRAGE +Ground to Flight Effects +PSF/OPD +Dynamic Errors +WFE Stability +Encircled +Strehl +Energy +Stability +Image Motion +JWST_00810 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 6: Each primary mirror segment and the secondary +mirror are supported by a hexapod as shown above. The pri- +mary mirror segments also have a radius of curvature actuator +that enables the focal lengths for each segment to be matched +during the alignment process. +Figure 7: Six of the flight mirror segments prepared to un- +dergo acceptance testing in the XRCF. The segment-level op- +tical performance and structural stability was precisely mea- +sured. +weighted by removing over 92% of the material in a honey- +comb shape by Axsys Technologies. Next, the mirror sub- +strates were polished by Tinsley Labs and each mounted +to its flexure and radius of curvature system. The mirrors +were then optically tested at ambient and cryogenic temper- +atures at X-ray and Cryogenic Facility (XRCF) at NASA’s +Marshall Space Flight Center, followed by another round of +cryo-polishing to ensure each mirror achieved the correct op- +tical figure at the intended cryogenic operating temperature +(Cole et al. 2006). The mirrors were gold coated using a vac- +uum vapor deposition process by QCI, Inc. The gold coating +provides high reflectivity across the operational wavelength +range of 0.6–28.1µm (Keski-Kuha et al. 2012). A protec- +tive SiOx overcoat was applied that improved the durability +of the coating and enabled cleaning at stages throughout the +I&T process (see Lobmeyer & Carey 2018). Finally, flight +acceptance testing for each segment was carried out in the +XRCF facility (see Figure 7). +4.2.2. Large Precision Cryogenic Structure +A large precision cryogenic structure was necessary to en- +able the passive optical stability without active control. The +telescope and the science instruments are supported by a +composite optomechanical structure that must withstand the +launch environment loads, deploy within the capture range of +the mirror actuators, survive the stresses induced from cool- +ing down to cryogenic temperatures, and have minimal ther- +mal distortion. +Early in the JWST development, it was recognized that the +materials database for composite structures was inadequate +over the operational temperature range, the ability to measure +deformations was inadequate for the JWST verification test- +ing, the engineering modeling tools needed development, and +manufacturing process controls needed improvement. The +coefficient of thermal expansion for the materials needed to +be measured to less than 30 ppb K−1 at temperatures of < +25 K, which was more than 100 times more precise than the +state of the art at the time (Atkinson et al. 2007). ATK imple- +mented a technique to measure the CTE for large structures +with the precision needed for JWST. With the materials char- +acterized, a prescription for the composite structure was de- +fined that used unidirectional prepreg made from M55J car- +bon fibers and resins into laminant mixtures tuned for the ap- +propriate strength and thermal performance. Manufacturing +controls were established to precisely align the fibers during +layup and closely manage the fiber to resin ratio necessary for +precise material properties. Controls were also put in place to +achieve the desired bonded joint adhesive thickness used to +connect the individual tubes into a truss. A prototype of the +composite structure, called the Backplane Stability Test Ar- +ticle (BSTA), was built by ATK and tested at MSFC’s XRCF +(Figure 8). Verification of the structure’s stability made use +of a new Electronic Speckle Pattern Interferometer (ESPI) +metrology approach (Saif et al. 2008), a technology develop- +ment in itself, to confirm the structure was TRL-6. +4.2.3. Wavefront Sensing & Control +The image-based phase retrieval methods used with JWST +have a heritage stretching back to the diagnosis and correc- +tion of the infamous spherical aberration in the Hubble pri- +mary mirror (Krist & Burrows 1995; Fienup et al. 1993). +Phase retrieval using the science instruments elegantly avoids +the need for substantial dedicated wavefront sensing hard- +ware, and ensures sensing of wavefronts directly at the sci- +ence focal planes. However, to accommodate the evolving +alignment of the mirrors (from initial deployment errors mea- +sured in millimeters to final alignments measured in nanome- +ters) these methods must operate over a tremendous dynamic +range, and must also sense dissimilar and degenerate degrees +of freedom. As a result several distinct wavefront sensing +methods must be used. The primary tool is focus-diverse +phase retrieval, using a hybrid diversity algorithm developed + +Actuator for ROC +Beryllium ROC strut (6x) +GSE handling ring +Backplane interface flexure (3x) +Actuator (6x, 3 bipods for 6 DOF) +Beryllium delta frame +Beryllium whiffle (3x) +Beryllium mirror substrate +JWST_003THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +11 +Figure 8: The Backplane Stability Test Article (BSTA) was +an engineering model for 1/6 of the JWST backplane, includ- +ing the structural elements used to create the entire backplane +assembly. This shows the BSTA being prepared for cryotest- +ing at operational temperatures in the XRCF. +specifically for JWST (Dean et al. 2006). Focus diversity +is provided at different stages by defocusing the secondary +mirror or by using weak lenses within NIRCam that can be +inserted into the beam path. The focus diversity method is +augmented with dispersed Hartmann sensing for the mea- +surement of segment piston (Shi et al. 2004). +The step-by-step sequence of sensing and control activi- +ties, as well as the associated algorithms and software, were +developed at Ball Aerospace. +To test and prove the im- +plementation, a 1:6 scale model and functionally accurate +Test Bed Telescope (TBT) was built (Kingsbury & Atche- +son (2004), see Figure 9). Using the TBT, the complete end- +to-end telescope alignment process was successfully demon- +strated, achieving TRL 6 in 2006 (Acton et al. 2007; Feinberg +et al. 2007). +Even with those fundamental tools proven, a decade of +work remained to mature them from lab-scale demonstra- +tion to flight-ready processes. Operational implementation +of the commissioning plan was complicated by the need to +begin operation of fine guidance control while still adjusting +mirrors (see Section 5.2), and to interweave telescope align- +ment with prerequisite steps of instrument commissioning +such as focal plane calibrations (see Section 5.3). The meth- +ods were refined and operational plans were prepared lead- +ing up to launch (Perrin et al. 2016), culminating in detailed +implementation plans, procedures, and observing programs. +Contingency plans were prepared in the event that nominal +plans could not be followed, for many distinct contingency +scenarios. Phase retrieval analysis software was similarly it- +eratively refined prior to launch; the initial delivery of flight +Figure 9: The JWST Test Bed Telescope (TBT) is a 1:6 scale +model of JWST’s telescope with the full sensing capabilities +and control authority, located at Ball Aerospace. +WSS software occurred in 2011, and regular improvements +continued thereafter as part of the I&T of the Science & and +Operations Center (SOC). +These processes for WFSC were repeatedly tested together +with the flight hardware at various stages of observatory I&T. +The initial test and operation of the integrated OTE electron- +ics and mechanisms took place in 2016 (see Section 4.3.1). +During the cryovacuum test of the telescope plus instrument +suite (Section 4.3.4) while the majority of that test program +used GSE for metrology of observatory alignments, specific +activities were included to test the flight scripts for wavefront +sensing and control using flight hardware. This was the first +and only time that NIRCam was used to sense OTE mirror +alignments on the ground (Acton et al. 2018). In parallel, +the data generated by that activity was flowed back to STScI +and used for a demonstration of sensing and control software +processes using the integrated SOC. This was the first ma- +jor demonstration of processing of JWST data in a flight-like +manner at the mission operations center (Lajoie et al. 2018). +The commanding for mirror moves and deployments was re- +peatedly exercised as part of regular OTE functional checks, +up to and including at the launch site. +A necessary input to the WFSC process was accurate +knowledge of instrument-specific wavefront errors, to allow +subtracting the instrument contributions from the results of +the image-based phase retrieval to perform OTE wavefront +control. This objective was met through precise measure- +ments at dozens of field points within all instruments, as part +of instrument cryo-vacuum testing completed by 2016. +The wavefront sensing and control activities demanded +human-in-the-loop controls and required training a large +wavefront team for round the clock operations during com- +missioning. The telescope was aligned start-to-finish over a +hundred times in simulation, individually by many members +of the team and in collaboration. As part of this training +process, there were iterative refinements of the methods, pro- + +12 +MCELWAIN, FEINBERG, & PERRIN ET AL. +cedures, and documentation. The individual simulations built +towards larger team rehearsals, including 20 internal wave- +front team practices and 25 mission operations or science +operations team wide rehearsals. Many of these rehearsals +were carried out throughout the COVID-19 pandemic under +work-from-home conditions, remotely. +The extensive re- +hearsal program was a critical, invaluable activity in building +a smooth-functioning cohesive wavefront team combining +staff from multiple organizations and skillsets. +4.3. Integration & Testing +The JWST integration and test program began at the com- +ponent level and continued as the hardware was integrated +into successively larger sub-assemblies, assemblies, and +eventually the entire observatory system. +At various lev- +els of assembly, testing took place to confirm the functional +performance and verify the workmanship. The testing ap- +proach followed a philosophy of independent, incremental +testing with predefined success criteria, crosschecks that in- +cluded end-to-end testing, comprehensive external and inter- +nal review, thorough risk management, and open transparent +communications and documentation. A strict requirement of +the optical test program was that the optical test equipment in +a verification test had to be independent of the test equipment +used to fabricate the optics (Feinberg & Geithner 2008). +In this section we highlight some key activities from the +later stages of the I&T program for the OTE. We then dis- +cuss the major I&T activities for the OTIS, which is the term +for the combination of the OTE plus the Integrated Science +Instrument Module (ISIM); OTIS = OTE + ISIM. We con- +clude the section with a summary of the activities involving +the OTIS after delivery back to Northrop for integration with +the rest of the observatory. A high-level flow of the OTE +activities through its integration and testing through Obser- +vatory readiness for launch is presented in Figure 10. +4.3.1. Flight OTIS Integration & Ambient Testing +The major structural elements of the OTE (see Figure 2), +as well as the associated electrical harnesses, were integrated +and tested at Northrop in 2014–2015. Major tests included +load testing of the mirror backplane and testing of the De- +ployable Tower Assembly (DTA). After precision integration +of the DTA, the two telescope wing structures, and the Sec- +ondary Mirror Support Structure (SMSS) to the backplane, +these subsystems were exercised for functionality and re- +peatability with ambient deployments (Glassman et al. 2016). +Modal surveys were also carried out in the stowed and de- +ployed configurations, with appropriate mass simulators for +hardware to be integrated later. +After shipping of this hardware to GSFC, the integration +of additional harnesses and small electronics boxes (e.g. for +the mirror actuators) and the OTE optics took place in a +dedicated assembly and alignment facility in GSFC’s largest +cleanroom. The PMSA shim prescription was determined us- +ing metrology from a coordinate measuring machine brought +from Tinsley Labs, now Coherent Inc., and laser tracker mea- +surements of the composite backplane structure. The PM +segments were mounted to the backplane with the assistance +of a traveling robot arm, as shown in Figure 11. Laser track- +ers measured the alignment state to guide the installation, +with laser radar independently measuring. Custom ground +shims and adhesive-filled pin gaps secured the location to +mechanical tolerances that were a small fraction of the range +budget for the PM actuators (see Atkinson et al. 2016 for de- +tails). +Integration of the Secondary Mirror Assembly and the Aft +Optics Subsystem (AOS) completed the OTE. With the in- +tegration of the Fixed and Aft Deployable ISIM Radiators +and the ISIM itself (the instrument module and the associ- +ated electronics compartment), and an array of thermal and +stray light control blankets (>900!), the OTIS was complete. +The ISIM of course had been through its own comprehen- +sive I&T program, including three cryo-vacuum tests totaling +nearly 300 days of round-the-clock operations (Kimble et al. +2016). This integration of the OTE + ISIM to form the OTIS +took place in May 2016. +Using the fully assembled OTIS, the integrated mirror con- +trol system hardware and software was exercised, first in +small steps starting in October 2016, and eventually in partial +deployments of all 18 PMSAs and the SM in preparation for +ambient optical testing. This included operation of PMSA +actuators and sensors (resolvers and linear variable differen- +tial transducer [LVDT] length sensors), operated by the Actu- +ator Drive Unit electronics and controlled by the Wavefront +Sensing System software. One minor anomaly discovered +during this time is that a small number of LVDT sensors do +not operate nominally. This was accepted to “use as is”, given +the availability of other telemetry to confirm mirror motions +for those segments (e.g. using resolver telemetry). For some +of the affected LVDT sensors, a modified operations concept +was developed that used temperature-dependent calibrations +to make the sensor information usable, later used success- +fully in flight. +Mirror control mechanisms and processes +continued to be exercised throughout the remainder of OTIS +I&T, in particular during the OTIS cryovacuum test. +The OTIS underwent proto-flight level ambient environ- +mental testing (vibration and acoustics) at GSFC in 2016– +2017. Ambient optical measurements were carried out before +and after those mechanical tests utilizing a Center of Curva- +ture test setup, including a null lens and Computer Generated +Hologram (CGH, for working with the aspheric PM surface), +as illustrated in Figure 12; see Saif et al. (2017). The Cen- +ter of Curvature setup incorporated a high-speed interferom- +eter for making figure measurements at rates up to 5.9 kHz. +With this equipment, the static wavefront of the PM segments + +THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +13 +Figure 10: A high level overview of the telescope integration and test sequence through launch. + +Structures +Integrate Structure. +Ship to +Backplane +Telescope Optics +at NGSS +Deployment & Modal Tests +GSFC +Integrate Optics +Integrate ISIM +CoC Test +Sine-Vibe & Acoustics +Cleanroom +OTIS Cryo +CoC Test +Ship to JSC +Ship to NGSS +Staging by JSC +Optical Test +Post Env Deployments +Observatory I&T Start +OTIS Install & Deployments +Stow +Sine-Vibe & Acoustics +Deployments +Stow +Ship to GSC +Launch +Legend +Component +Integration +Test +Prep & Transport +WST_01814 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 11: Installation of the final Primary Mirror segment, +assisted by a high-precision robotic arm, suspended from +traveling stages. Black covers were mounted on the mirrors +to protect from contamination. +Figure 12: Center of Curvature test setup for ambient opti- +cal measurements of the PM. The high-speed interferometer, +null lens, and CGH are mounted on the stable platform at +lower left. +was measured before and after the mechanical environmen- +tal testing, along with the dynamic response of the backplane +and mirror mounts. With a vibration stinger to excite the pay- +load, mechanical transfer functions were measured to look +for any signs of damage after the vibe and acoustics tests +– e.g. cracks in the structure; loosening of joints. No such +damage was seen, with the figure and dynamic measurements +repeating pre- and post-test within expected tolerances. Elec- +trical functional checks were also carried out before and after +the mechanical environmental testing, along with “first mo- +tion” (flinch) tests of deployment systems that couldn’t be +fully deployed at GSFC in the one-g environment. +An anomaly did arise during the OTIS-level vibration test, +when a loud “bang” was heard. This was determined to have +been caused by gapping at one of the preloaded interfaces of +a Launch Release Mechanism in a PM mirror wing. Mod- +ifications to the procedures to properly set and maintain the +preload of these interfaces resolved this issue, and no damage +had been done. In addition, excessive resonant response was +seen for the SMSS and the AOS at some frequencies due to +the low damping of the large composite structure. The test vi- +bration spectrum was notched at these frequencies to protect +the hardware during the OTIS-level test. Particle dampers +were subsequently designed and installed onto the AOS and +SMSS to reduce these responses; they operated successfully +later at Observatory-level testing and ultimately through the +actual launch. +4.3.2. Flight OTIS Cryo-Vacuum Test +The final phase of I&T for the OTIS was an extraordinarily +challenging cryo-vacuum test, previously described in Fein- +berg et al. (2011) and Kimble et al. (2018). This took place at +NASA’s Johnson Space Center in historic Chamber A, which +is a US national historic landmark from the Apollo program. +After shipment to JSC, the OTIS underwent electrical func- +tional testing, the SMSS was deployed (with assistance – the +necessary GSE for a powered deployment in one-g was only +at Northrop), and the DTA and PM wings were deployed. +After these activities, the payload was configured for the +cryo-vacuum test in Chamber A, which had been extensively +refurbished for the thermal and contamination requirements +of JWST and outfitted with optical and thermal GSE com- +prising an elegant test architecture, illustrated in Figure 13. +This architecture supported a rich array of operational, ther- +mal, and optical test goals, with 40 separate tests. End-to-end +optical tests were carried out using sub-apertures of the pri- +mary mirror. The test campaign applied lessons learned from +the Hubble program. +Key components of the optical test equipment included +the Center of Curvature Optical Assembly (COCOA, Wells +et al. 2010), whose interferometers had a view of the entire +PM; photogrammetry (PG) cameras on four rotating booms +(Lunt et al. 2020), which provided remarkably accurate rela- +tive positions (sub-100 µm) of optical targets over the many- +meter distances involved, through image triangulation; the +AOS Source Plate Assembly (ASPA), which mounted light +sources (fiber-fed or local) at the intermediate Cassegrain fo- +cus of the OTE – these provided downward (half-pass) im- +ages through the TM, FSM, and SIs and upward (pass-and- +a-half images) through the SM, PM and then, after reflec- +tion off sub-aperture Auto-Collimating Flats (ACFs), back +through the entire OTIS optical train; and fiducial light strips +straddling the edges of the PM. The position of the ASPA +sources made their images highly aberrated, but in a pre- +cisely known way, so that alignments and OTIS wavefront +measurements could be extracted nonetheless. The down- +ward sources were used to test the guiding control loop. +The optical test equipment (see Figure 14) worked together +to satisfy the critical optical verification goals (such as the +verification of the non-adjustable AOS to ISIM alignment, + +THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +15 +Figure 13: OTIS cryovacuum test configuration showing the telescope inside the chamber with optical metrology test equipment. +verification of the radius of curvature) as well as various +cross-check goals. A succinct description of the process is +as follows: +• PM segments were aligned and phased via photogram- +metry and COCOA interferometry. +• The SM was aligned via photogrammetry and checked +with Pass-and-a-Half imaging to the NIRCam instru- +ment. +• AOS to ISIM alignment was verified via Half-Pass +imaging using inward facing sources and all science +instruments. +• Fiducial lights above the primary mirror were used +for verifying pupil alignment, using NIRCam’s pupil +imaging capability. +• End-to-end imaging and field tilt was cross-checked +using Pass-and-a-Half imaging using outward facing +sources, the autocollimating flats, and all of the sci- +ence instruments. +• Wavefront Sensing & Control hardware checks and +demonstrations were performed via Pass-and-a-Half +testing and NIRCam. +4.3.3. The Pathfinder Program +The Pathfinder Program used spare and test equipment +as surrogates for the flight hardware in order to prove out +many of the integration and testing activities (see Feinberg +et al. 2014, Section 7.3). +The test article, referred to as +the “Pathfinder”, was comprised of two spare primary mirror +segments, a spare secondary mirror, and composite structure +representative of the center section and the secondary mirror +supports. The Pathfinder was transported and integrated us- +ing the protocols and procedures that would later be used on +the flight hardware, in some cases demonstrating the actual +capabilities of the test equipment. +Following integration, the Pathfinder was used to prepare +the cryovacuum testing facility and equipment at JSC in +2015/2016, while the OTE and OTIS were being integrated +and tested at Goddard. The Pathfinder cryotest program uti- +lized the thermal and optical GSE developed for the OTIS test +and the Pathfinder structure itself (Matthews et al. 2015a). +The first Optical GSE test (OGSE1) checked out the CO- +COA and PG operations with those systems, while the sec- +ond (OGSE2), incorporated the flight AOS (hence requir- +ing careful coordination with the flight I&T flow) and the +ASPA to dry-run the half-pass and pass-and-a-half tests as +well. A third Pathfinder test included thermal mock-ups of +the remaining center-section PM segments and validated the +cooldown and warmup procedures that would be needed, in- +cluding the requisite contamination control (Matthews et al. +2015b). +4.3.4. The Flight OTIS Cryo-Vacuum Test Results +Cryo-vacuum testing of the flight OTIS took place in 2017. +The 100-day, round-the-clock test campaign was remarkably +successful, despite the many challenges, including the pas- +sage of Hurricane Harvey through the JSC area in the mid- +dle of the test, which devastated the local community and +shut down JSC for normal operations with 55” of rainfall. +After a 5 day period of special hurricane operations, and + +Vibration Isolators (6x) +Center of Curvature Optical +Assembly (cOcOA) +Upper Suspension +Frame +Autocollimating Flat +(ACFs, 3x) +He Shroud +(14m diam, 24m tall) +Photogrammetry Cameras +on Windmill Booms (4x) +AOs Source +Plate Assembly +Space Vehicle Thermal +Simulator +The Flight OTIS +Support Structure with +Deep Space Environment +Radiation Sinks (DSERS) +JWST_00416 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 14: Simplified OTIS optical test schematic. +nearly running out of liquid nitrogen, the cryotest narrowly +missed being abruptly ended. Fortunately the hurricane did +not preclude safely continuing the test. All significant goals +for the test were achieved, including confirming the health of +the OTIS payload after its environmental test program (Wolf +et al. 2018), accomplishing the planned optical verifications +and cross-checks (e.g., Hadaway et al. 2018), validating the +OTIS thermal model and the OTIS thermal distortion model, +both required for the Integrated Modeling of the observatory +as a whole, and various operational validations and demon- +strations. +Performance of the OTIS was overall excellent, with pre- +dictions that satisfy the mission-level requirements (Light- +sey et al. 2018). But, there were several types of optical in- +stabilities identified. One, caused by over-tightness of the +soft-structure frill stray light blocker and PMSA closeouts at +the cryogenic operating temperatures (such that they exerted +temperature-dependent forces on the mirror backplane), was +mitigated with post-test modifications to restore the intended +slack where feasible. A second, which coupled temperature +variations in heater-controlled radiator panels on the instru- +ment electronics compartment (IEC) to the mirror backplane, +inducing oscillating structural distortions, was demonstrated +with ambient measurements and analysis to be caused by a +rigid non-flight mounting of the IEC for the OTIS CV test. +Both of these have been shown to have minimal wavefront +impact in flight (see Section 5.4.4) +A final instability, called “tilt events” referred to sudden, +stochastic changes in the piston/tip/tilt pose of individual +PM segments. Several such events were seen throughout the +OTIS test period. Though not fully explained, these were +ascribed to stick/slip release of stresses from cooldown ther- +mal deformation in the OTIS structure. These reduced dur- +ing the end of the test, and it was expected that these would +fade away with time after cooldown as the various stresses in +the system were gradually relieved. This behavior during the +OTIS test informed expectations that such events could be +seen in flight. This supposition appears to be confirmed by +the flight behavior (see Section 6.2.1), though with the ex- +cellent sensitivity of the flight wavefront sensing, most of the +observed tilt events in flight are actually below the detection +threshold of the OTIS cryo-vac analyses. +4.3.5. OTIS during Observatory I&T +After the deconfiguration of the OTIS from the test config- +uration, it was shipped to prime contractor Northrop Grum- +man’s Space Park in Redondo Beach, California. While at +Northrop, powered deployment of the SMSS and DTA took +place with appropriate one-g off-loading hardware. +Both +deployments were made from the spacecraft electronics to +demonstrate the connections and scripts were working prop- +erly. For the SMSS, this represented the only post-OTIS- +vibration powered deployment, confirming the health of the +deployment system after that proto-flight-level vibration ex- +posure. +In the summer of 2019, the OTIS was integrated with the +spacecraft and sunshield to form the full-up JWST observa- +tory. Alignment metrology was performed in the integrated +configuration to characterize the OTE to star tracker bore- +sights. +In the full-up observatory configuration, the payload un- +derwent various deployment tests (e.g., off-loaded deploy- +ment of the DTA and PM wings). The observatory as a whole +was then put through acceptance-level vibration and acoustic +tests, with subsequent deployment and electrical functional +tests. Like the OTIS, the spacecraft and sunshield had previ- +ously successfully undergone mechanical environmental test- +ing at proto-flight levels. +Both the PM and the SM were cleaned of particulates +at appropriate times in the Northrop flow, with a gentle + +Center of +Curvature +Interferometer +1.5m Diameter +Autocollimating +Flat (ACF) +Mirrors (3X) +Photogrammetry +Cameras on +windmill arms +(1 of 4) +nward and Outward +facing sources at +Intermediate Image +Fiducial lights +above PM edge +Absolute +Distance +ACF +Meter +ISIM +JWST_006THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +17 +brush technique described by Lobmeyer & Carey 2018. For +the SM, which had the most challenging particulate con- +tamination budget (so cleaning was desired as late as pos- +sible), this cleaning took place after the final stowing of +the observatory into its transport (and launch) configuration, +just before encapsulation of the observatory into a clean, +environmentally-controlled shipping container for transport +by sea to the Guiana Space Centre (GSC) in Kourou, French +Guiana. There, the observatory executed final ground func- +tional tests, was fueled, and was encapsulated in the Ariane +5 rocket fairing for launch. +5. On-Orbit Commissioning and Characterization +Following JWST’s launch, the telescope was deployed, +aligned, characterized, and readied for science observations. +In this section, we describe the overall sequence of activi- +ties executed during OTE commissioning as well as the re- +sulting performance of the optics, pointing, and focal plane +alignment. Many years of preparation, for both the observa- +tory hardware and the commissioning operations plans and +teamwork, resulted in a smooth and efficient commissioning +which completed successfully and as scheduled, and deliv- +ered an OTE performing at or above requirements. +5.1. OTE Commissioning Activities +To prepare the telescope for scientific observation, OTE +commissioning activities included the deployment of the mir- +ror segments from their launch restraints, the alignment of +the primary and secondary mirror segments, and the achieve- +ment of a finely phased telescope. The series of activities +was developed and rehearsed over many years and was al- +located ∼90 days to complete in the planned schedule. For +clarity, it is useful to break the whole sequence down into +a few major sets of activities: mirror segment deployment, +segment-level identification and alignment, co-phasing of the +segments, and multi-field alignment. Here, we summarize +these activities and describe the actual execution during flight +(see Figure 15). The detailed commissioning plan is pre- +sented in Acton et al. (2018) and additional details on the +as-run activities are presented in Feinberg et al. (2022) and +Acton et al. (2022). Briefly, the commissioning plan had +to accommodate initial PMSA and SM positional errors as +large as one millimeter, and progress to achieve fine align- +ments within a few tens of nanometers. This was achieved +using several different forms of wavefront sensing, most of +which were iterated multiple times, and which had to be in- +terspersed with early steps of instrument and guider commis- +sioning and focal plane calibrations. +5.1.1. OTE Deployments +The major structural deployments of the OTE (DTA, +SMSS, and PMBSS wing deployments) all completed suc- +cessfully and nominally, with no notable issues. +Subsequent results from telescope commissioning con- +firmed the precision of these deployments: for instance, the +8 m multi-hinged SMSS deployment placed the SM within +1.5 mm of its nominal position, well within the correction +range of the SM actuators. Further, the telescope boresight +offset relative to the spacecraft star trackers was found to be +3.4′, nicely consistent with the 3.05′ 1-sigma preflight pre- +diction. Similarly the corrections required to align the PM- +SAs were small (Table 4), with only one segment requiring a +corrective move larger than 1 millimeter in position. +5.1.2. Mirror Segment Deployment +For launch, all mirror segments were stowed in launch re- +straints in order to limit lateral displacements during launch +and ascent (see Figure 16). To begin to align the PMSAs and +SMA, the segments therefore had to be released from their +launch restraint, a pure piston move of ∼12.5 mm. +The deployment sequence of the PMSA and SMA was +carefully designed to verify the actuator stepper motor alive- +ness and responsiveness, as well as confirm proper manage- +ment of the segment envelope and workspace boundaries. +The sequence was tested on the ground on multiple occa- +sions (e.g., OTIS testing at JSC). As such, stepper motors +on segment A1 only were first incrementally commanded to +move 1 step, 1 revolution, 10 µm, 150 µm, and 340 µm. +The remaining segments (except A3 and A6, see below) were +then commanded the same sequence of steps, followed by 1- +mm increments until all the segments were fully deployed to +12.5 mm. Early in these deployments, some LVDT sensor +readings did not show as smooth a progression as expected, +which led to additional small ‘flinch’ moves to verify all ac- +tuators were moving. Once past the first few millimeters and +out of the launch restraints, the LVDTs showed the expected +linear response. The initial non-linearity was interpreted as +due to friction of surface contacts with the launch restraints, +which, however, posed no problem to the deployments. +Segments A3 and A6 were deployed separately and last +as a result of a faulty LVDT on one actuator each (as noted +above in Section 4.3.1). Although the sequence of moves +was identical to that of the other segments, the LVDT read- +ings, which provide a coarse direct measurement of actuator +length, had to be calibrated as the sensors cooled down. As a +result, they were deployed separately, without incident. +Finally, once at their deployed positions, the segments +were commanded to their intended nominal positions, based +on ground alignment test results corrected for 0-g via model- +ing. Mirror deployments completed successfully and with no +major issues. +5.1.3. Segment-level Identification & Alignment +The next series of activities aimed at finding, identifying, +and re-arranging the images produced by each of the mirror +segments. The deployed but not-yet-aligned segments were + +18 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 15: High-level overview of the sequence of events during OTE commissioning, along with examples of flight data. This +depiction greatly simplifies a complex process involving hundreds of individual steps and observations. +Figure 16: Rendering of the actuator in a stowed configura- +tion, with the snubber inside the launch restraint (in green). +each acting as its own aberrated ∼1.3 m telescope. When +first light on NIRCam was obtained on February 2nd 2022, +all celestial objects were indeed duplicated 18 times (see Fig- +ure 17). +To identify an image of each segment, an isolated bright +star was observed. Budgeted pre-flight uncertainties for mir- +ror segment initial deployments, as well as in the initial tele- +scope boresight offset (see Section 5.3.1), predicted that seg- +ments might be scattered by up to ∼15′. As a result, the tar- +get star was selected to have no similar-brightness neighbors +within such a distance, and a large half-degree-diameter mo- +saic around the target star was generated, taking around 25 +hours to complete. Figure 18 shows a cartoon of the planned +mosaic, overlaid on top of a catalog sky image around the +target. A subset of the flight data where the segment im- +ages were all found is also shown. The segment images were +found within ∼3.4′ of the nominal target location on aver- +age and the segment image scatter was of similar magnitude, +both better than requirements and expectations. The simplic- +ity of the plan for this initial mosaic step proved beneficial +to accommodate larger-than-expected coarse pointing uncer- +tainties at this time (prior to ACS tuning and optimization), +and increased levels of detector persistence due to operating +NIRCam well before it had fully cooled to its nominal tem- +perature. +Following the initial mosaic observation, a secondary mir- +ror focus sweep was performed in order to measure and move +the SM to a best focus position. This improved image quality +and enabled guiding later on during OTE commissioning (see +Section 5.2.1). The analysis led to an SM move of -427 µm +(i.e., away from the primary mirror). + +Initial Mosaic +Focus Sweep +1 . Mirror Deployment +2. Segment Image +Identification +Segment Global +Image Array +Alignment +3. Segment Alignment +4. Image Stacking +Telescope Alignment +Evaluation Image +5.Coarse Phasing +DHS Spectra +Webb Telescope Image Sharpness Check +6. Fine Phasing +NIRSpec +MIRI +8 +In Focus ++8 +waves +NIRCam +7.Multi-field Alignment +Fine Guidance Sensor +NIRISS +8. Iterate last three steps to converge +JWST_007THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +19 +Table 3: Timeline Summary for OTE Commissioning +Activity +Program IDa +Prelaunch Plan +Actual +Delta [days] +Mirror Deployments +n/a +2022-01-18 +2022-01-20 +2.0 +Segment Identification +1137 +2022-02-07 +2022-02-08 +0.2 +First Closed Loop Guiding +1410 +2022-02-12 +2022-02-13 +0.5 +Segment Alignment (iteration 1) +1141 +2022-02-12 +2022-02-19 +6.2 +Image stacking (iteration 1) +1143 +2022-02-14 +2022-02-22 +7.1 +Coarse Phasing (iteration 1) +1147 +2022-02-21 +2022-02-28 +6.8 +Coarse Multi Field +1148 +2022-02-22 +2022-03-03 +8.9 +Fine Phasing (iteration 1) +1155 +2022-03-01 +2022-03-08 +6.4 +Fine Phasing (iteration 3) +1160 +2022-03-05 +2022-03-11 +5.5 +Multi Field Multi Instrument Sensing 1 +1166 +2022-03-10 +2022-03-20 +9.7 +LOS Jitter Measurement +1170 +2022-03-17 +2022-03-21 +4.0 +Multi Field Multi Instrument Sensing 2 +1465 +2022-04-06 +2022-04-19 +12.8 +OTE Alignment Complete +n/a +2022-04-24 +2022-04-23 +-1.0 +aAPT program IDs, which may be used to retrieve these data, or any other commissioning data, from the +MAST archive. +NOTE—Timeline of events corresponding to Figure 15. Overall the OTE commissioning process proceeded +remarkably according to preflight plans. The major sources of schedule deviation were complications in +early guiding (delayed completion of segment alignment) and a revised, slower plan for MIRI cooldown +adopted by the MIRI team (delayed start of the second round of multi instrument sensing.) These were +balanced out by the budgeted third round of multi instrument sensing not being necessary, allowing the +completion of OTE alignment one day ahead of schedule. +Figure 17: First on-sky image using JWST and NIRCam, targeting a region of the Large Magellanic Cloud, obtained on 2022 +February 2. Because the mirror segments are not phased, each star is duplicated 18 times. Two copies of the core of globular +cluster NGC 1916 can be seen at upper left and lower right. + +20 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Table 4: PMSA Correction Magnitudes +Degree of Freedom +Unit +Typical +Maximum +Margin +Piston +µm +145 +281.7 +91% +Radial Translation +µm +450 +1205.5 +32% +Clocking +microradian +370 +766.0 +55% +Radius of Curvature +µm of surface sag +0.75 +2.117 +76% +NOTE—Typical and maximum corrective moves required to align the PMSAs. +OTE deployments initially placed segments typically within a few hundred +microns of their intended locations. The margin column gives the unused frac- +tion of the nominal correction range remaining after the maximum correction +moves for each degree of freedom. +Figure 18: Top: The search sequence of NIRCam observa- +tions centered on the bright target HD 84406. Bottom: The +inset shows a subset of the large mosaic from flight. All +18 images of the target star were found clustered together +in a relatively small area, due to good deployments preci- +sion. Also visible are some residual after images of stars +and curved trails from illumination during slews to differ- +ent pointings, due to increased persistence from higher-than- +nominal detector temperatures at the time of these observa- +tions. +Next, each mirror segment’s image in the initial mosaic +was identified by sequentially tilting each mirror. Once iden- +tified, segments were commanded to form the pre-defined +image array shown in Figure 15. +In the hexagonal array configuration, all segment images +could be observed on one NIRCam detector at once and +segment-level aberrations could be addressed as part of the +segment global alignment activities. To do so, the SM was +moved away from its nominal best focus by ±400 µm to col- +lect focus-diverse imagery for the purpose of phase retrieval +analysis. As a result of this analysis, the SM was corrected +in X- and Y-translation by 0.94 and 1.06 mm, respectively. +Focus corrections were applied to each segment, along with +corrections to the radius-of-curvature actuators of two PM- +SAs. +During a later stage of commissioning, a second iteration +of Global Alignment was executed as part of the iterative +alignment approach. At that time, additional small clocking, +radial translations, and radius of curvature corrections were +applied to the PMSAs to correct astigmatism and power. At +this point in the OTE phasing process, the observed wave- +front errors achieved excellent agreement with ground mea- +surements and preflight modeling of the higher spatial fre- +quency mirror maps (Figure 19). +5.1.4. Co-Phasing the Segments +Following each instance of Global Alignment, a sequence +of image stacking was executed to position segment images +on top of each other (see Figure 15). However, this stacking +does not mean the light paths from each segment are in phase +with one another, so Dispersed Hartmann Sensing measure- +ments (Coarse Phasing) were executed to establish phase er- +rors as a function of wavelength in order to measure pairwise +segment edge heights and derive an overall piston correction +to all of the PMSAs. Three iterations of this coarse phasing +sufficed to bring the PMSA piston offsets to less than 1 µm, +where fine phasing with NIRCam weak lenses could finalize +the OTE alignment. +Following these PMSA piston corrections, a fine phasing +activity took place where the NIRCam weak lenses were +used to collect focus-diverse measurements for phase re- +trieval analysis (see Figure 15). The results of such measure- +ments were then used to better (re-)stack the segment images +as well as correct residual piston offsets between segments. +On March 11 2022, the fine alignment process completed, +yielding a telescope aligned to roughly 50 nm RMS as seen at +the fiducial field point on NIRCam A3. Following this, a mo- +saic observation was carried out around the alignment star. +This multi-purpose observation tested science-like dithered +and mosaiced observations for the first time, confirmed ex- +cellent PSF quality over all of NIRCam’s field of view and +across NIRCam’s full wavelength range, provided early mea- +surements of observatory backgrounds, and yielded an early +glimpse of JWST’s sensitivity to the high redshift universe. +5.1.5. Multi-Field Alignment + +WISE W1 (3.6 μm) +Log, stretch +to emphasize faint stars +42 +41 +40 +39 +38 +(last) +Cyannumbers: +Ordering of +21 +20 +9 +observation sequence +22 +V3PA=195 +23 +(first) +2nd +13 +25 +30 +32 +33 +OTE-01 Qbs +Red oval shows approximate location +where segment spots were found +Yellow box = full mosaic search area for Obs 1 +Average offset ~ 3.4 arcmin from center +JWST_015THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +21 +Figure 19: On-orbit measured primary mirror segment wavefront errors following global alignment 2 (left) as compared to +the ground-test measurements at the XRCF with model backouts for 0-g gravity (right). The consistency of these two datasets +provides a striking in-flight validation of the OTE development program. It demonstrates the segments were fabricated to the +correct optical prescriptions, the 0-g gravity backouts were correct, there was no measurable thermal distortion during launch and +ascent, the segments were deployed to within the correction range of their actuators, and the WFSC processes worked to sense +and correct the initial misalignments. Note that the RMS WFE values labeled are for the primary mirror segments only, excluding +all the other optical surfaces which contribute to higher WFE for the observatory as a whole. +Aligning the telescope to only one field can lead to de- +generate solutions, where PMSA and SM misalignments bal- +ance each other out. Multi-field measurements are therefore +required in order to achieve optimal optical performances +across all the science instruments. +Two types of multi-field sensing were carried out: +a +NIRCam-only multi-field activity (using only Module A) +and two instances of multi-field, multi-instrument activities. +The NIRCam-only alignment activity was aimed at initially +removing most of the PMSA-to-SM misalignment using the +unstacked segments as a Hartmann sensor, analyzed with a +centroid-based approach to measure the field dependence of +aberrations. This “coarse” multi-field sensing proved to work +exceptionally well. As a result of this analysis, the SM was +moved in translation in X and Y (-210µm, 420µm) as well as +tilt in X and Y (-550 µrad, 34 µrad). The SM correction was +also accompanied by compensating PMSA moves in order to +maintain the hexagonal array configuration. +Finally, multi-field, multi-instrument measurements were +made in order to assess the field dependence over the whole +field of view, this time using focus diversity provided by +moving the SM in piston by ±100 µm. Two instances of +this activity were executed since, as expected, MIRI had not +yet reached its operational temperature at the time of the first +measurements. In both instances, the results indicated no +significant correctable field-dependent aberrations, in other +words the initial NIRCam-only multi-field activity had cor- +rected the telescope’s field dependent aberrations. Only a +small SM focus correction with minimal wavefront error gain +was applied, mostly to balance the relative focus terms of the +science instruments (see Figure 20). The observatory was +fully aligned on April 23 2022, and the commissioning of +the science instruments continued thereafter. +From that point, the operations team entered a new stage of +conducting routine OTE maintenance, which will be ongoing +throughout the mission, discussed further in Section 6. +5.2. Guiding and Line-of-Sight Pointing +To support all but the very earliest mirror alignment activ- +ities described in Section 5.1, the closed-loop FGS guiding +mode first needed to be commissioned and its performance +established. See Menzel PASP Observatory for a more de- +tailed discussion of the Attitude Control System. +5.2.1. Guiding during Commissioning +JWST uses star trackers, rate sensors, reaction wheels, and +a fine steering mirror to achieve a coarse pointing. To transi- +tion into guiding, one of two FGSs will attempt to identify the +intended guide star, whose position and fluxes are normally +provided by the Guide Star Catalog. The coarse position er- +ror of the guide star as seen by FGS is fed back to ACS for +correction. The guidestar will usually then be moved to a pre- +computed “science” location in the FGS field of view, after +which closed-loop guiding will be attempted and typically +engaged. During closed-loop guiding, FGS measures cen- +troids every 0.064 s and reports the position to ACS, which + +WAS Phase retrieval from GA2 obs 2+6 +Preflight Model: Uncorrectable high spatial +Measurement after GA2 corrections +frequencies from XRCF testing +200 +150 +Wavefront Error [nanometers] +100 +50 +50 +-100 +-150 +45.25 nm rms +43.32 nm rms +-200 +JWST_01722 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 20: Measured multi-field, multi-instrument wavefront residuals, here showing the end-to-end observatory (i.e. OTE plus +instruments) static wavefront residuals at the end of telescope alignment. The variation between field positions seen here is due +mostly to the inherent optical performance of the instruments, outside of the ability of OTE adjustments to correct. The final +adjustment after multi-instrument sensing was only a small focus shift of the secondary which brought the science instruments +into good average focus and positioned the global focus closer to MIRI. All instruments maintained precise confocality from +ground to space, such that it was unnecessary perform focus adjustments on the individual instruments. +then commands the fine steering mirror to move in order to +maintain the guide star at the appropriate position on the FGS +detector. +Early in OTE commissioning, when the OTE was still pro- +viding 18 images for every star in the field of view (includ- +ing on FGS), guiding operations had to be modified. Per our +plan, FGS used one of the segment images to guide on, along +with reference segment images. This was done by overriding +the guide star selection system to account for the segment +position offsets and flux differences compared to the guide +star’s catalog position and flux. Using this approach, closed- +loop guiding was successfully demonstrated during the FGS +LOS Initialization Activity (PID 1410). +Later in commissioning, guiding operations became in- +creasingly routine; once the PMSAs were stacked into a sin- +gle PSF, the guide and reference stars’ selection and locations +could be automatically provided by the ground system using +the operational catalog and only the associated stellar fluxes +had to be overridden. Once the PSF was phased, the fluxes as +well were being supplied by the system. By the end of OTE +commissioning, guiding required no special intervention. +The majority of instances of closed-loop guiding during +OTE commissioning were successful, although some failures +do occur for a variety of reasons, including bad pixels or mis- +cataloged guide stars. Nevertheless, success rates have risen +continuously throughout commissioning and into science op- +erations with now > 95% of planned visits being success- +fully executed. When guiding success is achieved (i.e., the +intended guidestar is identified, acquired, and tracked with a +settled closed loop), then the image stability of that pointing +becomes the performance metric of interest. +5.2.2. Characterizing Line of Sight Stability during +Commissioning +Special commissioning tests were included in the baseline +plan to obtain the power spectrum of the line-of-sight (LOS) +jitter and, in particular, to assess the contribution of vibra- +tions caused by the MIRI cryocooler (CC). +The dedicated test to probe excitations from vibrations in +the LOS data showed no evidence of significant contributors +and has revealed excellent stability performance at ∼1 mas +RMS radial, very close to the measurement noise floor and +significantly better than expectations and the 7 mas require- +ment. The typical results from the first commissioning LOS +jitter analysis (PID 1163, observation 2, from Hartig 2022) +are shown in Figure 21. In this case, jitter was measured at + +nm ++200 +133.0 +142.6 136.3 +158.7 +79.1 +84.4 +LW +147.3 +150.0 +88.2 +134.2 +161.3145.7 +135.6 +133.7 +133.9 +80.6 +NIRCam A +NIRCam B +60.6 +68.6 72.6 +59.2 +61.7 +69.8 +96.4 +79.9 +111.1 +NIRSpec +55.5 +52.2 +MIRI +0 +sw +99.1 +62.4 +67.7 +77.9 +67.1 +91.3 +63.8 72.9 +61.2 +72.7 +67.4 +89.5 +76.3 +68.2 +53.8 +79.5 +69.2 +77.2 +61.2 63.5 +NIRISS +Guider 1 +Guider 2 +77.0 +68.6 +67.8 +-200 +JWST_013THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +23 +1.04 mas RMS (radial). The same tests also revealed no need +to tune the MIRI cryocooler’s pulse frequency which remains +at its initial settings. +The analyses, which continue to be regularly performed as +part of the 2-day routine maintenance program (Section 6), +have consistently shown jitter levels around 1 mas radial +(moderately correlated with guidestar brightness). The anal- +ysis is quite sensitive and has revealed low frequency, low +power oscillations at ∼0.3 and 0.04 Hz. The 0.3 Hz fea- +ture has been attributed to bending modes at the 1 Hz isola- +tor at the SC-to-OTE interface, whereas the 0.04 Hz feature +has been shown to vary over time and might be attributed +to fuel slosh. During the commissioning period there were +no clear indications of any LOS jitter response to the reac- +tion wheel assemblies. However during the first months of +science operations, a handful of measurements have shown +minor LOS jitter contributions which appear correlated with +certain speeds of reaction wheel assembly #6; a resonance +appears to be excited in the vicinity of 16–17 Hz. +5.3. Focal Plane Calibration +In addition to guiding (Section 5.2), sufficiently accurate +target placement at each of the SIs was needed to support +late OTE commissioning activities. This involved two essen- +tial related areas of calibration: the first was the determina- +tion of the principal coordinate frame of FGS1 with respect +to the spacecraft’s, expressed as a direction cosine matrix and +stored on-board for use by the ACS. The second was the de- +termination and calibration of the SI fields of view relative +to FGS1, expressed in a convention similar to Euler angles +and stored in the SOC ground system for determining on-sky +pointings. These activities were performed in close coordi- +nation with WFS&C as part of the integrated flow through +OTE commissioning. +5.3.1. Calibrating the Telescope to Spacecraft +A key design feature of JWST is an OTE that is well sep- +arated and isolated, thermally and mechanically, from the +spacecraft by the Deployable Tower Assembly (DTA). Un- +certainties in the DTA deployment and other contributions +were expected to produce initial errors in the nominal align- +ment of the OTE V-frame with respect to the spacecraft’s fun- +damental coordinate system, called the J-frame, of ∼10–15′. +The slew of the observatory to its intended field is con- +trolled by the Star Tracker Assemblies (STAs), which reside +on the spacecraft side and are calibrated to the J-frame. How- +ever, the FGSs must then be able to acquire and identify the +intended guide star, and they reside on the OTE side. Cap- +turing the alignment between FGS and ACS requires the up- +dating of the FGS-to-J direction cosine matrix (DCM). This +matrix is updated to account for deployment uncertainties, as +stated above, as well as changes in telescope boresight, which +occur every time the SM is moved in translation or tilt (e.g. +Global Alignment, MIMF). +While the initial error in the OTE-to-spacecraft alignment +was expected to be close to 10′, the misalignment was found +to be only ∼3.4′ from the nominal ground values (see Fig- +ure 18). +The FGS-to-J DCM was subsequently manually +updated during commissioning after taking observations of +the sky with an SI or FGS and expressing that celestial +pointing in terms of the FGS1 frame, while obtaining from +the ACS/STAs the contemporaneous mapping of sky to the +J-frame, thus providing the information to relate FGS1 to +J-frame. +This operation was also successfully performed +during subsequent commissioning activities to maintain the +alignment. +During the science mission, this relationship will continue +to be dynamic at levels much lower than seen in commis- +sioning (i.e. a few arcsec), and the ACS will autonomously +update this calibration based on observed FGS guidestar lo- +cation error. +5.3.2. Calibrating the SIs to FGS +Using FGS1 to define the relationship of the OTE to the +spacecraft implies that it is also the reference for the OTE- +based frame to which the SI fields of view are calibrated. +This “V-frame” was defined nominally as a conventional 3- +axis coordinate system aligned with the OTE principal me- +chanical axes, having V1 pointing out along the Cassegrain +axis of symmetry (1). In flight, however, the V-frame is used +to specify SI and FGS fields and various fiducial field points +used for science targeting. So, in this application, it is treated +spherically as angles, with the axes V2 and V3 corresponding +to “field angles” within the OTE field. +In this scheme, the FGS1 field location and orientation +with respect to V2,V3 is fixed, and on-sky astrometric cal- +ibrations that determine the SIs’ and FGS2’s fields relative to +FGS1 in essence establish their locations, orientations, and +higher order distortions with respect to the V2,V3 field an- +gles. This astrometric calibration scheme, its tools and prod- +ucts, are thoroughly treated by Sahlmann (2019a). +These calibrations are required for successful target place- +ment and were performed during OTE commissioning to (1) +determine the post-launch changes to the ground-determined +relationships and (2) update this knowledge to ensure suc- +cessful multi-SI wavefront measurements. The commission- +ing team used a specially calibrated ∼15′ astrometric region +of the LMC for this purpose (Sahlmann 2019b). +The first measurements of SI relative locations showed +the ISIM to be stable, with ground-to-flight evolution in the +V2,V3 field angles to be at or below the ∼ 1′′, and orienta- +tion changes of the SI fields < 1′. Although precise scales +and distortion calibrations fall into the SI activities and con- +tinue into the science cycles, basic instrument scales were + +24 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 21: Left: Line-of-sight jitter distribution sampling every 2 ms over a 120 s interval. This “jitter ball” shows well-behaved +and very small variations in pointing using fine guidance control, with RMS variation only 1.1 mas. Right: The power spectrum +from the high-frequency jitter measurement. +tentatively measured during these OTE commissioning ac- +tivities, and were found to be < 0.15% different from ground +measurements. For comparison, Hubble’s SIs through the +generations typically saw 1-2′′ of V2,V3 shifts and 0.2-1.0% +scale change from ground to flight. +5.4. Optical Performance at the End of Commissioning +The optical performance as measured at the end of com- +missioning is better than the requirement values at the +system-level and for the most part, better than the sub-system +allocations as well. A summary of the optical performance +measurements and predictions is presented in this section. A +separate optical discussion is presented in the paper on JWST +Science Performance in this volume by Rigby PASP Science. +A detailed discussion of the telescope’s optical performance +against requirements at the end of commissioning is reported +in Knight & Lightsey (2022). +The total wavefront error combines the observed static +wavefront error with the dynamic stability and image mo- +tion terms, which is corrected when the NIRCam A field +point exceeds 80 nm rms. This means the end-to-end, tele- +scope and NIRCam SW has diffraction-limited image quality +at ∼1.1 µm. This is significantly better than the 150 nm RMS +requirement optical error budget total that enables diffraction +limited image quality (approximated by λ/14) at 2 µm. +5.4.1. Observed Wavefront Errors +Table 5 summarizes the end-to-end, referred to as ‘obser- +vatory’, static wavefront errors measured at the end of com- +missioning. The static wavefront errors are well below their +allocations in all channels, at all field points. +5.4.2. Area & Throughput +Table 5: WFE Table +Science Field +Static WFE +NIRCam A SW +61±8 +NIRCam B SW +69±11 +NIRCam A LW +134±38 +NIRCam B LW +134±39 +NIRISS +68±12 +FGS 1 +77±15 +FGS 2 +69±8 +MIRI +99±28 +NIRSpec +110±20 +NOTE—The +static +observatory +wavefront error measurements as +measured at the end of telescope +alignment in May 2022. +The +values reported are the average +across +the +points +measured +across the science field, with the +plus and minus values reporting +the peak to valley variations. +The +total +observatory +WFE +combines this static term with +the dynamic WFE stability and +image motion, typically ∼13 nm +rms equivalent, via RSS sum. +Units are nanometers rms. +JWST’s unobscured collecting area was measured using +the NIRCam pupil imaging lens to be 25.44 m2, exceed- +ing its requirement of 25 m2. The telescope’s wavelength- +dependent transmission ranges from 0.786 at 0.8 µm to 0.933 +at 28 µm, again better than requirements at each wavelength. +The transmission values were determined from final pre- +flight measurements of mirror witness samples, combined +with NIRCam grism measurements confirming the absence +of detectable ice deposits. + +X-jitter (RMS px): 0.d26 +0.10 +-N samples: 53100 +- Y-jitter (RMS px): 0.025 +-'RMS jjtter (mas): 1.12 +0.05 +Y offset (px) +0.00 +-0.05 +-0.10 +-0.10 +-0.05 +0.00 +0.05 +0.10 +X offset (px) +JWST00910-1 +10-2 +Power +10- +Relative +10- +10-5 +I SO +10-6 +10-7 +10-8 +0.01 +0.10 +1.00 +10.00 +100.00 +Period (s) +JWST010THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +25 +Table 6: Stability Performance +Contributor +Predicted Amplitude +Measured Amplitude +Measured Response +WFE (nm RMS) +WFE (nm RMS) +IEC Heater Cycling +3.5 +2.5 +224 s period oscillation +Frill & PMSA Closeout +9 +4.45±0.19 +0.77 hr time constant +Thermal Distortion +17 +17.94±0.39 +33.94 hr time constant +The product of the above observed values for OTE area and +transmission was also projected to end of life using modeled +degradation of the optics. This places the effective area × +transmission value of 19.58 m2 at 0.8 µm and 23.18 m2 at +20 µm, compared to the OTE requirements of 15.37 m2 and +22.00 m2, respectively. +5.4.3. Vignetting +Some of the OTE commissioning activities described ear- +lier obtained data that also supported a secondary goal of +probing various types of vignetting. Analyses of these data +show no indication of any field-of-view cropping, unex- +pected OTE structure incursion, or pupil vignetting. Estab- +lishing that the telescope was unobstructed (except for sec- +ondary mirror support structures) fulfilled a mission-level re- +quirement and was an exit criterion for OTE commissioning. +5.4.4. Thermal Stability +A dedicated thermal stability test was carried out following +the telescope alignment in order to characterize the wavefront +stability and image motion on various timescales following a +large, stressing thermal slew (i.e. early May 2022; PID 1445 +and 1446). This activity started by performing a 4-day ther- +mal soak at the hot (sun-normal) attitude and making baseline +measurements. Then, the telescope was slewed to the cold at- +titude where continuous wavefront measurements were made +for the first 24 hours and then every ∼8 hours for the follow- +ing 7 days. The thermal stability test confirmed three pre- +dicted wavefront drifts that were bounded by the modeling +predictions: short-timescale (2-4 min) oscillations from IEC +panel heater cycling, medium-timescale (∼1 hr) drift from +soft-structure induced thermal distortion, and long-timescale +(∼1.5 day) drift from the composite backplane induced ther- +mal distortion. These drifts are reported in Table 6. Tem- +perature sensors on the telescope were also monitored during +this test and confirmed that the temperature changes observed +were within the noise of the temperature sensors (< 40 mK). +Some science observation modes are, however, sensitive +to these levels of WFE drift. It is also important to note +that the worst-case delta-T induced as part of this thermal +slew test would rarely be realized during normal science. In +practice, science pointings across the sky are subject to much +smoother and smaller temperature changes. +Pointing stability immediately following the thermal slew +was measured when slewing from hot-to-cold and from cold +back to hot. As discussed in Section 3.1, thermal distortion at +the star tracker could result in uncorrected roll about the lo- +cation of the fine guide star on the fine guidance sensor. The +roll about the guide star was measured at the NIRCam field +location to be 0.0265 mas/hr in translation and comparable +to a measured radial displacement from the star of -0.0230 +mas/hr, well below the allocation of 6.3 mas. +During commissioning, there were many instances of sud- +den positional changes in one or more mirror segments, re- +ferred to as ‘tilt events’ (see Section 6.2.1). The largest of +these produce brief violations of the nominal stability values +reported in Table 1. These positional changes are typically +very small but detectable. These tilt events are generally as- +cribed to strain release within the OTE structures following +cooldown to cryogenic temperatures, although the sources of +the tilt events is not fully understood. The frequency and +magnitude of these events appear to be slowly declining. +6. Science Era Characterization +The maintenance and trending of the OTE in the science +mission era officially started in mid-July 2022 alongside the +start of the Cycle 1 science program. The telescope align- +ment state will be monitored and corrections will be made +as needed. Additionally, trending will be carried out across +the telescope performance. The telescope state will be made +available such that it can be used as part of the science anal- +ysis. +6.1. Wavefront Routine Maintenance Operations Concept +The baseline science operations concept for OTE main- +tenance uses wavefront sensing and control observations to +maintain the optical alignment near its optimal state. Wave- +front sensing observations are scheduled approximately ev- +ery 2 days and make use of the NIRCam ±8 wave weak lens +pair that is best matched to sensing aberrations at the nominal +sensing field point (on detector A3). These observations use +bright K∼7 target stars to minimize the exposure times and +the exact target is automatically drawn from a pool of targets +evenly distributed on the sky in order to minimize slew times +between science observations (see Figure 22). Also included +in those routine observations are line-of-sight jitter measure- +ment observations, which use the same target as the weak +lens observations and take about two minutes of NIRCam +8×8 data (see Section 5.2.2). In total, wavefront sensing ob- +servations take about 15 minutes of data, not including slews +and overheads (see PID 2586, 2724, 2725, and 2726). +Additionally, pupil images are also collected on a quar- +terly basis in order to monitor the state of the primary mir- +ror and, in particular, identify and characterize features due +to micrometeoroid degradation (see Section 6.2.2 for more + +26 +MCELWAIN, FEINBERG, & PERRIN ET AL. +Figure 22: Full sky map of the Cycle 1 wavefront routine +maintenance targets, which include 400 targets that were vet- +ted with binary and isolation criteria. The color and size of +each dot represent the sum of the time (in days) of Cycle 1 +science visits that are closest to that location. Small open pur- +ple targets have no nearby science visit in the Cycle 1 plan. +The large red circle is ∼41 days of observations in GOODS- +South and the green circle to the right is the ∼23 days of +observations in the COSMOS fields. The average distance of +a science pointing from a WFSC target is 4.7◦. +details). These observations take about 10 min of NIRCam +science time (see PID 2751). +The sensing data are automatically analyzed on the ground +using phase retrieval algorithms (see Acton et al. 2022) to +assess the state of the telescope and determine if a PMSA +control (or correction) is needed. The baseline frequency of +such correction was expected to be no more than once every +14 days, which was predicated on the expectation that even +worst-case thermal slews would not drift the telescope wave- +front error beyond the allocated values. +Wavefront corrections are typically scheduled when (1) +Observatory-level wavefront error exceeds 80 nm RMS or +(2) segment tip/tilt correction is larger than 0.05 µrad. Re- +call that the observatory-level WFE requirement at NIRCam +is 150 nm RMS, so the criterion used here is much tighter +thanks to the exceptional in-flight performances of the JWST +OTE. Also, these criteria were defined so as to not chase any +thermally-induced distortions but rather to correct any sys- +tematic alignment changes to the optical system. +6.2. Stability & Trending +The baseline operations concept outlined above has been +operational since mid-July 2022 and trending of the WFE, +a key performance metric, along with the occasional correc- +tions that were made to the PMSA are shown in Figure 23. +Since the start of the science mission, 47 wavefront sensing +observations have been executed, including 6 that included +PSMA control. This cadence averages out to correcting once +every 2.6 weeks, i.e. better than our expectations. In gen- +eral, the OTE has been stable to within ∼10 nm RMS until +a disturbance occurs and must be corrected. Figure 23 sug- +gests that corrections are grouped closer together, separated +by longer periods of stable WFE. Most of the sudden changes +in WFE seen in Figure 23 are referred to as tilt events and are +discussed below in Section 6.2.1. +Overall, the telescope’s performance has met the criteria +listed above about 84% of the time, and has met the mission +requirements 100% of the time since mid-July 2022. +6.2.1. Tilt Events +On many occasions, so-called tilt events, where sudden +and uncommanded tilts of individual or groups of segments +(e.g. wing segments), have been observed throughout com- +missioning and science operations. These tilt events were +first observed during OTIS cryo-vacuum testing at Johnson +Space Center in 2017 and they have been ascribed to the +stick/slip strain release stored in the OTE hardware and/or +structure during cooldown. They are expected to decrease in +numbers over time as the OTE structure and hardware relax +into their new environment. Tilt events continue to occur in +the science mission (e.g., Schlawin et al. 2022, in prepara- +tion), though less frequently and at a lesser magnitude than +during early commissioning. +As shown in Figure 23, tilt +events episodically punctuate weeks-long periods of wave- +front stability. In practice, the infrequent occurrences of large +tilt events have been the dominant source of WFE degrada- +tion requiring PMSA correction. Note however, that not all +tilt events have led to a PMSA correction, and those who did +were all corrected as part of our routine maintenance pro- +gram. Ongoing trending will track the nature and frequency +of these events. The last two months of 2023 did not have any +tilt events that drove excursions to 80 nm control threshold, +which supports the hypothesis that the OTE structure is relax- +ing to a stable state. An example of a tilt event that occurred +between wavefront sensing visits is shown in Figure 24. +6.2.2. Micrometeoroids +Impacts from micrometeoroids on the PMSA have been +observed since the middle of OTE commissioning. +Pupil +imaging first revealed localized surface changes to individual +mirror segments, and phase retrieval analysis has revealed, in +some cases, WFE changes on the impacted segments. How- +ever, not all micrometeoroid impacts have resulted in mea- +surable changes in WFE since some show up only on pupil +images or when averaging large numbers of optical path dif- +ference (OPD) maps (Figure 25). Moreover, the cumulative +effect of these micrometeoroids impacts has so far minimally +affected the overall telescope throughput. +Notably, however, a large impact on segment C3 was ob- +served in phase retrieval analysis from sensing visits cover- +ing the period 22-24 May UT. The impact was such that the +global WFE worsened by 9 nm RMS, after compensation by +applying segment corrections in all degrees of freedom. + +75° +days +60° +40 +45° +-35 +30° +-30 +15° +25 +.14h +16h +18h--20h- +22h +.Oh +2h - +4h . +6h. +"8h +.10h +.0 +20 +4 +-15° +15 +-10 +-30° +丰 ++ ++ +-5 +-45° +.09- +-0 +-75° +JWST_011THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +27 +Figure 23: Top: Observatory-level WFE as a function of time since the beginning of the science mission (mid July 2022), +showing every sensing visit along with the few instances where corrections to the PMSAs were applied. Occasional larger tilt +events are responsible for the larger the wavefront changes over time, as discussed in the text. Bottom: histogram of the WFE +so far showing that a large majority of observations have near-optimal image quality. A small fraction of the time has WFE +above the correction threshold, generally the time between a larger tilt event and its subsequent correction. PSFs during such +times can have modest but detectable differences from the average PSF, which can be calibrated during science analyses using +the wavefront data. The observatory-level WFE mission requirements have so far been met at all times within this period. +Figure 24: Example of a single-segment tilt event, as seen in optical path difference maps measured during wavefront sensing +observations. Only one segment, C6, moved much in this recent event; other events have shown correlated motions of several +segments, apparently related to release of tension within the wing hinge areas. When such events lead to the total WFE surpassing +the correction threshold, corrective mirror moves are scheduled for the subsequent WFS observation. Segments not affected by +any tilt event generally show superb stability, often below the 7 nm sensing noise, as seen above for the right half of the primary. + +ObservatoryWFEfrom2022-07-16to2023-01-03 +110 +Sensing visit +Corrections +Correction threshold +100 +WFE +06 +80 +70 +2022-07-15 +2022-08-012022-08-15 +2022-09-012022-09-15 +2022-10-01 +2022-10-15 +2022-11-01 +2022-11-15 +2022-12-012022-12-15 +2023-01-01 +Date +ObservatorywFEHistogramfrom2022-07-16to2023-01-03 +1.0 +0.200 J +FE +≤0.175手 +0.8三 +0.125 +0.100 +89.6%ofthetimehasmeasuredOTE+NiRCamWFE<80 +%0.075 +0.050 +0.2 +0.025 +0.000 +0.0 +60 +70 +80 +90 +100 +110 +RMS Wavefront Error [nm]Previous Obs. PID 02726, +Sensing Obs. PID 02726, +Obs 377 +Obs 403 +R2022102704 +R2022102903 +Drift between +(2022-10-27 18:47:45.269000) +(2022-10-29 21:17:41.119000) +Current and Previous +0.2 +B6 +B2 +0.1 +Iw +B5 +-0.1 +B3 +WFE RMS: +WFE RMS: +WFE RMS: +60.5nm +62.3nm +15.2nm +6.10 nm/hr +-0.2 +JWST_01428 +MCELWAIN, FEINBERG, & PERRIN ET AL. +The telescope wavefront error is still well below the nom- +inal requirement following the single C3-event. +However +models of similar events indicate that with about ten simi- +lar events, we could be at our end-of-life wavefront error re- +quirement of 150 nm RMS. Due to the precision launch of +JWST, the observatory has sufficient fuel for 20+ years of +mission life, considerably longer than the mission design re- +quirement minimum of 5 years. With the possibility of an +extended mission and the uncertain rate of C3-type events +(from only a single occurrence) and the unexpected result- +ing WFE, the project has implemented a meteoroid avoid- +ance zone (MAZ) for Cycle 2. Models produced by NASA’s +Meteoroid Environment Office show that the greatest impact +rate for higher energy micrometeoroid strikes from sporadic +sources, occurs in the so-called ram direction, the direction +of flight as JWST moves with the Earth around the Sun. +The Cycle 2 pointing restrictions favor observing in the wake +(anti-ram) direction whenever possible. The proposed MAZ +would reduce the instantaneous field of regard by about 40%. +Models which remove all pointings from the MAZ and redis- +tribute them over the allowed field of regard, can lower the +impact rate on the primary mirror by 55-65%. The reduc- +tion we expect to achieve in practice will likely be 30-40%, +because some high-priority and time-critical pointings in the +MAZ will be allowed. As the Cycle 2 detailed observing +plan is constructed, the expected impact reduction rate will +be determined and monitored throughout the Cycle. Detailed +damage models are also being constructed to provide a bet- +ter understanding of the true risk of further C3-events; pre- +liminary models suggest this was a higher than average en- +ergy impact on a sensitive area of C3. These models will be +supplemented by data from a series of ballistic tests on rel- +evant samples which should be completed over the next few +months, building upon the experimental testing carried out +early in the JWST development (e.g., Heaney et al. 2004). +The impact rate appears consistent with pre-flight expecta- +tions and mitigation strategies are being implemented. Mean- +while, micrometeoroids are being partially corrected as part +of our routine maintenance corrections using, in particular, +radius of curvature actuators. Since the end of commission- +ing, we have gained experience over a longer time baseline +and with larger-number statistics. +6.3. PSF Calibrations for Science Observations +Every wavefront sensing and control observation pro- +duces, by means of phase retrieval analysis, optical path +difference maps that can be used to model the point +spread function. +These maps are publicly available (see +https://mast.stsci.edu/) and can be readily imported using the +WebbPSF Python package (Perrin et al. 2014), which now +provides the capability to generate model PSFs that track the +measured variations in mirror alignment over time. Efforts +Figure 25: The NIRCam pupil imaging lens is used to +monitor the telescope mirrors on a quarterly basis. Local- +ized disturbances to the wavefront due to micrometeoroid +impacts are visible in the logarithmically stretched image +above. Some of the micrometeoroid events are detected by +wavefront changes during routine sensing observations, and +the corresponding pupil image features are marked here with +the blue arrows. The large C3 segment event, which had the +largest wavefront effect, can be seen next to the secondary +mirror support strut (bottom right). The other features in- +clude pre-flight contaminants along the optical path, most of +which have been stable throughout the ground test program. +The appearance of uneven segment gaps is due to pupil image +aberration and diffraction. +continue to further refine and improve PSF models based +on in-flight experience and detailed comparisons with flight +data. The package also includes trending features, some of +which were used to generate Figure 23. +Though science +observations inevitably encounter PSF variations over time, +sometimes more than others, for many science use cases the +impact of such variations can be reduced as long as those +variations are measured and characterized. +7. Discussion and Lessons Learned +JWST was a first-of-its-kind large segmented cryogenic +space telescope. This unique telescope brought with it many +challenges, requiring the team to investigate how to mount +and align a segmented mirror, how to test a large cryogenic +telescope, and how to commission this complicated system. +In the end, this highly complex, large, cryogenic, deploy- +able space optical system has worked extremely well. The +deployments and telescope alignment activities were exe- + +THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +29 +cuted smoothly, without incidents or invoking any contin- +gencies. +Several minor surprises were all handled within +the normal course of events as part of the commissioning +process and without any particular difficulty. +The optical +performance budgets were all met or exceeded, enabled by +following the systems engineering framework. +The tele- +scope hardware, wavefront sensing and control algorithms, +systems, processes, and our dedicated teams all worked as +planned. Future telescopes can benefit from and build upon +these lessons learned; some have already been published +during the development period (e.g., Stahl 2010, Feinberg +et al. 2018), while others are still being formulated and docu- +mented. It is an appropriate time to provide some discussion +as to what led to this success and what could be done better +next time. +7.1. Space-Based Deployable Optics and Active Wavefront +Control Flight Proven +It is now obvious, but still noteworthy, that the complex +sequence of events involved in deploying and aligning the +OTE was successful. Given the unprecedented nature of this +mission, this was not assured to be the case ahead of time. +Achieving this required state-of-the-art contributions across +a vast range of engineering disciplines from countless indi- +viduals and many organizations. The successes in flight are a +testament to the efficacy and rigor of these long engineering +processes. +JWST’s wavefront sensing achieved extraordinary preci- +sion. The wavefront sensing error budget included several +terms that were not possible to test to high precision and +the actual performance worked in our favor. +For exam- +ple, the fast wavefront sensing implemented for the IEC and +thermal stability tests demonstrated sub-nm fast differential +measurements, far better than expected. The NIRCam-only +multi field sensing measurement (coarse-MIMF) fully cor- +rected the multi-field alignment, which was confirmed when +no further corrections were warranted after multi-field sens- +ing measurements. As yet another example, in the final iter- +ation of the coarse phasing activity, the piston errors were so +small that our methods struggled to sense them. +The mirror control benefited from actuator moves that were +more precise than budgeted. This made the overall process +more efficient and effective, with the ultimate achieved align- +ment that is significantly better than the requirements. The +level of conservatism in the error budgeting and model un- +certainty factors was appropriate engineering given the first +of a kind nature of this process, but it is now clear how well +this all worked. +7.2. Piecewise Verification by Analysis Works +The driving telescope performance parameters could not +be directly tested and therefore needed to be verified by anal- +ysis. This was unprecedented in scope and required substan- +tial development of high-fidelity integrated models. The in- +tegrated modeling effort naturally grew in scale and fidelity +as the design matured, with initial efforts focused on design +trades and ultimately converging to a very accurate end-to- +end model used to verify performance for the pre-ship re- +view. The model size was complex as it required predictive +accuracy on nanometer scales for structures that were meters +in size and used novel materials operating over a wide range +of temperatures. These new models were managed with rig- +orous oversight for model construction, verification, and val- +idation at each level of integration. The team intentionally +adopted what were thought to be conservative model uncer- +tainty factors, which were in some cases found to be just right +for bounding flight performance, and in others overly conser- +vative. +The JSC cryotest was complex. Nevertheless it proved its +value, not just in validation of the overall performance and +requirement verification, but also by allowing the discovery +of several workmanship issues (particularly with the integra- +tion of the frill) that could be addressed prior to launch. +Piecewise verification by analysis, combined with ade- +quate testing to validate workmanship, was critical to the suc- +cess of JWST. +7.3. Application of Lessons from Hubble and Chandra +The JWST telescope implemented important lessons +learned from the Hubble and Chandra X-Ray Observatory +(Chandra) programs and these were considered critical to +the success of this mission. The integration and testing ap- +proach was developed in a way that considered the Hubble +program (Feinberg & Geithner 2008). For JWST, there were +pre-defined test criteria established and a process for report- +ing if those tests are not met. In the case of JWST, formal +processes were defined for problem reports, anomalies, and +failures. JWST also developed a matrix of crosschecks, both +in measurement and in analysis, that was used to catch errors +and provide more confidence in the results. The results were +transparently reported at data reviews and evaluated by the +project team, external review boards, and external advisory +groups such as the optical PIT (see Section 4). +We have previously noted the importance of independent +test equipment; here we highlight the importance of inde- +pendent analyses as well. +JWST designated a walled-off +group within the project team as an Independent Optical Ver- +ification Team. +This group was charged with conducting +independent analyses of key optical results, using indepen- +dently developed software tools and often different analysis +approaches from the primary optical verification team. This +independent check was critical in ensuring the validity of test +results, as well as of the numerous optical models and test +configuration metrology that underpinned those test results. + +30 +MCELWAIN, FEINBERG, & PERRIN ET AL. +The JWST Pathfinder Program (see Section 4.3.3) was a +hallmark of the JWST I&T program that provided invalu- +able risk reductions while proving out new systems for han- +dling, integration, and testing (Feinberg et al. 2010). The +Pathfinder Program was derived from a successful verifica- +tion engineering test article program on Chandra (Arenberg +et al. 2014). Although there were significant costs incurred +on the Pathfinder Program, it matured all of the processes +needed. This ultimately streamlined many activities by al- +lowing the evaluation of test environments prior to flight tests +and avoiding incidents on the flight hardware itself; the OTIS +cryotest met all of its test objectives the first time without in- +cident and without delays. +The Pathfinder program was particularly valuable for +Webb given the heightened challenges of the mission. Cryo- +vacuum testing always presents challenges, but JWST’s size, +wide range of payload operating temperatures, and infrared +wavelength coverage presented exceptional demands regard- +ing lengthy thermal transitions, contamination control, con- +trol of thermal background light, macroscopic motions and +interferences during cooldown, jitter control, and careful +management of atmospheric condensation onto chamber +shrouds. +The details of how these issues were managed +for JWST are outside the scope of this paper, but discus- +sions of such topics can be found in published overviews of +the major ISIM and OTIS cryo-vacuum tests (Kimble et al. +2016, 2018). +7.4. Balancing Development Risks and Science +Performance +Mission requirements are defined and rationalized through +a science traceability matrix, which then drives mission level +requirements and lower-level requirements. However, this +science-driven approach can sometimes lead to system-level +performances that are difficult to achieve, resulting in con- +siderable schedule and cost growth. For example, in 2005 the +JWST Science Assessment Team recommended, and the Sci- +ence Working Group approved, relaxation of strict contami- +nation requirements in favor of mirror cleaning procedures +(JWST Science Assessment Team 2005). These procedures +helped keep contamination budgets at manageable levels and +reduce cost. Future missions should set realistic contamina- +tion levels at the outset, which necessitates early discussion +of I&T. +The same review eliminated the 1 µm encircled energy re- +quirement (JWST Science Assessment Team 2005). There +were concerns with verifying the mirror performance at 1 µm +given factors such as the convergence rate to meet the polish- +ing specification, creep and micro yield mirror changes, and +individual deformation on mirrors from backplane deforma- +tion. The decision was to tighten the low-frequency wave- +front error allocation while increasing the mid- and high- +frequency allocations. This lead to significant risk mitigation +by providing mirror polishing schedule relief, relaxation of +challenging error budget terms, and relaxation of alignment +tolerances. As reported in Section 5.4.1, the image quality +observed at the end of commissioning was diffraction-limited +at 1.1 µm in spite of this requirement relaxation. +7.5. Stability of a Large Space Telescope +The telescope architecture relied on the support structure to +provide passive stability with wavefront sensing every other +day and control no more than every two weeks. This wave- +front sensing and control operations concept has proven to be +appropriate as demonstrated in Figure 23. +One of the larger uncertainties pre-launch was whether ob- +servatory stability levels would meet the predictions from the +integrated modeling. The observed drift sources and ampli- +tudes were anticipated and accurately modeled, as described +in Table 6. However, the thermal transient models were not +validated and for reasons not yet understood, some transient +thermal drifts were significantly over-predicted, and the ther- +mal distortion time constants came out significantly shorter +than predicted. +Although the IEC heaters’ on-off cycling with bang-bang +control results in a wavefront oscillation that is quite small +(only ∼2.5 nm), the effect is nonetheless easily sensed in +high-precision transit observations. This IEC heater insta- +bility was due to the cable harness connection to the tele- +scope being very stiff, which was not initially captured in the +telescope thermal distortion integrated model. The integrated +models did predict the observed level of drifts with remark- +able accuracy when their properties were included (see Ta- +ble 6). Modeling needs to carefully consider the harnesses, +and such bang-bang control heaters should be used with cau- +tion in future missions striving for ultra stability. +Tilt events have continued into the science mission, which +is not entirely unexpected, but they are still the dominant +source of wavefront changes. Tilt events are easily sensed +and corrected by the wavefront control system. The expecta- +tion is that the spontaneous tilt events will continue to subside +as the structure relieves stress, but continued monitoring will +help determine the nature of these events. JWST material +and cryogenic testing showed that the tilt events were infre- +quent and sufficiently small to satisfy JWST’s encircled en- +ergy stability requirements. For a future system that requires +picometer stability, it will be important to demonstrate that +picometer-class lurches are understood early (in the technol- +ogy phase) such that the active control systems can be de- +signed to sense and correct these displacements. +7.6. Team Cohesiveness and Rehearsals +The importance of an effective, well-trained, integrated +and “badgeless” team is well known, but worth repeating. + +THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION +31 +Several factors contributed to the development of such a team +for the JWST. The multi-institutional team comprised mem- +bers from the government, academia, and the aerospace in- +dustry spread across the United States. The telescope man- +agement team shared clear goals and created an environment +that encouraged cross-team interactions, communications, +and information sharing. +The management team worked +hard to foster trust and engender transparency across the dis- +tributed team, which we consider critical components of the +mission success. The regular interactions and open commu- +nications allowed the team to work effectively together, iden- +tify problems and formulate solutions. +The team was confident and well-prepared to execute the +mission, drawing on experience gained from the test program +and many dedicated commissioning rehearsals. +These re- +hearsals were operationally flight-like, incorporating proce- +dures and high-fidelity simulations for data analysis. Com- +prehensive, detailed operational processes and procedures, +critical to a mission of this complexity, were improved over +time given the lessons learned from rehearsals. These rigor- +ous rehearsals enabled the team to prepare and become com- +fortable with all aspects of the commissioning process prior +to launch. +The team also benefited from an unusually high degree of +continuity of staff on JWST, including key members in lead- +ership positions. +This continuity formed a base of multi- +institutional knowledge and long relationships that proved +valuable in problem solving. +Many had experience with +much of JWST mission’s life, with some being involved from +its inception through the end of commissioning. The team +also benefited from significant experience with other space +telescope projects, such as HST. +7.7. Guiding and Line of Sight Performance +JWST has demonstrated stable pointing with extremely +low jitter. JWST routinely achieves LOS jitter ∼1 mas RMS +through the use of a fine steering mirror in the control loop, +and via high cadence subarray readouts using a focal plane +instrument at pixel scales equivalent to the science instru- +ments. +For comparison, Hubble’s RMS jitter is typically +∼2.5 to 3 mas (Lallo 2012). Observational techniques to ob- +tain and analyze the jitter data were proven to be successful +(Hartig 2022), and results show the JWST design effectively +isolated vibration sources such as the cryocooler and sup- +pressed other potential contributors to jitter. +Complex attitude control systems are traditionally chal- +lenging to fully test pre-flight, and JWST’s was no excep- +tion. Even though a closed loop guiding demonstration was +executed during OTIS CV tests (Section 4.3.4), it was neces- +sarily lacking in full fidelity. Furthermore, early OTE com- +missioning activities required close-loop guiding with a mis- +aligned telescope, when the transformations from ACS to the +focal plane had not yet been established and guidestar PSFs +were not yet stacked. (Sections 5.2.1 and 5.3.1) +From early in the commissioning planning process, it was +recognized that this called for particularly close coordina- +tion among the teams responsible for Attitude Control, Wave- +front, and ISIM/FGS in developing the commissioning activ- +ities. The teams worked through the details of non-standard +guiding scenarios with control loop components that were +not yet fully calibrated, developed the operations concepts +for manually updating ACS transformations, produced the +tools for overriding the nominal guiding where necessary, +and wrote a number of contingency plans. As a result, in +flight, the sometimes subtle interdependencies of the ACS, +FGS, and the OTE optical alignment process came as little +surprise, and the team was able to effectively navigate along +the road to a successful and complete OTE commissioning. +The generalized lesson from this experience is that a space- +craft’s pointing and attitude control system is a key com- +ponent to its science performance, and benefits from being +treated as such, holistically, from early in the commissioning +plans. Yet, the integration and consideration of the ACS as a +system fundamental to science commissioning can be com- +plicated by cultural differences in approach, language, and +tools. This was anticipated from prior mission experience +and mitigated by an integrated inclusive approach to OTE +commissioning in general. +7.8. Importance of Determining Test Configurations Early +While a high-level summary of the I&T program is pro- +vided in Section 4.3, the test activities changed dramatically +in scope and implementation during the development phase. +An earlier definition of test configurations could have saved +resources. For example, the initial telescope cryotest concept +had the telescope pointing down, called “cup down”, on a +∼300,000 kg stainless steel tower with six spinning and rotat- +ing cryogenic autocollimating flats and cones of light through +instruments to measure the alignment by stitching interfero- +grams. The extreme complexity of the cup down test led to a +total redesign that resulted in a simplified, yet still challeng- +ing to execute, “cup up” configuration as described in Sec- +tion 4.3.4 (Atkinson et al. 2008). The cup down configuration +was originally baselined in order to prevent any contamina- +tion of the telescope mirrors, but the contamination was later +deemed manageable in the cup up configuration by imple- +menting a mirror-cleaning procedure. This mirror cleaning +procedure was used following the OTIS cryotest and before +shipping to the launch site (Lobmeyer & Carey 2018; Abeel +& Huang 2018). +7.9. Micrometeoroid Environment and Damage +The open architecture of the JWST telescope makes the +primary and secondary mirror optics particularly susceptible + +32 +MCELWAIN, FEINBERG, & PERRIN ET AL. +to micrometeoroid damage. As discussed in Section 6.2.2, +the effect from micrometeoroids was estimated based on the +environment and a damage model from impacts. The mi- +crometeor effects may have been underestimated, though the +bulk of the damage to date has come from the single event on +C3, so there is great statistical uncertainty in how the dam- +age rate should be extrapolated into the future. The impact +physics is also complex and uncertain; ground tests can’t +achieve micrometeoroid-like velocities and neglected how +cryogenic temperatures change the material stiffness proper- +ties. Because of the statistical and damage model uncertain- +ties, and with the prospect for a JWST mission lifetime far +exceeding requirements (due to available propellant), restric- +tions on the field of regard will be implemented to reduce ram +direction micrometeoroid impact rates. Missions under con- +sideration with even larger aperture optics and even tighter +wavefront requirements will need to consider this issue care- +fully and track JWST’s experience as it accumulates a longer +baseline of operations. +7.10. Optical Modeling Across Interfaces +In the development of the independent system elements, +there were separate optical and optomechanical models for +the telescope and each of the science instruments via the in- +terface shown in Figure 3. Early in the program as the prime +contractor and the NIRCam were selected, it was realized +that the optical designs were not compatible. To deal with +this, the telescope design was modified and the F-number +changed to accommodate the interface. However, it was later +uncovered that this design change increased susceptibility to +the rogue path that was described briefly in Section 3.1 and +more thoroughly in Lightsey et al. (2014). The rogue path +passes through the AOS entrance aperture and directly onto +the science instrument pick off mirrors, where it then en- +ters the science instrument optical path through reflections +or scattering. +The rogue path stray light was well known and modeled. +The models were used to confirm no direct optical paths to +the detector focal planes and to determine scattering from the +pick off mirrors was negligible. However, what was missed, +was the possibility of rogue path stray light causing graz- +ing angle scattering off instrument structure from the pickoff +mirror housing and downstream in the optical path of the in- +struments. After observing the stray light in flight (see Sec- +tion 5.3 of Rigby PASP Science), the optical and optome- +chanical models were used to reproduce the observed phe- +nomena, confirming the stray light paths. The area of sus- +ceptibility for the observed features that was observed is a +small subset of the total rogue path region of the sky. Now +that the grazing angle stray light paths are known, observa- +tion scheduling can largely prevent placing bright targets in +the susceptible region on the sky relative to the science target. +Not all of the full structural as well as optical characteris- +tics were captured in the pre-flight modeling. Programmatic +constraints impeded clear communication and modeling of +the full system interface. The optical prescription of the in- +struments was in the full system model, but not the detailed +structures that included the housing around the pick-off mir- +rors and the detailed optomechanical structures in the science +instruments. The full up system model only included NIR- +Cam and MIRI, in order to verify the stray light light require- +ments at the NIR and MIR wavelengths, and the instrument +teams were left to carry out the detailed analysis of their in- +struments. The lesson learned is that additional modeling of +the complete integrated system for all modes is needed to re- +duce risk of unexpected stray light phenomena. This includes +accurate details of mechanical structures as well as the opti- +cal prescriptions throughout the complete system. +8. Conclusion +The revolutionary JWST telescope is performing better +than all of its design objectives, enabling even higher sen- +sitivity and more stable observations than originally planned. +The telescope was made possible through the advancement +of several new technologies, all of which were developed and +flight proven to work as intended. The telescope has now em- +barked on its Cycle 1 science observations, beginning a sci- +entific journey that will answer some of the biggest questions +in astrophysics and planetary science. +We are grateful to the JWST photography team, espe- +cially Chris Gunn, for capturing the JWST development, and +Heather Ghannadian for graphics design. +The JWST mission is a joint project between the National +Aeronautics and Space Agency, European Space Agency, +and the Canadian Space Agency. The JWST telescope de- +velopment was led at NASA’s Goddard Space Flight Center +with a distributed team across Northrop Grumman Corpora- +tion, Ball Aerospace, L3Harris Technologies, the Space Tele- +scope Science Institute, and many other companies and insti- +tutions. This telescope was created by a large team of people +from many diverse backgrounds whose creativity, passion, +teamwork and endless sacrifices made this scientific dream a +reality. +References +Abeel, A. C., & Huang, Y. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Huntsville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Alabama 35812,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' USA 14L3Harris Corporation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Space and Intelligence Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 400 Initiative Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Box 60488, Rochester, NY, USA 14606, USA 15Thirty Meter Telescope, 100 West Walnut Street, Suite 300, Pasadena, CA 91124, USA 16Lambda Consulting LLC/Advanced NanoPhotonics, 4437 Windsor Farm Road, Harwood, MD 20776, USA 17Aerodyne Industries, 8910 Astronaut Blvd #208, Cape Canaveral, FL 32920, USA 18Genesis Engineering Solutions, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', 4501 Boston Way, Lanham, MD 20706, USA 19SAIC, 12010 Sunset Hills Road, Reston, VA 20190, USA 20KBR, 6200 Guardian Gateway, Suite 105, Aberdeen, MD 21005, USA 21ATA Aerospace, 7474 Greenway Center Drive, Greenbelt, MD 20770, USA Abstract The James Webb Space Telescope (JWST) is a large, infrared space telescope that has recently started its science program which will enable breakthroughs in astrophysics and planetary science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Notably, JWST will provide the very first observations of the earliest luminous objects in the Universe and start a new era of exoplanet atmospheric characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This transformative science is enabled by a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 m telescope that is passively cooled with a 5-layer sunshield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The primary mirror is comprised of 18 controllable, low areal density hexagonal segments, that were aligned and phased relative to each other in orbit using innovative image-based wavefront sensing and control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This revolutionary telescope took more than two decades to develop with a widely distributed team across engineering disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We present an overview arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='01779v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='IM] 4 Jan 2023 2 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' of the telescope requirements, architecture, development, superb on-orbit performance, and lessons learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST successfully demonstrates a segmented aperture space telescope and establishes a path to building even larger space telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Introduction In the mid-1990s, the Hubble Space Telescope (Hubble) observed the now-iconic Hubble Deep Field, which provided direct evidence that the universe evolved from the hot plasma left following the Big Bang to the galaxies and stars we see nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These observations revolutionized our understand- ing of our place in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, due to the intrin- sic faintness of very distant galaxies and their wavelength shift into the infrared, observations of the very early uni- verse remained beyond the capabilities of Hubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Observing the first stars and galaxies required infrared wavelength cov- erage and increased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This motivated the science community to conceive of the James Webb Space Telescope (JWST), a large, general-purpose infrared-optimized obser- vatory (Dressler & “HST and Beyond” Committee 1996) (Gardner PASP1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The early science motivations for JWST were organized into four broad themes: origins of the uni- verse (including first light and reionization), the assembly of galaxies, the birth of stars and protoplanetary systems, and the formation of planetary systems and the origins of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The 2000 Astrophysics Decadal Survey, “Astronomy and As- trophysics in the New Millenium” placed the JWST concept as the top priority for NASA (Council 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' An Indepen- dent Review Board in 2018 and the recent 2020 Astrophysics Decadal survey concluded the mission was as compelling two decades later as when it was originally conceived (Com- mittee for a Decadal Survey of Astronomy and Astrophysics 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the four science themes demanded a large (∼25 m2), infrared-space telescope covering a wavelength range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 µm (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The paradigm for space telescope architectures needed to change in order to make this space telescope with a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 m circum- scribed diameter primary mirror, as this physical size is larger than the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 m rocket fairing diameters available at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope and observatory needed to be stowed for launch and then undergo a series of major deployments to transform it into the operational configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The solution was to make the primary mirror segmented based on imple- mentations from the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Keck Observatory’s (Keck) two ground-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST primary mirror is com- prised of 18 hexagonal segments, each of which is supported by a common backplane opto-mechanical structure and ad- justable in 6 positional degrees of freedom, as well as radius 1 Citations followed by PASP indicate papers to be included in the JWST Overview Special Issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The primary mirror segments assemblies, large precision cryogenic structures, and the wavefront sensing and control did not exist and needed to be created specif- ically for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The infrared observations means that the entire telescope and instruments must operate at cryogenic temperatures, which are reached through passive cooling by a 5-layer sunshield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST Yardstick study established an architecture that demonstrated the science performance and technical feasi- bility (Bely 1998), which baselined passive cooling via the large sunshield while operating in orbit at L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early archi- tecture concepts were competed with a proposal down-select that was awarded to TRW Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', now Northrop Grumman Cor- poration (Northrop), and Ball Aerospace, along with a team at ATK, now Northrop, and Kodak, now L3Harris Technolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The architecture concept studies further refined the de- sign (Lightsey & Ebbets 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During the concept studies, the JWST primary mirror was 8 m in diameter with multi- ple mirror concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Once TRW Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' was selected in 2002, their initial design was based on a 36 segment architecture that was 7 m in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, each segment had only tip, tilt, piston and radius of curvature control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Later a trade was done to consider an 18 segment design that while only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 m in diameter would include seven degrees of freedom on each mirror using a hexapod with a radius of curvature actu- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A key consideration for the trade was that 18 segments reduced the amount of segment perimeters (edges) which are one of the key challenges of making mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Ultimately the decision was to baseline 18 segments and hexapods, which enabled astigmatism correction that was beneficial in the mir- ror manufacturing, alignment tolerances, and cryogenic test- ing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST observatory architecture, science instrument on-sky performance, backgrounds, and science performance are presented in this PASP JWST Overview Special Issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this paper, we focus on the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We start by describ- ing the driving requirements in Section 2, and the resulting design architecture in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Highlights from the devel- opment and integration and test phases are presented in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We present the commissioning and on-orbit perfor- mance of the telescope in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In Section 6, we report on the science era characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A discussion of lessons learned is presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We conclude in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The on-orbit performance of the telescope is better than the requirements of the telescope across the board (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2022), a result which was achieved through diligent systems engineering and a thorough test program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST’s success demonstrates that ambitious space telescopes can be built to THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 3 Table 1: Optical Requirements Optical Requirements Requirement Measured Optical Area in m2 >25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='44 Strehl ratio over NIRCam FOV at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 µm >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='84 Strehl ratio over MIRI FOV at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 µm >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='92 Encircled Energy Stability over 24 hoursa <2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2%c Encircled Energy Stability over 14 hoursb <3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='53%c Vignetting None None a Aperture radius of 80 mas at a wavelength of 2µm from the mean over a 24 hour period, evaluated for a worst-case 10◦pitch maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' b Aperture radius of 80 mas at a wavelength of 2µm from the mean over a 14 day period, evaluated for a worst-case hot-to-cold pitch maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' c Typical performance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Occasional larger instabilities are observed due to tilt events, discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' help answer some of the biggest questions facing humanity, such as how the universe works and how we got here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Telescope Requirements The telescope architecture was designed to meet just a few driving optical requirements derived by early science trade studies, as presented in Table 1 alongside the on-orbit perfor- mance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A large, primary mirror aperture was required to have a total unobscured collecting area greater than 25 m2 that operated over the spectral range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 µm to 27 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical area x transmission product was the metric used to specify the performance of the coatings, contamination, and micrometeoroid damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For this large aperture, the tele- scope image quality was specified using the Strehl ratio met- ric at 2 µm2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 µm, with a diffraction-limited image quality value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 over the NIRCam and MIRI fields of view, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Finally, a set of encircled energy stability requirements were defined at time intervals of 24 hours and 14 days, specifying less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 % and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 % RMS varia- tions of the energy within a 80 mas radius aperture at 2µm wavelength following a worst case hot-to-cold slew within the field of regard (FOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The radiometric sensitivity for the observatory required the observatory to have low backgrounds that enabled the detec- tion of faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The largest near-infrared background component was to be from zodiacal light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the longer wave- lengths of the mid-infrared channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' the requirements permit- ted the background to be dominated by thermal self-emission from the observatory rather than the natural sky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' but with that 2 A key requirements downscope was dropping the formal requirement for 1 µm performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' to reduce complexity in the integration and test (I&T) program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' with the expectation that an optical system meeting or exceeding requirements at 2 µm would necessarily also yield very good performance at 1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This has proven the case in flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' See Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 for lesson learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' thermal emission tightly constrained to yield the required un- precedented mid-infrared sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The near-infrared back- ground was limited by the observatory design and modeling, careful control of particulate levels, and the use of low reflec- tivity baffle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The low-background, mid-infrared en- vironment is enabled by passively cooling the telescope and science instruments to cryogenic operating temperatures for the life of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observatory operates in a Lis- sajous orbit about the Earth-Sun L2 point that permits a sun- shield to continuously keep the telescope and science instru- ment protected from the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (Rigby PASP Backgrounds) provides a detailed discussion of the backgrounds and the on-sky measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All of the performance requirements were defined to be met at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 yr after launch, referred to as “End of Life” (EOL), as that was the minimum required lifetime of the mission3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical performance properties are expected to degrade over time due exposure within the space environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical performance requirement predic- tions accounted for these degradations and used the worst- case prediction for each input category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For example, the encircled energy stability requirement used the worst-case wavefront error, thermal distortion, and image motion all happening at the same time, which is a condition that is known to be statistically unlikely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The worst case predictions also included model uncertainty factors (MUFs) to manage uncertainties and provide margin at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Therefore, the performance measured during com- missioning, the so-called “Beginning of Life” (BOL) optical properties, should not be directly compared to the require- ment EOL values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In addition to the driving performance requirements, the telescope had challenging design constraints in order for the observatory to meet its system-level requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope total mass was specified to be less than 2460 kg and the measured mass at launch was 2339kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The cen- ter of mass lateral location was specified to fit within a 200mm×200mm envelope and the Observatory fit within that envelope with 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 mm margin to the nearest boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The stowed telescope volume was set to be within an enve- lope 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='985 m × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='114 m × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='942 m (V1,V2,V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Both the mass and stowed telescope volume requirements were driven by the capabilities of the Ariane 5 launch vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The deployed telescope volume expanded considerably to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='386 m × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='100 m × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='971 m, which needed to be shaded by the sunshield while pointing within the designated field of regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Finally, the power consumption of the telescope was 3 The only consumable onboard JWST is the propellant, which is pre- dicted to have a lifetime of greater than 20 years (Menzel PASP Observa- tory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 1: The telescope and science instruments are sepa- rated from the spacecraft bus and shielded from direct sun- light by a 5-layer sunshield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope’s primary and sec- ondary mirrors are open to the celestial sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' When stowed (below), the observatory volume is significantly reduced to fit within the Ariane 5 fairing and pinned mechanisms enable the structure to withstand the launch environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The V co- ordinate system origin is at the vertex of the primary mirror surface, along the boresight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' set to be less than 50 W (measured 34 W) such that the total power budget for the observatory was maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Telescope Architecture The observatory architecture consists of three major sys- tems: the telescope and science instruments, the 5-layer sun- shield, and the spacecraft bus (Nella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Lightsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Renderings of the observatory in its operational and stowed configurations are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The tele- scope and the science instruments are passively cooled by the sunshield and thermally isolated from the spacecraft bus and solar array that are on the warm, sun-facing side of the obser- vatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope’s primary mirror (PM) and secondary mirror (SM) are open to the celestial sky, but the tertiary and fine steering mirror are enclosed within the aft optics system (AOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope Cassegrain focus is formed near the entrance to the AOS where an aperture mask is located for stray light suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There are four science instruments (SIs): the NIRCam (Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2005), NIRSpec (Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2022), MIRI (Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2015), and FGS/NIRISS (Doyon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All of the science in- struments are mounted within the same structure, called the integrated science instrument module (ISIM), which is blan- keted from the celestial sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The instrument drive electronics and radiators are mounted on the exterior of the ISIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fig- ure 2 shows an exploded view of the telescope components along with the science instruments, thermal management sys- tem and mechanical supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope field of regard is restricted to pointings that shield the telescope and science instruments from direct so- lar illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope boresight pitch constraints are between pitch angles of 85◦ and 130◦ (0◦ pointed towards the sun), roll about the telescope boresight is constrained to ±5◦, and yaw is unconstrained to a full 360◦ around the sunline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observatory is in an orbit around the Earth-Sun L2 La- grange libration point, such that it orbits the sun along with the Earth over one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' While the Earth and L2 Lagrange point sweep along the celestial sphere, the field of regard on- sky visibility changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' At any instantaneous moment, 40% of the sky is visible to the telescope, while over the course of the year the full sky is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The total visibility pe- riod over the course of a year increases with ecliptic latitude, ranging from ∼100 days in the ecliptic plane to continuous visibility in 5-degree-radius cones at the ecliptic poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST telescope was designed to be stowed in or- der to fit within the Ariane 5 fairing for launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Following launch, the observatory needed to be deployed into the oper- ational configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The 18 primary mirror segments are arranged in a close-packed configuration with twelve seg- ments in the center section and three segments on each of two wings which were folded back for launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' No segment is located at the nominal center position, as it is filled with the aft optical structure that holds the telescope tertiary and fine steering mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The secondary mirror was also stowed such that it was face down above the center section, shielding its surface from particulate redistribution during launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Front V3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='100 m V1 V2 Optical Telescope 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='600 m Element (OTE) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='625 m V3 Side Integrated Science Instrument Module (slM) Sunshield Spacecraft Bus 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='197 m Stowed Side V3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='661 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='472 m JWST_005THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 5 Figure 2: This exploded view shows telescope components, including the telescope mirrors, the optomechanical structures, control electronics, and the thermal management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The integrated science instrument module includes the four science instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' telescope deployments included separating from the space- craft bus using a deployable tower assembly, driving the sec- ondary mirror into position, rotating and latching the primary mirror wings into position, and finally deploying the primary mirror segment assemblies and the secondary mirror from their launch locks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope alignment used image- based sensing and control with the primary mirror segments and secondary mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Design Implementation The optical design of the telescope is a three-mirror anas- tigmat (TMA) that corrects spherical aberration, coma, astig- matism, and field curvature (see Korsch (1972)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In addition to the primary, secondary, and tertiary mirrors of the TMA, the telescope also includes a fine steering mirror (FSM) which is actively controlled to stabilize the line-of-sight opti- cal path into the science instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A raytrace through the telescope with annotations for the optical surface is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the 18 primary mirror segments are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='32 m (flat-to-flat) separated by 7 mm gaps (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The PM circumscribed diameter is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='64 m (segment flat-to-flat diam- eter is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='575 m, entrance pupil diameter is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 m) with an area equivalent to a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 m circular filled aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The tele- scope is f/20 with an effective focal length of 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The field is about ∼ 18′ × 9′ at the telescope focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the four science instruments has one or more pick-off mirrors near the focal plane of the telescope such that they observe offset parts of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope optical parameters are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 3: The JWST telescope is a three mirror anastigmat that has a primary mirror, secondary mirror, tertiary mirror, and a fine steering mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the four science instru- ments has one or more pickoff mirrors near the telescope’s focal surface such that they see different fields of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the 18 primary mirror segments is controlled in the six mechanical degrees of freedom (DoF) using a hexapod with six actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The segments are semi-rigid with a radius of curvature actuator at the center of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During the align- ment process, the radius of curvature for each primary mirror Aft Optics Telescope Secondary Mirror Support Subsystem (AOS) Electronic Structure (SMsS) Boxes ISIM Radiators ISIM Science Secondary Mirror Instruments Assembly (sMA) ISIM Harness Primary Mirror Segment Radiator Assemblies (PMsA) ISIM Electronics Backplane Support Structure Thermal Management Components Deployment Tower Assembly (DTA) JWST_012OTE!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='ISIM V3 (anti-spacecraft) (V1, V3 V2 origin) V1 Cassegrain Tertiary focus mirror Fine- steering Focal Secondary mirror surface Mirror Primary→ Mirror Sunshield Sunlight JWST_0016 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 4: The primary mirror dimensions and the tiling con- figuration with 18 hexagonal-shaped segments to form a tri- contagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The A-, B-, and C-segment prescriptions are sep- arate and take advantage of the six-fold symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' segment is matched and the primary and secondary mirrors are optimally aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The gaps between segments, nomi- nally 7 mm, were tracked during the deployment and align- ments by bookkeeping the actuator resolver counts and in- dependently monitoring the positions using linearly variable differential transformers (LVDT) electromechanical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All of the telescope optics are made from Beryllium based primarily on its low coefficient of thermal expansion (CTE) over the mirror’s operating temperature range of ∼35-55 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segments closer to the warm spacecraft core region have cor- respondingly higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mirror moves are controlled through the actuator drive unit (ADU) that is housed on an electronics panel within the spacecraft bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The ADU provides state-of-health telemetry for the telescope hardware, controls the 132 mirror actuators, and polls telemetry like motor revolutions and strain gauges on the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Those signals pass through cryogenic elec- tronics boxes: first the cold multiplexer units (CMUs) located on the telescope backplane, and a cold junction box (CJB) before returning to the ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In a separate mode, the ADU controls the fine steering mirror linear voice coil motors and provides telemetry for its x,y position, current, and tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Passive stability is achieved through mechanical ∼1 Hz isolators between the telescope and the spacecraft bus and a thermally stable backplane support for the telescope op- tics made from a lightweight composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The reaction wheel assemblies (RWA) and cryocooler compressor assemblies housed within the spacecraft include vibration attenuators within their subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The composite truss structure is comprised of a center section that supports 12 segments, and two wing sections, each supporting 3 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There is a separate backplane support frame that carries the load from the telescope structure and the ISIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The secondary mirror is Table 2: Telescope Optical Parameters Mirror RoC Surface Conic V1 V2 V3 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Size (mm) (mm) (mm) (mm) (mm) Primary 15879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 concave 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9967 0 0 0 6605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 Secondary 1778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 convex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6598 7169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 0 0 738 Tertiary 3016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 concave 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6595 796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='19 728×517 Fine Steering flat 1047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='36 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 NOTE—The primary-to-secondary effective focal length (EFL) is 59400 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', and the three- mirror telescope EFL is 131400 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' mounted to a tripod that consists of four composite tubes and connects to the primary center section in a four-bar linkage (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Active pointing stability is achieved using a control loop that senses line of sight pointing with a fine guidance sensor (FGS, ∼16 Hz) and corrects pointing errors using the FSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This controls line of sight pointing drifts with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='78 Hz con- trol loop bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In addition, as the fine guidance sen- sor can only provide pointing measurements about two axes (image tip and tilt), the coarse pointing roll controller uses star tracker and inertial reference unit (IRU) measurements to stabilize the roll (clocking) about the boresight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' How- ever, thermal distortions may cause drifts between the star- tracker-based spacecraft pointing frame and the line of sight frame, which cannot be measured and therefore would re- main uncorrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This could create a rotational drift about the guide star, which manifests as image smear elsewhere in the science instrument field of view (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Ad- ditionally, higher frequency image motion, due to reaction wheel and cryocooler disturbances, results in a Gaussian im- age blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The image motion has a negligible effect on the encircled energy, but may degrade the image quality defined by the Strehl ratio requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope open architecture for passive cooling makes it susceptible to stray light from the celestial sky and the ob- servatory emission itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There are successive layers of stray light protection used to shield the science instrument focal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The sunshield protects the optical path from direct sunlight and enables passive cooling of the telescope and sci- ence instruments that reduces the self emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The spreader bars that hold the sunshield layers in position have baffles on their caps, called epaulettes, that shield the optical path from the warm spreader bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There is an oversized, internal pupil stop near the FSM to transfer the maximum PM collected energy and provide a well defined pupil for wavefront sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A lightweight structure, called the ‘frill’ (see Figure 1 top), extends from the perimeter of the PM and substantially fills in the gap be- tween the oversized pupil stop and the image of the PM at the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='32 meters flat to flat B1 C6 C1 B6 A1 B2 A6 A2 C5 C2 A5 A3 B5 A4 B3 C4 C3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 meters B4 flat to flat JWST_002THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 7 stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' By filling in this gap, the frill blocks celestial light from behind and around the PM (‘truant’ stray light path) from be- coming a source of background stray light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A similar baffle, called the bib, extends below the frill and blocks the direct path to the warm spacecraft core area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The AOS enclosure blocks stray light around the tertiary and FSM mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The AOS entrance aperture is near the Cassegrain focus of the telescope and provides an aperture stop for stray light suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A baffle extending above the FSM combined with careful telescope alignment, prevents light from the sky passing through the AOS entrance aperture and striking any instrument optics directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This stray light path, called the rogue path, was identified early in the de- sign phase and precautions were taken to keep light from this path from propagating directly through the SI optical trains to their detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In flight, however, scatter paths off SI mechanical structure were found to produce unwanted back- grounds from sources in this small region of sky (see Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 in Rigby PASP Science, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 lesson learned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fortunately, the rogue path stay light can be largely managed through observation scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The integrated science instrument module is enclosed and blanketed for stray light and thermal purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each of the science instruments also has an enclosure and the optical paths include internal baffles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Telescope Development The telescope development was a long and complex pro- cess that took place between 2002 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this section, we provide an overview of the telescope development phase with discussions of the systems engineering, the new tech- nologies developed, and the integration and testing needed to verify the design and workmanship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Prime contractor Northrop Grumman was responsible for the telescope design and built a team that included optics lead Ball Aerospace, composite lead ATK (later to be bought by Northrop Grumman), and the L3Harris Corporation (orig- inally part of Kodak, then ITT Exelis) for their large op- tics integration heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The lead government organization was NASA’s Goddard Space Flight Center, which took full responsibility for the telescope and provided key facilities where the mirrors and science instruments were integrated with the backplane structure and tested at ambient temper- atures for vibration and acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Jet Propulsion Lab- oratory and NASA’s Marshall Space Flight Center helped with technology development and mirror testing, while cryo- genic tests took place at NASA’s Johnson Space Flight Cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Space Telescope Science Institute provided lead- ership in developing telescope operations plans and helped lead the commissioning efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Throughout the development phase, groups of scientists and engineers worked collabora- tively across these organizations to jointly design, integrate, test, and commission JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Further, the telescope development team sought input from external expertise for technical decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The project formed an independent Product Integrity Team (PIT) to pro- vide technical engineering advice on the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The PIT was led by Professor Duncan Moore of University of Rochester and Professor James Wyant of University of Ari- zona and included space- and ground-telescope experts from Hubble, the Spitzer Space Telescope (Spitzer), Keck and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Throughout the development phase, the PIT independently reviewed all aspects of the optics program, especially the integration and test campaign with its direct ties to the verification plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The early test concepts ma- tured and improved significantly over time with input from the optical PIT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In cases where performance predictions threatened requirements, the JWST science requirements advisory board was convened, with rep- resentation from the JWST Science Working Group, to eval- uate the science impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope design process began with the mirror sub- strate and moved outwards to the backplane and ultimately to the whole telescope and observatory design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' New tech- nologies were developed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Sub-system develop- ment schedules were managed such that the telescope would meet the larger observatory schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early engineering de- sign units (EDUs) of the primary mirror segment and sec- ondary mirror were important to prove out the manufactur- ing process for these critical-path components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The primary mirror EDU demonstrated technology readiness, TRL-6, by carrying out successful acoustic and vibration tests that were not completed in the earlier technology development period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' An engineering design unit partial version of the OTE center section, called Pathfinder, allowed verification of assembly, handling, and testing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Systems Engineering for the JWST Telescope The JWST telescope development relied on systems en- gineering principles, tools, and practices as described in Menzel PASP Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The systems engineering ap- proach for JWST is presented thoroughly in Lightsey & Arenberg (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope performance requirements relied on detailed budgets for optical performance (Lightsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2010), alignments (Glassman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016), and actuator ranges (Barto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The budgets were used through- out the mission development: to make design decisions and carry out concept trades, to design the integration and test program, to verify the requirements before launch, for oper- ations planning, and to assess the on-orbit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical performance requirements were managed through the wavefront error budget that was based on the image quality metrics of Strehl ratio and encircled energy 8 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (see Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The top-level image quality requirements put direct constraints on the system RMS wavefront error (WFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Specifically, the driving requirement of a Strehl ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 at 2 µm wavelength required a system-wide RMS WFE < 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A portion of this WFE budget was then allocated to each subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This top-level system-level performance was used to make allocations to each of the sub-systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', telescope, integrating structure, science instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The driving encircled energy stability requirement was <3% in a 80 mas radius aperture at 2µm wavelength over 14 days following a worst case hot-to-cold slew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The encircled en- ergy stability was budgeted by assigning wavefront error al- locations in different spatial frequency composition regimes based on capability of the active wavefront control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The low- frequency modes were allocated based on the active global- alignment aberration control, while the mid-frequency modes were allocated based on the active primary mirror figure con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The high-frequency modes were based on the static high spatial frequency errors that were controlled during the fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each sub-system’s WFE allocation was further divided into WFE static residuals, WFE stability, and image motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a system, each allocation could either be tested (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g, mirror static high-frequency errors) or assessed via integrated modeling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', OTE stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST’s performance verification could not follow the tra- ditional NASA paradigm to “test as you fly” (TAYF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The test facilities were not available for end-to-end testing of an observatory of this size operating at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Rather, the integration and test program developed for JWST relied on incremental performance verification with testing at the sub-system level and, to the extent possible, testing at higher levels of assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The test conditions across the observatory were dramatically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For example, the telescope and science instruments were cryovacuum tested at their operational temperatures of ∼40 K in Chamber A at NASA’s Johnson Space Center (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4) whereas the sunshield and spacecraft bus were cryovacuum tested at temperatures ranging between 150–330 K in a more conven- tional thermal-vacuum environment at Northrop Grumman (McElwain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This piecewise verification relied on subsystem test data to anchor the high-fidelity integrated models used to make performance predictions for many of the system-level requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Integrated Modeling Highly complex integrated modeling was a key enabling capacity throughout all stages of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical performance estimates made use of ground test data, inte- grated models, and simulations including the uncertainties in wavefront sensing and control to verify the pre-launch requirements (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Component design and ground test performance was used to provide predictions for optical alignments, component-level wavefront error, and ground-to- flight effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The dynamic components of the error budget used test data as inputs to an extensive structural-thermal- optical (STOP) integrated modeling process that predicted wavefront stability (Knight et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012) and line-of-sight im- age motion (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The integrated modeling for telescope performance made use of models of the struc- ture, deployed thermal, thermal distortion, optical perfor- mance, dynamics and attitude control, and stray light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each model was validated upon test data and conservative model uncertainty factors were applied to bound the worst case per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Image motion predictions made use of exported vibrations and a model of the deployed dynamics of the ob- servatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope thermal distortion and pointing sta- bility following a worst-case slew used a thermal model that balanced the steady state at the hot attitude (pitched towards the sun) and the cold attitude (pitched away from the sun) with a worst case roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The small temperature changes, less than 15 mK, from those thermal extremes was predicted for each of the thermal nodes on the telescope and used to de- termine the mechanical displacements on the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The repository of test data and integrated modeling results were used as inputs to the Integrated Telescope Model (ITM) sim- ulator developed by Ball Aerospace in order to predict the optical performance, simulate data products for the develop- ment of analysis tools, and to rehearse the telescope align- ment process (Knight et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The same optical models were also used to inform pre- flight modeling of point spread functions (in particular using the software package WebbPSF, Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2014), which were used extensively in science planning, in development of proposal planning tools such as the exposure time calcu- lator, and in development of data analyses pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The core Fourier optical simulations of PSFs were augmented over time to become part of comprehensive high-fidelity data simulators, such as MIRAGE for NIRCam, NIRISS, and FGS data (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Hilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2022) and MIRISim for MIRI (Klaassen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These, along with ITM, became critical enabling tools for the long campaign of preflight rehearsals of the WFSC alignment process (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' New Technologies Needed Early in the mission development, three new telescope technologies were identified that needed to be developed ex- plicitly for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The project made significant early invest- ments in these enabling technologies to ensure they would be at a technology readiness level (TRL) of 6 (TRL-6 = fully functional model or prototype, demonstrated in a rel- evant environment) or higher prior to the mission’s prelimi- nary design review (PDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope technology matu- ration program included primary mirror segment assemblies, THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 9 Figure 5: JWST’s optical performance estimates were made using a combination of test data and integrated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For require- ment verifications, there were conservative model uncertainty factors applied and the end of life performance during worst case conditions were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' large precision cryogenic structures, and wavefront sensing and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In addition, new metrology capabilities were needed to verify the performance, and a series of new inter- ferometric techniques were developed to test the stability of the composite structure and verify the optical performance of the telescope at ambient and cryogenic temperatures (Saif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The following discussion provides highlights from the technology development program that was com- pleted in 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Primary Mirror Segment Assemblies Low areal density mirrors were recognized as a key tech- nology gap to enabling a ∼25 m2 aperture space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The areal density of the Hubble primary mirror is 240 kg m−2, while JWST’s objective was < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 kg m−2, which was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A mirror technology development program was convened to evaluate and advance mirror technologies through multiple programs, including the subscale Beryl- lium mirror demonstrator (SBMD, Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2001) and the advanced mirror system demonstrator (AMSD, Stahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The AMSD program evaluated ULE and Beryllium with a wide variety of parameters such as the optical perfor- mances achieved, control authority, mounting, and fabrica- tion schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' While ULE was deemed to have programmatic advantages, it was found to have an astigmatism as it cooled that was non-deterministic and would have added uncertainty to the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The decision to select O30 Beryllium, a more isotropic form of Beryllium not previously used in space telescopes, was made following the recommendation from the Mirror Recommendation Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Beryllium was se- lected largely due to its small coefficient of thermal expan- sion (CTE) within the telescope’s operational temperature range, making it particularly advantageous during the cryo- polishing fabrication process and achieving the telescope’s passive stability objectives by not requiring active thermal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Beryllium is also light weight, advantageous given the very tight mass constraints for the telescope (Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Beryllium mirrors have flight heritage from pre- vious space missions, including the Spitzer, the Infrared As- tronomical Satellite (IRAS), and the Cosmic Background Ex- plorer (COBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Active control of primary mirror segment position was achieved using actuators mounted in a hexapod arrangement, plus a center actuator for active control of radius of curva- ture (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Specialized actuator mechanisms were developed specifically for JWST in order to enable the active positioning of the large segmented mirrors and to support the mirrors during ground test and the launch environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Each mechanism makes use of a fine stage flexure and coarse drive coupling to control the linear displacement (Warden 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The actuators themselves have remarkable performance pa- rameters including a fine step size of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 nm resolution, with 2 nm of fine repeatability, over a 10 µm fine range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A coarse drive coupling in the same mechanism provides a 58 nm step size over a full 21 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Further, unlike ground telescope active and adaptive optics, which often use electrostatic or piezoelectric actuators, JWST’s actuators operate mechani- cally via a gear train and flexures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' the mechanical gear trains hold position stiffly and precisely even when the actuator is entirely unpowered, which is necessary to avoid undesired waste heat into the cryogenic telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The primary mirror segment development process required new facilitization and made use of economies of scale to fab- ricate the multiple segments in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The mirror blanks were made from O30 Beryllium through a hot isostatic press- ing process by Brush Wellman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The blanks were then light StaticErrors WFSC Algorithms Optical Alignments Simulations WFSC Component WFE Realizations Processing (PM, SM, TM, FSM, SIs) with ITM / MIRAGE Ground to Flight Effects PSF/OPD Dynamic Errors WFE Stability Encircled Strehl Energy Stability Image Motion JWST_00810 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 6: Each primary mirror segment and the secondary mirror are supported by a hexapod as shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The pri- mary mirror segments also have a radius of curvature actuator that enables the focal lengths for each segment to be matched during the alignment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 7: Six of the flight mirror segments prepared to un- dergo acceptance testing in the XRCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The segment-level op- tical performance and structural stability was precisely mea- sured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' weighted by removing over 92% of the material in a honey- comb shape by Axsys Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Next, the mirror sub- strates were polished by Tinsley Labs and each mounted to its flexure and radius of curvature system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The mirrors were then optically tested at ambient and cryogenic temper- atures at X-ray and Cryogenic Facility (XRCF) at NASA’s Marshall Space Flight Center, followed by another round of cryo-polishing to ensure each mirror achieved the correct op- tical figure at the intended cryogenic operating temperature (Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The mirrors were gold coated using a vac- uum vapor deposition process by QCI, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The gold coating provides high reflectivity across the operational wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1µm (Keski-Kuha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A protec- tive SiOx overcoat was applied that improved the durability of the coating and enabled cleaning at stages throughout the I&T process (see Lobmeyer & Carey 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Finally, flight acceptance testing for each segment was carried out in the XRCF facility (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Large Precision Cryogenic Structure A large precision cryogenic structure was necessary to en- able the passive optical stability without active control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope and the science instruments are supported by a composite optomechanical structure that must withstand the launch environment loads, deploy within the capture range of the mirror actuators, survive the stresses induced from cool- ing down to cryogenic temperatures, and have minimal ther- mal distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early in the JWST development, it was recognized that the materials database for composite structures was inadequate over the operational temperature range, the ability to measure deformations was inadequate for the JWST verification test- ing, the engineering modeling tools needed development, and manufacturing process controls needed improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The coefficient of thermal expansion for the materials needed to be measured to less than 30 ppb K−1 at temperatures of < 25 K, which was more than 100 times more precise than the state of the art at the time (Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' ATK imple- mented a technique to measure the CTE for large structures with the precision needed for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' With the materials char- acterized, a prescription for the composite structure was de- fined that used unidirectional prepreg made from M55J car- bon fibers and resins into laminant mixtures tuned for the ap- propriate strength and thermal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Manufacturing controls were established to precisely align the fibers during layup and closely manage the fiber to resin ratio necessary for precise material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Controls were also put in place to achieve the desired bonded joint adhesive thickness used to connect the individual tubes into a truss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A prototype of the composite structure, called the Backplane Stability Test Ar- ticle (BSTA), was built by ATK and tested at MSFC’s XRCF (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Verification of the structure’s stability made use of a new Electronic Speckle Pattern Interferometer (ESPI) metrology approach (Saif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2008), a technology develop- ment in itself, to confirm the structure was TRL-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wavefront Sensing & Control The image-based phase retrieval methods used with JWST have a heritage stretching back to the diagnosis and correc- tion of the infamous spherical aberration in the Hubble pri- mary mirror (Krist & Burrows 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fienup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Phase retrieval using the science instruments elegantly avoids the need for substantial dedicated wavefront sensing hard- ware, and ensures sensing of wavefronts directly at the sci- ence focal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, to accommodate the evolving alignment of the mirrors (from initial deployment errors mea- sured in millimeters to final alignments measured in nanome- ters) these methods must operate over a tremendous dynamic range, and must also sense dissimilar and degenerate degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result several distinct wavefront sensing methods must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The primary tool is focus-diverse phase retrieval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' using a hybrid diversity algorithm developed Actuator for ROC Beryllium ROC strut (6x) GSE handling ring Backplane interface flexure (3x) Actuator (6x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 3 bipods for 6 DOF) Beryllium delta frame Beryllium whiffle (3x) Beryllium mirror substrate JWST_003THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 11 Figure 8: The Backplane Stability Test Article (BSTA) was an engineering model for 1/6 of the JWST backplane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' includ- ing the structural elements used to create the entire backplane assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This shows the BSTA being prepared for cryotest- ing at operational temperatures in the XRCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' specifically for JWST (Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Focus diversity is provided at different stages by defocusing the secondary mirror or by using weak lenses within NIRCam that can be inserted into the beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The focus diversity method is augmented with dispersed Hartmann sensing for the mea- surement of segment piston (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The step-by-step sequence of sensing and control activi- ties, as well as the associated algorithms and software, were developed at Ball Aerospace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To test and prove the im- plementation, a 1:6 scale model and functionally accurate Test Bed Telescope (TBT) was built (Kingsbury & Atche- son (2004), see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Using the TBT, the complete end- to-end telescope alignment process was successfully demon- strated, achieving TRL 6 in 2006 (Acton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Even with those fundamental tools proven, a decade of work remained to mature them from lab-scale demonstra- tion to flight-ready processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Operational implementation of the commissioning plan was complicated by the need to begin operation of fine guidance control while still adjusting mirrors (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2), and to interweave telescope align- ment with prerequisite steps of instrument commissioning such as focal plane calibrations (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The meth- ods were refined and operational plans were prepared lead- ing up to launch (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016), culminating in detailed implementation plans, procedures, and observing programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Contingency plans were prepared in the event that nominal plans could not be followed, for many distinct contingency scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Phase retrieval analysis software was similarly it- eratively refined prior to launch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' the initial delivery of flight Figure 9: The JWST Test Bed Telescope (TBT) is a 1:6 scale model of JWST’s telescope with the full sensing capabilities and control authority, located at Ball Aerospace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' WSS software occurred in 2011, and regular improvements continued thereafter as part of the I&T of the Science & and Operations Center (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These processes for WFSC were repeatedly tested together with the flight hardware at various stages of observatory I&T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The initial test and operation of the integrated OTE electron- ics and mechanisms took place in 2016 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During the cryovacuum test of the telescope plus instrument suite (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4) while the majority of that test program used GSE for metrology of observatory alignments, specific activities were included to test the flight scripts for wavefront sensing and control using flight hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was the first and only time that NIRCam was used to sense OTE mirror alignments on the ground (Acton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In parallel, the data generated by that activity was flowed back to STScI and used for a demonstration of sensing and control software processes using the integrated SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was the first ma- jor demonstration of processing of JWST data in a flight-like manner at the mission operations center (Lajoie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The commanding for mirror moves and deployments was re- peatedly exercised as part of regular OTE functional checks, up to and including at the launch site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A necessary input to the WFSC process was accurate knowledge of instrument-specific wavefront errors, to allow subtracting the instrument contributions from the results of the image-based phase retrieval to perform OTE wavefront control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This objective was met through precise measure- ments at dozens of field points within all instruments, as part of instrument cryo-vacuum testing completed by 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The wavefront sensing and control activities demanded human-in-the-loop controls and required training a large wavefront team for round the clock operations during com- missioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope was aligned start-to-finish over a hundred times in simulation, individually by many members of the team and in collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As part of this training process, there were iterative refinements of the methods, pro- 12 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' cedures, and documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The individual simulations built towards larger team rehearsals, including 20 internal wave- front team practices and 25 mission operations or science operations team wide rehearsals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Many of these rehearsals were carried out throughout the COVID-19 pandemic under work-from-home conditions, remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The extensive re- hearsal program was a critical, invaluable activity in building a smooth-functioning cohesive wavefront team combining staff from multiple organizations and skillsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Integration & Testing The JWST integration and test program began at the com- ponent level and continued as the hardware was integrated into successively larger sub-assemblies, assemblies, and eventually the entire observatory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' At various lev- els of assembly, testing took place to confirm the functional performance and verify the workmanship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The testing ap- proach followed a philosophy of independent, incremental testing with predefined success criteria, crosschecks that in- cluded end-to-end testing, comprehensive external and inter- nal review, thorough risk management, and open transparent communications and documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A strict requirement of the optical test program was that the optical test equipment in a verification test had to be independent of the test equipment used to fabricate the optics (Feinberg & Geithner 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this section we highlight some key activities from the later stages of the I&T program for the OTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We then dis- cuss the major I&T activities for the OTIS, which is the term for the combination of the OTE plus the Integrated Science Instrument Module (ISIM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTIS = OTE + ISIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We con- clude the section with a summary of the activities involving the OTIS after delivery back to Northrop for integration with the rest of the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A high-level flow of the OTE activities through its integration and testing through Obser- vatory readiness for launch is presented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Flight OTIS Integration & Ambient Testing The major structural elements of the OTE (see Figure 2), as well as the associated electrical harnesses, were integrated and tested at Northrop in 2014–2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Major tests included load testing of the mirror backplane and testing of the De- ployable Tower Assembly (DTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After precision integration of the DTA, the two telescope wing structures, and the Sec- ondary Mirror Support Structure (SMSS) to the backplane, these subsystems were exercised for functionality and re- peatability with ambient deployments (Glassman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Modal surveys were also carried out in the stowed and de- ployed configurations, with appropriate mass simulators for hardware to be integrated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After shipping of this hardware to GSFC, the integration of additional harnesses and small electronics boxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' for the mirror actuators) and the OTE optics took place in a dedicated assembly and alignment facility in GSFC’s largest cleanroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The PMSA shim prescription was determined us- ing metrology from a coordinate measuring machine brought from Tinsley Labs, now Coherent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', and laser tracker mea- surements of the composite backplane structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The PM segments were mounted to the backplane with the assistance of a traveling robot arm, as shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Laser track- ers measured the alignment state to guide the installation, with laser radar independently measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Custom ground shims and adhesive-filled pin gaps secured the location to mechanical tolerances that were a small fraction of the range budget for the PM actuators (see Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016 for de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Integration of the Secondary Mirror Assembly and the Aft Optics Subsystem (AOS) completed the OTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' With the in- tegration of the Fixed and Aft Deployable ISIM Radiators and the ISIM itself (the instrument module and the associ- ated electronics compartment), and an array of thermal and stray light control blankets (>900!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' ), the OTIS was complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The ISIM of course had been through its own comprehen- sive I&T program, including three cryo-vacuum tests totaling nearly 300 days of round-the-clock operations (Kimble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This integration of the OTE + ISIM to form the OTIS took place in May 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Using the fully assembled OTIS, the integrated mirror con- trol system hardware and software was exercised, first in small steps starting in October 2016, and eventually in partial deployments of all 18 PMSAs and the SM in preparation for ambient optical testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This included operation of PMSA actuators and sensors (resolvers and linear variable differen- tial transducer [LVDT] length sensors), operated by the Actu- ator Drive Unit electronics and controlled by the Wavefront Sensing System software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' One minor anomaly discovered during this time is that a small number of LVDT sensors do not operate nominally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was accepted to “use as is”, given the availability of other telemetry to confirm mirror motions for those segments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' using resolver telemetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For some of the affected LVDT sensors, a modified operations concept was developed that used temperature-dependent calibrations to make the sensor information usable, later used success- fully in flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mirror control mechanisms and processes continued to be exercised throughout the remainder of OTIS I&T, in particular during the OTIS cryovacuum test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The OTIS underwent proto-flight level ambient environ- mental testing (vibration and acoustics) at GSFC in 2016– 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Ambient optical measurements were carried out before and after those mechanical tests utilizing a Center of Curva- ture test setup, including a null lens and Computer Generated Hologram (CGH, for working with the aspheric PM surface), as illustrated in Figure 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' see Saif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Cen- ter of Curvature setup incorporated a high-speed interferom- eter for making figure measurements at rates up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' With this equipment, the static wavefront of the PM segments THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 13 Figure 10: A high level overview of the telescope integration and test sequence through launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Structures Integrate Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Ship to Backplane Telescope Optics at NGSS Deployment & Modal Tests GSFC Integrate Optics Integrate ISIM CoC Test Sine-Vibe & Acoustics Cleanroom OTIS Cryo CoC Test Ship to JSC Ship to NGSS Staging by JSC Optical Test Post Env Deployments Observatory I&T Start OTIS Install & Deployments Stow Sine-Vibe & Acoustics Deployments Stow Ship to GSC Launch Legend Component Integration Test Prep & Transport WST_01814 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 11: Installation of the final Primary Mirror segment, assisted by a high-precision robotic arm, suspended from traveling stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Black covers were mounted on the mirrors to protect from contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 12: Center of Curvature test setup for ambient opti- cal measurements of the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The high-speed interferometer, null lens, and CGH are mounted on the stable platform at lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' was measured before and after the mechanical environmen- tal testing, along with the dynamic response of the backplane and mirror mounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' With a vibration stinger to excite the pay- load, mechanical transfer functions were measured to look for any signs of damage after the vibe and acoustics tests – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' cracks in the structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' loosening of joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' No such damage was seen, with the figure and dynamic measurements repeating pre- and post-test within expected tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Elec- trical functional checks were also carried out before and after the mechanical environmental testing, along with “first mo- tion” (flinch) tests of deployment systems that couldn’t be fully deployed at GSFC in the one-g environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' An anomaly did arise during the OTIS-level vibration test, when a loud “bang” was heard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was determined to have been caused by gapping at one of the preloaded interfaces of a Launch Release Mechanism in a PM mirror wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mod- ifications to the procedures to properly set and maintain the preload of these interfaces resolved this issue, and no damage had been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In addition, excessive resonant response was seen for the SMSS and the AOS at some frequencies due to the low damping of the large composite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The test vi- bration spectrum was notched at these frequencies to protect the hardware during the OTIS-level test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Particle dampers were subsequently designed and installed onto the AOS and SMSS to reduce these responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' they operated successfully later at Observatory-level testing and ultimately through the actual launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Flight OTIS Cryo-Vacuum Test The final phase of I&T for the OTIS was an extraordinarily challenging cryo-vacuum test, previously described in Fein- berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2011) and Kimble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This took place at NASA’s Johnson Space Center in historic Chamber A, which is a US national historic landmark from the Apollo program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After shipment to JSC, the OTIS underwent electrical func- tional testing, the SMSS was deployed (with assistance – the necessary GSE for a powered deployment in one-g was only at Northrop), and the DTA and PM wings were deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After these activities, the payload was configured for the cryo-vacuum test in Chamber A, which had been extensively refurbished for the thermal and contamination requirements of JWST and outfitted with optical and thermal GSE com- prising an elegant test architecture, illustrated in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This architecture supported a rich array of operational, ther- mal, and optical test goals, with 40 separate tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' End-to-end optical tests were carried out using sub-apertures of the pri- mary mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The test campaign applied lessons learned from the Hubble program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Key components of the optical test equipment included the Center of Curvature Optical Assembly (COCOA, Wells et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2010), whose interferometers had a view of the entire PM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' photogrammetry (PG) cameras on four rotating booms (Lunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2020), which provided remarkably accurate rela- tive positions (sub-100 µm) of optical targets over the many- meter distances involved, through image triangulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' the AOS Source Plate Assembly (ASPA), which mounted light sources (fiber-fed or local) at the intermediate Cassegrain fo- cus of the OTE – these provided downward (half-pass) im- ages through the TM, FSM, and SIs and upward (pass-and- a-half images) through the SM, PM and then, after reflec- tion off sub-aperture Auto-Collimating Flats (ACFs), back through the entire OTIS optical train;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' and fiducial light strips straddling the edges of the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The position of the ASPA sources made their images highly aberrated, but in a pre- cisely known way, so that alignments and OTIS wavefront measurements could be extracted nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The down- ward sources were used to test the guiding control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical test equipment (see Figure 14) worked together to satisfy the critical optical verification goals (such as the verification of the non-adjustable AOS to ISIM alignment, THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 15 Figure 13: OTIS cryovacuum test configuration showing the telescope inside the chamber with optical metrology test equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' verification of the radius of curvature) as well as various cross-check goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A succinct description of the process is as follows: PM segments were aligned and phased via photogram- metry and COCOA interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The SM was aligned via photogrammetry and checked with Pass-and-a-Half imaging to the NIRCam instru- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' AOS to ISIM alignment was verified via Half-Pass imaging using inward facing sources and all science instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fiducial lights above the primary mirror were used for verifying pupil alignment, using NIRCam’s pupil imaging capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' End-to-end imaging and field tilt was cross-checked using Pass-and-a-Half imaging using outward facing sources, the autocollimating flats, and all of the sci- ence instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wavefront Sensing & Control hardware checks and demonstrations were performed via Pass-and-a-Half testing and NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Pathfinder Program The Pathfinder Program used spare and test equipment as surrogates for the flight hardware in order to prove out many of the integration and testing activities (see Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2014, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The test article, referred to as the “Pathfinder”, was comprised of two spare primary mirror segments, a spare secondary mirror, and composite structure representative of the center section and the secondary mirror supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Pathfinder was transported and integrated us- ing the protocols and procedures that would later be used on the flight hardware, in some cases demonstrating the actual capabilities of the test equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Following integration, the Pathfinder was used to prepare the cryovacuum testing facility and equipment at JSC in 2015/2016, while the OTE and OTIS were being integrated and tested at Goddard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Pathfinder cryotest program uti- lized the thermal and optical GSE developed for the OTIS test and the Pathfinder structure itself (Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The first Optical GSE test (OGSE1) checked out the CO- COA and PG operations with those systems, while the sec- ond (OGSE2), incorporated the flight AOS (hence requir- ing careful coordination with the flight I&T flow) and the ASPA to dry-run the half-pass and pass-and-a-half tests as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A third Pathfinder test included thermal mock-ups of the remaining center-section PM segments and validated the cooldown and warmup procedures that would be needed, in- cluding the requisite contamination control (Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Flight OTIS Cryo-Vacuum Test Results Cryo-vacuum testing of the flight OTIS took place in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The 100-day, round-the-clock test campaign was remarkably successful, despite the many challenges, including the pas- sage of Hurricane Harvey through the JSC area in the mid- dle of the test, which devastated the local community and shut down JSC for normal operations with 55” of rainfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After a 5 day period of special hurricane operations, and Vibration Isolators (6x) Center of Curvature Optical Assembly (cOcOA) Upper Suspension Frame Autocollimating Flat (ACFs, 3x) He Shroud (14m diam, 24m tall) Photogrammetry Cameras on Windmill Booms (4x) AOs Source Plate Assembly Space Vehicle Thermal Simulator The Flight OTIS Support Structure with Deep Space Environment Radiation Sinks (DSERS) JWST_00416 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 14: Simplified OTIS optical test schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' nearly running out of liquid nitrogen, the cryotest narrowly missed being abruptly ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fortunately the hurricane did not preclude safely continuing the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All significant goals for the test were achieved, including confirming the health of the OTIS payload after its environmental test program (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018), accomplishing the planned optical verifications and cross-checks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Hadaway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018), validating the OTIS thermal model and the OTIS thermal distortion model, both required for the Integrated Modeling of the observatory as a whole, and various operational validations and demon- strations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Performance of the OTIS was overall excellent, with pre- dictions that satisfy the mission-level requirements (Light- sey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' But, there were several types of optical in- stabilities identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' One, caused by over-tightness of the soft-structure frill stray light blocker and PMSA closeouts at the cryogenic operating temperatures (such that they exerted temperature-dependent forces on the mirror backplane), was mitigated with post-test modifications to restore the intended slack where feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A second, which coupled temperature variations in heater-controlled radiator panels on the instru- ment electronics compartment (IEC) to the mirror backplane, inducing oscillating structural distortions, was demonstrated with ambient measurements and analysis to be caused by a rigid non-flight mounting of the IEC for the OTIS CV test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Both of these have been shown to have minimal wavefront impact in flight (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4) A final instability, called “tilt events” referred to sudden, stochastic changes in the piston/tip/tilt pose of individual PM segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Several such events were seen throughout the OTIS test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Though not fully explained, these were ascribed to stick/slip release of stresses from cooldown ther- mal deformation in the OTIS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These reduced dur- ing the end of the test, and it was expected that these would fade away with time after cooldown as the various stresses in the system were gradually relieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This behavior during the OTIS test informed expectations that such events could be seen in flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This supposition appears to be confirmed by the flight behavior (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1), though with the ex- cellent sensitivity of the flight wavefront sensing, most of the observed tilt events in flight are actually below the detection threshold of the OTIS cryo-vac analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTIS during Observatory I&T After the deconfiguration of the OTIS from the test config- uration, it was shipped to prime contractor Northrop Grum- man’s Space Park in Redondo Beach, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' While at Northrop, powered deployment of the SMSS and DTA took place with appropriate one-g off-loading hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Both deployments were made from the spacecraft electronics to demonstrate the connections and scripts were working prop- erly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For the SMSS, this represented the only post-OTIS- vibration powered deployment, confirming the health of the deployment system after that proto-flight-level vibration ex- posure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the summer of 2019, the OTIS was integrated with the spacecraft and sunshield to form the full-up JWST observa- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Alignment metrology was performed in the integrated configuration to characterize the OTE to star tracker bore- sights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the full-up observatory configuration, the payload un- derwent various deployment tests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', off-loaded deploy- ment of the DTA and PM wings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observatory as a whole was then put through acceptance-level vibration and acoustic tests, with subsequent deployment and electrical functional tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Like the OTIS, the spacecraft and sunshield had previ- ously successfully undergone mechanical environmental test- ing at proto-flight levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Both the PM and the SM were cleaned of particulates at appropriate times in the Northrop flow, with a gentle Center of Curvature Interferometer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5m Diameter Autocollimating Flat (ACF) Mirrors (3X) Photogrammetry Cameras on windmill arms (1 of 4) nward and Outward facing sources at Intermediate Image Fiducial lights above PM edge Absolute Distance ACF Meter ISIM JWST_006THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 17 brush technique described by Lobmeyer & Carey 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For the SM, which had the most challenging particulate con- tamination budget (so cleaning was desired as late as pos- sible), this cleaning took place after the final stowing of the observatory into its transport (and launch) configuration, just before encapsulation of the observatory into a clean, environmentally-controlled shipping container for transport by sea to the Guiana Space Centre (GSC) in Kourou, French Guiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There, the observatory executed final ground func- tional tests, was fueled, and was encapsulated in the Ariane 5 rocket fairing for launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' On-Orbit Commissioning and Characterization Following JWST’s launch, the telescope was deployed, aligned, characterized, and readied for science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this section, we describe the overall sequence of activi- ties executed during OTE commissioning as well as the re- sulting performance of the optics, pointing, and focal plane alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Many years of preparation, for both the observa- tory hardware and the commissioning operations plans and teamwork, resulted in a smooth and efficient commissioning which completed successfully and as scheduled, and deliv- ered an OTE performing at or above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTE Commissioning Activities To prepare the telescope for scientific observation, OTE commissioning activities included the deployment of the mir- ror segments from their launch restraints, the alignment of the primary and secondary mirror segments, and the achieve- ment of a finely phased telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The series of activities was developed and rehearsed over many years and was al- located ∼90 days to complete in the planned schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For clarity, it is useful to break the whole sequence down into a few major sets of activities: mirror segment deployment, segment-level identification and alignment, co-phasing of the segments, and multi-field alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Here, we summarize these activities and describe the actual execution during flight (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The detailed commissioning plan is pre- sented in Acton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2018) and additional details on the as-run activities are presented in Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2022) and Acton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Briefly, the commissioning plan had to accommodate initial PMSA and SM positional errors as large as one millimeter, and progress to achieve fine align- ments within a few tens of nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was achieved using several different forms of wavefront sensing, most of which were iterated multiple times, and which had to be in- terspersed with early steps of instrument and guider commis- sioning and focal plane calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTE Deployments The major structural deployments of the OTE (DTA, SMSS, and PMBSS wing deployments) all completed suc- cessfully and nominally, with no notable issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Subsequent results from telescope commissioning con- firmed the precision of these deployments: for instance, the 8 m multi-hinged SMSS deployment placed the SM within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 mm of its nominal position, well within the correction range of the SM actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Further, the telescope boresight offset relative to the spacecraft star trackers was found to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4′, nicely consistent with the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05′ 1-sigma preflight pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Similarly the corrections required to align the PM- SAs were small (Table 4), with only one segment requiring a corrective move larger than 1 millimeter in position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mirror Segment Deployment For launch, all mirror segments were stowed in launch re- straints in order to limit lateral displacements during launch and ascent (see Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To begin to align the PMSAs and SMA, the segments therefore had to be released from their launch restraint, a pure piston move of ∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The deployment sequence of the PMSA and SMA was carefully designed to verify the actuator stepper motor alive- ness and responsiveness, as well as confirm proper manage- ment of the segment envelope and workspace boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The sequence was tested on the ground on multiple occa- sions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', OTIS testing at JSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As such, stepper motors on segment A1 only were first incrementally commanded to move 1 step, 1 revolution, 10 µm, 150 µm, and 340 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The remaining segments (except A3 and A6, see below) were then commanded the same sequence of steps, followed by 1- mm increments until all the segments were fully deployed to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early in these deployments, some LVDT sensor readings did not show as smooth a progression as expected, which led to additional small ‘flinch’ moves to verify all ac- tuators were moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Once past the first few millimeters and out of the launch restraints, the LVDTs showed the expected linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The initial non-linearity was interpreted as due to friction of surface contacts with the launch restraints, which, however, posed no problem to the deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segments A3 and A6 were deployed separately and last as a result of a faulty LVDT on one actuator each (as noted above in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Although the sequence of moves was identical to that of the other segments, the LVDT read- ings, which provide a coarse direct measurement of actuator length, had to be calibrated as the sensors cooled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result, they were deployed separately, without incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Finally, once at their deployed positions, the segments were commanded to their intended nominal positions, based on ground alignment test results corrected for 0-g via model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mirror deployments completed successfully and with no major issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segment-level Identification & Alignment The next series of activities aimed at finding, identifying, and re-arranging the images produced by each of the mirror segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The deployed but not-yet-aligned segments were 18 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 15: High-level overview of the sequence of events during OTE commissioning, along with examples of flight data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This depiction greatly simplifies a complex process involving hundreds of individual steps and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 16: Rendering of the actuator in a stowed configura- tion, with the snubber inside the launch restraint (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' each acting as its own aberrated ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' When first light on NIRCam was obtained on February 2nd 2022, all celestial objects were indeed duplicated 18 times (see Fig- ure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To identify an image of each segment, an isolated bright star was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Budgeted pre-flight uncertainties for mir- ror segment initial deployments, as well as in the initial tele- scope boresight offset (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1), predicted that seg- ments might be scattered by up to ∼15′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result, the tar- get star was selected to have no similar-brightness neighbors within such a distance, and a large half-degree-diameter mo- saic around the target star was generated, taking around 25 hours to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 18 shows a cartoon of the planned mosaic, overlaid on top of a catalog sky image around the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A subset of the flight data where the segment im- ages were all found is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The segment images were found within ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4′ of the nominal target location on aver- age and the segment image scatter was of similar magnitude, both better than requirements and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The simplic- ity of the plan for this initial mosaic step proved beneficial to accommodate larger-than-expected coarse pointing uncer- tainties at this time (prior to ACS tuning and optimization), and increased levels of detector persistence due to operating NIRCam well before it had fully cooled to its nominal tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Following the initial mosaic observation, a secondary mir- ror focus sweep was performed in order to measure and move the SM to a best focus position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This improved image quality and enabled guiding later on during OTE commissioning (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The analysis led to an SM move of -427 µm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', away from the primary mirror).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Initial Mosaic Focus Sweep 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mirror Deployment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segment Image Identification Segment Global Image Array Alignment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segment Alignment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Image Stacking Telescope Alignment Evaluation Image 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='Coarse Phasing DHS Spectra Webb Telescope Image Sharpness Check 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Fine Phasing NIRSpec MIRI 8 In Focus +8 waves NIRCam 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='Multi-field Alignment Fine Guidance Sensor NIRISS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Iterate last three steps to converge JWST_007THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 19 Table 3: Timeline Summary for OTE Commissioning Activity Program IDa Prelaunch Plan Actual Delta [days] Mirror Deployments n/a 2022-01-18 2022-01-20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 Segment Identification 1137 2022-02-07 2022-02-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 First Closed Loop Guiding 1410 2022-02-12 2022-02-13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 Segment Alignment (iteration 1) 1141 2022-02-12 2022-02-19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 Image stacking (iteration 1) 1143 2022-02-14 2022-02-22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 Coarse Phasing (iteration 1) 1147 2022-02-21 2022-02-28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 Coarse Multi Field 1148 2022-02-22 2022-03-03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 Fine Phasing (iteration 1) 1155 2022-03-01 2022-03-08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 Fine Phasing (iteration 3) 1160 2022-03-05 2022-03-11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 Multi Field Multi Instrument Sensing 1 1166 2022-03-10 2022-03-20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 LOS Jitter Measurement 1170 2022-03-17 2022-03-21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 Multi Field Multi Instrument Sensing 2 1465 2022-04-06 2022-04-19 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 OTE Alignment Complete n/a 2022-04-24 2022-04-23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 aAPT program IDs, which may be used to retrieve these data, or any other commissioning data, from the MAST archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' NOTE—Timeline of events corresponding to Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Overall the OTE commissioning process proceeded remarkably according to preflight plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The major sources of schedule deviation were complications in early guiding (delayed completion of segment alignment) and a revised, slower plan for MIRI cooldown adopted by the MIRI team (delayed start of the second round of multi instrument sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=') These were balanced out by the budgeted third round of multi instrument sensing not being necessary, allowing the completion of OTE alignment one day ahead of schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 17: First on-sky image using JWST and NIRCam, targeting a region of the Large Magellanic Cloud, obtained on 2022 February 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Because the mirror segments are not phased, each star is duplicated 18 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Two copies of the core of globular cluster NGC 1916 can be seen at upper left and lower right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 20 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Table 4: PMSA Correction Magnitudes Degree of Freedom Unit Typical Maximum Margin Piston µm 145 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 91% Radial Translation µm 450 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 32% Clocking microradian 370 766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 55% Radius of Curvature µm of surface sag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='117 76% NOTE—Typical and maximum corrective moves required to align the PMSAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTE deployments initially placed segments typically within a few hundred microns of their intended locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The margin column gives the unused frac- tion of the nominal correction range remaining after the maximum correction moves for each degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 18: Top: The search sequence of NIRCam observa- tions centered on the bright target HD 84406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Bottom: The inset shows a subset of the large mosaic from flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All 18 images of the target star were found clustered together in a relatively small area, due to good deployments preci- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Also visible are some residual after images of stars and curved trails from illumination during slews to differ- ent pointings, due to increased persistence from higher-than- nominal detector temperatures at the time of these observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Next, each mirror segment’s image in the initial mosaic was identified by sequentially tilting each mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Once iden- tified, segments were commanded to form the pre-defined image array shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the hexagonal array configuration, all segment images could be observed on one NIRCam detector at once and segment-level aberrations could be addressed as part of the segment global alignment activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To do so, the SM was moved away from its nominal best focus by ±400 µm to col- lect focus-diverse imagery for the purpose of phase retrieval analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result of this analysis, the SM was corrected in X- and Y-translation by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='94 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='06 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Focus corrections were applied to each segment, along with corrections to the radius-of-curvature actuators of two PM- SAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During a later stage of commissioning, a second iteration of Global Alignment was executed as part of the iterative alignment approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' At that time, additional small clocking, radial translations, and radius of curvature corrections were applied to the PMSAs to correct astigmatism and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' At this point in the OTE phasing process, the observed wave- front errors achieved excellent agreement with ground mea- surements and preflight modeling of the higher spatial fre- quency mirror maps (Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Co-Phasing the Segments Following each instance of Global Alignment, a sequence of image stacking was executed to position segment images on top of each other (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, this stacking does not mean the light paths from each segment are in phase with one another, so Dispersed Hartmann Sensing measure- ments (Coarse Phasing) were executed to establish phase er- rors as a function of wavelength in order to measure pairwise segment edge heights and derive an overall piston correction to all of the PMSAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Three iterations of this coarse phasing sufficed to bring the PMSA piston offsets to less than 1 µm, where fine phasing with NIRCam weak lenses could finalize the OTE alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Following these PMSA piston corrections, a fine phasing activity took place where the NIRCam weak lenses were used to collect focus-diverse measurements for phase re- trieval analysis (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The results of such measure- ments were then used to better (re-)stack the segment images as well as correct residual piston offsets between segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' On March 11 2022, the fine alignment process completed, yielding a telescope aligned to roughly 50 nm RMS as seen at the fiducial field point on NIRCam A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Following this, a mo- saic observation was carried out around the alignment star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This multi-purpose observation tested science-like dithered and mosaiced observations for the first time, confirmed ex- cellent PSF quality over all of NIRCam’s field of view and across NIRCam’s full wavelength range, provided early mea- surements of observatory backgrounds, and yielded an early glimpse of JWST’s sensitivity to the high redshift universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Multi-Field Alignment WISE W1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 μm) Log, stretch to emphasize faint stars 42 41 40 39 38 (last) Cyannumbers: Ordering of 21 20 9 observation sequence 22 V3PA=195 23 (first) 2nd 13 25 30 32 33 OTE-01 Qbs Red oval shows approximate location where segment spots were found Yellow box = full mosaic search area for Obs 1 Average offset ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 arcmin from center JWST_015THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 21 Figure 19: On-orbit measured primary mirror segment wavefront errors following global alignment 2 (left) as compared to the ground-test measurements at the XRCF with model backouts for 0-g gravity (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The consistency of these two datasets provides a striking in-flight validation of the OTE development program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' It demonstrates the segments were fabricated to the correct optical prescriptions, the 0-g gravity backouts were correct, there was no measurable thermal distortion during launch and ascent, the segments were deployed to within the correction range of their actuators, and the WFSC processes worked to sense and correct the initial misalignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Note that the RMS WFE values labeled are for the primary mirror segments only, excluding all the other optical surfaces which contribute to higher WFE for the observatory as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Aligning the telescope to only one field can lead to de- generate solutions, where PMSA and SM misalignments bal- ance each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Multi-field measurements are therefore required in order to achieve optimal optical performances across all the science instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Two types of multi-field sensing were carried out: a NIRCam-only multi-field activity (using only Module A) and two instances of multi-field, multi-instrument activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The NIRCam-only alignment activity was aimed at initially removing most of the PMSA-to-SM misalignment using the unstacked segments as a Hartmann sensor, analyzed with a centroid-based approach to measure the field dependence of aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This “coarse” multi-field sensing proved to work exceptionally well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result of this analysis, the SM was moved in translation in X and Y (-210µm, 420µm) as well as tilt in X and Y (-550 µrad, 34 µrad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The SM correction was also accompanied by compensating PMSA moves in order to maintain the hexagonal array configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Finally, multi-field, multi-instrument measurements were made in order to assess the field dependence over the whole field of view, this time using focus diversity provided by moving the SM in piston by ±100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Two instances of this activity were executed since, as expected, MIRI had not yet reached its operational temperature at the time of the first measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In both instances, the results indicated no significant correctable field-dependent aberrations, in other words the initial NIRCam-only multi-field activity had cor- rected the telescope’s field dependent aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Only a small SM focus correction with minimal wavefront error gain was applied, mostly to balance the relative focus terms of the science instruments (see Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observatory was fully aligned on April 23 2022, and the commissioning of the science instruments continued thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' From that point, the operations team entered a new stage of conducting routine OTE maintenance, which will be ongoing throughout the mission, discussed further in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Guiding and Line-of-Sight Pointing To support all but the very earliest mirror alignment activ- ities described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1, the closed-loop FGS guiding mode first needed to be commissioned and its performance established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' See Menzel PASP Observatory for a more de- tailed discussion of the Attitude Control System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Guiding during Commissioning JWST uses star trackers, rate sensors, reaction wheels, and a fine steering mirror to achieve a coarse pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To transi- tion into guiding, one of two FGSs will attempt to identify the intended guide star, whose position and fluxes are normally provided by the Guide Star Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The coarse position er- ror of the guide star as seen by FGS is fed back to ACS for correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The guidestar will usually then be moved to a pre- computed “science” location in the FGS field of view, after which closed-loop guiding will be attempted and typically engaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During closed-loop guiding, FGS measures cen- troids every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='064 s and reports the position to ACS, which WAS Phase retrieval from GA2 obs 2+6 Preflight Model: Uncorrectable high spatial Measurement after GA2 corrections frequencies from XRCF testing 200 150 Wavefront Error [nanometers] 100 50 50 100 150 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='25 nm rms 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='32 nm rms 200 JWST_01722 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 20: Measured multi-field, multi-instrument wavefront residuals, here showing the end-to-end observatory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' OTE plus instruments) static wavefront residuals at the end of telescope alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The variation between field positions seen here is due mostly to the inherent optical performance of the instruments, outside of the ability of OTE adjustments to correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The final adjustment after multi-instrument sensing was only a small focus shift of the secondary which brought the science instruments into good average focus and positioned the global focus closer to MIRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' All instruments maintained precise confocality from ground to space, such that it was unnecessary perform focus adjustments on the individual instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' then commands the fine steering mirror to move in order to maintain the guide star at the appropriate position on the FGS detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early in OTE commissioning, when the OTE was still pro- viding 18 images for every star in the field of view (includ- ing on FGS), guiding operations had to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Per our plan, FGS used one of the segment images to guide on, along with reference segment images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was done by overriding the guide star selection system to account for the segment position offsets and flux differences compared to the guide star’s catalog position and flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Using this approach, closed- loop guiding was successfully demonstrated during the FGS LOS Initialization Activity (PID 1410).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Later in commissioning, guiding operations became in- creasingly routine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' once the PMSAs were stacked into a sin- gle PSF, the guide and reference stars’ selection and locations could be automatically provided by the ground system using the operational catalog and only the associated stellar fluxes had to be overridden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Once the PSF was phased, the fluxes as well were being supplied by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' By the end of OTE commissioning, guiding required no special intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The majority of instances of closed-loop guiding during OTE commissioning were successful, although some failures do occur for a variety of reasons, including bad pixels or mis- cataloged guide stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Nevertheless, success rates have risen continuously throughout commissioning and into science op- erations with now > 95% of planned visits being success- fully executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' When guiding success is achieved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', the intended guidestar is identified, acquired, and tracked with a settled closed loop), then the image stability of that pointing becomes the performance metric of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Characterizing Line of Sight Stability during Commissioning Special commissioning tests were included in the baseline plan to obtain the power spectrum of the line-of-sight (LOS) jitter and, in particular, to assess the contribution of vibra- tions caused by the MIRI cryocooler (CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The dedicated test to probe excitations from vibrations in the LOS data showed no evidence of significant contributors and has revealed excellent stability performance at ∼1 mas RMS radial, very close to the measurement noise floor and significantly better than expectations and the 7 mas require- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The typical results from the first commissioning LOS jitter analysis (PID 1163, observation 2, from Hartig 2022) are shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this case, jitter was measured at nm +200 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 LW 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 NIRCam A NIRCam B 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 NIRSpec 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 MIRI 0 sw 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 NIRISS Guider 1 Guider 2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 200 JWST_013THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='04 mas RMS (radial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The same tests also revealed no need to tune the MIRI cryocooler’s pulse frequency which remains at its initial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The analyses, which continue to be regularly performed as part of the 2-day routine maintenance program (Section 6), have consistently shown jitter levels around 1 mas radial (moderately correlated with guidestar brightness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The anal- ysis is quite sensitive and has revealed low frequency, low power oscillations at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='04 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 Hz fea- ture has been attributed to bending modes at the 1 Hz isola- tor at the SC-to-OTE interface, whereas the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='04 Hz feature has been shown to vary over time and might be attributed to fuel slosh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During the commissioning period there were no clear indications of any LOS jitter response to the reac- tion wheel assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However during the first months of science operations, a handful of measurements have shown minor LOS jitter contributions which appear correlated with certain speeds of reaction wheel assembly #6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' a resonance appears to be excited in the vicinity of 16–17 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Focal Plane Calibration In addition to guiding (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2), sufficiently accurate target placement at each of the SIs was needed to support late OTE commissioning activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This involved two essen- tial related areas of calibration: the first was the determina- tion of the principal coordinate frame of FGS1 with respect to the spacecraft’s, expressed as a direction cosine matrix and stored on-board for use by the ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The second was the de- termination and calibration of the SI fields of view relative to FGS1, expressed in a convention similar to Euler angles and stored in the SOC ground system for determining on-sky pointings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These activities were performed in close coordi- nation with WFS&C as part of the integrated flow through OTE commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Calibrating the Telescope to Spacecraft A key design feature of JWST is an OTE that is well sep- arated and isolated, thermally and mechanically, from the spacecraft by the Deployable Tower Assembly (DTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Un- certainties in the DTA deployment and other contributions were expected to produce initial errors in the nominal align- ment of the OTE V-frame with respect to the spacecraft’s fun- damental coordinate system, called the J-frame, of ∼10–15′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The slew of the observatory to its intended field is con- trolled by the Star Tracker Assemblies (STAs), which reside on the spacecraft side and are calibrated to the J-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' How- ever, the FGSs must then be able to acquire and identify the intended guide star, and they reside on the OTE side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Cap- turing the alignment between FGS and ACS requires the up- dating of the FGS-to-J direction cosine matrix (DCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This matrix is updated to account for deployment uncertainties, as stated above, as well as changes in telescope boresight, which occur every time the SM is moved in translation or tilt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Global Alignment, MIMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' While the initial error in the OTE-to-spacecraft alignment was expected to be close to 10′, the misalignment was found to be only ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4′ from the nominal ground values (see Fig- ure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The FGS-to-J DCM was subsequently manually updated during commissioning after taking observations of the sky with an SI or FGS and expressing that celestial pointing in terms of the FGS1 frame, while obtaining from the ACS/STAs the contemporaneous mapping of sky to the J-frame, thus providing the information to relate FGS1 to J-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This operation was also successfully performed during subsequent commissioning activities to maintain the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During the science mission, this relationship will continue to be dynamic at levels much lower than seen in commis- sioning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' a few arcsec), and the ACS will autonomously update this calibration based on observed FGS guidestar lo- cation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Calibrating the SIs to FGS Using FGS1 to define the relationship of the OTE to the spacecraft implies that it is also the reference for the OTE- based frame to which the SI fields of view are calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This “V-frame” was defined nominally as a conventional 3- axis coordinate system aligned with the OTE principal me- chanical axes, having V1 pointing out along the Cassegrain axis of symmetry (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In flight, however, the V-frame is used to specify SI and FGS fields and various fiducial field points used for science targeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' So, in this application, it is treated spherically as angles, with the axes V2 and V3 corresponding to “field angles” within the OTE field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In this scheme, the FGS1 field location and orientation with respect to V2,V3 is fixed, and on-sky astrometric cal- ibrations that determine the SIs’ and FGS2’s fields relative to FGS1 in essence establish their locations, orientations, and higher order distortions with respect to the V2,V3 field an- gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This astrometric calibration scheme, its tools and prod- ucts, are thoroughly treated by Sahlmann (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These calibrations are required for successful target place- ment and were performed during OTE commissioning to (1) determine the post-launch changes to the ground-determined relationships and (2) update this knowledge to ensure suc- cessful multi-SI wavefront measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The commission- ing team used a specially calibrated ∼15′ astrometric region of the LMC for this purpose (Sahlmann 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The first measurements of SI relative locations showed the ISIM to be stable, with ground-to-flight evolution in the V2,V3 field angles to be at or below the ∼ 1′′, and orienta- tion changes of the SI fields < 1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Although precise scales and distortion calibrations fall into the SI activities and con- tinue into the science cycles, basic instrument scales were 24 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 21: Left: Line-of-sight jitter distribution sampling every 2 ms over a 120 s interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This “jitter ball” shows well-behaved and very small variations in pointing using fine guidance control, with RMS variation only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Right: The power spectrum from the high-frequency jitter measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' tentatively measured during these OTE commissioning ac- tivities, and were found to be < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='15% different from ground measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For comparison, Hubble’s SIs through the generations typically saw 1-2′′ of V2,V3 shifts and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0% scale change from ground to flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Optical Performance at the End of Commissioning The optical performance as measured at the end of com- missioning is better than the requirement values at the system-level and for the most part, better than the sub-system allocations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A summary of the optical performance measurements and predictions is presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A separate optical discussion is presented in the paper on JWST Science Performance in this volume by Rigby PASP Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A detailed discussion of the telescope’s optical performance against requirements at the end of commissioning is reported in Knight & Lightsey (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The total wavefront error combines the observed static wavefront error with the dynamic stability and image mo- tion terms, which is corrected when the NIRCam A field point exceeds 80 nm rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This means the end-to-end, tele- scope and NIRCam SW has diffraction-limited image quality at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This is significantly better than the 150 nm RMS requirement optical error budget total that enables diffraction limited image quality (approximated by λ/14) at 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Observed Wavefront Errors Table 5 summarizes the end-to-end, referred to as ‘obser- vatory’, static wavefront errors measured at the end of com- missioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The static wavefront errors are well below their allocations in all channels, at all field points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Area & Throughput Table 5: WFE Table Science Field Static WFE NIRCam A SW 61±8 NIRCam B SW 69±11 NIRCam A LW 134±38 NIRCam B LW 134±39 NIRISS 68±12 FGS 1 77±15 FGS 2 69±8 MIRI 99±28 NIRSpec 110±20 NOTE—The static observatory wavefront error measurements as measured at the end of telescope alignment in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The values reported are the average across the points measured across the science field, with the plus and minus values reporting the peak to valley variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The total observatory WFE combines this static term with the dynamic WFE stability and image motion, typically ∼13 nm rms equivalent, via RSS sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Units are nanometers rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST’s unobscured collecting area was measured using the NIRCam pupil imaging lens to be 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='44 m2, exceed- ing its requirement of 25 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope’s wavelength- dependent transmission ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='786 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 µm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='933 at 28 µm, again better than requirements at each wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The transmission values were determined from final pre- flight measurements of mirror witness samples, combined with NIRCam grism measurements confirming the absence of detectable ice deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' X-jitter (RMS px): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='d26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 N samples: 53100 Y-jitter (RMS px): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content="025 'RMS jjtter (mas): 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05 Y offset (px) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 X offset (px) JWST00910-1 10-2 Power 10- Relative 10- 10-5 I SO 10-6 10-7 10-8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 Period (s) JWST010THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 25 Table 6: Stability Performance Contributor Predicted Amplitude Measured Amplitude Measured Response WFE (nm RMS) WFE (nm RMS) IEC Heater Cycling 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 224 s period oscillation Frill & PMSA Closeout 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='77 hr time constant Thermal Distortion 17 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='39 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='94 hr time constant The product of the above observed values for OTE area and transmission was also projected to end of life using modeled degradation of the optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This places the effective area × transmission value of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='58 m2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8 µm and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='18 m2 at 20 µm, compared to the OTE requirements of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='37 m2 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='00 m2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Vignetting Some of the OTE commissioning activities described ear- lier obtained data that also supported a secondary goal of probing various types of vignetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Analyses of these data show no indication of any field-of-view cropping, unex- pected OTE structure incursion, or pupil vignetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Estab- lishing that the telescope was unobstructed (except for sec- ondary mirror support structures) fulfilled a mission-level re- quirement and was an exit criterion for OTE commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Thermal Stability A dedicated thermal stability test was carried out following the telescope alignment in order to characterize the wavefront stability and image motion on various timescales following a large, stressing thermal slew (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' early May 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' PID 1445 and 1446).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This activity started by performing a 4-day ther- mal soak at the hot (sun-normal) attitude and making baseline measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Then, the telescope was slewed to the cold at- titude where continuous wavefront measurements were made for the first 24 hours and then every ∼8 hours for the follow- ing 7 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The thermal stability test confirmed three pre- dicted wavefront drifts that were bounded by the modeling predictions: short-timescale (2-4 min) oscillations from IEC panel heater cycling, medium-timescale (∼1 hr) drift from soft-structure induced thermal distortion, and long-timescale (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 day) drift from the composite backplane induced ther- mal distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These drifts are reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Tem- perature sensors on the telescope were also monitored during this test and confirmed that the temperature changes observed were within the noise of the temperature sensors (< 40 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Some science observation modes are, however, sensitive to these levels of WFE drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' It is also important to note that the worst-case delta-T induced as part of this thermal slew test would rarely be realized during normal science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In practice, science pointings across the sky are subject to much smoother and smaller temperature changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Pointing stability immediately following the thermal slew was measured when slewing from hot-to-cold and from cold back to hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1, thermal distortion at the star tracker could result in uncorrected roll about the lo- cation of the fine guide star on the fine guidance sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The roll about the guide star was measured at the NIRCam field location to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0265 mas/hr in translation and comparable to a measured radial displacement from the star of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0230 mas/hr, well below the allocation of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' During commissioning, there were many instances of sud- den positional changes in one or more mirror segments, re- ferred to as ‘tilt events’ (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The largest of these produce brief violations of the nominal stability values reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These positional changes are typically very small but detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These tilt events are generally as- cribed to strain release within the OTE structures following cooldown to cryogenic temperatures, although the sources of the tilt events is not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The frequency and magnitude of these events appear to be slowly declining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Science Era Characterization The maintenance and trending of the OTE in the science mission era officially started in mid-July 2022 alongside the start of the Cycle 1 science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope align- ment state will be monitored and corrections will be made as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Additionally, trending will be carried out across the telescope performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope state will be made available such that it can be used as part of the science anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wavefront Routine Maintenance Operations Concept The baseline science operations concept for OTE main- tenance uses wavefront sensing and control observations to maintain the optical alignment near its optimal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wave- front sensing observations are scheduled approximately ev- ery 2 days and make use of the NIRCam ±8 wave weak lens pair that is best matched to sensing aberrations at the nominal sensing field point (on detector A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These observations use bright K∼7 target stars to minimize the exposure times and the exact target is automatically drawn from a pool of targets evenly distributed on the sky in order to minimize slew times between science observations (see Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Also included in those routine observations are line-of-sight jitter measure- ment observations, which use the same target as the weak lens observations and take about two minutes of NIRCam 8×8 data (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In total, wavefront sensing ob- servations take about 15 minutes of data, not including slews and overheads (see PID 2586, 2724, 2725, and 2726).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Additionally, pupil images are also collected on a quar- terly basis in order to monitor the state of the primary mir- ror and, in particular, identify and characterize features due to micrometeoroid degradation (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 for more 26 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 22: Full sky map of the Cycle 1 wavefront routine maintenance targets, which include 400 targets that were vet- ted with binary and isolation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The color and size of each dot represent the sum of the time (in days) of Cycle 1 science visits that are closest to that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Small open pur- ple targets have no nearby science visit in the Cycle 1 plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The large red circle is ∼41 days of observations in GOODS- South and the green circle to the right is the ∼23 days of observations in the COSMOS fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The average distance of a science pointing from a WFSC target is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These observations take about 10 min of NIRCam science time (see PID 2751).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The sensing data are automatically analyzed on the ground using phase retrieval algorithms (see Acton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2022) to assess the state of the telescope and determine if a PMSA control (or correction) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The baseline frequency of such correction was expected to be no more than once every 14 days, which was predicated on the expectation that even worst-case thermal slews would not drift the telescope wave- front error beyond the allocated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Wavefront corrections are typically scheduled when (1) Observatory-level wavefront error exceeds 80 nm RMS or (2) segment tip/tilt correction is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='05 µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Re- call that the observatory-level WFE requirement at NIRCam is 150 nm RMS, so the criterion used here is much tighter thanks to the exceptional in-flight performances of the JWST OTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Also, these criteria were defined so as to not chase any thermally-induced distortions but rather to correct any sys- tematic alignment changes to the optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Stability & Trending The baseline operations concept outlined above has been operational since mid-July 2022 and trending of the WFE, a key performance metric, along with the occasional correc- tions that were made to the PMSA are shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Since the start of the science mission, 47 wavefront sensing observations have been executed, including 6 that included PSMA control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This cadence averages out to correcting once every 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6 weeks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' better than our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In gen- eral, the OTE has been stable to within ∼10 nm RMS until a disturbance occurs and must be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 23 sug- gests that corrections are grouped closer together, separated by longer periods of stable WFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Most of the sudden changes in WFE seen in Figure 23 are referred to as tilt events and are discussed below in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Overall, the telescope’s performance has met the criteria listed above about 84% of the time, and has met the mission requirements 100% of the time since mid-July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Tilt Events On many occasions, so-called tilt events, where sudden and uncommanded tilts of individual or groups of segments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' wing segments), have been observed throughout com- missioning and science operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These tilt events were first observed during OTIS cryo-vacuum testing at Johnson Space Center in 2017 and they have been ascribed to the stick/slip strain release stored in the OTE hardware and/or structure during cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' They are expected to decrease in numbers over time as the OTE structure and hardware relax into their new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Tilt events continue to occur in the science mission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2022, in prepara- tion), though less frequently and at a lesser magnitude than during early commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As shown in Figure 23, tilt events episodically punctuate weeks-long periods of wave- front stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In practice, the infrequent occurrences of large tilt events have been the dominant source of WFE degrada- tion requiring PMSA correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Note however, that not all tilt events have led to a PMSA correction, and those who did were all corrected as part of our routine maintenance pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Ongoing trending will track the nature and frequency of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The last two months of 2023 did not have any tilt events that drove excursions to 80 nm control threshold, which supports the hypothesis that the OTE structure is relax- ing to a stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' An example of a tilt event that occurred between wavefront sensing visits is shown in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Micrometeoroids Impacts from micrometeoroids on the PMSA have been observed since the middle of OTE commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Pupil imaging first revealed localized surface changes to individual mirror segments, and phase retrieval analysis has revealed, in some cases, WFE changes on the impacted segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' How- ever, not all micrometeoroid impacts have resulted in mea- surable changes in WFE since some show up only on pupil images or when averaging large numbers of optical path dif- ference (OPD) maps (Figure 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Moreover, the cumulative effect of these micrometeoroids impacts has so far minimally affected the overall telescope throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Notably, however, a large impact on segment C3 was ob- served in phase retrieval analysis from sensing visits cover- ing the period 22-24 May UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The impact was such that the global WFE worsened by 9 nm RMS, after compensation by applying segment corrections in all degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 75° days 60° 40 45° 35 30° 30 15° 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='14h 16h 18h--20h- 22h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='Oh 2h - 4h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' "8h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 20 4 15° 15 10 30° 丰 + + 5 45° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='09- 0 75° JWST_011THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 27 Figure 23: Top: Observatory-level WFE as a function of time since the beginning of the science mission (mid July 2022), showing every sensing visit along with the few instances where corrections to the PMSAs were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Occasional larger tilt events are responsible for the larger the wavefront changes over time, as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Bottom: histogram of the WFE so far showing that a large majority of observations have near-optimal image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' A small fraction of the time has WFE above the correction threshold, generally the time between a larger tilt event and its subsequent correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' PSFs during such times can have modest but detectable differences from the average PSF, which can be calibrated during science analyses using the wavefront data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observatory-level WFE mission requirements have so far been met at all times within this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Figure 24: Example of a single-segment tilt event, as seen in optical path difference maps measured during wavefront sensing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Only one segment, C6, moved much in this recent event;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' other events have shown correlated motions of several segments, apparently related to release of tension within the wing hinge areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' When such events lead to the total WFE surpassing the correction threshold, corrective mirror moves are scheduled for the subsequent WFS observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Segments not affected by any tilt event generally show superb stability, often below the 7 nm sensing noise, as seen above for the right half of the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' ObservatoryWFEfrom2022-07-16to2023-01-03 110 Sensing visit Corrections Correction threshold 100 WFE 06 80 70 2022-07-15 2022-08-012022-08-15 2022-09-012022-09-15 2022-10-01 2022-10-15 2022-11-01 2022-11-15 2022-12-012022-12-15 2023-01-01 Date ObservatorywFEHistogramfrom2022-07-16to2023-01-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='200 J FE ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='175手 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6%ofthetimehasmeasuredOTE+NiRCamWFE<80 %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='0 60 70 80 90 100 110 RMS Wavefront Error [nm]Previous Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' PID 02726, Sensing Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' PID 02726, Obs 377 Obs 403 R2022102704 R2022102903 Drift between (2022-10-27 18:47:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='269000) (2022-10-29 21:17:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='119000) Current and Previous 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 B6 B2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 Iw B5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 B3 WFE RMS: WFE RMS: WFE RMS: 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5nm 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3nm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2nm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10 nm/hr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2 JWST_01428 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope wavefront error is still well below the nom- inal requirement following the single C3-event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However models of similar events indicate that with about ten simi- lar events, we could be at our end-of-life wavefront error re- quirement of 150 nm RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Due to the precision launch of JWST, the observatory has sufficient fuel for 20+ years of mission life, considerably longer than the mission design re- quirement minimum of 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' With the possibility of an extended mission and the uncertain rate of C3-type events (from only a single occurrence) and the unexpected result- ing WFE, the project has implemented a meteoroid avoid- ance zone (MAZ) for Cycle 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Models produced by NASA’s Meteoroid Environment Office show that the greatest impact rate for higher energy micrometeoroid strikes from sporadic sources, occurs in the so-called ram direction, the direction of flight as JWST moves with the Earth around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Cycle 2 pointing restrictions favor observing in the wake (anti-ram) direction whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The proposed MAZ would reduce the instantaneous field of regard by about 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Models which remove all pointings from the MAZ and redis- tribute them over the allowed field of regard, can lower the impact rate on the primary mirror by 55-65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The reduc- tion we expect to achieve in practice will likely be 30-40%, because some high-priority and time-critical pointings in the MAZ will be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As the Cycle 2 detailed observing plan is constructed, the expected impact reduction rate will be determined and monitored throughout the Cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Detailed damage models are also being constructed to provide a bet- ter understanding of the true risk of further C3-events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' pre- liminary models suggest this was a higher than average en- ergy impact on a sensitive area of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These models will be supplemented by data from a series of ballistic tests on rel- evant samples which should be completed over the next few months, building upon the experimental testing carried out early in the JWST development (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Heaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The impact rate appears consistent with pre-flight expecta- tions and mitigation strategies are being implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Mean- while, micrometeoroids are being partially corrected as part of our routine maintenance corrections using, in particular, radius of curvature actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Since the end of commission- ing, we have gained experience over a longer time baseline and with larger-number statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' PSF Calibrations for Science Observations Every wavefront sensing and control observation pro- duces, by means of phase retrieval analysis, optical path difference maps that can be used to model the point spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These maps are publicly available (see https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='edu/) and can be readily imported using the WebbPSF Python package (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2014), which now provides the capability to generate model PSFs that track the measured variations in mirror alignment over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Efforts Figure 25: The NIRCam pupil imaging lens is used to monitor the telescope mirrors on a quarterly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Local- ized disturbances to the wavefront due to micrometeoroid impacts are visible in the logarithmically stretched image above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Some of the micrometeoroid events are detected by wavefront changes during routine sensing observations, and the corresponding pupil image features are marked here with the blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The large C3 segment event, which had the largest wavefront effect, can be seen next to the secondary mirror support strut (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The other features in- clude pre-flight contaminants along the optical path, most of which have been stable throughout the ground test program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The appearance of uneven segment gaps is due to pupil image aberration and diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' continue to further refine and improve PSF models based on in-flight experience and detailed comparisons with flight data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The package also includes trending features, some of which were used to generate Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Though science observations inevitably encounter PSF variations over time, sometimes more than others, for many science use cases the impact of such variations can be reduced as long as those variations are measured and characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Discussion and Lessons Learned JWST was a first-of-its-kind large segmented cryogenic space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This unique telescope brought with it many challenges, requiring the team to investigate how to mount and align a segmented mirror, how to test a large cryogenic telescope, and how to commission this complicated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the end, this highly complex, large, cryogenic, deploy- able space optical system has worked extremely well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The deployments and telescope alignment activities were exe- THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 29 cuted smoothly, without incidents or invoking any contin- gencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Several minor surprises were all handled within the normal course of events as part of the commissioning process and without any particular difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical performance budgets were all met or exceeded, enabled by following the systems engineering framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The tele- scope hardware, wavefront sensing and control algorithms, systems, processes, and our dedicated teams all worked as planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Future telescopes can benefit from and build upon these lessons learned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' some have already been published during the development period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', Stahl 2010, Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2018), while others are still being formulated and docu- mented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' It is an appropriate time to provide some discussion as to what led to this success and what could be done better next time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Space-Based Deployable Optics and Active Wavefront Control Flight Proven It is now obvious, but still noteworthy, that the complex sequence of events involved in deploying and aligning the OTE was successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Given the unprecedented nature of this mission, this was not assured to be the case ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Achieving this required state-of-the-art contributions across a vast range of engineering disciplines from countless indi- viduals and many organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The successes in flight are a testament to the efficacy and rigor of these long engineering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST’s wavefront sensing achieved extraordinary preci- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The wavefront sensing error budget included several terms that were not possible to test to high precision and the actual performance worked in our favor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For exam- ple, the fast wavefront sensing implemented for the IEC and thermal stability tests demonstrated sub-nm fast differential measurements, far better than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The NIRCam-only multi field sensing measurement (coarse-MIMF) fully cor- rected the multi-field alignment, which was confirmed when no further corrections were warranted after multi-field sens- ing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As yet another example, in the final iter- ation of the coarse phasing activity, the piston errors were so small that our methods struggled to sense them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The mirror control benefited from actuator moves that were more precise than budgeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This made the overall process more efficient and effective, with the ultimate achieved align- ment that is significantly better than the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The level of conservatism in the error budgeting and model un- certainty factors was appropriate engineering given the first of a kind nature of this process, but it is now clear how well this all worked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Piecewise Verification by Analysis Works The driving telescope performance parameters could not be directly tested and therefore needed to be verified by anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was unprecedented in scope and required substan- tial development of high-fidelity integrated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The in- tegrated modeling effort naturally grew in scale and fidelity as the design matured, with initial efforts focused on design trades and ultimately converging to a very accurate end-to- end model used to verify performance for the pre-ship re- view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The model size was complex as it required predictive accuracy on nanometer scales for structures that were meters in size and used novel materials operating over a wide range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These new models were managed with rig- orous oversight for model construction, verification, and val- idation at each level of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The team intentionally adopted what were thought to be conservative model uncer- tainty factors, which were in some cases found to be just right for bounding flight performance, and in others overly conser- vative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JSC cryotest was complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Nevertheless it proved its value, not just in validation of the overall performance and requirement verification, but also by allowing the discovery of several workmanship issues (particularly with the integra- tion of the frill) that could be addressed prior to launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Piecewise verification by analysis, combined with ade- quate testing to validate workmanship, was critical to the suc- cess of JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Application of Lessons from Hubble and Chandra The JWST telescope implemented important lessons learned from the Hubble and Chandra X-Ray Observatory (Chandra) programs and these were considered critical to the success of this mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The integration and testing ap- proach was developed in a way that considered the Hubble program (Feinberg & Geithner 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For JWST, there were pre-defined test criteria established and a process for report- ing if those tests are not met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' In the case of JWST, formal processes were defined for problem reports, anomalies, and failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST also developed a matrix of crosschecks, both in measurement and in analysis, that was used to catch errors and provide more confidence in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The results were transparently reported at data reviews and evaluated by the project team, external review boards, and external advisory groups such as the optical PIT (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We have previously noted the importance of independent test equipment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' here we highlight the importance of inde- pendent analyses as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST designated a walled-off group within the project team as an Independent Optical Ver- ification Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This group was charged with conducting independent analyses of key optical results, using indepen- dently developed software tools and often different analysis approaches from the primary optical verification team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This independent check was critical in ensuring the validity of test results, as well as of the numerous optical models and test configuration metrology that underpinned those test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 30 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST Pathfinder Program (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3) was a hallmark of the JWST I&T program that provided invalu- able risk reductions while proving out new systems for han- dling, integration, and testing (Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Pathfinder Program was derived from a successful verifica- tion engineering test article program on Chandra (Arenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Although there were significant costs incurred on the Pathfinder Program, it matured all of the processes needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This ultimately streamlined many activities by al- lowing the evaluation of test environments prior to flight tests and avoiding incidents on the flight hardware itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' the OTIS cryotest met all of its test objectives the first time without in- cident and without delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The Pathfinder program was particularly valuable for Webb given the heightened challenges of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Cryo- vacuum testing always presents challenges, but JWST’s size, wide range of payload operating temperatures, and infrared wavelength coverage presented exceptional demands regard- ing lengthy thermal transitions, contamination control, con- trol of thermal background light, macroscopic motions and interferences during cooldown, jitter control, and careful management of atmospheric condensation onto chamber shrouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The details of how these issues were managed for JWST are outside the scope of this paper, but discus- sions of such topics can be found in published overviews of the major ISIM and OTIS cryo-vacuum tests (Kimble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Balancing Development Risks and Science Performance Mission requirements are defined and rationalized through a science traceability matrix, which then drives mission level requirements and lower-level requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, this science-driven approach can sometimes lead to system-level performances that are difficult to achieve, resulting in con- siderable schedule and cost growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For example, in 2005 the JWST Science Assessment Team recommended, and the Sci- ence Working Group approved, relaxation of strict contami- nation requirements in favor of mirror cleaning procedures (JWST Science Assessment Team 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These procedures helped keep contamination budgets at manageable levels and reduce cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Future missions should set realistic contamina- tion levels at the outset, which necessitates early discussion of I&T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The same review eliminated the 1 µm encircled energy re- quirement (JWST Science Assessment Team 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' There were concerns with verifying the mirror performance at 1 µm given factors such as the convergence rate to meet the polish- ing specification, creep and micro yield mirror changes, and individual deformation on mirrors from backplane deforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The decision was to tighten the low-frequency wave- front error allocation while increasing the mid- and high- frequency allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This lead to significant risk mitigation by providing mirror polishing schedule relief, relaxation of challenging error budget terms, and relaxation of alignment tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As reported in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1, the image quality observed at the end of commissioning was diffraction-limited at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 µm in spite of this requirement relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Stability of a Large Space Telescope The telescope architecture relied on the support structure to provide passive stability with wavefront sensing every other day and control no more than every two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This wave- front sensing and control operations concept has proven to be appropriate as demonstrated in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' One of the larger uncertainties pre-launch was whether ob- servatory stability levels would meet the predictions from the integrated modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The observed drift sources and ampli- tudes were anticipated and accurately modeled, as described in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, the thermal transient models were not validated and for reasons not yet understood, some transient thermal drifts were significantly over-predicted, and the ther- mal distortion time constants came out significantly shorter than predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Although the IEC heaters’ on-off cycling with bang-bang control results in a wavefront oscillation that is quite small (only ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 nm), the effect is nonetheless easily sensed in high-precision transit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This IEC heater insta- bility was due to the cable harness connection to the tele- scope being very stiff, which was not initially captured in the telescope thermal distortion integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The integrated models did predict the observed level of drifts with remark- able accuracy when their properties were included (see Ta- ble 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Modeling needs to carefully consider the harnesses, and such bang-bang control heaters should be used with cau- tion in future missions striving for ultra stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Tilt events have continued into the science mission, which is not entirely unexpected, but they are still the dominant source of wavefront changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Tilt events are easily sensed and corrected by the wavefront control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The expecta- tion is that the spontaneous tilt events will continue to subside as the structure relieves stress, but continued monitoring will help determine the nature of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST material and cryogenic testing showed that the tilt events were infre- quent and sufficiently small to satisfy JWST’s encircled en- ergy stability requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For a future system that requires picometer stability, it will be important to demonstrate that picometer-class lurches are understood early (in the technol- ogy phase) such that the active control systems can be de- signed to sense and correct these displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Team Cohesiveness and Rehearsals The importance of an effective, well-trained, integrated and “badgeless” team is well known, but worth repeating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' THE TELESCOPE FOR THE JAMES WEBB SPACE TELESCOPE MISSION 31 Several factors contributed to the development of such a team for the JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The multi-institutional team comprised mem- bers from the government, academia, and the aerospace in- dustry spread across the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope man- agement team shared clear goals and created an environment that encouraged cross-team interactions, communications, and information sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The management team worked hard to foster trust and engender transparency across the dis- tributed team, which we consider critical components of the mission success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The regular interactions and open commu- nications allowed the team to work effectively together, iden- tify problems and formulate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The team was confident and well-prepared to execute the mission, drawing on experience gained from the test program and many dedicated commissioning rehearsals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These re- hearsals were operationally flight-like, incorporating proce- dures and high-fidelity simulations for data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Com- prehensive, detailed operational processes and procedures, critical to a mission of this complexity, were improved over time given the lessons learned from rehearsals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' These rigor- ous rehearsals enabled the team to prepare and become com- fortable with all aspects of the commissioning process prior to launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The team also benefited from an unusually high degree of continuity of staff on JWST, including key members in lead- ership positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This continuity formed a base of multi- institutional knowledge and long relationships that proved valuable in problem solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Many had experience with much of JWST mission’s life, with some being involved from its inception through the end of commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The team also benefited from significant experience with other space telescope projects, such as HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Guiding and Line of Sight Performance JWST has demonstrated stable pointing with extremely low jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' JWST routinely achieves LOS jitter ∼1 mas RMS through the use of a fine steering mirror in the control loop, and via high cadence subarray readouts using a focal plane instrument at pixel scales equivalent to the science instru- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For comparison, Hubble’s RMS jitter is typically ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='5 to 3 mas (Lallo 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Observational techniques to ob- tain and analyze the jitter data were proven to be successful (Hartig 2022), and results show the JWST design effectively isolated vibration sources such as the cryocooler and sup- pressed other potential contributors to jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Complex attitude control systems are traditionally chal- lenging to fully test pre-flight, and JWST’s was no excep- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Even though a closed loop guiding demonstration was executed during OTIS CV tests (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4), it was neces- sarily lacking in full fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Furthermore, early OTE com- missioning activities required close-loop guiding with a mis- aligned telescope, when the transformations from ACS to the focal plane had not yet been established and guidestar PSFs were not yet stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1) From early in the commissioning planning process, it was recognized that this called for particularly close coordina- tion among the teams responsible for Attitude Control, Wave- front, and ISIM/FGS in developing the commissioning activ- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The teams worked through the details of non-standard guiding scenarios with control loop components that were not yet fully calibrated, developed the operations concepts for manually updating ACS transformations, produced the tools for overriding the nominal guiding where necessary, and wrote a number of contingency plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As a result, in flight, the sometimes subtle interdependencies of the ACS, FGS, and the OTE optical alignment process came as little surprise, and the team was able to effectively navigate along the road to a successful and complete OTE commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The generalized lesson from this experience is that a space- craft’s pointing and attitude control system is a key com- ponent to its science performance, and benefits from being treated as such, holistically, from early in the commissioning plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Yet, the integration and consideration of the ACS as a system fundamental to science commissioning can be com- plicated by cultural differences in approach, language, and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This was anticipated from prior mission experience and mitigated by an integrated inclusive approach to OTE commissioning in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Importance of Determining Test Configurations Early While a high-level summary of the I&T program is pro- vided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3, the test activities changed dramatically in scope and implementation during the development phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' An earlier definition of test configurations could have saved resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' For example, the initial telescope cryotest concept had the telescope pointing down, called “cup down”, on a ∼300,000 kg stainless steel tower with six spinning and rotat- ing cryogenic autocollimating flats and cones of light through instruments to measure the alignment by stitching interfero- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The extreme complexity of the cup down test led to a total redesign that resulted in a simplified, yet still challeng- ing to execute, “cup up” configuration as described in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='4 (Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The cup down configuration was originally baselined in order to prevent any contamina- tion of the telescope mirrors, but the contamination was later deemed manageable in the cup up configuration by imple- menting a mirror-cleaning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This mirror cleaning procedure was used following the OTIS cryotest and before shipping to the launch site (Lobmeyer & Carey 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Abeel & Huang 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Micrometeoroid Environment and Damage The open architecture of the JWST telescope makes the primary and secondary mirror optics particularly susceptible 32 MCELWAIN, FEINBERG, & PERRIN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' to micrometeoroid damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' As discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='2, the effect from micrometeoroids was estimated based on the environment and a damage model from impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The mi- crometeor effects may have been underestimated, though the bulk of the damage to date has come from the single event on C3, so there is great statistical uncertainty in how the dam- age rate should be extrapolated into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The impact physics is also complex and uncertain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' ground tests can’t achieve micrometeoroid-like velocities and neglected how cryogenic temperatures change the material stiffness proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Because of the statistical and damage model uncertain- ties, and with the prospect for a JWST mission lifetime far exceeding requirements (due to available propellant), restric- tions on the field of regard will be implemented to reduce ram direction micrometeoroid impact rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Missions under con- sideration with even larger aperture optics and even tighter wavefront requirements will need to consider this issue care- fully and track JWST’s experience as it accumulates a longer baseline of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Optical Modeling Across Interfaces In the development of the independent system elements, there were separate optical and optomechanical models for the telescope and each of the science instruments via the in- terface shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Early in the program as the prime contractor and the NIRCam were selected, it was realized that the optical designs were not compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' To deal with this, the telescope design was modified and the F-number changed to accommodate the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, it was later uncovered that this design change increased susceptibility to the rogue path that was described briefly in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='1 and more thoroughly in Lightsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The rogue path passes through the AOS entrance aperture and directly onto the science instrument pick off mirrors, where it then en- ters the science instrument optical path through reflections or scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The rogue path stray light was well known and modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The models were used to confirm no direct optical paths to the detector focal planes and to determine scattering from the pick off mirrors was negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' However, what was missed, was the possibility of rogue path stray light causing graz- ing angle scattering off instrument structure from the pickoff mirror housing and downstream in the optical path of the in- struments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' After observing the stray light in flight (see Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content='3 of Rigby PASP Science), the optical and optome- chanical models were used to reproduce the observed phe- nomena, confirming the stray light paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The area of sus- ceptibility for the observed features that was observed is a small subset of the total rogue path region of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Now that the grazing angle stray light paths are known, observa- tion scheduling can largely prevent placing bright targets in the susceptible region on the sky relative to the science target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Not all of the full structural as well as optical characteris- tics were captured in the pre-flight modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Programmatic constraints impeded clear communication and modeling of the full system interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The optical prescription of the in- struments was in the full system model, but not the detailed structures that included the housing around the pick-off mir- rors and the detailed optomechanical structures in the science instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The full up system model only included NIR- Cam and MIRI, in order to verify the stray light light require- ments at the NIR and MIR wavelengths, and the instrument teams were left to carry out the detailed analysis of their in- struments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The lesson learned is that additional modeling of the complete integrated system for all modes is needed to re- duce risk of unexpected stray light phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This includes accurate details of mechanical structures as well as the opti- cal prescriptions throughout the complete system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Conclusion The revolutionary JWST telescope is performing better than all of its design objectives, enabling even higher sen- sitivity and more stable observations than originally planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope was made possible through the advancement of several new technologies, all of which were developed and flight proven to work as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The telescope has now em- barked on its Cycle 1 science observations, beginning a sci- entific journey that will answer some of the biggest questions in astrophysics and planetary science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' We are grateful to the JWST photography team, espe- cially Chris Gunn, for capturing the JWST development, and Heather Ghannadian for graphics design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST mission is a joint project between the National Aeronautics and Space Agency, European Space Agency, and the Canadian Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' The JWST telescope de- velopment was led at NASA’s Goddard Space Flight Center with a distributed team across Northrop Grumman Corpora- tion, Ball Aerospace, L3Harris Technologies, the Space Tele- scope Science Institute, and many other companies and insti- tutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' This telescope was created by a large team of people from many diverse backgrounds whose creativity, passion, teamwork and endless sacrifices made this scientific dream a reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' References Abeel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=', & Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfz_7W/content/2301.01779v1.pdf'} 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South Dakota, +Vermillion, SD 57069, USA. +We use a variational method to calculate the spectrum and the parton distribution function of +ground state hadrons of various gauge theories in 1+1 dimensions. The template functions in our +method minimize a free energy functional defined as a combination of free valence partons’ kinetic +energy on the lightcone and the Renyi entanglement entropy of biparton subsystems. Our results +show that hadrons in these theories minimize the proposed free energy. The success of this technique +motivates applying it to confining gauge theories in higher dimensions. +I. +INTRODUCTION +One of the enduring questions in modern physics is +a description of the spectrum of strongly coupled theo- +ries such as Quantum Chromodynamics (QCD), which +describes the strong nuclear force that holds quarks and +gluons together in colorless hadrons. +Our information +about a hadron structure comes from experimentally ac- +cessible structure functions such as the Parton Distribu- +tion Function (PDF), which can be isolated using factor- +ization theorems. While no analytical technique exists +for calculating these functions, numerical computations +in certain regimes are possible with lattice QCD [1–7]. +However, these tools are computationally and economi- +cally expensive and do not offer any simple insight into +the mechanism of strong interactions. +This has motivated novel approaches (e.g. [8–21]) for +understanding confining theories, many of which were +first proposed in 1+1 dimensions (1+1D). Here, gluons +are not propagating degrees of freedom, there is no spin, +and since the gauge coupling is dimensionful, the running +effects are power suppressed. +The PDF of the quarks +inside hadrons of 1+1D theories is simply equal to the +absolute square of their wavefunction and does not run +with the energy scale, see Ref. [22] for further details. +In this letter, we suggest that since a strongly coupled +bound state is a complex system with numerous inter- +actions between its partons, notions from complex sys- +tems, statistical mechanics, and information theory (see +Ref. [23] for connections between the latter two subjects) +can be elevated to a more central role in describing its +properties. (Notions from quantum information sciences +have already been used in studying other aspects of con- +fining gauge theories, e.g. see Refs. [24–54].) Similar to +generic systems studied in classical statistical mechanics, +e.g. a container of gas, that are governed by a minimum +free energy principle, we conjecture partons’ distribution +inside hadrons are governed by a minimum free energy +principle that includes a measure of entanglement be- +∗ pasadi@uoregon.edu +† varun.vaidya@usd.edu +tween pairs of partons. Such a principle should not rely +on the special properties of a theory, such as its symme- +tries, so that it can be readily applied to higher dimen- +sional systems. +In Ref. [22], we tested whether the hadron wavefunc- +tion in these theories could be described as a thermal +ensemble. The ansatz for the wavefunction was derived +from minimizing a proposed free energy +F = E − TS, +(1) +where E (S) is free parton kinetic energy on the light- +cone (the von Neumann entanglement entropy of bipar- +ton subsystems). We observed that this description was +exact in the limit of infinite parton mass. The descrip- +tion deviates from exact numerical results as we move to +lower quark masses and is no longer even approximately +applicable in the deep non-perturbative regime, i.e. when +mq/g ≪ 1 with mq denoting the quark mass and g denot- +ing the coupling of the confining gauge group in 1+1D. +This motivates us to explore modifications of the free en- +ergy principle, keeping in mind that it should asymptote +to the free energy of Ref. [22] in the large quark mass +limit. +Our proposal in this letter is to replace the von Neu- +mann entropy of Eq. (1) with either Tsallis [55] or Renyi +[56] entanglement entropy (both these entropy functions +work for our problem); for simplicity, we use the Renyi +entropy throughout this letter. This family of entropy +functions finds a variety of applications in physics in- +cluding quantum gravity (e.g. Refs. [57–59]), cosmology +[60], topological phases [61], non-equilibrium many body +systems [62], and condensed matter physics [63]. +We can interpret our proposal as a variational method +for approximating the wavefunction and mass spectrum +of the shallowest bound states of 1+1D gauge theories. +We find that our proposed free energy correctly pre- +dicts the hadrons’ mass spectrum and PDF for all quark +masses. The success of this approximation in reproduc- +ing the existing results motivates us to propose our mini- +mum free energy principle as the first principle governing +the properties of bound states of confining gauge groups, +including those in higher dimensions. +arXiv:2301.03611v1 [hep-th] 9 Jan 2023 + +2 +II. +A VARIATIONAL METHOD FOR HADRON +WAVEFUNCTION +We propose replacing the von Neumann entropy in the +biparton free energy of Ref. [22] by one of its generaliza- +tions, namely the Renyi entropy [56]1 +Sα(ρ) = +1 +1 − α ln +� +Tr +� +ρα�� +, +(2) +where ρ is a reduced density matrix and α is the order. +In the limit of α → 1, the Renyi entropy Sα reduces to +the von Neumann entropy. We therefore propose a new +biparton Renyi free energy functional +Fα = E − TSα, +(3) +where E denotes the kinetic energy of free valence par- +tons on the lightcone and Sα is the order α Renyi entropy +of the biparton subsystems of the hadron, see Ref. [22] +for further details. We propose to test whether minimiz- +ing this functional will lead to an improved description +of a hadron structure. +For a meson, in 1+1D the wavefunction is described +by a q¯q state since the higher Fock state contribution is +suppressed for the ground state. Following Ref. [22], our +modified free energy for the q¯q biparton system is given +by +Fα = m2 +q +P − +� +dx|φ(x)|2 +� 1 +x + +1 +1 − x +� +− +T +1 − α ln +�� +dx|φ(x)|2α +� +, +(4) +where P − is the large lightcone momentum, φ(x) is the +meson wavefunction, x is the momentum fraction of P − +carried by the quark, and T is a Lagrange multiplier. +This functional is minimized by the following ansatz +|φ(x)|2 = +� +m2 +q +T 2 +� +1 +α−1 � +x(1 − x) +� +1 +1−α , +(5) +where +T 2 = TP − +α +(1 − α) +� +dx|φ(x)|2α , +(6) +whose numerical value will be determined by the normal- +ization condition on the PDF. +Similarly, following the prescription from Ref. [22], +after minimizing the biparton Renyi free energy of a +baryon, we arrive at the ansatz +|φ(x, z1, z2, · · · , zN−2)|2 = +� +m2 +q +T 2 +� +1 +α−1 � +x +� +1 − x − +N−2 +� +k=1 +zk +� N−2 +� +k=1 +zk +� +1 +1−α +, +(7) +where N is number of colors and again T will be determined numerically by demanding the right normalization for +the wavefunction. We can now write an expression for the quark PDF inside a baryon (see Ref. [22] for further details) +fq(x) = N +� +m2 +q +T 2 +� +1 +α−1 �N−2 +� +i=1 +� 1−x−�i−1 +j=1 zj +0 +dzi +� � +x +� +1 − x − +N−2 +� +k=1 +zk +� N−2 +� +k=1 +zk +� +1 +1−α +. +(8) +We will use Eqs. (5) and (8) as the template functions +in a variational method for finding eigenvalues and eigen- +functions of ground state hadrons in various confining +theories in 1+1D. The variational parameter is the or- +der α and will be calculated numerically by demanding +that the wavefunction minimizes the expectation value +of the Hamiltonian. The specifics of a particular theory +will therefore be reflected in how α varies as a function +of mq. We can now look at specific theories in 1+1D and +carry out this calculation. Since the gauge coupling g +is dimensionful in 1+1D theories, we normalize all other +dimensionful quantities by g. +1 One can show that for our study, Renyi and Tsallis [55] entropy +give rise to exactly the same results. Throughout the text we +work with the Renyi entropy, but all our conclusions remain in- +tact if we use Tsallis entropy instead. +III. +SCHWINGER MODEL +This is a U(1) gauge theory in 1+1D with a fermion +of mass mq coupled to a photon [10]. The photon can be +eliminated using gauge redundancy and equations of mo- +tion, i.e. it is not a propagating mode in 1+1D. We work +in the Infinite Momentum Frame (IMF) with P − → ∞, +where P + acts as the Hamiltonian. +In this frame the +right-handed component of the fermion field, ψR, is not +a propagating degree of freedom either and can be elim- +inated using equations of motion. Having integrated out +these fields, we find an effective four-fermion interac- +tion term in the Hamiltonian written in position space +as [64, 65] +Hint(x0) = g2 +� +dx1dy1J(x0, x1)|x1 − y1|J(x0, y1),(9) + +3 +��������� +�� ����� (�=�)� ������ +�� ����� (�=�)� ����� +��� +��� +��� +��� +-�� +-� +-� +-� +-� +� +��/� +α +FIG. 1. +The order α for the Schwinger model (blue), mesons +of the ’t Hooft model with N = 3 (purple), and baryons +(orange) of the ’t Hooft model with N = 3. We find that at +large mq/g values α → 1, i.e. our biparton Renyi free energy +approaches the thermal free energy of Ref. [22]. +where J is the bilinear left handed fermion current ψ† +LψL. +The expectation value of the Hamiltonian for the ground +state meson in this model is given by +M 2 +hadron = ⟨φ|P −P +|φ⟩ = ⟨φ|P −(H0 + Hint)|φ⟩ +(10) += m2 +q +� +dx|φ(x)|2 +� 1 +x + +1 +1 − x +� ++ g2 +π +� +dxdyφ(x)φ∗(y) ++ g2 +2π +� +dxdy |φ(x) − φ(y)|2 +(x − y)2 +. +Here M 2 +hadron is the invariant mass squared of the meson +bound state. We now plug in the ansatz from Eq. (5) +and minimize M 2 +hadron over α to find the mass and wave- +function of the meson. +We show the value of order α for different quark masses +in Fig. 1. The order is a monotonically increasing func- +tion of mq, asymptoting to −∞ as mq → 0 and approach- +ing 1 as mq → ∞. What is peculiar is the negative or- +der at low quark masses. We observe that the shift into +negative order happens as we transition into the non- +perturbative (mq ≪ g) regime. +Putting the calculated value of α into the ansatz of +Eq. (5), we find the meson mass (Fig. 2) and PDF (Fig. 3) +for different values of mq. We find excellent agreement +for all values of quark masses (compared to Ref. [66]) +which supports our free energy conjecture. +IV. +’T HOOFT MODEL +Non-abelian gauge groups SU(N) in 1+1D are collec- +tively referred to as the ’t Hooft model, in recognition +of ’t Hooft’s contribution in studying their properties +in the large N limit [11, 14]. The Hamiltonian of the +bound states in the lightcone is derived and, by solving a +time-independent Schr¨odinger equation, the hadron mass +spectrum (as a function of the quark mass) and the wave- +function of the bound state are calculated [11, 14, 67]. +FIG. 2. Our prediction for the ground state meson mass in +the Schwinger model (solid blue) that agrees perfectly with +the existing results (blue squares) [66]. In the mq/g ≫ 1 limit, +α → 1 and our result converges to the thermal description of +Ref. [22] that used the von Neumann entropy instead of the +Renyi entropy (golden). +FIG. 3. PDF of the Schwinger model ground state meson for +two different values of quark mass. Our results (solid) agree +perfectly with the existing results (dashed) [66]. +For the rest of this work, we focus on N = 3; extending +our results to higher values of N is straightforward. For +one flavor of quarks in the fundamental representation of +SU(N), the expectation value of the Hamiltonian of the +meson state is +M 2 +hadron = m2 +q +� +dx|φ(x)|2 +� 1 +x + +1 +1 − x +� +(11) ++ g2 +2π +32 − 1 +3 +� +dxdy |φ(x) − φ(y)|2 +(x − y)2 +, +while for the baryon it is +M 2 +hadron = m2 +q +� +dx +� 1−x +0 +dz|φ(x, z)|2 +� 1 +x + +1 +1 − x − z + 1 +z +� ++ g2 +2π +3 +2 +32 − 1 +3 +� +dz +� 1−z +0 +dxdy |φ(x, z) − φ(y, z)|2 +(x − y)2 +. +(12) +Here φ(x, z) is the joint wavefunction of the three valence +partons. + +4.0 +3.5 +_ Renyi Ansatz +3.0 +. von Neumann Ansatz +2.5 +Hamer et.al. +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +ma/g2.0 +mq/g=0.08 +mg/g=0.28 +1.5 +1.0 +0.5 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x4 + + + + + + + + + +������ ����� ������ +������ ����� ����� +��� +��� +��� +��� +� +� +� +� +� +� +� +��/� +�������/� +FIG. 4. Our prediction for the ground state meson (purple) +and baryon (orange) mass as a function of the quark mass +mq for N = 3 ’t Hooft model. We find perfect agreement be- +tween our results and existing ones from solving the lightcone +Schr¨odinger equations (denoted by □) for both mesons [11] +and baryons [67]. +We now put the template ansatz from Eq. (5) or (8) +in Eqs. (11)–(12) and minimize the eigenvalue as a func- +tion of the order α. The resulting α, as a function of +the quark mass, is shown in Fig. 1. Similar to the case +of the Schwinger model, we find that α grows monotoni- +cally with mq/g and in the limit of mq ≫ g it asymptotes +to 1. Note that α → 1 corresponds to the von Neumann +entropy, thus in this limit our free energy functional be- +comes identical to the free energy used in Ref. [22]. +We can now use these values of α and calculate our ap- +proximation for the mass spectrum of the ground state +hadrons for different quark masses, see Fig. 4. We are not +aware of numerically accurate results for baryons in the +low mass quark limit; our findings are in perfect agree- +ment with the existing results. We also show the PDF of +the ground state mesons (baryons) for different values of +mq/g on the top (bottom) plot in Fig. 5. For the case of +mesons we again find perfect agreement with the results +from exactly solving the lightcone Schr¨odinger equation, +further corroborating our biparton Renyi free energy con- +jecture. We are not aware of any existing results in the +literature for baryons. +V. +DISCUSSION +In this letter, we have proposed a variational method +for calculating ground state wavefunction and spectrum +of hadrons in 1+1D. We derive template functions from a +physically motivated free energy functional made of free +partons’ kinetic energy in the lightcone frame and the +Tsallis/Renyi entanglement entropy of fixed momentum +biparton subsystems in the bound state. The variation +parameter is the order of the entropy and is calculated +by minimizing the expectation value of the Hamiltonian +of a given theory. We found that our method correctly +reproduces the existing results for the mass spectrum and +for the PDF of hadrons in the Schwinger and the ’t Hooft +��/� = ���� +��/� = ���� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +� +��(�) +��/� = ���� +��/� = ���� +��/� = ��� +��� +��� +��� +��� +��� +��� +� +� +� +� +� +� +��(�) +FIG. 5. Top: The PDF of the ground state meson of ’t Hooft +model with N = 3 and for a single flavor of quark in the +fundamental representation for different quark masses. Our +results (solid lines) are in agreement with the existing results, +see Ref. [68]. +Bottom: PDF of the ground state baryon +in the same theory for a few different quark masses. In the +mq → 0 limit, we approach the analytic results of Ref. [67]. +We are not aware of any numerical calculation of these PDFs +for finite masses. +(with N = 3 and one flavor of quarks in the fundamental +representation) model. +Our results show that hadrons in these models mini- +mize our proposed biparton Renyi free energy functional. +We also observe that, irrespective of the specific theory or +bound state, the order of the entropy increases monoton- +ically with the bare parton mass from α → −∞ at zero +quark mass to α → 1 at infinite quark mass limit. We find +that the order becomes negative at low parton masses +just as we enter the non-perturbative regime. Statisti- +cally, we can understand this as a switch from maximiz- +ing high probability configurations to minimizing those +with low probability. +We do not yet have any deeper +physical intuition about α beyond this interpretation. +Given our construction, we conjecture that in a bi- +parton subsystem (in any bound state of any confining +theory) carrying a fixed total momentum, a single par- +ton reduced density matrix is effectively described by an +ensemble which is maximally entangled at zero mass and +which approaches a thermal state at large masses. +Our method can be extended to other 1+1D models. +Our ultimate goal is to extend our method to models +in higher dimensions, where a host of new complica- + +5 +tions (e.g. +renormalization group evolution, spin, and +gluon degrees of freedom) need to be accommodated. 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Xiong, JHEP 11, 151 +(2017), arXiv:1708.09379 [hep-ph]. + diff --git a/S9E2T4oBgHgl3EQfCQZQ/content/tmp_files/load_file.txt b/S9E2T4oBgHgl3EQfCQZQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7db066050d79b3f1cc6e08474ac86b1efa4e9a3 --- /dev/null +++ b/S9E2T4oBgHgl3EQfCQZQ/content/tmp_files/load_file.txt @@ -0,0 +1,692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf,len=691 +page_content='1+1D Hadrons Minimize their Biparton Renyi Free Energy Pouya Asadi1, ∗ and Varun Vaidya2, † 1Institute for Fundamental Science and Department of Physics, University of Oregon, Eugene, OR 97403, USA 2Department of Physics, University of South Dakota, Vermillion, SD 57069, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We use a variational method to calculate the spectrum and the parton distribution function of ground state hadrons of various gauge theories in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The template functions in our method minimize a free energy functional defined as a combination of free valence partons’ kinetic energy on the lightcone and the Renyi entanglement entropy of biparton subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our results show that hadrons in these theories minimize the proposed free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The success of this technique motivates applying it to confining gauge theories in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' INTRODUCTION One of the enduring questions in modern physics is a description of the spectrum of strongly coupled theo- ries such as Quantum Chromodynamics (QCD), which describes the strong nuclear force that holds quarks and gluons together in colorless hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our information about a hadron structure comes from experimentally ac- cessible structure functions such as the Parton Distribu- tion Function (PDF), which can be isolated using factor- ization theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' While no analytical technique exists for calculating these functions, numerical computations in certain regimes are possible with lattice QCD [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' However, these tools are computationally and economi- cally expensive and do not offer any simple insight into the mechanism of strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' This has motivated novel approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [8–21]) for understanding confining theories, many of which were first proposed in 1+1 dimensions (1+1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Here, gluons are not propagating degrees of freedom, there is no spin, and since the gauge coupling is dimensionful, the running effects are power suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The PDF of the quarks inside hadrons of 1+1D theories is simply equal to the absolute square of their wavefunction and does not run with the energy scale, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In this letter, we suggest that since a strongly coupled bound state is a complex system with numerous inter- actions between its partons, notions from complex sys- tems, statistical mechanics, and information theory (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [23] for connections between the latter two subjects) can be elevated to a more central role in describing its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (Notions from quantum information sciences have already been used in studying other aspects of con- fining gauge theories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [24–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=') Similar to generic systems studied in classical statistical mechanics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' a container of gas, that are governed by a minimum free energy principle, we conjecture partons’ distribution inside hadrons are governed by a minimum free energy principle that includes a measure of entanglement be- ∗ pasadi@uoregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='edu † varun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='vaidya@usd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='edu tween pairs of partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Such a principle should not rely on the special properties of a theory, such as its symme- tries, so that it can be readily applied to higher dimen- sional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22], we tested whether the hadron wavefunc- tion in these theories could be described as a thermal ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The ansatz for the wavefunction was derived from minimizing a proposed free energy F = E − TS, (1) where E (S) is free parton kinetic energy on the light- cone (the von Neumann entanglement entropy of bipar- ton subsystems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We observed that this description was exact in the limit of infinite parton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The descrip- tion deviates from exact numerical results as we move to lower quark masses and is no longer even approximately applicable in the deep non-perturbative regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' when mq/g ≪ 1 with mq denoting the quark mass and g denot- ing the coupling of the confining gauge group in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' This motivates us to explore modifications of the free en- ergy principle, keeping in mind that it should asymptote to the free energy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] in the large quark mass limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our proposal in this letter is to replace the von Neu- mann entropy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (1) with either Tsallis [55] or Renyi [56] entanglement entropy (both these entropy functions work for our problem);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' for simplicity, we use the Renyi entropy throughout this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' This family of entropy functions finds a variety of applications in physics in- cluding quantum gravity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [57–59]), cosmology [60], topological phases [61], non-equilibrium many body systems [62], and condensed matter physics [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We can interpret our proposal as a variational method for approximating the wavefunction and mass spectrum of the shallowest bound states of 1+1D gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We find that our proposed free energy correctly pre- dicts the hadrons’ mass spectrum and PDF for all quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The success of this approximation in reproduc- ing the existing results motivates us to propose our mini- mum free energy principle as the first principle governing the properties of bound states of confining gauge groups, including those in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='03611v1 [hep-th] 9 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' A VARIATIONAL METHOD FOR HADRON WAVEFUNCTION We propose replacing the von Neumann entropy in the biparton free energy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] by one of its generaliza- tions, namely the Renyi entropy [56]1 Sα(ρ) = 1 1 − α ln � Tr � ρα�� , (2) where ρ is a reduced density matrix and α is the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In the limit of α → 1, the Renyi entropy Sα reduces to the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We therefore propose a new biparton Renyi free energy functional Fα = E − TSα, (3) where E denotes the kinetic energy of free valence par- tons on the lightcone and Sα is the order α Renyi entropy of the biparton subsystems of the hadron, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We propose to test whether minimiz- ing this functional will lead to an improved description of a hadron structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' For a meson, in 1+1D the wavefunction is described by a q¯q state since the higher Fock state contribution is suppressed for the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22], our modified free energy for the q¯q biparton system is given by Fα = m2 q P − � dx|φ(x)|2 � 1 x + 1 1 − x � − T 1 − α ln �� dx|φ(x)|2α � , (4) where P − is the large lightcone momentum, φ(x) is the meson wavefunction, x is the momentum fraction of P − carried by the quark, and T is a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' This functional is minimized by the following ansatz |φ(x)|2 = � m2 q T 2 � 1 α−1 � x(1 − x) � 1 1−α , (5) where T 2 = TP − α (1 − α) � dx|φ(x)|2α , (6) whose numerical value will be determined by the normal- ization condition on the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Similarly, following the prescription from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22], after minimizing the biparton Renyi free energy of a baryon, we arrive at the ansatz |φ(x, z1, z2, · · · , zN−2)|2 = � m2 q T 2 � 1 α−1 � x � 1 − x − N−2 � k=1 zk � N−2 � k=1 zk � 1 1−α , (7) where N is number of colors and again T will be determined numerically by demanding the right normalization for the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We can now write an expression for the quark PDF inside a baryon (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] for further details) fq(x) = N � m2 q T 2 � 1 α−1 �N−2 � i=1 � 1−x−�i−1 j=1 zj 0 dzi � � x � 1 − x − N−2 � k=1 zk � N−2 � k=1 zk � 1 1−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (8) We will use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (5) and (8) as the template functions in a variational method for finding eigenvalues and eigen- functions of ground state hadrons in various confining theories in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The variational parameter is the or- der α and will be calculated numerically by demanding that the wavefunction minimizes the expectation value of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The specifics of a particular theory will therefore be reflected in how α varies as a function of mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We can now look at specific theories in 1+1D and carry out this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Since the gauge coupling g is dimensionful in 1+1D theories, we normalize all other dimensionful quantities by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 1 One can show that for our study, Renyi and Tsallis [55] entropy give rise to exactly the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Throughout the text we work with the Renyi entropy, but all our conclusions remain in- tact if we use Tsallis entropy instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' SCHWINGER MODEL This is a U(1) gauge theory in 1+1D with a fermion of mass mq coupled to a photon [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The photon can be eliminated using gauge redundancy and equations of mo- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' it is not a propagating mode in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We work in the Infinite Momentum Frame (IMF) with P − → ∞, where P + acts as the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In this frame the right-handed component of the fermion field, ψR, is not a propagating degree of freedom either and can be elim- inated using equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Having integrated out these fields, we find an effective four-fermion interac- tion term in the Hamiltonian written in position space as [64, 65] Hint(x0) = g2 � dx1dy1J(x0, x1)|x1 − y1|J(x0, y1),(9) 3 ��������� �� ����� (�=�)� ������ �� ����� (�=�)� ����� ��� ��� ��� ��� �� � � � � � ��/� α FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The order α for the Schwinger model (blue), mesons of the ’t Hooft model with N = 3 (purple), and baryons (orange) of the ’t Hooft model with N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We find that at large mq/g values α → 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' our biparton Renyi free energy approaches the thermal free energy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' where J is the bilinear left handed fermion current ψ† LψL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The expectation value of the Hamiltonian for the ground state meson in this model is given by M 2 hadron = ⟨φ|P −P +|φ⟩ = ⟨φ|P −(H0 + Hint)|φ⟩ (10) = m2 q � dx|φ(x)|2 � 1 x + 1 1 − x � + g2 π � dxdyφ(x)φ∗(y) + g2 2π � dxdy |φ(x) − φ(y)|2 (x − y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Here M 2 hadron is the invariant mass squared of the meson bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We now plug in the ansatz from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (5) and minimize M 2 hadron over α to find the mass and wave- function of the meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We show the value of order α for different quark masses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The order is a monotonically increasing func- tion of mq, asymptoting to −∞ as mq → 0 and approach- ing 1 as mq → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' What is peculiar is the negative or- der at low quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We observe that the shift into negative order happens as we transition into the non- perturbative (mq ≪ g) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Putting the calculated value of α into the ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (5), we find the meson mass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 2) and PDF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 3) for different values of mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We find excellent agreement for all values of quark masses (compared to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [66]) which supports our free energy conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' ’T HOOFT MODEL Non-abelian gauge groups SU(N) in 1+1D are collec- tively referred to as the ’t Hooft model, in recognition of ’t Hooft’s contribution in studying their properties in the large N limit [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The Hamiltonian of the bound states in the lightcone is derived and, by solving a time-independent Schr¨odinger equation, the hadron mass spectrum (as a function of the quark mass) and the wave- function of the bound state are calculated [11, 14, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our prediction for the ground state meson mass in the Schwinger model (solid blue) that agrees perfectly with the existing results (blue squares) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In the mq/g ≫ 1 limit, α → 1 and our result converges to the thermal description of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22] that used the von Neumann entropy instead of the Renyi entropy (golden).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' PDF of the Schwinger model ground state meson for two different values of quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our results (solid) agree perfectly with the existing results (dashed) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' For the rest of this work, we focus on N = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' extending our results to higher values of N is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' For one flavor of quarks in the fundamental representation of SU(N), the expectation value of the Hamiltonian of the meson state is M 2 hadron = m2 q � dx|φ(x)|2 � 1 x + 1 1 − x � (11) + g2 2π 32 − 1 3 � dxdy |φ(x) − φ(y)|2 (x − y)2 , while for the baryon it is M 2 hadron = m2 q � dx � 1−x 0 dz|φ(x, z)|2 � 1 x + 1 1 − x − z + 1 z � + g2 2π 3 2 32 − 1 3 � dz � 1−z 0 dxdy |φ(x, z) − φ(y, z)|2 (x − y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (12) Here φ(x, z) is the joint wavefunction of the three valence partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 _ Renyi Ansatz 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' von Neumann Ansatz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 Hamer et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 ma/g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 mq/g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='08 mg/g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='0 x4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 \uf3a4 ������ ����� ������ ������ ����� ����� ��� ��� ��� ��� � � � � � � � ��/� �������/� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our prediction for the ground state meson (purple) and baryon (orange) mass as a function of the quark mass mq for N = 3 ’t Hooft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We find perfect agreement be- tween our results and existing ones from solving the lightcone Schr¨odinger equations (denoted by □) for both mesons [11] and baryons [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We now put the template ansatz from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (5) or (8) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (11)–(12) and minimize the eigenvalue as a func- tion of the order α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The resulting α, as a function of the quark mass, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Similar to the case of the Schwinger model, we find that α grows monotoni- cally with mq/g and in the limit of mq ≫ g it asymptotes to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Note that α → 1 corresponds to the von Neumann entropy, thus in this limit our free energy functional be- comes identical to the free energy used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We can now use these values of α and calculate our ap- proximation for the mass spectrum of the ground state hadrons for different quark masses, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We are not aware of numerically accurate results for baryons in the low mass quark limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' our findings are in perfect agree- ment with the existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We also show the PDF of the ground state mesons (baryons) for different values of mq/g on the top (bottom) plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' For the case of mesons we again find perfect agreement with the results from exactly solving the lightcone Schr¨odinger equation, further corroborating our biparton Renyi free energy con- jecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We are not aware of any existing results in the literature for baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' DISCUSSION In this letter, we have proposed a variational method for calculating ground state wavefunction and spectrum of hadrons in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We derive template functions from a physically motivated free energy functional made of free partons’ kinetic energy in the lightcone frame and the Tsallis/Renyi entanglement entropy of fixed momentum biparton subsystems in the bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The variation parameter is the order of the entropy and is calculated by minimizing the expectation value of the Hamiltonian of a given theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We found that our method correctly reproduces the existing results for the mass spectrum and for the PDF of hadrons in the Schwinger and the ’t Hooft ��/� = ���� ��/� = ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � ��(�) ��/� = ���� ��/� = ���� ��/� = ��� ��� ��� ��� ��� ��� ��� � � � � � � ��(�) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Top: The PDF of the ground state meson of ’t Hooft model with N = 3 and for a single flavor of quark in the fundamental representation for different quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our results (solid lines) are in agreement with the existing results, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Bottom: PDF of the ground state baryon in the same theory for a few different quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' In the mq → 0 limit, we approach the analytic results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We are not aware of any numerical calculation of these PDFs for finite masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' (with N = 3 and one flavor of quarks in the fundamental representation) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our results show that hadrons in these models mini- mize our proposed biparton Renyi free energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We also observe that, irrespective of the specific theory or bound state, the order of the entropy increases monoton- ically with the bare parton mass from α → −∞ at zero quark mass to α → 1 at infinite quark mass limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We find that the order becomes negative at low parton masses just as we enter the non-perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Statisti- cally, we can understand this as a switch from maximiz- ing high probability configurations to minimizing those with low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We do not yet have any deeper physical intuition about α beyond this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Given our construction, we conjecture that in a bi- parton subsystem (in any bound state of any confining theory) carrying a fixed total momentum, a single par- ton reduced density matrix is effectively described by an ensemble which is maximally entangled at zero mass and which approaches a thermal state at large masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our method can be extended to other 1+1D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Our ultimate goal is to extend our method to models in higher dimensions, where a host of new complica- 5 tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' renormalization group evolution, spin, and gluon degrees of freedom) need to be accommodated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' We leave such studies, and many other phenomenology ex- plorations, for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' ACKNOWLEDGEMENT We thank Chris Akers and Tim Cohen for helpful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' The work of PA is supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Department of Energy under Grant Number DE- SC0011640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' VV is supported by startup funds from the University of South Dakota.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' D 97, 014505 (2018), arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='01295 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Shifman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Vainshtein, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Zakharov, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} +page_content=' B 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfCQZQ/content/2301.03611v1.pdf'} diff --git a/StFKT4oBgHgl3EQfjy7A/content/tmp_files/2301.11847v1.pdf.txt b/StFKT4oBgHgl3EQfjy7A/content/tmp_files/2301.11847v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..296c8aabacf1e0d5903e807ab978e1d3b2756f1e --- /dev/null +++ b/StFKT4oBgHgl3EQfjy7A/content/tmp_files/2301.11847v1.pdf.txt @@ -0,0 +1,950 @@ + +1 +A Comparative Study of Pretrained Language +Models for Long Clinical Text + + +Yikuan Li 1, Ramsey M. Wehbe, MD 2,3, Faraz S. Ahmad, MD 1,2,3, Hanyin Wang 1, +Yuan Luo, PhD 1 * + +Emails: +yikuan.li@northwestern.edu +ramsey.wehbe@northwestern.edu +faraz.ahmad@northwestern.edu +hanyin.wang@northwestern.edu +yuan.luo@northwestern.edu + + +1 Division of Health and Biomedical Informatics, Department of Preventive +Medicine, Feinberg School of Medicine, Northwestern University, Chicago, +Illinois, USA +2 Division of Cardiology, Department of Medicine, Feinberg School of Medicine, +Northwestern University, Chicago, Illinois, USA +3 Bluhm Cardiovascular Institute Center for Artificial Intelligence, Northwestern +Medicine, Chicago, Illinois, USA + +* Corresponding author + + +Keywords: Clinical Natural Language Processing, Text Classification, Named Entity +Recognition, Question Answering, Natural Language Inference + + + +Word Count = 3,320 + + + + + +2 +Abstract +Objective: Clinical knowledge enriched transformer models (e.g., ClinicalBERT) have state-of- +the-art results on clinical NLP (natural language processing) tasks. One of the core limitations of +these transformer models is the substantial memory consumption due to their full self-attention +mechanism, which leads to the performance degradation in long clinical texts. To overcome this, +we propose to leverage long-sequence transformer models (e.g., Longformer and BigBird), +which extend the maximum input sequence length from 512 to 4096, to enhance the ability to +model long-term dependencies in long clinical texts. +Materials and Methods: Inspired by the success of long sequence transformer models and the +fact that clinical notes are mostly long, we introduce two domain enriched language models, +Clinical-Longformer and Clinical-BigBird, which are pre-trained on a large-scale clinical corpus. +We evaluate both language models using 10 baseline tasks including named entity recognition, +question answering, natural language inference, and document classification tasks. +Results: The results demonstrate that Clinical-Longformer and Clinical-BigBird consistently and +significantly outperform ClinicalBERT and other short-sequence transformers in all 10 +downstream tasks and achieve new state-of-the-art results. +Discussion: Our pre-trained language models provide the bedrock for clinical NLP using long +texts. We have made our source code available at https://github.com/luoyuanlab/Clinical- +Longformer, and the pre-trained models available for public download at: +https://huggingface.co/yikuan8/Clinical-Longformer. +Conclusion: This study demonstrates that clinical knowledge enriched long-sequence +transformers are able to learn long-term dependencies in long clinical text. Our methods can also +inspire the development of other domain-enriched long-sequence transformers. + + +3 +Introduction +Transformer-based models have been wildly successful in setting state-of-the-art benchmarks on +a broad range of natural language processing (NLP) tasks, including question answering, +document classification, machine translation, text summarization, and others [1-3]. These +successes have been replicated in the clinical and biomedical domain via pre-training language +models using large-scale clinical or biomedical corpora, then fine-tuning on a variety of clinical +or biomedical downstream tasks, including computational phenotyping [4], automatical ICD +coding [5], knowledge graph completion [6] and clinical question answering [7]. +The self-attention mechanism [8] is one of the most critical components that lead to the success +of transformer-based models, which allows each token in the input sequence to independently +interact with every other token in the sequence in parallel. However, the memory consumption of +the self-attention mechanism grows quadratically with sequence length, resulting in +impracticable training time, and easily reaching the memory limits of modern GPUs. +Consequently, transformer-based models that leverage a complete self-attention mechanism, +such as BERT and RoBERTa, typically have an input sequence length limit of 512 tokens. To +deal with this limit when modeling long texts using transformer-based models, the input +sequence shall be either truncated to the first 512 tokens or processed via a sliding window of +512 tokens with or without overlap. If the latter method is applied to a document-level +classification task, an aggregation operation will be added to yield the final output from multiple +snippets. Both methods ignore long-term dependencies spanning over 512 tokens and may +achieve suboptimal results due to information loss. Additionally, this input token limitation of +the self-attention mechanism could impact language model pre-training and then be amplified to +downstream tasks. In clinical NLP, transformer-based modeling approaches have also + + +4 +encountered this limitation [9]. For example, the discharge summaries in MIMIC-III, which are +often used to predict clinically meaningful events like hospital re-admission [10] or mortality +[11], have 2,984 tokens (1,435 words) on average, far exceeding the 512 token limit of most full +attention-based transformer models. +Recently, investigators have developed novel variants of transformers specifically for long +sequences that reduce memory usage from quadratic to linear scale of the sequence length [12- +14]. The core idea behind these models is to replace the full attention mechanism with a sparse +attention mechanism, which is typically a blend of sliding windows and reduced global attention. +These models are capable of processing up to 4,096 tokens and have empirically boosted +performance on NLP tasks, including question answering as well as text summarization. +However, to the best of our knowledge, long sequence transformers in the clinical and +biomedical domain have not yet been systematically explored. The purpose of this manuscript is +to examine the adaptability of these long sequence models to a series of clinical NLP tasks. We +make the following contributions: +• We leverage large-scale clinical notes to pre-train two new language models, namely +Clinical-Longformer and Clinical-BigBird. +• We demonstrate that both Clinical-Longformer and Clinical-BigBird improve the +performance of a variety of downstream clinical NLP datasets, including question answering, +named entity recognition, and document classification tasks. + +Background and Significance +Clinical and Biomedical Transformers + + +5 +Transformer-based models, especially BERT [2], can be enriched with clinical and biomedical +knowledge through pre-training on large-scale clinical and biomedical corpora. These domain- +enriched models, for example, BioBERT [15] pre-trained on biomedical publications and +ClinicalBERT [16] pre-trained on clinical narratives, set state-of-the-art benchmarks on +downstream clinical and biomedical NLP tasks. Inspired by the success of these domain- +enriched models, more pre-trained models were released to boost the performance of NLP +models when applied to specific clinical scenarios. For example, Smit et al. [17] proposed +CheXbert to annotate thoracic disease findings from radiology reports, which outperformed +previous rule-based labelers with statistical significance. The model was pre-trained using a +combination of human-annotated and machine-annotated radiology reports. He et al. [18] +introduced DiseaseBERT, which infused disease knowledge to the BERT model by pre-training +on a series of disease description passages that were constructed from Wikipedia and MeSH +terms. DiseaseBERT achieved superior results on consumer health question answering tasks +compared with BERT and ClinicalBERT. Michalopoulos et al. [19] proposed UmlsBERT, which +used the Unified Medical Language System (UMLS) Metathesaurus to augment the domain +knowledge learning ability of ClinicalBERT. Zhou et al. [20] developed CancerBERT to extract +breast cancer-related concepts from clinical notes and pathology reports. Agrawal et al. [21] +leveraged order contrastive pre-training on longitudinal data to tackle the difficulty when only a +small proportion of the clinical notes were annoated. However, all models mentioned above were +built on the vanilla BERT architecture, which has a limitation of 512 tokens in the input +sequence length. This limitation may result in the information loss of long-term dependencies in +the training processes. +Transformers for Long Sequences + + +6 +Various attention mechanisms have been proposed to handle the large memory consumption of +the attention operations in the vanilla transformer architecture. Transformer-XL [22] segmented +a long sequence into multiple small chunks and then learned long-term dependencies with a left- +to-right segment-level recurrence mechanism. Transformer-XL learns 5.5 times longer +dependencies than the vanilla transformer models but loses the advantage of bidirectional +representation of BERT-like models. In another study, Reformer [23] applied two techniques to +reduce the complexity of transformer architecture by replacing dot-product attention operation +with locality-sensitive hashing and sharing the activation function among layers. Reformer was +able to process longer sequences at a faster speed and be more memory efficient. However, this +enhancement improves space, time, and memory efficiency, but not accuracy on specific tasks. +Almost simultaneously, Longformer [13] and BigBird [14] were proposed to drastically alleviate +the memory consumption of transformer models by replacing the pairwise full attention +mechanisms with a combination of sliding window attention and global attention mechanisms. +They are slightly different regarding implementation and configuration of the global and local +attention mechanism, where BigBird introduces additional contrastive predictive coding to train +global tokens [14]. Both models support input sequences up to 4.096 tokens long (8 times the +input sequence limit of BERT) and significantly improve performance on long-text question +answering and summarization tasks. However, the adaptability of these long sequence +transformers to the clinical and biomedical fields, where document length mostly exceeds the +limits of BERT-like models, has not been investigated. +Materials and Methods +In this section, we first introduce the clinical dataset we use as the pre-training corpus, followed +by the pre-training processes for Clinical-Longformer and Clinical-BigBird. Next, we enumerate + + +7 +the downstream tasks we use to compare our long sequence models with the short sequence +models. We also provide the technical details of pre-training and fine-tuning for the purposes of +reproducing our results. The entire pipeline can be found in Figure 1. +Figure 1: The pipeline for pre-training and fine-tuning transformer-based language models. + +Datasets +Similar to Huang et al. [10] and Alsentzer et al. [16], we leverage approximately 2 million +clinical notes extracted from the MIMIC-III [24] dataset, which is the most extensive publicly +available electronic health records (EHR) dataset that contains clinical narratives of over 40,000 +patients admitted to the intensive care units. We only apply minimal pre-processing steps, +including 1) to remove all de-identification placeholders from the clinical notes that were +generated to protect the PHI (protected health information); 2) to remove all characters other +than alphanumerics and punctuation marks; 3) to convert all alphabetical characters to lower +cases, and 4) to strip extra white spaces. We believe that complicated pre-processing in the pre- +training stage may not improve downstream performance but will sacrifice the generalizability of +language models and significantly increase training time. +Pre-training +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Longformer +BigBird +BERT +MIMIC III +Clinical +Notes +English Wiki +PubMed +MIMIC III +Clinical- +Longformer +Clinical- +BigBird +BERT +BioBERT +ClinicalBERT +Transformer +Models +Pre-training +Corpora +Knowledge-enriched +Transformers +Long-sequence +Models +Short-sequence +Models +medNLI, +an inference task, +eval. by accuracy +emrQA (relation subset), +a QA task, +eval. by EM score +i2b2 - 2014, +an NER task, +eval. by F1 score +Downstream +Clinical NLP Tasks + +0.75 +0.8 +0.85 +0.9 +0.95 +1 +8 +Longformer [13] and BigBird [14] are the two best-performing transformer models that are +designed for long input sequences. Both models extend the maximum input sequence length to +4,096 tokens, which is 8× the limit of conventional transformer-based models, by introducing +localized sliding windows and global attention mechanisms to reduce the computational +expenses of full self-attention mechanisms. The differences between the two models are how the +global attention is realized and the selection of loss function in fine-tuning [13]. BigBird also +contains some random localized attention operations. The reported performance difference +between the two models on downstream tasks is minimal [14]. Therefore, we seek to pre-train +both models and compare their performance on clinical NLP tasks. We refer readers to the +original papers of Longformer [13] and BigBird [14] for more technical details. +We initialize Clinical-Longformer and Clinical-BigBird from the pre-trained weights of the base +version of Longformer and the ITC (internal transformer construction) version of BigBird, +respectively. Although the ETC (extended transformer construction) version of BigBird may +have superior performance, HuggingFace (the largest community for sharing open-source pre- +trained transformer models) only provides the implementation and the pre-trained checkpoints of +the ITC version. The difference between ITC and ETC version is that in ITC version some +existing tokens are made “global” and attend over the entire sequence, while ETC version +introduces additionally “global” tokens such as CLS. Byte-level BPE (Byte-Pair-Encoding) [25] +is applied to tokenize the clinical corpus. Both models are distributed in parallel to six 32GB +Tesla V100 GPUs. FP16 precision is enabled to accelerate training. Batch size is 18 for Clinical- +Longformer and 12 for Clinical-Bigbird, which are the upper limits under 6 32GB GPUs. We +pre-train Clinical-Longformer for 200,000 steps and Clinical-BigBird for 300,000 steps, which +ensures that each clinical note is seen equal times by the two models. The learning rates are 3e-5 + + +9 +for both models, the same as the learning rate used in the pretraining of Longformer. The entire +pre-training process takes more than two weeks for each model. +To evaluate the performance of pre-training, we create a testing set that contains 1,000 +documents that are also from MIMIC-III but have not been used as the pre-training corpora. +Each document in the testing set is truncated to 512 tokens long. We randomly select 10% tokens +from each document and replace them with a mask token. We compare our two pre-trained +models with the short-sequence models in filling in the masked tokens using context. We report +the perplexity score and top 5 accuracy in filling in the masked tokens of each model. +Downstream Tasks +In this study, we fine-tune the pre-trained Clinical-Longformer and Clinical-BigBird on 10 +clinical NLP datasets. These 10 NLP datasets broadly cover various NLP tasks, including +extractive question answering, named entity recognition, natural language inference, and +document classification. We rely on these NLP tasks to validate the performance improvement of +long sequence models compared to their short sequence counterparts. The statistics and +descriptions of all datasets can be found in Table 1. +Question Answering +Question answering (QA) is a common NLP task that aims to automatically answer questions +asked in natural language [26]. In the clinical context, QA systems answer clinicians' questions +by understanding the clinical narratives extracted from electronic health record + + + + + + + + +10 +Table 1: Description and statistics of downstream clinical NLP tasks +Dataset +Task +Source +Sample +Size +Avg. Seq. +Length +Max Seq. +Length +MedNLI +Inference +MIMIC +14,049 +39 +409 +i2b2 2006 +NER +i2b2 +66,034 +867 +3,986 +i2b2 2010 +NER +i2b2 +43,947 +1,459 +6,052 +i2b2 2012 +NER +i2b2 +13,108 +794 +2,900 +i2b2 2014 +NER +i2b2 +83,466 +5,134 +14,370 +emrQA-Relation +QA +i2b2 +255,908 +1,880 +6,109 +emrQA-Medication +QA +i2b2 +141,243 +1,460 +6,050 +emrQA-HeartDisease QA +i2b2 +30,731 +5,293 +14,060 +openI +Multilabel Classif. +IndianaU +3,684 +70 +294 +MIMIC-CXR +Multilabel Classif. +MIMIC- +CXR +222,713 +119 +874 +MIMIC-AKI +Binary Classif. +MIMIC +16,536 +1,463 +20,857 + +systems to support decision-making. emrQA [27] is the most frequently used benchmark dataset +in clinical QA, which contains more than 400,000 question-answer pairs semi-automatically +generated from past Informatics for Integrating Biology and the Bedside (i2b2) challenges. +emrQA falls into the category of extractive question answering, aiming to identify answer spans +from reference texts instead of generating new answers in a word-by-word fashion. Researchers +have attempted to solve emrQA tasks by using word embedding models [28], conditional random +fields (CRFs) [29] and transformer-based models [30], among which transformer-based models +performed best. In our experiments, we investigate the performance of our pre-trained models +using the three largest emrQA subsets: Medication, Relation, and Heart Disease. We evaluate +QA performance with two commonly used metrics: exact match (EM) and F1-score. Exact match +evaluates whether entire predicted spans match exactly with the ground-truth annotations. F1- +score is a looser metric derived from token-level precision and recall, which measures the + + +11 +overlap between the predictions and the targets. We generate train-dev-test splits by following +the instruction of Yue et al. [28]. The training set of relation and medication subsets are +randomly under-sampled to reduce training time. Based on their experience, performance was +not compromised by under-sampling. Of note, the emrQA dataset has some known issues, e.g., +incomplete answers, it is template-based, and the annotation were generated semi- +automatically[28]. We consider the usage of emrQA as a proof-of-concept experiment to +compare the performance of the transformer-based model on the QA task. +Named Entity Recognition +Named entity recognition is a token-level classification task that seeks to identify the named +entities and classify them in predefined categories. This genre of NLP tasks has broad +applications in the clinical and biomedical domains. e.g., de-identification of PHI and medical +concept extraction from clinical notes. Prior studies have shown that transformer-based models +[15] significantly outperformed the models built on pre-trained static word embeddings [31] or +LSTM networks [32]. We compare our pre-trained models using four data challenges: 1) i2b2 +2006 [33] to de-identify PHI from medical discharge notes; 2) i2b2 2010 [34] to extract and +annotate medical concepts from patient reports; 3) i2b2 2012 [35] to identify both clinical +concepts and events relevant to the patient's clinical timeline from discharge summaries, and 4) +i2b2 2014 [36] to identify PHI information from longitudinal clinical narratives. We follow the +processing steps of Alsentzer et al. [16], which converts the raw data from all four tasks to the +IOB (inside–outside–beginning) tagging format proposed by Ramshaw et al. [37], and then +create train-dev-test splits. We evaluate the model performance with F1 score similarly to QA +tasks. +Document Classification + + +12 +Document classification is one of the most common NLP tasks, where a sentence or document is +assigned to one or more classes or categories. In the clinical domain, document classification can +be used to identify the onset of a particular disease process or predict patient prognosis using +entire clinical notes. We use the following three document classification datasets to evaluate the +pre-trained models from different perspectives. +MIMIC-AKI [38, 39] MIMIC-AKI is a binary classification task, where we aim to predict the +possibility of AKI (acute kidney injury) for critically ill patients using the clinical notes within +the first 24 hours following intensive care unit (ICU) admission. We follow Li et al. [38] to +extract the cohort from MIMIC-III. We evaluate the model performance using AUC and F1 +score. +OpenI [40] OpenI is a publicly available chest x-ray (CXR) dataset collected by Indiana +University. The dataset provides around 4,000 radiology reports and their associated human +annotated Medical Subject Headings (MeSH) terms. In our experiments, the task is to detect the +presence of the annotated thoracic findings from CXR reports, which is considered a multi-label +classification task. Given the small sample size, we will only use OpenI as the testing set. The +pre-trained language models are fine-tuned using MIMIC-CXR [41], another publicly available +chest x-ray dataset that contains more than 200,000 CXR reports. Unlike openI, the ground-truth +labels for MIMIC-CXR were automatically generated using NLP approaches. The overlapping +findings between the two CXR data sources are:: Cardiomegaly, Edema, Consolidation, +Pneumonia, Atelectasis, Pneumothorax and Pleural Effusion. We report the sample number +weighted average of the area under the receiver operating characteristic curve (AUC) as +proposed and used in Li et al. [42]and Wang et al.[43]. + + +13 +MedNLI [44] Natural language inference (NLI) is the task of determining the relationship +between sentence pairs. MedNLI is a collection of sentence pairs extracted from the MIMIC-III +[24] and annotated by two board-certified radiologists. The relationship between the premise +sentence and the hypothesis sentence could be entailment, contradiction, or neutral. Transformer- +based models process NLI tasks also as document classification by merging the sentence pair and +placing a delimiter token right after the end of the first sentence. We follow the original splits as +Romanov et al. [44] and use accuracy to evaluate the performance. +Baseline Models and Comparisons +Both Clinical-Longformer and Clinical-BigBird are compared with the short sequence models, +including BERT, ClinicalBERT, RoBERTa, and BioBERT. We do not include the static word +embedding models, e.g., Word2Vec and FastText, in the comparisons, because those models +yield less competitive performance compared to the transformer-based models and cannot easily +handle token-level classification tasks. The BERT [2] model is the first-of-its-kind transformer +architecture that achieved state-of-the-art results on eleven NLP tasks. Both masked language +modeling and next sentence prediction were used to learn contextualized word representation +from BooksCorpus and English Wikipedia in the pre-training stage. BioBERT [15] is the first +biomedical domain-specific BERT variant pre-trained from PubMed abstracts and PMC full-text +articles. The weights of BioBERT were initialized from BERT. BioBERT yielded optimal +performance in biomedical QA, NER, and relation extraction tasks. ClinicalBERT [16], +initialized from BioBERT, was further pre-trained using clinical notes also extracted from +MIMIC-III. ClinicalBERT boosted the performance for MedNLI and four i2b2 NER tasks that +are also included in our study. BioBERT and ClinicalBERT use the next sentence prediction and +masked language modeling as pre-training strategies. RoBERTa [3] is an improved variant of + + +14 +BERT model, which is trained with larger corpus, bigger batch size and gets rid of the next +sentence prediction objectives. Both Longformer and BigBird initialize their training weights +from RoBERTa checkpoint. We also try hierarchical transformers [45] in the experiment of +MIMIC-AKI. The hierarchical transformer model uses the BERT model to learn outputs from +each small chunk of text. Then, the outputs of small chunks are fed into the recurrent neural +network. Given that the hierarchical transformer model is not explicitly developed for clinical +NLP, we load the weights of ClinicalBERT to initialize the BERT layers. +Experimental setup +For the token-level classification, including QA and NER, a classification head is added to the +output of each token obtained from the transformer-based architecture. The sequences are split +into chunks in the length of 4,096 for Clinical-Longformer and Clinical-BigBird, and 512 for all +the other three baseline models. 1,024 strides are taken between the chunks of long-sequence +models; 128 strides are taken between the chunks of short-sequence models. +For the document classification tasks, the predicted outcomes are derived from the [CLS] token +added to the beginning of each sentence or document. The maximum sequence lengths of OpenI +and MedNLI are less than 512 tokens. Therefore, no truncation or sliding window approaches are +needed for these two datasets. In MIMIC-AKI, given that some clinical notes are extremely long, +which may even exceed the length limits of all models, we first truncate each document to the +first 4096 tokens, which meets the length limits of Clinical-Longformer and Clinical-BigBird. +The predicted outcomes are directly derived from the [CLS] output when using both long- +sequence models. When dealing with short-sequence models, the documents are further +segmented to snippets of 512 tokens in order to accommodate for the length requirement of short +sequence models. A pooling strategy, which was introduced by Huang et al. [10] to predict ICU + + +15 +readmission from discharge summaries, is applied to aggregate the probability outputs from +short snippets. The probability of AKI onset for a patient with 𝑛 short snippets is computed by: +𝑃!"# = +[%&' +!∈# +(!]#*# +$∗,% +# ∑ (! +# +! +. +# +/*# +$ + , where 𝑝0 is the probability output of the 𝑖12 snippet from the short- +sequence model. Our preliminary experiments show that this pooling strategy slightly +outperforms the maximum pooling method. +We conduct our experiments using four 32GB GPUs. We maximize the batch size for each +experiment given the memory limits of GPUs to save training time. The batch size during +training is 16 for Clinical-Longformer, 12 for Clinical-BigBird, and 64 for all other models. +Batch sizes are doubled when evaluating or testing. Half precision is applied to both Clinical- +Longformer and Clinical-BigBird. We try learning rates: {1e-5, 2e-5, and 5e-5} for the +experiments of each model on each task. We fine-tune 6 epochs for each set-up. All experiments +converge within 6 epochs. The best-performing model parameters are determined by the +performance of the development split. The experiments are implemented with python 3.8.0, +PyTorch 1.9.0 and Transformer 4.9.0. The versions and downloadable links for all models can be +found in Table S1. + +Results and Discussion +The evaluation of pre-training can be found in Table 2. The results demonstrate that Clinical- +Longformer and Clinical-Bigbird can learn more useful contextualized relationships from +clinical notes in the pre-training when compared to other baseline models, which provides the +foundation for performance improvement in downstream tasks. BERT, BioBERT, and RoBERTa +which are not pre-trained using clinical notes, yield very poor perplexity scores and MLM + + +16 +accuracies. This confirms that pre-training using domain-specific corpus is essential for learning +the domain-specific contextualized relationships. We also visualize an example in +Supplementary Figure 1. When [Stroke] is replaced with a mask token, Clinical-Longformer can +infer this word from [infarct], [hemorrhagic], [epilepticus], and [hemorrhage], that are more than +1,000 tokens away from the mask token. This example demonstrates that our models can learn +long-term dependencies from clinical narratives. +Full results for QA, NER, and classification tasks are presented in Table 3, 4, and 5, respectively +(for full results with variance measurements, please see Table S2, S3, and S4). In question +answering tasks, both Clinical-Longformer and Clinical-BigBird outperform the short-sequence +transformer models by around 2 percent across all three emrQA subsets when evaluated by F1 +score. When considering the stricter EM metric, Clinical-Longformer, and Clinical-BigBird +improve ~ 5 percent on the relations subset but yield similar results to ClinicalBERT in the other +two subsets. In NER tasks, Clinical-Longformer consistently leads the short-sequence +transformers by more than 2 percent in all 4 i2b2 datasets. Clinical-BigBird also performs better +than ClinicalBERT and BioBERT in all NER experiments. In document classification tasks, our +two long-sequence transformers achieve superior results compared to prior models on OpenI, +MIMIC-AKI, and medNLI tasks. +We observe that Clinical-Longformer and Clinical-BigBird not only improve the performance of +long sequences tasks but also short sequences. The maximum sequences of MedNLI and OpenI +are smaller than 512 tokens, which can be entirely fed into the BERT-like models. However, the +long sequence models still achieve better results. We attribute these improvements to the pre- +training stages of Clinical-Longformer and Clinical-BigBird, where the language models can +learn more long-term dependencies by extending the sequence length limit, thereby learning a + + +17 +richer contextualization of clinical concepts. We find more significant gains, however, when +applying our two long-sequence models to the datasets with longer sequences. For example, the +performance improvement is most dramatic on the i2b2 2014 dataset, which has the largest +average sequence length in all 4 NER tasks (almost twice the other three subsets). Likewise, +Clinical-Longformer more strongly improves the F1 score of the heart disease subset from +emrQA. This suggests that Clinical-Longformer and Clinical-BigBird are also better at modeling +long-term dependencies in downstream tasks. Moreover, in i2b2 2006 dataset, the models +achieve superior results in identifying the PHI information from the clinical notes. However, all +PHI placeholders are completely removed in the pre-processing step of pre-training. This +confirms that the language models can be generalized to new tokens in downstream tasks that are +unseen in pre-training stage. Finally, we also find that Clinical-Longformer yields slightly better +results when compared to Clinical-BigBird, although the differences in most experiments are not +statistically significant. Given that Clinical-BigBird also requires more fine-tuning time and +memory costs, we recommend that future investigators apply our Clinical-Longformer +checkpoint to their own tasks when resources are limited. +Our study has several limitations. Firstly, we only apply Longformer and BigBird to large-scale +clinical corpus. In future iterations, we plan to release more pre-trained models for long +sequences enriched with other biomedical corpora, e.g., PubMed and PMC publications. Also, +we only pre-train the base cased version of Clinical-Longformer and Clinical-BigBird. We will +publish the un-cased and large version at the next step. Secondly, another recent approach +developed to address the memory problem of long sequences is simplifying or compressing the +transformer architecture. In future work, we will compare this genre of transformers, e.g., +TinyBERT [46], to our current long sequence models. Thirdly, we do not integrate Clinical- + + +18 +Longformer or Clinical-BigBird into an encoder-decoder framework due to the memory limits of +our GPU cards. Therefore, experiments on generative tasks like text generation or document +summarization are not included in this study. We intend to incorporate these tasks into future +versions of these models as our computational capability evolves. Fourthly, the emrQA was +annotated in a semiautomatic way without expert calibration. There are incorrect NER labels as +mentioned in Yue et al. [28]. We will conduct the experiments on a large-scale human-annotated +NER dataset should there be any availability. Finally, the vocabularies of Clinical-Longformer +and Clinical-BigBird are inherited from the 5,000 sub-word units used in the RoBERTa [3] +model that was developed for non-clinical corpus. We have no idea if other types of tokenizers +or a clinical-adaptive vocabulary can boost the performance. Therefore, we will examine more +combinations in future studies. +Table 2: The evaluation of transformer-based models after language modeling (LM) pretraining +Pre-trained Models +Perplexity Score +MLM Accuracy +BERT +52807.11 +0.633 +BioBERT +131176.11 +0.001 +ClinicalBERT +8.67 +0.803 +RoBERTa +1378.71 +0.693 +Clinical-Longformer +1.61 +0.940 +Clinical-BigBird +1.41 +0.936 +The best scores are in bold, and the second-best scores are underlined. + +Table 3: The performance of transformer-based pre-trained models on question answering tasks. +Pre-trained Models +emrQA-Medication +emrQA-Relation +emrQA-Heart Disease +metrics +EM +F1 +EM +F1 +EM +F1 +BERT +0.240 +0.675 +0.833 +0.924 +0.650 +0.698 +BioBERT +0.247 +0.700 +0.836 +0.926 +0.647 +0.702 +ClinicalBERT +0.297 +0.698 +0.849 +0.929 +0.666 +0.711 +RoBERTa +0.280 +0.706 +0.825 +0.917 +0.655 +0.682 +Clinical-Longformer +0.302 +0.716 +0.911 +0.948 +0.698 +0.734 +Clinical-BigBird +0.300 +0.715 +0.898 +0.944 +0.664 +0.711 + + +19 +The best scores are in bold, and the second-best scores are underlined. + + +Table 4: The performance of transformer-based pre-trained models on NER tasks. +Pre-trained Models +i2b2 2006 +i2b2 2010 +i2b2 2012 +i2b2 2014 +metrics +F1 +F1 +F1 +F1 +BERT +0.939 +0.835 +0.759 +0.928 +BioBERT +0.948 +0.865 +0.789 +0.930 +ClinicalBERT +0.951 +0.861 +0.773 +0.929 +RoBERTa +0.956 +0.851 +0.767 +0.930 +Clinical-Longformer +0.974 +0.887 +0.800 +0.961 +Clinical-BigBird +0.967 +0.872 +0.787 +0.952 +The best scores are in bold, and the second-best scores are underlined. + +Table 5: The performance of transformer-based models on document classification tasks. +Pre-trained Models +OpenI +MIMIC-AKI +medNLI +metrics Accuracy +AUC +F1 +Accuracy +BERT +0.952 +0.545 +0.296 +0.776 +BioBERT +0.954 +0.717 +0.372 +0.808 +ClinicalBERT +0.967 +0.747 +0.468 +0.812 +RoBERTa +0.963 +0.708 +0.358 +0.808 +Hierarchical Transformer +- +0.726 +0.462 +- +Clinical-Longformer +0.977 +0.762 +0.484 +0.842 +Clinical-BigBird +0.972 +0.755 +0.480 +0.827 +The best scores are in bold, and the second-best scores are underlined. + +Conclusion +In this study, we introduce Clinical-Longformer and Clinical-BigBird, two pre-trained language +models designed specifically for long clinical text NLP tasks. We compare these two models +with the BERT-variant short-sequence transformer-based models, e.g., ClinicalBERT, in named +entity recognition, question answering, and document classification tasks. Results demonstrate +that Clinical-Longformer and Clinical-BigBird achieve better results on both long sequence and + + +20 +short sequence benchmark datasets. Future studies will investigate the generalizability of our +proposed models to clinical text generation and summarization tasks, and the comparison with +other modeling approaches that are also developed to solve the memory consumption of long +text. +Disclosure +All Authors declare no Competing Financial or Non-Financial Interests. +Author Contributions +YLi., RMW, FSA and YLuo conceived of the presented idea. YLi and HW carried out the +experiments. YLi., RMW, FSA, and YLuo contributed to interpreting the results. YLi wrote the +manuscript in consultation with YLuo. All authors provided critical feedback and helped shape +the research, analysis, and manuscript. +Funding +This work was supported by the National Institutes of Health [U01TR003528 and +R01LM013337]. +Data Availability +The benchmark datasets are derived from multiple publicly available datasets, including MIMIC +III from https://physionet.org/content/mimiciii/1.4/; MIMIC-CXR from +https://physionet.org/content/mimic-cxr/2.0.0/; i2b2 from https://portal.dbmi.hms.harvard.edu/; +openI from https://openi.nlm.nih.gov/; and MedNLI from +https://physionet.org/content/mednli/1.0.0/. To officially gain access, the authors should apply +and sign data user agreement with the data owner. We provide codes to pre-process and generate +splits at: https://github.com/luoyuanlab/Clinical-Longformer. + + +21 +Reference +1. +Brown, T., et al., Language models are few-shot learners. 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Shivade, Lessons from natural language inference in the clinical domain. +arXiv preprint arXiv:1808.06752, 2018. +45. +Pappagari, R., et al. Hierarchical transformers for long document classification. in 2019 IEEE +Automatic Speech Recognition and Understanding Workshop (ASRU). 2019. IEEE. +46. +Jiao, X., et al., Tinybert: Distilling bert for natural language understanding. arXiv preprint +arXiv:1909.10351, 2019. + + + + + + diff --git a/StFKT4oBgHgl3EQfjy7A/content/tmp_files/load_file.txt b/StFKT4oBgHgl3EQfjy7A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dce6dba45b6d457c8704daed8fabaa8c78a5e75e --- /dev/null +++ b/StFKT4oBgHgl3EQfjy7A/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf,len=773 +page_content='1 A Comparative Study of Pretrained Language Models for Long Clinical Text Yikuan Li 1, Ramsey M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Wehbe, MD 2,3, Faraz S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Ahmad, MD 1,2,3, Hanyin Wang 1, Yuan Luo, PhD 1 * Emails: yikuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='li@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu ramsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='wehbe@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu faraz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='ahmad@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu hanyin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='wang@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu yuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='luo@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu 1 Division of Health and Biomedical Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Department of Preventive Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Feinberg School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' USA 2 Division of Cardiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Department of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Feinberg School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' USA 3 Bluhm Cardiovascular Institute Center for Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Northwestern Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' USA Corresponding author Keywords: Clinical Natural Language Processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Text Classification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Question Answering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Natural Language Inference Word Count = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='320 2 Abstract Objective: Clinical knowledge enriched transformer models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', ClinicalBERT) have state-of- the-art results on clinical NLP (natural language processing) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' One of the core limitations of these transformer models is the substantial memory consumption due to their full self-attention mechanism, which leads to the performance degradation in long clinical texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' To overcome this, we propose to leverage long-sequence transformer models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', Longformer and BigBird), which extend the maximum input sequence length from 512 to 4096, to enhance the ability to model long-term dependencies in long clinical texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Materials and Methods: Inspired by the success of long sequence transformer models and the fact that clinical notes are mostly long, we introduce two domain enriched language models, Clinical-Longformer and Clinical-BigBird, which are pre-trained on a large-scale clinical corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We evaluate both language models using 10 baseline tasks including named entity recognition, question answering, natural language inference, and document classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Results: The results demonstrate that Clinical-Longformer and Clinical-BigBird consistently and significantly outperform ClinicalBERT and other short-sequence transformers in all 10 downstream tasks and achieve new state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Discussion: Our pre-trained language models provide the bedrock for clinical NLP using long texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We have made our source code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='com/luoyuanlab/Clinical- Longformer, and the pre-trained models available for public download at: https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='co/yikuan8/Clinical-Longformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Conclusion: This study demonstrates that clinical knowledge enriched long-sequence transformers are able to learn long-term dependencies in long clinical text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Our methods can also inspire the development of other domain-enriched long-sequence transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 3 Introduction Transformer-based models have been wildly successful in setting state-of-the-art benchmarks on a broad range of natural language processing (NLP) tasks, including question answering, document classification, machine translation, text summarization, and others [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' These successes have been replicated in the clinical and biomedical domain via pre-training language models using large-scale clinical or biomedical corpora, then fine-tuning on a variety of clinical or biomedical downstream tasks, including computational phenotyping [4], automatical ICD coding [5], knowledge graph completion [6] and clinical question answering [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The self-attention mechanism [8] is one of the most critical components that lead to the success of transformer-based models, which allows each token in the input sequence to independently interact with every other token in the sequence in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, the memory consumption of the self-attention mechanism grows quadratically with sequence length, resulting in impracticable training time, and easily reaching the memory limits of modern GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Consequently, transformer-based models that leverage a complete self-attention mechanism, such as BERT and RoBERTa, typically have an input sequence length limit of 512 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' To deal with this limit when modeling long texts using transformer-based models, the input sequence shall be either truncated to the first 512 tokens or processed via a sliding window of 512 tokens with or without overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' If the latter method is applied to a document-level classification task, an aggregation operation will be added to yield the final output from multiple snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both methods ignore long-term dependencies spanning over 512 tokens and may achieve suboptimal results due to information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Additionally, this input token limitation of the self-attention mechanism could impact language model pre-training and then be amplified to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In clinical NLP, transformer-based modeling approaches have also 4 encountered this limitation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' For example, the discharge summaries in MIMIC-III, which are often used to predict clinically meaningful events like hospital re-admission [10] or mortality [11], have 2,984 tokens (1,435 words) on average, far exceeding the 512 token limit of most full attention-based transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Recently, investigators have developed novel variants of transformers specifically for long sequences that reduce memory usage from quadratic to linear scale of the sequence length [12- 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The core idea behind these models is to replace the full attention mechanism with a sparse attention mechanism, which is typically a blend of sliding windows and reduced global attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' These models are capable of processing up to 4,096 tokens and have empirically boosted performance on NLP tasks, including question answering as well as text summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, to the best of our knowledge, long sequence transformers in the clinical and biomedical domain have not yet been systematically explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The purpose of this manuscript is to examine the adaptability of these long sequence models to a series of clinical NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We make the following contributions: We leverage large-scale clinical notes to pre-train two new language models, namely Clinical-Longformer and Clinical-BigBird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We demonstrate that both Clinical-Longformer and Clinical-BigBird improve the performance of a variety of downstream clinical NLP datasets, including question answering, named entity recognition, and document classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Background and Significance Clinical and Biomedical Transformers 5 Transformer-based models, especially BERT [2], can be enriched with clinical and biomedical knowledge through pre-training on large-scale clinical and biomedical corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' These domain- enriched models, for example, BioBERT [15] pre-trained on biomedical publications and ClinicalBERT [16] pre-trained on clinical narratives, set state-of-the-art benchmarks on downstream clinical and biomedical NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Inspired by the success of these domain- enriched models, more pre-trained models were released to boost the performance of NLP models when applied to specific clinical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' For example, Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [17] proposed CheXbert to annotate thoracic disease findings from radiology reports, which outperformed previous rule-based labelers with statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The model was pre-trained using a combination of human-annotated and machine-annotated radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [18] introduced DiseaseBERT, which infused disease knowledge to the BERT model by pre-training on a series of disease description passages that were constructed from Wikipedia and MeSH terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' DiseaseBERT achieved superior results on consumer health question answering tasks compared with BERT and ClinicalBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Michalopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [19] proposed UmlsBERT, which used the Unified Medical Language System (UMLS) Metathesaurus to augment the domain knowledge learning ability of ClinicalBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [20] developed CancerBERT to extract breast cancer-related concepts from clinical notes and pathology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [21] leveraged order contrastive pre-training on longitudinal data to tackle the difficulty when only a small proportion of the clinical notes were annoated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, all models mentioned above were built on the vanilla BERT architecture, which has a limitation of 512 tokens in the input sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This limitation may result in the information loss of long-term dependencies in the training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Transformers for Long Sequences 6 Various attention mechanisms have been proposed to handle the large memory consumption of the attention operations in the vanilla transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Transformer-XL [22] segmented a long sequence into multiple small chunks and then learned long-term dependencies with a left- to-right segment-level recurrence mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Transformer-XL learns 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='5 times longer dependencies than the vanilla transformer models but loses the advantage of bidirectional representation of BERT-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In another study, Reformer [23] applied two techniques to reduce the complexity of transformer architecture by replacing dot-product attention operation with locality-sensitive hashing and sharing the activation function among layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Reformer was able to process longer sequences at a faster speed and be more memory efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, this enhancement improves space, time, and memory efficiency, but not accuracy on specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Almost simultaneously, Longformer [13] and BigBird [14] were proposed to drastically alleviate the memory consumption of transformer models by replacing the pairwise full attention mechanisms with a combination of sliding window attention and global attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' They are slightly different regarding implementation and configuration of the global and local attention mechanism, where BigBird introduces additional contrastive predictive coding to train global tokens [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both models support input sequences up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='096 tokens long (8 times the input sequence limit of BERT) and significantly improve performance on long-text question answering and summarization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, the adaptability of these long sequence transformers to the clinical and biomedical fields, where document length mostly exceeds the limits of BERT-like models, has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Materials and Methods In this section, we first introduce the clinical dataset we use as the pre-training corpus, followed by the pre-training processes for Clinical-Longformer and Clinical-BigBird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Next, we enumerate 7 the downstream tasks we use to compare our long sequence models with the short sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We also provide the technical details of pre-training and fine-tuning for the purposes of reproducing our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The entire pipeline can be found in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Figure 1: The pipeline for pre-training and fine-tuning transformer-based language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Datasets Similar to Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [10] and Alsentzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [16], we leverage approximately 2 million clinical notes extracted from the MIMIC-III [24] dataset, which is the most extensive publicly available electronic health records (EHR) dataset that contains clinical narratives of over 40,000 patients admitted to the intensive care units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We only apply minimal pre-processing steps, including 1) to remove all de-identification placeholders from the clinical notes that were generated to protect the PHI (protected health information);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 2) to remove all characters other than alphanumerics and punctuation marks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 3) to convert all alphabetical characters to lower cases, and 4) to strip extra white spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We believe that complicated pre-processing in the pre- training stage may not improve downstream performance but will sacrifice the generalizability of language models and significantly increase training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Pre-training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='95 1 Longformer BigBird BERT MIMIC III Clinical Notes English Wiki PubMed MIMIC III Clinical- Longformer Clinical- BigBird BERT BioBERT ClinicalBERT Transformer Models Pre-training Corpora Knowledge-enriched Transformers Long-sequence Models Short-sequence Models medNLI, an inference task, eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' by accuracy emrQA (relation subset), a QA task, eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' by EM score i2b2 - 2014, an NER task, eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' by F1 score Downstream Clinical NLP Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='95 1 8 Longformer [13] and BigBird [14] are the two best-performing transformer models that are designed for long input sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both models extend the maximum input sequence length to 4,096 tokens, which is 8× the limit of conventional transformer-based models, by introducing localized sliding windows and global attention mechanisms to reduce the computational expenses of full self-attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The differences between the two models are how the global attention is realized and the selection of loss function in fine-tuning [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' BigBird also contains some random localized attention operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The reported performance difference between the two models on downstream tasks is minimal [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Therefore, we seek to pre-train both models and compare their performance on clinical NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We refer readers to the original papers of Longformer [13] and BigBird [14] for more technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We initialize Clinical-Longformer and Clinical-BigBird from the pre-trained weights of the base version of Longformer and the ITC (internal transformer construction) version of BigBird, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Although the ETC (extended transformer construction) version of BigBird may have superior performance, HuggingFace (the largest community for sharing open-source pre- trained transformer models) only provides the implementation and the pre-trained checkpoints of the ITC version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The difference between ITC and ETC version is that in ITC version some existing tokens are made “global” and attend over the entire sequence, while ETC version introduces additionally “global” tokens such as CLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Byte-level BPE (Byte-Pair-Encoding) [25] is applied to tokenize the clinical corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both models are distributed in parallel to six 32GB Tesla V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' FP16 precision is enabled to accelerate training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Batch size is 18 for Clinical- Longformer and 12 for Clinical-Bigbird, which are the upper limits under 6 32GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We pre-train Clinical-Longformer for 200,000 steps and Clinical-BigBird for 300,000 steps, which ensures that each clinical note is seen equal times by the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The learning rates are 3e-5 9 for both models, the same as the learning rate used in the pretraining of Longformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The entire pre-training process takes more than two weeks for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' To evaluate the performance of pre-training, we create a testing set that contains 1,000 documents that are also from MIMIC-III but have not been used as the pre-training corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Each document in the testing set is truncated to 512 tokens long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We randomly select 10% tokens from each document and replace them with a mask token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We compare our two pre-trained models with the short-sequence models in filling in the masked tokens using context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We report the perplexity score and top 5 accuracy in filling in the masked tokens of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Downstream Tasks In this study, we fine-tune the pre-trained Clinical-Longformer and Clinical-BigBird on 10 clinical NLP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' These 10 NLP datasets broadly cover various NLP tasks, including extractive question answering, named entity recognition, natural language inference, and document classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We rely on these NLP tasks to validate the performance improvement of long sequence models compared to their short sequence counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The statistics and descriptions of all datasets can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Question Answering Question answering (QA) is a common NLP task that aims to automatically answer questions asked in natural language [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=" In the clinical context, QA systems answer clinicians' questions by understanding the clinical narratives extracted from electronic health record 10 Table 1: Description and statistics of downstream clinical NLP tasks Dataset Task Source Sample Size Avg." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Length Max Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Length MedNLI Inference MIMIC 14,049 39 409 i2b2 2006 NER i2b2 66,034 867 3,986 i2b2 2010 NER i2b2 43,947 1,459 6,052 i2b2 2012 NER i2b2 13,108 794 2,900 i2b2 2014 NER i2b2 83,466 5,134 14,370 emrQA-Relation QA i2b2 255,908 1,880 6,109 emrQA-Medication QA i2b2 141,243 1,460 6,050 emrQA-HeartDisease QA i2b2 30,731 5,293 14,060 openI Multilabel Classif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' IndianaU 3,684 70 294 MIMIC-CXR Multilabel Classif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' MIMIC- CXR 222,713 119 874 MIMIC-AKI Binary Classif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' MIMIC 16,536 1,463 20,857 systems to support decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' emrQA [27] is the most frequently used benchmark dataset in clinical QA, which contains more than 400,000 question-answer pairs semi-automatically generated from past Informatics for Integrating Biology and the Bedside (i2b2) challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' emrQA falls into the category of extractive question answering, aiming to identify answer spans from reference texts instead of generating new answers in a word-by-word fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Researchers have attempted to solve emrQA tasks by using word embedding models [28], conditional random fields (CRFs) [29] and transformer-based models [30], among which transformer-based models performed best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In our experiments, we investigate the performance of our pre-trained models using the three largest emrQA subsets: Medication, Relation, and Heart Disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We evaluate QA performance with two commonly used metrics: exact match (EM) and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Exact match evaluates whether entire predicted spans match exactly with the ground-truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' F1- score is a looser metric derived from token-level precision and recall, which measures the 11 overlap between the predictions and the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We generate train-dev-test splits by following the instruction of Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The training set of relation and medication subsets are randomly under-sampled to reduce training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Based on their experience, performance was not compromised by under-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Of note, the emrQA dataset has some known issues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', incomplete answers, it is template-based, and the annotation were generated semi- automatically[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We consider the usage of emrQA as a proof-of-concept experiment to compare the performance of the transformer-based model on the QA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Named Entity Recognition Named entity recognition is a token-level classification task that seeks to identify the named entities and classify them in predefined categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This genre of NLP tasks has broad applications in the clinical and biomedical domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', de-identification of PHI and medical concept extraction from clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Prior studies have shown that transformer-based models [15] significantly outperformed the models built on pre-trained static word embeddings [31] or LSTM networks [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We compare our pre-trained models using four data challenges: 1) i2b2 2006 [33] to de-identify PHI from medical discharge notes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 2) i2b2 2010 [34] to extract and annotate medical concepts from patient reports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=" 3) i2b2 2012 [35] to identify both clinical concepts and events relevant to the patient's clinical timeline from discharge summaries, and 4) i2b2 2014 [36] to identify PHI information from longitudinal clinical narratives." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We follow the processing steps of Alsentzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [16], which converts the raw data from all four tasks to the IOB (inside–outside–beginning) tagging format proposed by Ramshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [37], and then create train-dev-test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We evaluate the model performance with F1 score similarly to QA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Document Classification 12 Document classification is one of the most common NLP tasks, where a sentence or document is assigned to one or more classes or categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In the clinical domain, document classification can be used to identify the onset of a particular disease process or predict patient prognosis using entire clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We use the following three document classification datasets to evaluate the pre-trained models from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' MIMIC-AKI [38, 39] MIMIC-AKI is a binary classification task, where we aim to predict the possibility of AKI (acute kidney injury) for critically ill patients using the clinical notes within the first 24 hours following intensive care unit (ICU) admission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We follow Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [38] to extract the cohort from MIMIC-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We evaluate the model performance using AUC and F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' OpenI [40] OpenI is a publicly available chest x-ray (CXR) dataset collected by Indiana University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The dataset provides around 4,000 radiology reports and their associated human annotated Medical Subject Headings (MeSH) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In our experiments, the task is to detect the presence of the annotated thoracic findings from CXR reports, which is considered a multi-label classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Given the small sample size, we will only use OpenI as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The pre-trained language models are fine-tuned using MIMIC-CXR [41], another publicly available chest x-ray dataset that contains more than 200,000 CXR reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Unlike openI, the ground-truth labels for MIMIC-CXR were automatically generated using NLP approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The overlapping findings between the two CXR data sources are:: Cardiomegaly, Edema, Consolidation, Pneumonia, Atelectasis, Pneumothorax and Pleural Effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We report the sample number weighted average of the area under the receiver operating characteristic curve (AUC) as proposed and used in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [42]and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 13 MedNLI [44] Natural language inference (NLI) is the task of determining the relationship between sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' MedNLI is a collection of sentence pairs extracted from the MIMIC-III [24] and annotated by two board-certified radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The relationship between the premise sentence and the hypothesis sentence could be entailment, contradiction, or neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Transformer- based models process NLI tasks also as document classification by merging the sentence pair and placing a delimiter token right after the end of the first sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We follow the original splits as Romanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [44] and use accuracy to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Baseline Models and Comparisons Both Clinical-Longformer and Clinical-BigBird are compared with the short sequence models, including BERT, ClinicalBERT, RoBERTa, and BioBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We do not include the static word embedding models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', Word2Vec and FastText, in the comparisons, because those models yield less competitive performance compared to the transformer-based models and cannot easily handle token-level classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The BERT [2] model is the first-of-its-kind transformer architecture that achieved state-of-the-art results on eleven NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both masked language modeling and next sentence prediction were used to learn contextualized word representation from BooksCorpus and English Wikipedia in the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' BioBERT [15] is the first biomedical domain-specific BERT variant pre-trained from PubMed abstracts and PMC full-text articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The weights of BioBERT were initialized from BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' BioBERT yielded optimal performance in biomedical QA, NER, and relation extraction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' ClinicalBERT [16], initialized from BioBERT, was further pre-trained using clinical notes also extracted from MIMIC-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' ClinicalBERT boosted the performance for MedNLI and four i2b2 NER tasks that are also included in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' BioBERT and ClinicalBERT use the next sentence prediction and masked language modeling as pre-training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' RoBERTa [3] is an improved variant of 14 BERT model, which is trained with larger corpus, bigger batch size and gets rid of the next sentence prediction objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Both Longformer and BigBird initialize their training weights from RoBERTa checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We also try hierarchical transformers [45] in the experiment of MIMIC-AKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The hierarchical transformer model uses the BERT model to learn outputs from each small chunk of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Then, the outputs of small chunks are fed into the recurrent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Given that the hierarchical transformer model is not explicitly developed for clinical NLP, we load the weights of ClinicalBERT to initialize the BERT layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Experimental setup For the token-level classification, including QA and NER, a classification head is added to the output of each token obtained from the transformer-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The sequences are split into chunks in the length of 4,096 for Clinical-Longformer and Clinical-BigBird, and 512 for all the other three baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 1,024 strides are taken between the chunks of long-sequence models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 128 strides are taken between the chunks of short-sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' For the document classification tasks, the predicted outcomes are derived from the [CLS] token added to the beginning of each sentence or document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The maximum sequence lengths of OpenI and MedNLI are less than 512 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Therefore, no truncation or sliding window approaches are needed for these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In MIMIC-AKI, given that some clinical notes are extremely long, which may even exceed the length limits of all models, we first truncate each document to the first 4096 tokens, which meets the length limits of Clinical-Longformer and Clinical-BigBird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The predicted outcomes are directly derived from the [CLS] output when using both long- sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' When dealing with short-sequence models, the documents are further segmented to snippets of 512 tokens in order to accommodate for the length requirement of short sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' A pooling strategy, which was introduced by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [10] to predict ICU 15 readmission from discharge summaries, is applied to aggregate the probability outputs from short snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The probability of AKI onset for a patient with 𝑛 short snippets is computed by: 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' "# = [%&\' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='∈# (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' ]#*# $∗,% # ∑ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' # !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' # /*# $ , where 𝑝0 is the probability output of the 𝑖12 snippet from the short- sequence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Our preliminary experiments show that this pooling strategy slightly outperforms the maximum pooling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We conduct our experiments using four 32GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We maximize the batch size for each experiment given the memory limits of GPUs to save training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The batch size during training is 16 for Clinical-Longformer, 12 for Clinical-BigBird, and 64 for all other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Batch sizes are doubled when evaluating or testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Half precision is applied to both Clinical- Longformer and Clinical-BigBird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We try learning rates: {1e-5, 2e-5, and 5e-5} for the experiments of each model on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We fine-tune 6 epochs for each set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' All experiments converge within 6 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The best-performing model parameters are determined by the performance of the development split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The experiments are implemented with python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0, PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0 and Transformer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The versions and downloadable links for all models can be found in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Results and Discussion The evaluation of pre-training can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The results demonstrate that Clinical- Longformer and Clinical-Bigbird can learn more useful contextualized relationships from clinical notes in the pre-training when compared to other baseline models, which provides the foundation for performance improvement in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' BERT, BioBERT, and RoBERTa which are not pre-trained using clinical notes, yield very poor perplexity scores and MLM 16 accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This confirms that pre-training using domain-specific corpus is essential for learning the domain-specific contextualized relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We also visualize an example in Supplementary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' When [Stroke] is replaced with a mask token, Clinical-Longformer can infer this word from [infarct], [hemorrhagic], [epilepticus], and [hemorrhage], that are more than 1,000 tokens away from the mask token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This example demonstrates that our models can learn long-term dependencies from clinical narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Full results for QA, NER, and classification tasks are presented in Table 3, 4, and 5, respectively (for full results with variance measurements, please see Table S2, S3, and S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In question answering tasks, both Clinical-Longformer and Clinical-BigBird outperform the short-sequence transformer models by around 2 percent across all three emrQA subsets when evaluated by F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' When considering the stricter EM metric, Clinical-Longformer, and Clinical-BigBird improve ~ 5 percent on the relations subset but yield similar results to ClinicalBERT in the other two subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In NER tasks, Clinical-Longformer consistently leads the short-sequence transformers by more than 2 percent in all 4 i2b2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Clinical-BigBird also performs better than ClinicalBERT and BioBERT in all NER experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In document classification tasks, our two long-sequence transformers achieve superior results compared to prior models on OpenI, MIMIC-AKI, and medNLI tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We observe that Clinical-Longformer and Clinical-BigBird not only improve the performance of long sequences tasks but also short sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' The maximum sequences of MedNLI and OpenI are smaller than 512 tokens, which can be entirely fed into the BERT-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, the long sequence models still achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We attribute these improvements to the pre- training stages of Clinical-Longformer and Clinical-BigBird, where the language models can learn more long-term dependencies by extending the sequence length limit, thereby learning a 17 richer contextualization of clinical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We find more significant gains, however, when applying our two long-sequence models to the datasets with longer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' For example, the performance improvement is most dramatic on the i2b2 2014 dataset, which has the largest average sequence length in all 4 NER tasks (almost twice the other three subsets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Likewise, Clinical-Longformer more strongly improves the F1 score of the heart disease subset from emrQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This suggests that Clinical-Longformer and Clinical-BigBird are also better at modeling long-term dependencies in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Moreover, in i2b2 2006 dataset, the models achieve superior results in identifying the PHI information from the clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' However, all PHI placeholders are completely removed in the pre-processing step of pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' This confirms that the language models can be generalized to new tokens in downstream tasks that are unseen in pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Finally, we also find that Clinical-Longformer yields slightly better results when compared to Clinical-BigBird, although the differences in most experiments are not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Given that Clinical-BigBird also requires more fine-tuning time and memory costs, we recommend that future investigators apply our Clinical-Longformer checkpoint to their own tasks when resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Our study has several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Firstly, we only apply Longformer and BigBird to large-scale clinical corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In future iterations, we plan to release more pre-trained models for long sequences enriched with other biomedical corpora, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', PubMed and PMC publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Also, we only pre-train the base cased version of Clinical-Longformer and Clinical-BigBird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We will publish the un-cased and large version at the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Secondly, another recent approach developed to address the memory problem of long sequences is simplifying or compressing the transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' In future work, we will compare this genre of transformers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', TinyBERT [46], to our current long sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Thirdly, we do not integrate Clinical- 18 Longformer or Clinical-BigBird into an encoder-decoder framework due to the memory limits of our GPU cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Therefore, experiments on generative tasks like text generation or document summarization are not included in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We intend to incorporate these tasks into future versions of these models as our computational capability evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Fourthly, the emrQA was annotated in a semiautomatic way without expert calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' There are incorrect NER labels as mentioned in Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We will conduct the experiments on a large-scale human-annotated NER dataset should there be any availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Finally, the vocabularies of Clinical-Longformer and Clinical-BigBird are inherited from the 5,000 sub-word units used in the RoBERTa [3] model that was developed for non-clinical corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We have no idea if other types of tokenizers or a clinical-adaptive vocabulary can boost the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Therefore, we will examine more combinations in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Table 2: The evaluation of transformer-based models after language modeling (LM) pretraining Pre-trained Models Perplexity Score MLM Accuracy BERT 52807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='633 BioBERT 131176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='001 ClinicalBERT 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='803 RoBERTa 1378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='693 Clinical-Longformer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='940 Clinical-BigBird 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='936 The best scores are in bold, and the second-best scores are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Table 3: The performance of transformer-based pre-trained models on question answering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Pre-trained Models emrQA-Medication emrQA-Relation emrQA-Heart Disease metrics EM F1 EM F1 EM F1 BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='698 BioBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='647 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='702 ClinicalBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='711 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='682 Clinical-Longformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='734 Clinical-BigBird 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='711 19 The best scores are in bold, and the second-best scores are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Table 4: The performance of transformer-based pre-trained models on NER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Pre-trained Models i2b2 2006 i2b2 2010 i2b2 2012 i2b2 2014 metrics F1 F1 F1 F1 BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='928 BioBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='930 ClinicalBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='929 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='930 Clinical-Longformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='961 Clinical-BigBird 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='952 The best scores are in bold, and the second-best scores are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Table 5: The performance of transformer-based models on document classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Pre-trained Models OpenI MIMIC-AKI medNLI metrics Accuracy AUC F1 Accuracy BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='776 BioBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='808 ClinicalBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='812 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='808 Hierarchical Transformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='462 Clinical-Longformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='842 Clinical-BigBird 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='827 The best scores are in bold, and the second-best scores are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Conclusion In this study, we introduce Clinical-Longformer and Clinical-BigBird, two pre-trained language models designed specifically for long clinical text NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We compare these two models with the BERT-variant short-sequence transformer-based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', ClinicalBERT, in named entity recognition, question answering, and document classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Results demonstrate that Clinical-Longformer and Clinical-BigBird achieve better results on both long sequence and 20 short sequence benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Future studies will investigate the generalizability of our proposed models to clinical text generation and summarization tasks, and the comparison with other modeling approaches that are also developed to solve the memory consumption of long text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Disclosure All Authors declare no Competing Financial or Non-Financial Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Author Contributions YLi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', RMW, FSA and YLuo conceived of the presented idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' YLi and HW carried out the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' YLi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=', RMW, FSA, and YLuo contributed to interpreting the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' YLi wrote the manuscript in consultation with YLuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' All authors provided critical feedback and helped shape the research, analysis, and manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Funding This work was supported by the National Institutes of Health [U01TR003528 and R01LM013337].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Data Availability The benchmark datasets are derived from multiple publicly available datasets, including MIMIC III from https://physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='org/content/mimiciii/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='4/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' MIMIC-CXR from https://physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='org/content/mimic-cxr/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' i2b2 from https://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='dbmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='hms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='edu/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' openI from https://openi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='gov/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' and MedNLI from https://physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='org/content/mednli/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' To officially gain access, the authors should apply and sign data user agreement with the data owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' We provide codes to pre-process and generate splits at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='com/luoyuanlab/Clinical-Longformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' 21 Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content=' Brown, T.' metadata={'source': 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+page_content=' arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} +page_content='10351, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFKT4oBgHgl3EQfjy7A/content/2301.11847v1.pdf'} diff --git a/T9AyT4oBgHgl3EQfufnB/content/tmp_files/2301.00615v1.pdf.txt b/T9AyT4oBgHgl3EQfufnB/content/tmp_files/2301.00615v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e0c9222f249b5e243a89e16ff8f0ce811aff6d8 --- /dev/null +++ b/T9AyT4oBgHgl3EQfufnB/content/tmp_files/2301.00615v1.pdf.txt @@ -0,0 +1,3892 @@ +ChameleMon: Shifting Measurement Attention as Network +State Changes +Kaicheng Yang†, Yuhan Wu†, Ruijie Miao†, Tong Yang†, Zirui Liu†, +Zicang Xu†, Rui Qiu†, Yikai Zhao†, Hanglong Lv†, Zhigang Ji§, Gaogang Xie¶ +†Department of Computer Science, Peking University, China +§Huawei Technologies Co., Ltd, China +¶CNIC CAS; UCAS, China +Abstract +Flow-level network measurement is critical to many network +applications. Among various measurement tasks, packet loss +detection and heavy-hitter detection are two most important +measurement tasks, which we call the two key tasks. In prac- +tice, the two key tasks are often required at the same time, +but existing works seldom handle both tasks. In this paper, +we design ChameleMon to support the two key tasks simulta- +neously. One key design/novelty of ChameleMon is to shift +measurement attention as network state changes, through two +dimensions of dynamics: 1) dynamically allocating memory +between the two key tasks; 2) dynamically monitoring the +flows of importance. To realize the key design, we propose a +key technique, leveraging Fermat’s little theorem to devise a +flexible data structure, namely FermatSketch. FermatSketch +is dividable, additive, and subtractive, supporting the two key +tasks. We have fully implemented a ChameleMon prototype +on a testbed with a Fat-tree topology. We conduct extensive +experiments and the results show ChameleMon supports the +two key tasks with low memory/bandwidth overhead, and +more importantly, it can automatically shift measurement at- +tention as network state changes. +1 +Introduction +Network measurement provides critical statistics for various +network operations, such as traffic engineering [1,2], conges- +tion control [3], network accounting and billing [4], anomaly +detection [5–8], and failure troubleshooting [9,10]. Recently, +the emerging programmable switches inject new vitality into +network measurement, and designing novel sketches for flow- +level measurement capabilities on programmable switches +becomes a hot topic [11–16]. +Among all flow-level measurement tasks, packet loss de- +tection and heavy-hitter detection attract the most attention, +which we call two key tasks. However, the two key tasks +are seldom considered and supported simultaneously in one +solution. One reason behind is that solutions designed for +packet loss detection and solutions designed for heavy-hitter +detection usually differ significantly in data structures and +operations. +However, in practice, the two key tasks are often required +at the same time, and there are only limited resources for +measurement in programmable switches. Therefore, the first +requirement for a practical measurement system is versatile +to support the two key tasks with high accuracy using limited +resources, where limited resources refer to sub-linear space +complexity. +Based on the first requirement, the second requirement is +to pay attention to different key tasks for different network +states. When the network state is healthy and there are only +few packet losses in the network, the system should pay more +attention (e.g., allocate more memory) to heavy-hitter detec- +tion. When the network state is ill and there are lots of packet +losses in the network, the system should pay more attention +to packet loss detection, especially for those flows which ex- +perience a great number of packet losses. +In summary, it is desired to design a practical measurement +system with the following requirements: [R1.1] versatility +requirement: supporting both packet loss detection and heavy- +hitter detection simultaneously; [R1.2] efficiency requirement: +achieving high accuracy with sub-linear space complexity; +[R2] attention requirement: paying attention to different key +tasks for different network states. +Existing solutions can be mainly classified into three cate- +gories according to supported measurement tasks: +• 1) Solutions for packet loss detection: Typical solutions +including LossRadar [17] based on Invertible Bloom fil- +ter [18], NetSight [9] based on in-band network telemetry +(INT), Netseer [19] based on the advanced features of pro- +grammable switches, and more. These solutions fail to meet +[R1.1] because they do not handle heavy-hitter detection. +• 2) Solutions for heavy-hitter detection: These solutions +are usually based on sketches, including HashPipe [20], +UnivMon [13], ElasticSketch [16], BeauCoup [12], and +more. These solutions fail to meet [R1.1] because they do +not handle heavy-hitter detection. +1 +arXiv:2301.00615v1 [cs.NI] 2 Jan 2023 + +• 3) Solutions for both tasks: These solutions support +both tasks by recording exact IDs of all flows, including +FlowRadar [21], OmniMon [22], Counter Braids [23], and +more. However, recording exact IDs of all flows requires +memory/bandwidth overhead linear with the number of +flows. Therefore, these solutions fail to meet [R1.2]. +In summary, the first two categories of solutions cannot +meet [R1.1] due to their limited measurement capabilities, +and the third category of solutions cannot meet [R1.2] due +to their linear space complexities. A naive solution meeting +both [R1.1] and [R1.2] is to combine the first two categories +of solutions: choosing one solution in each category to de- +tect heavy-hitters (HHs for short) and packet losses, respec- +tively. However, such a combination fails to achieve [R2] on +programmable switches. The reason behind is that the data +structures and operations of different categories of solutions +usually differ significantly1, and therefore the solutions in +different categories can only utilize their resources allocated +at compile time, which prohibits flexible allocation of mem- +ory resources between packet loss detection and heavy-hitter +detection. Therefore, the naive solution cannot pay attention +to different key tasks for different network states. +In this paper, we design ChameleMon, which meets all +the above requirements simultaneously. Besides the two key +tasks, i.e., packet loss detection and heavy-hitter detection, +ChameleMon can support five common measurement tasks. +Compared to the state-of-the-art solutions, for packet loss +detection, ChameleMon reduces the memory overhead from +proportional to the number of all flows (FlowRadar) or lost +packets (LossRadar), to proportional to the number of flows +experiencing packet losses, which we call victim flows; for +heavy-hitter detection and the other five tasks, ChameleMon +achieves at least comparable accuracy. Our ChameleMon has +a key design and a key technique as follows. +The key design of ChameleMon is to shift measurement at- +tention as network state changes, which is just like the process +of the chameleons changing their skin coloration, through two +dimensions of dynamics: 1) dynamically allocating memory +between the two key tasks; 2) dynamically monitoring the +flows of importance. First, ChameleMon monitors the net- +work state and allocates memory between the two key tasks +accordingly. When the network state is healthy and only a few +packet losses occur in the network, ChameleMon pays most +attention to and allocates most of the memory for heavy-hitter +detection. As the network state degrades and packet losses +increase, ChameleMon gradually shifts measurement atten- +tion to and allocates more and more memory for packet loss +detection. Second, ChameleMon ranks the flows according +to their importance, and selects those of most importance to +monitor. When the network state is ill and there are too many +victim flows in the network, ChameleMon selects those flows +1For example, LossRadar [17] records the IDs and existences of packets +using XOR operation and addition, while HashPipe [20] records the IDs and +sizes of flows using comparison, substitution, and addition. +experiencing many packet losses (called heavy-losses, HLs +for short) to monitor, instead of monitoring all victim flows. +Overall, when the network state continuously degrades +from the healthy state to the ill state, ChameleMon runs as +follows. 1) As the number of victim flows increases, Chamele- +Mon leverages the first dimension of dynamic: gradually shift- +ing measurement attention to and allocating more and more +memory for packet loss detection; 2) When the victim flows +are too many to monitor, ChameleMon leverages the second +dimension of dynamic: focusing measurement attention on +HLs while monitoring a small portion of other packet losses +(named light-losses, LLs for short) through sampling. +To realize the key design, ChameleMon incorporates a key +technique, leveraging Fermat’s little theorem2 to devise a flex- +ible data structure, namely FermatSketch. The data structure +of FermatSketch is made of many same units. FermatSketch is +dividable, additive, and subtractive, supporting packet loss de- +tection and heavy hitter detection simultaneously. By dividing +FermatSketch into three parts to detect HLs, LLs, and HHs, +ChameleMon can flexibly move the division points to shift +attention and allocate memory between the two key tasks as +network state changes. For each incoming packet, We further +use a flow classifier (TowerSketch [24]) to determine which +of the three parts to insert. For packet loss detection, owing to +Fermat’s little theorem, FermatSketch only requires memory +proportional to the number of victim flows. Differently, the +state-of-the-art solutions require memory proportional to the +number of all flows (FlowRadar) or lost packets (LossRadar). +Thanks to the visibility to per-flow size provided by Tow- +ersketch, ChameleMon can support five other common mea- +surement tasks, including flow size estimation, heavy-change +detection, flow size distribution estimation, entropy estima- +tion, and cardinality estimation. We have fully implemented a +ChameleMon prototype on a testbed with a Fat-tree topology +composed of 10 Tofino switches and 8 end-hosts. We conduct +extensive experiments and the results show that ChameleMon +supports both kinds of tasks with low memory/bandwidth +overhead, and more importantly, it can automatically shift +measurement attention as network state changes at run-time. +We have released all related source codes at Github3. +2 +Overview of ChameleMon +ChameleMon monitors the network in four steps (Figure 1). +1) Capturing flow-level statistics on edge switches: To cap- +ture desired flow-level statistics, ChameleMon deploys three +sketches on the data plane of each edge switch, including +a flow classifier (TowerSketch), an upstream flow encoder +(our FermatSketch), and a downstream flow encoder (our Fer- +matSketch). To detect HHs, HLs, and LLs, the upstream and +downstream flow encoders are divided into multiple parts: +1) the upstream flow encoder is divided into an upstream +2Fermat’s little theorem states that if p is a prime, then for any integer a +that is indivisible by p, we have ap−1 ≡ 1 mod p. +3https://github.com/ChameleMoncode/ChameleMon +2 + +Step 1: Capture flow-level statistics +Sample +0 +0 +1 +1 0 1 0 1 0 1 +Step 2: Collect from edge +monitor 𝑆1 +Step 3: Perform analysis +Step 4: Shift attention +• Reallocate memory +• Adjust thresholds +UpStream +DownStream +Edge +Analyzer +UpStream +DownStream +Collect +HL Encoder +HH Encoder +LL Encoder +Reconfigure data plane +Flow size +Heavy-change +Entropy +Flow size distr. +Packet loss +Heavy-hitter +Cardinality +Key tasks +Other tasks +collect 𝑆0 +monitor 𝑆0 +collect 𝑆1 +HL Encoder LL Encoder +Classifier +Figure 1: Overview of ChameleMon. +HH encoder, an upstream HL encoder, and an upstream LL +encoder; 2) the downstream flow encoder is divided into a +downstream HL encoder and a downstream LL encoder. For +every packet with flow ID f entering the network, according +to the size of flow f, the flow classifier classifies flow f into +one of three hierarchies: 1) HH candidate, 2) HL candidate, +or 3) LL candidate. The LL candidate is further classified into +sampled LL candidate or non-sampled LL candidate through +sampling. Based on the hierarchy of flow f, the packet is then +inserted into the corresponding part of the upstream flow en- +coder and downstream flow encoder when it enters and exits +the network, respectively. +2) Collecting sketches from edge switches: A central con- +troller periodically collects sketches from each edge switch +to support persistent measurement. To avoid colliding with +packet insertion when collecting sketches, each edge switch +divides the timeline into consecutive fixed-length time in- +tervals (called epochs), and copies a group of sketches for +rotation. Every time an epoch ends, the central controller col- +lects the group of sketches monitoring this epoch, and the +other group of sketches starts to monitor the current epoch. +3) Performing network-wide analysis: Every epoch, the +central controller performs network-wide analysis of the col- +lected sketches to support seven measurement tasks. By ana- +lyzing the upstream and downstream flow encoders, the cen- +tral controller can support packet loss detection. By analyzing +the flow classifier and the upstream HH encoder, the central +controller can support heavy-hitter detection and five common +tasks. +4) Shifting measurement attention as network state +changes: Every epoch, the central controller monitors the +real-time network state by analyzing the collected sketches. +Then, the central controller reconfigures the data plane of +edge switches at run-time according to the real-time network +state, shifting measurement attention through two dimensions +of dynamics. In the first dimension, the central controller dy- +namically allocates memory between packet loss detection +and heavy-hitter detection by reallocating the memory of the +upstream and downstream encoders between their different +parts. In the second dimension, the central controller dynami- +cally selects the most important flows (HH/HL/sampled LL +candidates) to monitor by adjusting the thresholds for flow +classification and the sample rate for sampling LL candidates. +3 +Design of ChameleMon Data Plane +The ChameleMon data plane consists of the flow classifier, +the upstream flow encoder, and the downstream flow encoder +deployed on each edge switch. In this section, we detail the +design of the ChameleMon data plane. First, we propose +the key technique of ChameleMon, namely FermatSketch. +Second, we detail each component of the ChameleMon data +plane in sequence. +3.1 +The FermatSketch Algorithm +Rationale: Our primary goal is to detect packet losses with +low memory overhead. Existing solutions focus on either per- +packet loss (LossRadar [17]) or all-flow visibility (FlowRadar +[21]), incurring unacceptable memory overhead. To reduce +overhead, we hope to aggregate all the lost packets of the +same flow to detect per-flow packet losses. It is very challeng- +ing because existing solutions commonly use XOR operation +for high memory efficiency and hardware-friendliness, but +simply using XOR operation to aggregate flow IDs of lost +packets causes every two lost packets of the same flow to +cancel each other out. While invertible Bloom lookup table +(IBLT) [25] can overcome this challenge as IBLT uses addi- +tion to aggregate flow IDs, such design requires computation +over large numbers, and thus complicates the implementation +of IBLT on programmable switches. To address this challenge +while maintaining hardware-friendliness, we devise FermatS- +ketch, which uses modular addition to aggregate flow IDs and +leverages Fermat’s little theorem to extract flow IDs. +Data structure (Figure 2): FermatSketch has d equal-sized +bucket arrays B1,··· ,Bd, each of which consists of m buck- +ets. Each bucket array Bi is associated with a pairwise- +independent hash function hi(·) that maps each incoming +packet into one bucket (called mapped bucket) in it. Each +bucket Bi[j] contains two fields: 1) a count field Bc +i [j] record- +ing the number of packets mapped into the bucket; 2) an +IDsum field BID +i [j] recording the result of the sum of flow +IDs of packets mapped into the bucket modulo a prime p. At +initialization, we set all fields of all buckets in FermatSketch +to zero, and p to a prime that must be larger than any avail- +able flow ID f and the size of any flow, so as to make use of +Fermat’s little theorem. +Encoding/Insertion operation (Figure 2): To encode +an incoming packet with flow ID f, we first calculate +the d hash functions to locate the d mapped buck- +ets: B1[h1( f)],B2[h2(f)],··· ,Bd[hd(f)]. For each mapped +bucket Bi[hi( f)], we update it as follows. First, we in- +crement its count field Bc +i [hi( f)] by one. Second, we up- +date its IDsum field through modular addition: BID +i [f)] ← +��BID +i [hi(f)]+ f +� +mod p +� +. The pseudo-code of encoding +operation is shown in Algorithm 1 in Appendix A.1. +Decoding operation: The decoding operation, which can ex- +tract exact flow IDs and flow sizes from FermatSketch, has +two important suboperations: 1) pure bucket verification that +verifies whether a bucket only records packets of a single +3 + +HH HIHHTH+ 1 +8 +0 +1 +0 +0 +0 +4 +0 +0 +1 +0 +0 +0 +4 +0 +0 +0 +ℬ1 +𝐶 +ℬ1 +𝐼𝐷 +ℬ1 1 ~ℬ1 4 +ℬ2 1 ~ℬ2 4 +Modular +Addition +ℬ2 +𝐶 +ℬ2 +𝐼𝐷 +0 +2 +0 +0 +0 +1 +0 +0 +1 +0 +1 +0 +4 +0 +8 +0 +ℬ1 +𝐶 +ℬ1 +𝐼𝐷 +ℬ2 +𝐶 +ℬ2 +𝐼𝐷 ++ 1 +8 +𝑝 = 11 +𝑓 +Packet with 𝑓 = 8 +Figure 2: An example of encoding/insertion. +flow (pure bucket); 2) single flow extraction that extracts and +deletes a single flow and its size from all its mapped buckets. +Next, we first propose the workflow of the decoding operation, +and then detail the two suboperations. The pseudo-code of +decoding operation is shown in Algorithm 2 in Appendix A.1. +• Decoding workflow (Figure 3): The decoding operation +proceeds as follows. +1 +Traverse FermatSketch and push all non-zero buckets to +the decoding queue. +2 +Pop a bucket from the queue. +3 +For the popped bucket, we perform pure bucket verifica- +tion to verify whether it is a pure bucket. If not, we simply +ignore the bucket. +4 +If so, we perform single flow extraction to extract and +delete a single flow and its size from the pure bucket as well +as the other mapped buckets of the single flow. +5 +We insert the extracted single flow and its size into a hash +table, namely Flowset, which is used to record all the extracted +flows and their sizes. We regard all flows recorded in the +Flowset as the flows previously encoded into FermatSketch. +6 Except the popped pure bucket, we push the other mapped +non-zero buckets of the extracted flow into the queue. +7 +Check whether the queue is empty. If so, the decoding +stops. Otherwise, go back to step +2 . After stopping, if there +are still non-zero buckets in FermatSketch, the decoding is +considered as failed. Otherwise, the decoding is considered +as successful. +•Pure bucket verification: The pure bucket verification re- +ports whether one given bucket is pure (i.e., only records a +single flow), but it may misjudge a non-pure bucket as a pure +one with a small probability 1 +m. Suppose a bucket Bi[j] only +records a single flow f ′, it should satisfy that (Bc +i [j] × f ′) +mod p = BID +i [j]. Leveraging Fermat’s little theorem, we can +get that f ′ = (BID +i [j]×(Bc +i [j])p−2) mod p. Considering that +bucket Bi[j] should be one of the d mapped buckets of flow f ′, +to verify whether Bi[j] is a pure bucket, we propose a verifica- +tion method namely rehashing verification. First, we calculate +the ith hash function hi(·) to locate the ith mapped bucket of +f ′, i.e., we calculate hi( f ′). Then we check whether hi( f ′) is +equal to j. If so, we consider Bi[j] as a pure bucket recording +flow f ′ with size Bc +i [j]. Note that the false positive rate of pure +bucket verification, i.e., the probability of misjudging a non- +pure bucket as a pure one, is 1 +m, which is calculated as follows. +For any non-pure bucket, we can calculate its flow ID, which +− 1 +4 +0 +2 +0 +0 +0 +1 +0 +0 +1 +0 +1 +0 +4 +0 +8 +0 +ℬ1 +𝐶 +ℬ1 +𝐼𝐷 +𝑓′ = ℬ2 +𝐼𝐷 1 × (ℬ2 +𝐶[1])𝑝−2 = 4 +Pure Bucket +Verification +ℎ2 𝑓′ = 1 +ℬ1 1 ~ℬ1 4 +ℬ2 1 ~ℬ2 4 +Modular +Subtraction +ℬ2 +𝐶 +ℬ2 +𝐼𝐷 +0 +1 +0 +0 +0 +8 +0 +0 +0 +0 +1 +0 +0 +0 +8 +0 +ℬ1 +𝐶 +ℬ1 +𝐼𝐷 +ℬ2 +𝐶 +ℬ2 +𝐼𝐷 +− 1 +4 +ℬ2 1 +ℬ1 2 +v +v +3 +POP +6 +PUSH +v +v +··· +Single Flow +Extraction +Rehashing +𝑝 = 11 +CHECK: if non-zero bucket exists +𝑓′ +7 +v +Insert into +Flowset +Initialization: push all non-zero buckets +1 +4 +5 +2 +Figure 3: An example of decoding. +should be considered as a random value. The probability that +a random ID is hashed to the same bucket is 1 +m. +• Single flow extraction: To extract/delete flow f ′ from Bi[j] +as well as its other mapped buckets, first, we locate its other +(d − 1) mapped buckets. Second, for each mapped bucket +Bi′[hi′( f ′)], we decrement its count field Bc +i′[hi′(f ′)] by Bc +i [j], +and update its IDsum field to ((BID +i′ [hi′( f ′)]−BID +i [j]) mod p) +through modular subtraction. +Addition/Subtraction operations: Adding/Subtracting Fer- +matSketch FS1 to/from FermatSketch FS2. FS1 and FS2 must +use the same parameters including the hash functions, number +of arrays, number of buckets, and primes. For each bucket of +FS2, we update it as follows. First, we locate the bucket of +FS1 that is in the same position as it. Second, we add/subtract +the count field of the located bucket of FS1 to/from its count +field. Third, we modular add/subtract the IDsum field of the +located bucket of FS1 to/from its IDsum field. +[Optional] fingerprint verification: To reduce the false pos- +itive rate of pure bucket verification, we can perform an extra +verification method, namely fingerprint verification, by ex- +tending the IDsum field in each bucket by w bits and using +the extra w bits as a fingerprint. For simplicity and accuracy, +we recommend implementing FermatSketch without finger- +prints in most cases. Only if there is some memory that can +hardly be utilized due to hardware constraints unless used +as fingerprints4, we recommend implementing FermatSketch +with fingerprints. Please refer to Appendix A.2 for details. +Discussion: We further discuss the space and time complexity +of FermatSketch in Appendix A.3, where we also show how +the decoding operation of FermatSketch automatically elimi- +nates the false positives generated in pure bucket verification. +In summary: 1) FermatSketch achieves the highest memory +efficiency when d is set to 3, that on average 1.23 buckets can +record a flow and its size; 2) when used as a hash table, its +memory overhead is proportional to the number of inserted +flows; 3) when used for packet loss detection, its memory +4An example is the implementation of FermatSketch in P4. Please refer +to Appendix D.1 for details. +4 + +overhead is proportional to the number of victim flows; 4) the +time complexity of decoding is O(md2). +Mathematical analysis: Let FermatSketch consists of d +bucket arrays, each of which consists of m buckets. Let M be +the number of flows inserted into that FermatSketch. We use +Theorem 3.1 to show that when FermatSketch is configured +correctly and M is not too small, the decoding of FermatS- +ketch has an extremely small probability of failure. For the +detailed proof, please refer to Appendix A.4. +Theorem 3.1 Suppose +md > cdM + ε +and +M ⩾ +Ω(d4dlogd(M)). +the +decoding +of FermatSketch fails +with probability O( +1 +Md−2 ), where both ε and cd are small +constants. +cd = +� +sup +� +α +���α ∈ (0,1),∀x ∈ (0,1),1−e−dαxd−1��−1 +For example, c3 = 1.222,c4 = 1.295,c5 = 1.425. +3.2 +ChameleMon Data Plane Components +As shown in Figure 1, every packet entering the network +undergoes the three components of the ChameleMon data +plane in sequence: 1) the flow classifier, 2) the upstream flow +encoder, and 3) the downstream flow encoder. +3.2.1 +Flow Classifier +Rationale: To detect HHs, HLs, and LLs, ChameleMon de- +ploys the flow classifier in the ingress of each edge switch, +so as to classify flows into different hierarchies. While it is +easy to select HHs to monitor according to flow sizes, it is not +easy to select HLs to monitor because we can hardly predict +how many packets a flow will lose. Our observation is that for +each flow, the number of its lost packets cannot exceed its size. +Therefore, the sizes of HLs should have a minimum value. +ChameleMon selects flows whose sizes exceed this value +to monitor, so as to approximate the monitoring of HLs. In +summary, the flow classifier classifies flows purely according +to flow sizes. Thus, we choose TowerSketch [24], a simple, +accurate, and hardware-friendly sketch, as the flow classifier. +Data Structure: The flow classifier consists of l equal-sized +arrays. The ith array Ai consists of wi δi-bit counters, where +wi ×δi is a constant and δi−1 < δi. Also, array Ai is associated +with a pairwise-independent hash function si(·). For each δi- +bit counter, its maximum value 2δi −1 is used to represent the +state that it is overflowed, and thus be regarded as +∞. +Insertion: To insert a packet with flow ID f, we first +calculate the l +hash functions to locate l +counters: +A1[s1( f)],A2[s2( f)],··· ,Al[sl( f)]. We call these counters the +l mapped counters. Then, for each of the l mapped counters, +we increment it by one unless it is overflowed. +Online query: To query the size of flow f online, we sim- +ply report the minimum value among the l mapped counters. +Packet processing: For a packet with flow ID f entering the +network, the flow classifier processes it as follows. First, we +insert it into the flow classifier and query the size of flow f. +Then, with the queried flow size, we classify flow f into the +corresponding hierarchy according to two thresholds Th and +Tl, where Th is used for selecting HH candidates, and Tl is +used for selecting HL candidates. In general, it satisfies that +Tl <= Th. If the flow size is larger than or equal to Th, flow f +is classified as a HH candidate. If the flow size is less than +Tl, flow f is classified as a LL candidate. If the flow size is +between Tl and Th, flow f is classified as a HL candidate. The +LL candidate is further classified into sampled LL candidate +or non-sample LL candidate through sampling. +3.2.2 +Upstream Flow Encoder +Rationale: To support packet loss detection, ChameleMon +deploys the upstream flow encoder in the ingress of each edge +switch just after the flow classifier, so as to encode the packets +entering the network. Therefore, the upstream flow encoder +should contain two FermatSketches to encode HL candidates +and sampled LL candidates individually. Here, for better mon- +itoring of the network state, ChameleMon monitors a portion +of LLs to maintain an overview of all victim flows. Besides, +to support heavy-hitter detection, the upstream flow encoder +should contain a FermatSketch to encode HH candidates. In +summary, the upstream flow encoder should consist of three +FermatSketches. +Data structure: The upstream flow encoder is a d-array Fer- +matSketch divided into three d-array FermatSketches: 1) an +upstream HH encoder for encoding HH candidates; 2) an +upstream HL encoder for encoding HL candidates; 3) an +upstream LL encoder for encoding sampled LL candidates. +We denote the number of buckets per array of the upstream +flow encoder, HH encoder, HL encoder, and LL encoder by +muf , mhh, mhl, and mll, respectively. Obviously, it satisfies that +muf = mhh +mhl +mll. +Packet processing: For a packet with flow ID f entering the +network, the upstream flow encoder processes it by encoding +the packet into one of the encoders corresponding to the hier- +archy of flow f unless flow f is a non-sampled LL candidate. +Here, the hierarchy of flow f can be directly obtained because +the upstream flow encoder and the flow classifier are deployed +on the same edge switch. +3.2.3 +Downstream Flow Encoder +Rationale: To support packet loss detection, ChameleMon de- +ploys the downstream flow encoder in the egress of each edge +switch, so as to encode the packets exiting the network. As the +downstream flow encoder is not responsible for heavy-hitter +detection, it should consist of two FermatSketches to encode +HL candidates and sampled LL candidates individually. +Data structure: The downstream flow encoder is a d-array +FermatSketch divided into two d-array FermatSketches: 1) +a downstream HL encoder; 2) a downstream LL encoder. +To support packet loss detection, the number of buckets per +array of the downstream HL encoder and LL encoder must +also be mhl and mll, respectively, so as to support addition +and subtraction operations with the corresponding upstream +5 + +encoder. We denote the number of buckets per array of the +downstream flow encoder by md f . In general, it satisfies that +md f < muf , and therefore satisfies that md f ⩾ mhl +mll. +Packet processing: For a packet with flow ID f exiting the +network, the downstream flow encoder processes it by en- +coding the packet into one of the encoders corresponding to +the hierarchy of flow f unless flow f is a non-sampled LL +candidate. Here, packets of HH candidates are also encoded +into the downstream HL encoder. Different from the upstream +flow encoder, the downstream flow encoder cannot directly +obtain the flow hierarchy from the flow classifier, as a flow +could enter and exit the network at different edge switches. To +address this issue, first, considering that there are four flow hi- +erarchies, we can use ⌈log(4)⌉ = 2 bits in the original packet +header to transmit this information. For example, for IPv4 +protocol, we can use the unused bits in the type of service +(ToS) field. If there are not enough unused bits, second, we +can transmit the flow hierarchy in an INT-like [26] manner. +4 +Design of ChameleMon Control Plane +The ChameleMon control plane consists of a central con- +troller, as well as the control plane of each edge switch. In +this section, we detail the design of the ChameleMon con- +trol plane. We begin by laying out how the ChameleMon +control plane collects sketches from the ChameleMon data +plane, then introduce how to support seven measurement tasks +with the collected sketches, and finally propose how to shift +measurement attention as network state changes. +4.1 +Collection from ChameleMon Data Plane +The central controller needs to periodically collect sketches, +i.e., the flow classifier, the upstream flow encoder, and the +downstream flow encoder, from the ChameleMon data plane, +so as to support persistent measurement. However, the collec- +tion cannot be completed in an instant, and thus inevitably col- +lide with packet insertion if there is only a group of sketches. +Specifically, if the central controller wants to collect sketches +at time t, it will inevitably collect some counters inserted +by packets after t, which could result in decoding failure of +FermatSketch. To address this issue, ChameleMon takes two +steps: 1) timeline division and 2) clock synchronization. Next, +we just briefly cover the two steps. We detail the two steps in +Appendix B, where we further analyze the appropriate time +for the central controller to collect sketches. +Timeline division: Each edge switch periodically flips a 1- +bit timestamp to divide the timeline into fixed-length time +intervals (called epochs) with interleaved 0/1 timestamp, and +copies a group of sketches for rotation. Each group of sketches +corresponds to a distinct timestamp value, and monitors the +epochs with that timestamp value. +Clock synchronization: The central controller also main- +tains a 1-bit periodically flipping timestamp, and periodically +synchronizes its clock with the control plane of each edge +switch, so as to make opportunities for collection. +Every time the locally maintained 1-bit timestamp flips, an +epoch ends, the central controller starts to collect the group +of sketches monitoring this epoch, and the other group of +sketches starts to monitor the current epoch. +4.2 +Measurement Tasks +With the collected sketches, the central controller can support +packet loss detection, heavy-hitter detection, and five other +common measurement tasks. +Packet loss detection: reporting each victim flow and the +number of its lost packets in the network. The central con- +troller can support packet loss detection by analyzing the +upstream and downstream flow encoders collected from +each edge switch. First, for each edge switch, we decode +the upstream HH encoder to obtain the HH Flowset, and +then reinsert each flow with its size in the HH Flowset +into the upstream HL encoder. Second, we add up the up- +stream/downstream HL/LL encoder of each edge switch +through addition operation to obtain the cumulative up- +stream/downstream HL/LL encoder. Third, we subtract the +cumulative downstream HL/LL encoder from the cumula- +tive upstream HL/LL encoder to obtain the delta HL/LL en- +coder. Fourth, we decode the delta HL/LL encoder to obtain +the HL/LL Flowset. Finally, we report the flows in the HL +Flowset as HLs, and the flows in the LL Flowset but not in +the HL Flowset as LLs. For each of these flows, its estimated +number of lost packets is the sum of its size in the HL Flowset +and the LL Flowset. +For each edge switch, the central controller can support +the following six tasks by analyzing the flow classifier and +upstream HH encoder collected from it. Then, by synthesizing +the results of each edge switch, the central controller can +easily support these tasks in a network-wide manner. We +detail these six tasks from the perspective of an edge switch. +Heavy-hitter detection: reporting flows whose sizes exceed +∆h. First, we decode the upstream HH encoder to obtain the +HH Flowset, which records flows with ID fi and size qi. For +any flow f j in the HH Flowset, if its estimated flow size +Th +qj is larger than ∆h, we report it as a HH. Note that Th is +the threshold used for selecting HH candidates. +Flow size estimation: reporting flow size of flow f j. Sim- +ilarly, we obtain the HH Flowset. If flow f j is in the HH +Flowset, we report its flow size as Th + qj. Otherwise, we +report its flow size as query result from the flow classifier. +Heavy-change detection: reporting flows whose sizes +change beyond ∆c in two adjacent epochs. Similarly, we ob- +tain the HH Flowset. For any flow f j in the HH Flowset of +either epoch, we estimate its flow size in the two epochs. If +the difference between the two estimated flow sizes is larger +than ∆c, we report flow f j as a heavy-change. +Cardinality estimation: reporting number of flows. We ap- +ply linear-counting algorithm [27] to the counter array with +the most counters in the flow classifier to estimate cardinality. +Flow size distribution estimation: reporting distribution of +flow sizes. We apply MRAC algorithm [28] to each counter +6 + +array in the flow classifier. Array Ai provides the estimated +distribution of flow size in range [2δi−1 −1,2δi −1). The re- +maining distribution of flow size in range [2δi −1,+∞) is es- +timated from the flows larger than 2δi −2 in the HH Flowset. +Entropy estimation: reporting entropy of flow sizes. Based +on the estimated flow size distribution, we can easily com- +pute the entropy as follows: −∑ +� +ni · i +N log i +N +� +, where ni is the +number of flows of size i, and N = ∑(i·ni). +4.3 +Shifting Measurement Attention +A practical measurement system should pay attention to dif- +ferent tasks for different network states. When there are only +rare packet losses in network, the system should pay more +attention to and allocate more memory for heavy-hitter de- +tection. In contrast, when there are lots of packet losses in +network, the system should pay more attention to and allocate +more memory for packet loss detection. +Aiming at this target, ChameleMon decides to shift mea- +surement attention as network changes. Every time all +the sketches monitoring the previous epoch are collected, +ChameleMon takes two phases to shift measurement attention. +First, the central controller monitors the real-time network +state, including the number and flow size distribution of flows +and victim flows, by analyzing the collected sketches. Sec- +ond, the central controller reconfigures the ChameleMon data +plane according to the real-time network state, so as to support +packet loss detection and heavy-hitter detection simultane- +ously while maintaining high memory utilization. The central +controller not only reallocates memory of the upstream and +downstream encoders between their different parts, but also +adjusts the thresholds for flow classification and the sample +rate for sampling LL candidates. To avoid interference with +the monitoring of the current epoch, the reconfiguration will +not function immediately, but in the next epoch. +For ChameleMon, the network state could be clearly clas- +sified into two levels: 1) healthy network state that Chamele- +Mon can allocate sufficient memory to monitor all victim +flows; 2) ill network state that ChameleMon cannot allo- +cate sufficient memory to monitor all victim flows, and thus +must select HLs to monitor. For each level of network state, +ChameleMon behaves almost the same in shifting measure- +ment attention, and we detail how it behaves in this section. +4.3.1 +Healthy Network State +Suppose the previously monitored network state is healthy, +and now the central controller starts to shift measurement +attention. Currently, the LL encoders are not allocated any +memory as ChameleMon can monitor all victim flows, and Tl +must be 1 as no flows should be classified into LL candidates. +The memory allocation between the upstream HH encoder +and the upstream HL encoder is flexible. +Monitoring real-time network state: The monitoring pro- +ceeds as follows. First, for each edge switch, the central con- +troller estimates the number of flows and flow size distribution +as described above (§ 4.2). Second, for each edge switch, the +central controller obtains the number of HH candidates by +decoding the upstream HH encoder. After all decoding stops, +if the decoding of any upstream HH encoder fails, the central +controller stops the monitoring as the decoding of the delta +HL encoder requires reinserting the decoded HH candidates +into the upstream HL encoders. Third, the central controller +obtains the number of HLs (equals to victim flows for healthy +network state) by decoding the delta HL encoder as described +above (§ 4.2). If the decoding fails, the central controller +estimates the number of HLs by applying linear-counting +algorithm to any bucket array of the delta HL encoder. +Reconfiguring ChameleMon data plane: The core idea of +reconfiguration is to first ensure the successful decoding +of FermatSketches for supporting packet loss detection and +heavy-hitter detection, while maintaining high memory uti- +lization. The reconfiguration proceeds as follows. +Step 1: We focus on the successful decoding of the upstream +HH encoders as they are decoded first. For each edge switch, +if the decoding of the upstream HH encoder fails, the central +controller turns up Th according to the number of flows and +flow size distribution, controlling the expected load factor5 of +the upstream HH encoder at around 70%6, so as to maintain +high memory utilization. After turning up Th, the central con- +troller stops the reconfiguration as the decoding of the delta +HL encoder cannot proceed. +Step 2: We focus on the successful decoding and high mem- +ory utilization of the delta HL encoder. If the decoding of the +delta HL encoder fails, according to the estimated number +of HLs, the central controller estimates the required mem- +ory for 70% load factor. If the maximum memory that the +HL encoders can be allocated to, i.e., all the memory of the +downstream flow encoder, cannot cover the required memory, +the healthy network state transitions to the ill network state. +In this case, the central controller 1) reallocates the memory +inside the upstream and downstream flow encoders as the +fixed allocation described in the ill network state (§ 4.3.2), +2) sets Tl to Th, and 3) adjusts the sample rate for 70% load +factor of the delta LL encoder assuming that each HL will be +a LL. Otherwise, the central controller just expands the HL +encoders to the required memory. If the decoding of the delta +HL encoder succeeds and its load factor is lower than 60%, +the central controller tries to compress the HL encoders to ap- +proach 70% load factor for high memory utilization. Here, we +reserve the minimum memory for the HL encoders to handle +the potential small burst of victim flows. +Step 3: After all the memory reallocation, we focus on the suc- +cessful decoding and high memory utilization of the upstream +HH encoders. For each edge switch, with the number of HH +5Load factor refers to the ratio of the number of recorded flows to the num- +ber of buckets of FermatSketch. The maximum load factor of FermatSketch +is around 81.3% = +1 +1.23. +6Here, we decide not to pursue the maximum load factor for two reasons: +1) the potential increase of HH candidates in the current epoch and 2) the +inevitable estimation error in linear-counting. +7 + +candidates and the memory of the upstream HH encoder, the +central controller further estimates the expected load factor of +the upstream HH encoder. if the expected load factor of the +upstream HH encoder is lower than 60% or larger than 70%, +the central controller turns down or up Th to approach 70% +load factor. +4.3.2 +Ill Network State +Suppose the previously monitored network state is ill, and +now the central controller starts to shift measurement attention. +Currently, all the HH, HL and LL encoders are allocated fixed +memory, and Tl must be larger than 1 to select HL candidates. +Specifically, the upstream HH encoder is compressed to the +minimum memory, which is the memory difference between +the upstream flow encoder and the downstream flow encoder. +Monitoring real-time network state: The monitoring pro- +ceeds in a similar way to that of the healthy network state. In +addition, the central controller obtains the number of LLs by +decoding the delta LL encoder as described above (§ 4.2). If +the decoding fails, the central controller estimates the number +of LLs by applying linear-counting algorithm to the delta +LL encoder, and then stops the monitoring. If both decod- +ing of the delta HL and LL encoders succeeds, the central +controller estimates the number and flow size distribution of +victim flows as follows. First, the central controller samples +the HLs with the same sampling method and rate as LLs. +Second, the central controller merges sampled HLs and sam- +pled LLs to obtain sampled victim flows. Third, the central +controller estimates the flow size distribution of victim flows +through querying the flow size of each sampled victim flow, +and the number of victim flows through dividing the number +of sampled victim flows by sample rate. If the decoding of +the delta HL encoder fails, the central controller regards the +estimated flow size distribution of sampled LLs, which is also +estimated by querying flow sizes, as the flow size distribution +of victim flows. +Reconfiguring ChameleMon data plane: The core idea of +reconfiguration is the same as that of the healthy network +state. The reconfiguration proceeds as follows. +Step 1: We focus on the successful decoding of the upstream +HH encoders, and the reconfiguration proceeds the same as +the first step of the healthy network state. In addition, we +focus on the successful decoding of the delta LL encoder. If +the decoding of the delta LL encoder fails, according to the +estimated number of LLs, the central controller adjusts the +sample rate to make the delta LL encoder approach 70% load +factor, and then stops the reconfiguration. +Step 2: We focus on the successful decoding of the delta +HL encoder. If the decoding of the delta HL encoder fails, +according to the estimated flow size distribution of victim +flows, assuming that each victim flow larger than Tl will be +a HL, the central controller turns up Tl to make the delta HL +encoder approach 70% load factor. +Step 3: we focus on the high memory utilization of the HL +and LL encoders. If both the decoding of the delta HL and +LL encoders succeeds, according to the estimated number +of victim flows, the central controller estimates the required +memory for monitoring all the victim flows with 70% load +factor. If the downstream flow encoder can cover the required +memory, the ill network state transitions to the healthy net- +work state. In this case, the central controller 1) eliminates the +LL encoders, 2) allocates the required memory (at least the +reserved minimum memory) to the HL encoders, and 3) sets +Tl to 1. If the downstream flow encoder cannot cover the re- +quired memory, and the load factor of the delta HL encoder or +the delta LL encoder is lower than 60%, the central controller +turns up Tl or the sample rate according to the estimated flow +size distribution of victim flows or the estimated number of +LLs, respectively, so as to approach 70% load factor. +Step 4: After all the memory reallocation, we focus on the +successful decoding and high memory utilization of the up- +stream HH encoders, and the reconfiguration proceeds the +same as the third step of the healthy network state. +5 +Evaluation +We conduct various experiments on CPU platform and our +testbed, and focus on the following five key questions. We +place the experimental results that can answer the second +question in Appendix C. +• How much memory/time can ChameleMon save in +packet loss detection? (Figure 4 - 6) We implement Fer- +matSketch and its competitors in C++, and use CAIDA +dataset [29] to evaluate their memory and time overhead +for packet loss detection on CPU platform. Results show +that FermatSketch can save memory in all cases and time +in most cases. +• How accurately can ChameleMon support heavy-hitter +detection and other five tasks? (Figure 14) We imple- +ment the combination of TowerSketch and FermatSketch +and its competitors in C++, and use CAIDA dataset to eval- +uate their accuracy for these six tasks on CPU platform. +Results show that the combination can achieve at least com- +parable accuracy in all six tasks. +• Can ChameleMon automatically shift measurement at- +tention? (Figure 7-8) We generate workloads according to +widely used traffic distributions (e.g., DCTCP [30]) for eval- +uation. We use the above workloads to evaluate Chamele- +Mon by generating different network states on our testbed. +Results show that ChameleMon can always automatically +shift measurement attention between packet loss detection +and heavy-hitter detection at run-time, and maintains high +memory utilization in most cases. +• How fast can ChameleMon shift measurement atten- +tion? (Figure 9) We use the above workloads to evaluate +ChameleMon over a large time window, in which the net- +work state changes 8 times. Results show that ChameleMon +can shift measurement attention within at most 3 epochs. +• How fast can ChameleMon monitor the network? (Fig- +ure 10-12) We use the above workloads to evaluate various +8 + +factors that can affect the epoch length. Results show that +ChameleMon can monitor the network every 50ms on our +testbed, using only two CPU cores and consuming only +320Mbps bandwidth. We believe ChameleMon can easily +scale to monitor a much larger network in a faster manner. +5.1 +Evaluation on Packet Loss Detection +Dataset: We use the anonymized IP traces collected in 2018 +from CAIDA [29] as dataset, and use the 32-bit source IP +address as the flow ID. We use the first 100K flows containing +5.3M packets for evaluation. +Setup: We set up a simulation with a simple topology con- +sisting of only a link on CPU platform. We compare Fer- +matSketch with FlowRadar [21] and LossRadar [17]. For +FermatSketch, we set its count field and ID field to 32bits, +and the number of hash functions to 3. For FlowRadar, we +allocates 10% memory to the flow filter and 90% memory +to the counting table. For the flow filter, which is actually a +Bloom filter [31], we sets its number of hash functions to 10. +For the counting table, we set its FlowXOR field, FlowCount +field, and PacketCount field to 32bits, and its number of hash +functions to 3. For LossRadar, we set its count field to 32bits, +xorSum field to 48bits, and number of hash functions to 3. +Here, the xorSum field of LossRadar encodes a 32-bit flow ID +as well as a 16-bit packet-specific information that represents +the order of a packet in a flow. For each solution, we deploy +it upstream and downstream of the link to encode the packets +entering and exiting the link. +Memory/Time overhead7 vs. number of victim flows (Fig- +ure 4): Experimental results show that the memory/time over- +head of FermatSketch is proportional to the number of victim +flows. We let the largest 10K flows pass through the link, +among which a part of flows are victim flows. The packet +loss rate of victim flows is set to 1%. As the number of vic- +tim flows increases, the memory/time overhead of FlowRadar +remains unchanged, while that of FermatSketch increases +almost linearly. We find when the number of victim flows +exceeds 6000, the decoding time of FermatSketch exceeds +that of FlowRadar. This is because the decoding operation of +FermatSketch is more complex than FlowRadar. Compared to +FlowRadar/LossRadar, FermatSketch saves up to 15.9/23.2 +times memory and up to 3.0/4.6 times decoding time. +Memory/Time overhead vs. packet loss rate (Figure 5): +Experimental results show that the memory/time overhead +of FermatSketch is independent of the number of lost pack- +ets. We let the largest 10K flows pass through the link, +among which the largest 100 flows are victim flows. As +the packet loss rate of victim flows increases, the mem- +ory/time overhead of FermatSketch and FlowRadar remains +unchanged, while that of LossRadar increases linearly. Com- +pared to FlowRadar/LossRadar, FermatSketch saves up to +7The memory overhead refers to the minimum memory required to +achieve 99.9% decoding success rate, and the time overhead refers to the +corresponding decoding time with the minimum memory. +2 +4 +6 +8 +10 +# Victim flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +Minimum memory (MB) +Fermat +LossRadar +FlowRadar +(a) Memory overhead. +2 +4 +6 +8 +10 +# Victim flows (K) +0 +2 +4 +6 +8 +10 +Decoding time (ms) +Fermat +LossRadar +FlowRadar +(b) Time overhead. +Figure 4: Memory/Time overhead vs. number of victim flows. +10 +20 +30 +40 +50 +Loss rate (%) +0 +2 +4 +6 +8 +Minimum memory (MB) +Fermat +LossRadar +FlowRadar +(a) Memory overhead. +10 +20 +30 +40 +50 +Loss rate (%) +0 +20 +40 +60 +80 +100 +Decoding time (ms) +Fermat +LossRadar +FlowRadar +(b) Time overhead. +Figure 5: Memory/Time overhead vs. packet loss rate. +10 +3 +10 +4 +10 +5 +# Total flows +10 +3 +10 +2 +10 +1 +10 +0 +10 +1 +Minimum memory (MB) +Fermat +LossRadar +FlowRadar +(a) Memory overhead. +10 +3 +10 +4 +10 +5 +# Total flows +10 +2 +10 +1 +10 +0 +10 +1 +10 +2 +Decoding time (ms) +Fermat +LossRadar +FlowRadar +(b) Time overhead. +Figure 6: Memory/Time overhead vs. number of flows. +276.1/6411.2 times memory and up to 64.5/1585.6 times de- +coding time. +Memory/Time overhead vs. number of flows (Figure 6): +Experimental results show that the memory/time overhead of +FermatSketch is independent of the number of flows. We let a +certain number of flows pass through the link, among which +the largest 100 flows are victim flows. The packet loss rate of +victim flows is set to 1%. As the number of flows increases, +the memory/time overhead of FermatSketch and LossRadar re- +mains unchanged, while that of FlowRadar increases linearly. +Compared to FlowRadar/LossRadar, FermatSketch saves up +to 1535.0/128.8 times memory and up to 821.3/23.7 times +decoding time. +5.2 +Evaluation on Testbed +Testbed setup: We have fully implemented a ChameleMon +prototype on a testbed with a Fat-tree topology composed +of 10 Tofino switches and 8 servers, with 1400 lines of P4 +[32] code and 2400 lines of C/C++ code. Each server has 48 +2.1GHz CPU cores, 256 GB RAM, and a 40Gb Mellanox +Connectx-3 Pro NIC. Switches and servers are interconnected +with 40Gb links. We deploy the ChameleMon data plane on +all four ToR/edge switches. An additional server linked with +a certain edge switch works as the central controller. For +implementation details of the ChameleMon data plane and +control plane, please refer to Appendix D. +Workloads: We generate workloads consisting of UDP flows +according to four widely used distribution: DCTCP [30], +9 + +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +20 +40 +60 +80 +100 +# Total flows (K) +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Threshold +Th +Tl +(c) Threshold. +0 +20 +40 +60 +80 +100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 7: Measurement attention vs. number of flows. Figure 7(a) depicts the memory division of HH encoder (HHE), HL encoder (HLE), and LL +encoder (LLE) inside the upstream flow encoder. Figure 7(b) depicts the number of HH candidates of an edge switch, the number of HLs in the network, and the +number of sampled LLs in the network. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Threshold +Th +Tl +(c) Threshold. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 8: Measurement attention vs. ratio of victim flows. +HADOOP [33], VL2 [34] and CACHE [35]. We use the +104-bit 5-tuple as the flow ID. For each flow, We choose +its source and destination IP address uniformly, and therefore +each server sends and receives almost the same number of +flows. The packet sender and packet receiver are integrated +into a program written in DPDK [36]. To manually control +packet losses, we let switches proactively drop packets whose +ECN fields are set to 1. In this way, we can flexibly specify +any flow as a victim flow and control its packet loss rate. To +simulate the complex and changeable network state, we set +the packet loss rate of each victim flow to a random value be- +tween 1% and 20%. To avoid packet losses due to congestion, +when sending packets, we set the size of every packet to 64 +bytes regardless of its original size, so as to significantly re- +duce the traffic load in the network and eliminate congestion. +Such operation does not change the number of packets of each +flow, and thus has no impact on the behavior of ChameleMon. +Parameter settings: We set the epoch length to 50ms by +default8. For the flow classifier, we set it to consist of an 8-bit +counter array and a 16-bit counter array. We set the number of +8-bit counters w1 to 32768 and the number of 16-bit counters +w2 to 16384. For the upstream flow encoder and downstream +flow encoder, we set them to consist of 3 bucket arrays for +the highest memory efficiency. We set the number of buckets +per array of the upstream flow encoder mu f to 4096, and that +of the downstream flow encoder md f to 3072. For the healthy +network state, we fix the minimum memory reserved for HL +encoders to a 3-array FermatSketch with 512 buckets per +array. For the ill network state, we fix the upstream HH, HL, +LL encoders to a 3-array FermatSketch with 1024, 2560, and +512 buckets per array, respectively. Please refer to Table 1 in +Appendix D.1 for resource usage on Tofino switches. +8For some workloads that cannot run out in 50ms, we extend the epoch +length appropriately. +First, on DCTCP workload, we evaluate whether Chamele- +Mon can automatically shift measurement attention for dif- +ferent network states9. For experimental results on the other +three workloads, please refer to Appendix E. +Measurement attention vs. number of flows (Figure 7): +Experimental results show that ChameleMon can automat- +ically shift measurement attention to packet loss detection +while maintaining high memory utilization, as the number of +flows increases and the number of victim flows increases. We +vary the number of flows in the network from 10K to 100K, +and fix the ratio of victim flows to 10%. At first, the network +state is healthy. As the number of flows increases from 10K +to 20K, ChameleMon can record all flows and victim flows, +and therefore sets both Th and Tl to 1. As the number of flows +increases from 30K to 70K, ChameleMon records all victim +flows by allocating more and more memory to HL encoders. +However, ChameleMon cannot record all flows, and thus in- +creases Th to decrease the number of HH candidates. As the +number of flows increases from 80K to 100K, ChameleMon +cannot record all victim flows, and thus the network state +transitions to the ill network state. ChameleMon allocates +fixed memory to LL encoders, increases Tl, and decreases the +sample rate, so as to control the number of HLs and sampled +LLs. Meanwhile, ChameleMon keeps increasing Th to control +the number of HH candidates. Throughout the experiment, +ChameleMon maintains high memory utilization. The sum +of decoded flows (Figure 7(b)) always exceeds 8K unless +ChameleMon can record all flows and victim flows, repre- +senting a load factor larger than 65% considering that the +9For each data point of Figure 7-8, we randomly initialize the configura- +tion of the ChameleMon data plane, and collect the data point after Chamele- +Mon successfully shifts measurement attention and the configuration of the +ChameleMon data plane is stable. +10 + +upstream flow encoder has 12288 buckets and the maximum +load factor is 81.3%. +Measurement attention vs. ratio of victim flows (Fig- +ure 8): Experimental results show that ChameleMon can +automatically shift measurement attention to packet loss de- +tection while maintaining high memory utilization, as the +ratio of victim flows increases and the number of victim flows +increases. We fix the number of flows to 50K, and vary the +ratio of victim flows from 2.5% to 25%. At first, the network +state is healthy. As the ratio of victim flows increases from +2.5% to 12.5%, ChameleMon records all victim flows by allo- +cating more and more memory to HL encoders, and increases +Th to decrease the number of HH candidates. As the ratio +of victim flows increases from 15% to 25%, ChameleMon +cannot record all victim flows, and thus the network state +transitions to the ill network state. ChameleMon allocates +fixed memory to LL encoders, increases Tl, and decreases the +sample rate, so as to control the number of HLs and sampled +LLs. Meanwhile, because the memory of upstream HH en- +coder and the number of flows remain unchanged, Th also +remains unchanged. Throughout the experiment, Chamele- +Mon maintains high memory utilization. The sum of decoded +flows (Figure 8(b)) always exceeds 8K, representing a load +factor larger than 65%. +Second, on DCTCP workload, we evaluate how fast can +ChameleMon shift measurement attention over a large time +window, in which the network state changes 8 times. +Measurement attention vs. epoch (Figure 9): Experimen- +tal results show that ChameleMon can shift measurement +attention within at most 3 epochs. Figure 9 plots the shift of +measurement attention in a large time window consisting of +45 consecutive epochs. We change the network state (either +the number of flows or the victim flow ratio) every 5 epochs, +and the detailed settings are shown in the top sub-figure. Over- +all, the network state first degrades from the healthy network +state to the ill network state, and then improves back to the +healthy network state. For the eight changes, ChameleMon +shifts measurement attention within 1 (6->7), 2 (11->13), 3 +(16->19), 2 (21->23), 2 (26->28), 1 (31->32), 1 (36->37), and +1 (41->42) epochs, respectively. +To evaluate how fast can ChameleMon monitor the net- +work, we evaluate various factors that could affect the setting +of epoch length: 1) the time and bandwidth required to collect +sketches from edge switches, 2) the time required to respond +to different network states, and 3) the time required to recon- +figure the ChameleMon data plane. The central controller +only uses two CPU cores in evaluation. +Time/Bandwidth overhead for collection (Figure 11): Ex- +perimental results show that ChameleMon consumes only a +small amount of time and bandwidth in collecting all the data +structures deployed on edge switches. ChameleMon takes a +total of 11.33ms to collect sketches (refer to Appendix D.2 +for details). As for bandwidth, when the epoch length is set +to 50ms, the bandwidth overhead for collection is 317Mbps, +0 +30 +60 +# Flows (K) +0 +0.5 +1 +Memory division +0 +5 +10 +# Decoded flows (K) +10 +10� +10 +Threshold +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +Epoch +0 +0.5 +1 +Sample rate +Total flows +Victim flows +HHE +HH +HLE +HL +LLE +LL +Th +Tl +Figure 9: Measurement attention vs. epoch. +0 +20 +40 +60 +80 +100 +# Total flows (K) +5 +10 +15 +20 +25 +Response time (ms) +DCTCP +CACHE +VL2 +HADOOP +(a) Varying number of flows. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +5 +10 +15 +20 +25 +30 +Response time (ms) +DCTCP +CACHE +VL2 +HADOOP +(b) Varying ratio of victim flows. +Figure 10: Response time to different network states. +consuming only 0.8% bandwidth for the central controller +equipped with a 40Gb NIC. +Response time to different network states (Figure 10): Ex- +perimental results show that ChameleMon can always respond +to different network states within 30ms. We count the re- +sponse time of ChameleMon to each network state previously +appeared in Figure 7-8, where the response time refers to +the time interval between the central controller finishing the +collection of sketches and the central controller generating the +reconfiguration packet10 for the ChameleMon data plane. Al- +though the response time does not seem to show a clear trend +with the network state, it is mainly determined by the number +of HH candidates, because the central controller needs to first +extract them from the upstream HH encoders and then rein- +sert them to the upstream HL encoders. As shown in Figure +10(b), as the ratio of victim flows increases, the response time +on all the four workloads decreases because the number of +HH candidates decreases. The response time finally stabilizes +because the fixed memory allocation in the ill network state +always decodes a similar number of flows. +10The central controller sends the reconfiguration packets to edge switches +to reconfigure their data planes. Please refer to Appendix D.2 for details. +11 + +0 +200 +400 +600 +800 +1000 +Epoch length (ms) +0 +100 +200 +300 +400 +Bandwidth (Mbps) +Figure 11: Bandwidth. +2 +3 +4 +5 +6 +7 +Reconfiguration time (ms) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Figure 12: Reconfiguration. +CDF of reconfiguration time (Figure 12): Experimental re- +sults show that it takes 2∼7ms to reconfigure the Chamele- +Mon data plane. The central controller sends 10K reconfigu- +ration packets with random configuration of the ChameleMon +data plane to each edge switch, and we count the time for an +edge switch to execute the reconfiguration. We find 60% of +reconfigurations take less than 5ms. The difference in time +consumption is mainly because different reconfigurations re- +quire updating different numbers of TCAM entries to the +switch data plane for supporting dynamic memory allocation +(refer to Appendix D.1 for details). +Adding up the above all time consumption, we find that the +overall time consumption is less than 50ms. This verifies that +ChameleMon can monitor the network every 50ms on our +testbed. Considering that 1) the central controller only uses +two CPU cores in experiments and 2) monitoring the network +every 50ms only consumes 0.8% bandwidth of a 40Gb NIC, +we believe ChameleMon can easily scale to monitor a much +larger network with a shorter epoch length, requiring only one +server as the central controller. +6 +Related Work +Prior art can be classified into four categories based on sup- +ported measurement tasks. +Prior art for packet loss detection: They can be classi- +fied into two kinds of solutions. The first kind is algorithm- +oriented solutions, including LossRadar [17] based on In- +vertible Bloom filter (IBF) [18]. LossRadar can pinpoint +the location of every lost packet and infer the root causes +of packet losses by deploying IBF to monitor every link in +the network. The second kind is system-oriented solutions, +including Netseer [19] and PacketScope [37] that are based on +programmable switches, NetSight [9] based on In-band net- +work telemetry (INT), and more [38]. Among them, NetSight +mirrors desired statistics of every packet at every hop to a cen- +tral controller, incurring 7% bandwidth overhead. Both the +above kinds of solutions fail to meet versatility requirement +as they do not handle heavy-hitter detection. +Prior art for heavy-hitter detection: They can be classi- +fied into two kinds of solutions. The first kind of solutions is +sketches specifically designed for only heavy-hitter detection, +including HashPipe [20], R-HHH [39], and more [40–42]. +Among them, HashPipe designs a multi-stage data struc- +ture and kicks out small flows through comparison. The sec- +ond kind of solutions is sketches that support many mea- +surement tasks, including UnivMon [13], ElasticSketch [16], +CocoSketch [11], SketchLib [43], FlyMon [44], and more +[4, 12, 14, 15, 45–52]. Among them, CocoSketch proposes +a key technique, namely stochastic variance minimization +technique, to provide unbiased estimation for arbitrary partial +key. Both the above kinds of solutions fail to meet versatility +requirement as they do not handle packet loss detection. +Prior art for both tasks: These solutions record the IDs and +sizes of all flows in a zero-error manner. Typical solutions +include FlowRadar [21], OmniMon [22], Counter Braids [23], +and more [53]. Among them, FlowRadar encodes the IDs and +sizes of all flows into a variant of IBLT [25] in switches, and +then executes well-designed decoding schemes to retrieve ex- +act flow IDs and sizes. These solutions fail to meet efficiency +requirement as they record the exact IDs of all flows, incurring +memory/bandwidth overhead linear with the number of flows. +Besides, INT-based solutions that carry desired statistics in +packet headers can potentially support both tasks with packet- +level visibility. Typical solutions include INT [26], PINT [54], +LightGuardian [55], and more [24, 56–58]. However, INT- +based solutions suffer from granularity-cost trade-off, and +thus fail to meet either versatility requirement or efficiency +requirement. +Other prior art: Other than the above solutions, there are +still three kinds of solutions relevant to network measurement. +• Sampling-based solutions: These solutions collect desired +statistics from a subset of network traffic through packet +sampling, including Csamp [59], NetFlow [60], sFlow [61], +EverFlow [62], and more [33,63–70]. While sampling solu- +tions significantly reduce the bandwidth overhead through +sampling, they cannot well handle packet loss detection as +only sampled packets are measured, and thus fail to meet +efficiency requirement. +• Programmable-switch-assisted solutions: Besides packet +loss detection, these solutions leverage the advanced fea- +tures and capabilities of programmable switches to monitor +micro-bursts [71], perform queue measurement [72–74], +and more [75–78]. +• Host-based solutions: Due to the flexibility, abundant re- +sources, and high visibility to flow-level statistics of end- +hosts, these solutions are typically designed for inferring +the existences, locations, and root causes of specific net- +work events or network failures. Typical solutions either +send tailored probes into the network [10,79–85] or analyze +the performance of protocol stack or other I/O [86–95]. Be- +sides, some solutions further leverage switches to perform +measurement [96,97] or record forwarding paths [98,99]. +ChameleMon can complement these solutions as Chamele- +Mon provides flow-level statistics with high accuracy. Take +007 [94] as an instance. Network operators can replace the +TCP monitoring agent that detects TCP retransmissions +in 007 with ChameleMon. After the replacement, 007 can +monitor packet losses of TCP flows as well as packet losses +of flows of other protocols. Such extra visibility can help +007 better locate the link failures. +12 + +7 +Conclusion +In this paper, we present ChameleMon, which can automat- +ically shift measurement attention as network state changes +at run-time. To achieve this, ChameleMon designs FermatS- +ketch, a dividable, additive, and subtractive data structure that +supports both packet loss detection and heavy-hitter detection +simultaneously. We have fully implemented a ChameleMon +prototype on a testbed consisting of 10 Tofino switches and 8 +end-hosts. Extensive experimental results on the testbed ver- +ify that 1) ChameleMon can achieve high accuracy in packet +loss detection, heavy-hitter detection, and five common mea- +surement tasks; 2) ChameleMon can monitor the network +every 50ms and shift measurement attention within at most 3 +epochs as network changes. +13 + +References +[1] Theophilus Benson, Ashok Anand, Aditya Akella, and +Ming Zhang. Microte: Fine grained traffic engineer- +ing for data centers. In Proceedings of the seventh +conference on emerging networking experiments and +technologies, pages 1–12, 2011. +[2] Anja Feldmann, Albert Greenberg, Carsten Lund, Nick +Reingold, Jennifer Rexford, and Fred True. Deriving +traffic demands for operational ip networks: Method- +ology and experience. 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In Proceedings of the ACM SIGCOMM +Symposium on SDN Research (SOSR), pages 176–182, +2021. +19 + +A +The FermatSketch Algorithm +A.1 +Pseudo-Code +Algorithm 1: Encoding/Insertion operation of Fer- +matSketch +Input: Flow ID f +1 for i ∈ [1,d] do +2 +j = hi(f); +3 +BID +i [j] = (BID +i [j]+ f) mod p; +4 +Bc +i [j]++; +5 end +Algorithm 2: Decoding operation of FermatSketch +1 Function IsPure(i, j): +2 +f = (BID +i [j]×(Bc +i [j])(p−2)) mod p; +3 +return j == hi( f); +4 Function Delete(Bi′[j′],Bi[j]): +5 +Bc +i′[j′] = Bc +i′[j′]−Bc +i [j]; +6 +BID +i′ [j′] = (BID +i′ [j′]−BID +i [j]) mod p; +7 Function Decode(): +8 +Queue is an empty queue; +9 +Flowset is an empty map; +10 +for i ∈ [1,d], j ∈ [1,w] do +11 +if Bc +i [j]! = 0 then +12 +Queue.push(Bi[j]); +13 +end +14 +end +15 +while !Queue.empty() do +16 +Bi[j] = Queue.front(); +17 +Queue.pop(); +18 +if IsPure(i, j) then +19 +f ′ = (BID +i [j]×(Bc +i [j])(p−2)) mod p; +20 +Flowset[f ′] = Flowset[f ′]+Bc +i [j]; +21 +for i′ ∈ [1,d] do +22 +Delete (Bi′[hi′( f ′)],Bi[j]); +23 +if Bc +i′[hi′( f ′)]! = 0 then +24 +Queue.push(Bi′[hi′(f ′)]); +25 +end +26 +end +27 +end +28 +end +29 +return Flowset +A.2 +Fingerprint Verification +To reduce the false positive rate of pure bucket verification, we +can perform an extra verification method, namely fingerprint +verification, by extending the IDsum field in each bucket by +w bits and using the extra w bits as a fingerprint. For each +incoming packet with flow ID f, a new hash function hf p(·) +gives it a w-bit fingerprint hf p( f) for checking whether a +bucket is pure. For encoding operation, instead of inserting +flow ID f, we insert an extended ID concatenated by flow ID +f and fingerprint h f p(f), and the extended IDsum field stores +the result of the sum of the extended IDs modulo prime p. +Note that p must be a prime larger than any available extended +ID. For decoding operation, obviously, we can still perform +rehashing verification with the extended ID. Our fingerprint +verification works as follows. Suppose a bucket is pure. First, +we reuse the the extended ID of the single flow calculated in +rehashing verification. Then, we divide the extended ID to +get the flow ID and its fingerprint. If the divided fingerprint +equals to the fingerprint of the divided flow ID, we consider +the bucket passes fingerprint verification. Only buckets pass +both rehashing and fingerprint verification will be considered +as pure. The false positive rate of only fingerprint verification +is obviously +1 +2w . Considering that rehashing verification and +fingerprint verification are independent, the false positive rate +of pure bucket verification could be reduced to +1 +2wm with w-bit +fingerprint. +We conduct experiments to demonstrate the effect of 8-bit +fingerprint on improving the decoding success rate. As shown +in Figure 13(a), when the number of flows is 1K, with the +same number of buckets, 8-bit fingerprint can improve the +decoding success rate by at most 6.73%. However, when the +number of flows is 10K, the improvement falls to at most +2.26%. This is because as the number of buckets increases, m +increases, and the false positive rate of pure bucket verification +quickly drops, and thus further reducing the false positive rate +with fingerprint yields less improvement on the decoding suc- +cess rate. As shown in Figure 13(b), under the same memory +usage, 8-bit fingerprint actually reduces the decoding success +rate. This is because fingerprint consumes additional memory, +while this memory could be used as buckets to reduce the +probability of 2-core of the random hypergraph and improve +the decoding success rate. Figure 13(a)-(b) also demonstrate +that the memory overhead of FermatSketch is proportional to +the number of inserted flows. +In summary, for simplicity and accuracy, we recommend +implementing FermatSketch without fingerprints in most +cases. Only if there is some memory that can hardly be uti- +lized due to hardware constraints unless used as fingerprints, +we recommend implementing FermatSketch with fingerprints. +A.3 +Discussion of FermatSketch +Space complexity: Suppose FermatSketch is large enough, +and the pure bucket verification has negligible false positive +rate. The decoding operation is almost the same as that of +IBLT [25], which is exactly the procedure used to find the 2- +core of a random hypergraph [100,101]. Therefore, the mem- +ory overhead of FermatSketch is proportional to the number +of inserted flows. FermatSketch also shares similar properties +with IBLT: the number of hash functions, i.e., the number of +the bucket arrays d, is recommended to set to 3 for the highest +20 + +1.17 +1.2 +1.23 +1.26 +1.29 +# Buckets Per Flow +0 +20 +40 +60 +80 +100 +Decoding Success Rate (%) +10K flows w/o fp +10K flows w/ fp +1K flows w/o fp +1K flows w/ fp +(a) Same number of buckets per flow. +9 +10 +11 +12 +Memory Per Flow (Byte) +0 +20 +40 +60 +80 +100 +Decoding Success Rate (%) +10K flows w/o fp +10K flows w/ fp +1K flows w/o fp +1K flows w/ fp +(b) Same memory per flow. +Figure 13: Experiments on 8-bit fingerprints. We use the +anonymized IP traces collected in 2018 from CAIDA [29] +as dataset. We use the 32-bit source IP address as the flow +ID, and choose the first 10K flows for experiments. Here, fp +represents 8-bit fingerprint. +memory efficiency, that on average 1.23 buckets can record a +flow and its size.. +Packet loss detection: To support packet loss detection, we +can deploy a group of FermatSketches on edge switches to +encode the packets entering the network, and another group +of FermatSketches to encode the packets exiting the network. +Thanks to the additivity and subtractivity of FermatSketch, +for each group, we add up the FermatSketches in it to obtain +a cumulative FermatSketch encoding all the packets enter- +ing or exiting the network. Then, we subtract the cumulative +FermatSketch encoding all the packets exiting the network +from the other one, and the FermatSketch after subtraction +just encodes all the victim flows in the network. This Fer- +matSketch just requires memory proportional to the number +of victim flows for successful decoding. In other words, Fer- +matSketch can support packet loss detection with memory +overhead proportional to the number of victim flows. +Eliminating false positives in rehashing verification: Due +to hash collisions, the rehashing verification will inevitably +misjudge some impure buckets as pure buckets with false +positive rate 1 +m. Such misjudgement will lead to extraction +of flows that are not inserted into FermatSketch, and finally +could hinder the decoding. From another point of view, ex- +tracting a flow from such a misjudged impure bucket, i.e., +false positive, equals to inserting a wrong flow with a neg- +ative size into FermatSketch. The decoding operation can +automatically eliminate these false positives: in the decoding +procedure, these inserted wrong flows could also be extracted +and deleted from FermatSketch, and then the impact caused +by the false positives disappears. +Time complexity of decoding operation: Suppose FermatS- +ketch is large enough and the false positive rate in pure bucket +verification is negligible. In step 1 , we traverse FermatS- +ketch and push all non-zero buckets into the decoding queue. +The number of these buckets is at most md, and thus the time +complexity of step 1 +is O(md). In the rest steps, we process +all the buckets pushed into the queue, which consists of two +parts: 1) the md buckets pushed into in step 1 , and 2) the +mapped buckets except the popped pure bucket of each ex- +tracted flow. Considering that the number of extracted flows is +bounded by the number of buckets of FermatSketch, i.e., md, +the number of buckets of the second part is O(md2). There- +fore, the time complexity of the rest steps is O(md2). Adding +up the time complexities of all steps, the time complexity of +decoding operation is O(md2). +A.4 +Proof of Theorem A.1 +Theorem A.1 Let FermatSketch consists of d bucket arrays, +each of which consists of m buckets. Let M be the number of +flows inserted into that FermatSketch. Suppose md > cdM +ε +and M ⩾ Ω(d4dlogd(M)). the decoding of FermatSketch fails +with probability O( +1 +Md−2 ), where both ε and cd are small +constants. +cd = +� +sup +� +α +���α ∈ (0,1),∀x ∈ (0,1),1−e−dαxd−1��−1 +For example, c3 = 1.222,c4 = 1.295,c5 = 1.425. +Proof A.1 This is an analysis based on the theory of the 2- +core in random hypergraph [100,101] and IBLT. Compared +with 2-core or IBLT, we only introduce a kind of additional +error, which is the false positives when we use pure bucket +verification to verify the pure buckets. The IBLT assumes there +is no error when verifying buckets because IBLT uses addi- +tional hashkeySum field that can be long enough. The results +of 2-core and IBLT show that the failure probability without +wrong verification is O( +1 +Md−2 ). Here we aim at proving that +the consequences of our false positives are negligible when +M is not too small. +In the decoding procedure, the pure bucket verification +runs at most O(Md) times, and the false positive rate is +O( 1 +m) with only rehashing verification. By Chernoff bound, +when M = O(md) and δ = O( +1 +Md−2 ), the number of false +positives will not exceed F = O(d3log(M)) in most cases +(i.e., 1 − O(δ)). A false positive will incur a wrong flow ID +with a wrong single flow deletion that influences d buckets. +There is at most Fd buckets can be influenced, called poi- +soned buckets. The existing study [25] of poisoned bucket +shows that a small number of poisoned buckets will be au- +tomatically recovered, and the probability of failure due +to poisoned bucket is O( +� Fd +M +�d) = O( d4dlogd(M) +M(d−1) +). When set- +ting M = Ω(d4dlogd(M)), the overall failure probability is +δ = O( +1 +Md−2 ). In practice, M = Ω(d4dlogd(M)) is easy to +meet because M is large and d is a small constant. Here, we +only use rehashing verification for pure bucket verification. +The theorem can also be easily extended if we further use +fingerprint verification. +B +Collection from ChameleMon Data Plane +Timeline split: For each edge switch, we maintain a 1-bit +timestamp in its ingress, which is periodically flipped by the +21 + +switch control plane, so as to split the timeline into consecu- +tive fixed-length epochs with interleaved 0/1 timestamp value. +Further, we copy an additional group of sketches in the switch +data plane for rotation. Each group of sketches corresponds +to a distinct timestamp value (0/1), and monitors the epochs +with that timestamp value. Specifically, at each edge switch, +every packet entering the network first obtains the current +timestamp value, and then is inserted into the flow classifier +and upstream flow encoder corresponding to the obtained +timestamp value. When the packet exits the network, it is also +inserted into the downstream flow encoder corresponding to +the timestamp value it obtained when it entered the network. +To maintain the timestamp value during the packet transmis- +sion, we can use one unused bit in the original packet header +as discussed above (§ 3.2.3). +Clock synchronization: Through maintaining a 1-bit times- +tamp and copying a group of sketches, we successfully split +the timeline and insert packets of different epochs to their +corresponding groups of sketches, laying a solid foundation +for subsequent collection. However, if the clocks of the con- +trol planes of edge switches are out of synchronization to +some extent, we still can not find opportunities to collect the +sketches without colliding with packet insertion. Consider- +ing such an extreme situation. There are three edge switches +in a given network, and the transmission time between any +two edge switches is the same. The time offset between the +control planes of two of the edge switches is exactly the size +of the epoch, i.e., at any time, the flipping timestamps of the +two edge switches are different (0<->1). There are continuous +packets entering the network at the above two edge switches, +and exiting the network at the third switch. As a result, both +groups of sketches of the third switch are continuously in- +serted, and thus can never be collected. To address this issue, +the central controller synchronizes the clocks of the control +planes of all edge switches with itself, trying to keep only a +group of sketches being inserted at any time, so as to make +opportunities to collect the other group. +Then, we further discuss the appropriate time for the central +controller to collect sketches. +Appropriate time for collection: The central controller also +maintains a 1-bit timestamp, trying to collect the group of +sketches monitoring the previous epoch after it ends. Before +collection, the central controller should ensure that all the +packets in the previous epoch have been inserted into sketches +or lost in the network, so as to guarantee the correctness of +measurement. First, we analyze an ideal situation, that the +clock synchronization is zero-error. For ingress sketches, i.e., +the flow classifier and the upstream flow encoder, as soon as +the locally maintained 1-bit timestamp flips, the central con- +troller can collect the group of ingress sketches monitoring +the previous epoch from each edge switch, because all the +packets in that epoch have already been inserted into ingress +sketches. For egress sketches, i.e., the downstream flow en- +coder, every time the locally maintained timestamp flips, the +central controller must first wait an additional period of time, +so as to ensure that all the packets in the previous epoch have +either been lost in the network, or passed through the network +and been inserted into egress sketches. Then, the central con- +troller can collect the group of egress sketches monitoring +the previous epoch from each edge switch. Obviously, the +additional period of time should be longer than the maximum +time for packet transmission in the network. Considering that +the buffer sizes of DCN switches are at 10MB-level [102], +with 100Gb link speed, the queuing delay in a single switch +is at most 1ms in most cases. Therefore, for typical data cen- +ter networks that usually have at most five hops, setting the +additional time to 10ms can cope with most cases. However, +in practice, the clock synchronization can never be zero-error. +Therefore, before collecting both ingress and egress sketches, +the central controller needs to wait for another additional pe- +riod of time, which should be longer than the precision of +synchronization, so as to guarantee the correctness of mea- +surement. In addition, the central controller should end the +collection some time before its 1-bit timestamp flips again, +which should also be longer than the precision of synchro- +nization, in case the packets in the next epoch are inserted +into the group of sketches being collected. +C +Heavy-hitter Detection and Other Tasks +Metrics: +• Average Relative Error (ARE): +1 +|Ω| ∑fi∈Ω +|vi−ˆni| +vi +, where Ω is +the set including all flows, vi is the true size of flow fi, and +ˆvi is the estimated size of flow fi. +• F1 Score: 2·PR·RR +PR+RR , where PR (Precision Rate) refers to the +ratio of the number of the correctly reported instances to +the number of all reported instances, and RR (Recall Rate) +refers to the ratio of the number of the correctly reported +instances to the number of all correct instances. +• Relative Error (RE): |True−Est| +True +, where True and Est are the +true and estimated statistics, respectively. +• Weighted Mean Relative Error (WMRE) [15]: ∑z +i=1 |ni− ˆni| +∑z +i=1 +� ni+ ˆni +2 +�, +where z is the maximum flow size, ni and ˆni are the true and +estimated numbers of the flows of size i, respectively. +Dataset: We also use the IP traces from CAIDA [29] as our +dataset, and use the 32-bit source IP address as the flow ID. +We use four traces for evaluation, each of which monitors +the traffic in five seconds. Each trace contains 63K flows and +2.3M packets on average. We report the average accuracy that +each algorithm achieves on each CAIDA trace. +Setup: We compare the combination of TowerSketch and +FermatSketch (Tower+Fermat) with 9 algorithms: CM [51], +CU [4], CountHeap [52], UnivMon [13], ElasticSketch [16], +FCM-sketch [103], HashPipe [20], CocoSketch [11], and +MRAC [28]. We do not compare with FlowRadar because +FlowRadar can hardly perform successful decoding with the +memory sizes we used for evaluation (200KB-600KB). For +heavy-hitter detection and heavy-change detection, we set +22 + +their thresholds ∆h and ∆c to about 0.02% and 0.01% of the +total packets, i.e., 500 and 250, respectively. We configure +Tower+Fermat and its competitors as follows. Overall, the +configurations of these competitors are recommended in liter- +ature. +• Tower+Fermat: For Tower, we set it to consist of an 8-bit +counter array and a 16-bit counter array. For Fermat, We +set its count field and ID field to 32bits, and allocate 2500 +buckets to it for 99.9% decoding success rate. +we set the threshold Th for identifying heavy-hitter candi- +dates to the heavy-change threshold ∆c, i.e., 250, for detect- +ing most heavy-hitters and heavy-changes. +• CM/CU/CountHeap: We use 3 hash functions as recom- +mended in [104]. We set the counter size to 32bits. For +CountHeap, we additionally set its heap capacity to 4096 +for heavy-hitter detection. +• UnivMon: We use 14 levels and each level can record 1000 +heavy hitters. +• Elastic: We use the hardware version of Elastic. For the +heavy part, we use 4 stages and each stage has 3072 buck- +ets. For the light part, we use a one-layer CM with 8-bit +counters. +• FCM: We use the top-k version of FCM. It is almost the +same as Elastic except the light part is substituted by a +16-ary FCM whose depth is set to 2. +• Hashpipe: We set the number of stages to 6. +• Coco: We use the hardware version of Coco that only uses +one hash function. +Heavy-hitter detection (Figure 14(a)): Experimental results +show that Tower+Fermat achieves comparable accuracy with +HashPipe, and higher accuracy than other algorithms. When +using only 200KB memory, the F1 score of Tower+Fermat is +99.8%, while that of Elastic and FCM is lower than 99%. +Flow size estimation (Figure 14(b)): Experimental results +show that Tower+Fermat achieves comparable accuracy with +FCM, and higher accuracy than other algorithms. When us- +ing only 200KB memory, the ARE of Tower+Fermat is 4.51 +times, 3.19 times, 2.09 times, and 1.59 times smaller than that +of CM, CU, Elastic, and FCM, respectively. +Heavy-change detection (Figure 14(c)): Experimental re- +sults show that the Tower+Fermat achieves higher accuracy +than other algorithms. Tower+Fermat achieves 99.6% F1 +score when using only 400KB memory, while that of the +other algorithms is below 99.0%. +Flow size distribution estimation (Figure 14(d)): Exper- +imental results show that Tower+Fermat achieves higher +accuracy than Elastic and FCM, and comparable accuracy +with MRAC. When using 600KB memory, the WMRE of +Tower+Fermat is 0.039, 1.09 and 1.42 times smaller than that +of Elastic and FCM, respectively. +Entropy estimation (Figure 14(e)): Experimental results +show that Tower+Fermat achieves higher accuracy than Univ- +Mon, and comparable accuracy with Elastic and FCM. When +using 600KB memory, the ARE of Tower+Fermat is 0.003, +3.3 times smaller than that of UnivMon. +Cardinality estimation (Figure 14(f)): Experimental results +show that the Tower+Fermat achieves higher accuracy than +other algorithms. When using 600KB memory, the RE of +Tower+Fermat is 0.0016, 13.125 times, 10.08 times, and 4.57 +times smaller than that of UnivMon, Elastic, and FCM, re- +spectively. +200 +300 +400 +500 +600 +Memory (KB) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F1 score +Tower + Fermat +FCM +UnivMon +CountHeap +Elastic +Hashpipe +Coco +(a) Heavy-hitter +200 +300 +400 +500 +600 +Memory (KB) +0 +2 +4 +6 +ARE +Tower + Fermat +FCM +CM +CU +Elastic +(b) Flow size +200 +300 +400 +500 +600 +Memory (KB) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F1 score +Tower + Fermat +FCM +UnivMon +CountHeap +Elastic +Coco +(c) Heavy-change. +200 +300 +400 +500 +600 +Memory (KB) +0.00 +0.07 +0.14 +0.21 +0.28 +WMRE +Tower + Fermat +FCM +MRAC +Elastic +(d) Flow size distribution. +200 +300 +400 +500 +600 +Memory (KB) +0.00 +0.02 +0.04 +0.06 +RE +Tower + Fermat +FCM +UnivMon +Elastic +MRAC +(e) Entropy. +200 +300 +400 +500 +600 +Memory (KB) +0.00 +0.03 +0.06 +0.09 +0.12 +RE +Tower + Fermat +FCM +UnivMon +Elastic +(f) Cardinality. +Figure 14: Accuracy comparison for six tasks. +D +Prototype Implementation +In this section, we present the important details of Chamele- +Mon prototype. We lay out important implementation details +of the ChameleMon data plane and control plane in sequence. +D.1 +Data Plane Implementation +We have fully implemented the ChameleMon data plane on +the switch data planes of four edge Tofino switches in P4 [32]. +In this section, we detail the implementation logic of data +plane along the workflow (Figure 15). +Hash: First, a packet with flow ID f enters the network at an +edge switch. With its flow ID (5-tuple) as input, the packet +is hashed to multiple indexes through pairwise-independent +hash functions generated from different CRC polynomials, +which are deployed at stage 0 in ingress. These hash indexes +are either used as base indexes for locating the mapped coun- +ters/buckets in the subsequent insertions, or used for sampling +LL candidates, or used as fingerprints for improving decoding +success rate of FermatSketch. Note that due to the limitation +of Tofino switches, each hash index is uniformly distributed +on [0, 2t −1], where t is an arbitrary positive integer. +1-bit flipping timestamp: Second, the packet reads the cur- +rent 1-bit flipping timestamp and from a match-action table, +23 + +Hash +Stages: #0 +#1 #2-3 #4 #8-11 +Time +Stamp +Flow +Classifier +Sampling +Blackbox +Network +Stages: #4-7 +Downstream +Flow Encoder +Ingress Pipeline +Egress Pipeline +Upstream +Flow Encoder +Figure 15: Implementation logic of ChameleMon. +which is deployed at stage 1 in ingress. The 1-bit timestamp +is used to indicate the corresponding group of sketches for +the subsequent insertions. +Flow classifier: Third, the packet is inserted into the flow +classifier, which is deployed at stage 2-3 in ingress. The flow +classifier is a TowerSketch consisting of an 8-bit counter array +and a 16-bit counter array. The 8-bit and 16-bit counter arrays +consist of w1 8-bit and w2 16-bit counters, respectively. Each +counter array is built on a register and accessed by a stateful +arithmetic logic unit (SALU). To save SALU resources, we +simulate the two flow classifiers by doubling the number of +counters of the 8-bit and 16-bit counter arrays instead of build- +ing additional registers. The left/right w1 8-bit and w2 16-bit +counters form the flow classifier corresponding to timestamp +0/1, respectively. We use the base indexes calculated by hash +functions as the relative positions of the mapped counters in +the flow classifier, and add offsets corresponding to the 1-bit +timestamp to the base indexes, so as to locate the mapped +counters. Specifically, when the timestamp is 0, the offset is +0; when the timestamp is 1, the offset is w1 for 8-bit counter +array or w2 for 16-bit counter array. During insertion, the +SALU adopts saturated addition operation for each mapped +counter, which can increment the counter to its maximum +value but never overflow it, and reports the value recorded in +the counter, so as to simulate the behavior of TowerSketch. +After insertion, we take the minimum value among the re- +ported values as the size of flow f, and then input the flow +size to a match-action Table that uses range matching on the +flow size, so as to obtain the hierarchy of flow f. +Sampling: Fourth, if flow f is classified as a LL candidate, +the packet reads a value from a match-action table, which is +deployed at stage 4 in ingress. We then compare the read value +with a 16-bit value, which is calculated by a hash function +with the 5-tuple of the flow and a random seed11 as input. If +the read value is equal to or larger than the 16-bit value, flow +f is classified as a sampled LL candidate. Otherwise, the flow +is classified as a non-sampled LL candidate. Obviously, to +simulate a sample rate R, the value should be set to ⌈65536× +R⌉. +FermatSketch: Before detailing the implementation of up- +stream and downstream flow encoders, we present the imple- +mentation of FermatSketch that they are based on. To encode +the 104-bit flow ID (5-tuple) of each packet, an ideal bucket +in FermatSketch should contain a 105-bit IDsum field and +a 32-bit count field. However, because each SALU can ac- +cess up to a pair of 32-bit counters, the IDsum field cannot +11The random seed is used to randomize the sampled LL candidates, and +we change the seed every epoch. +srcPort[14:0] +dstPort[15:0] +32-bit register +FingerPrint[19:0] Rest[10:0] +32-bit register +srcIP[30:0] +32-bit register +32-bit register +dstIP[30:0] +Rest[10:0] : srcIP[31] + dstIP[31] + srcPort[15] + protocol[7:0] +: 1-bit reserved bit +Figure 16: Division of the 5-tuple. +Algorithm 3: Simulated modular addition. +Input: An ID fragment f, a counter reg for encoding +the ID fragment, and a prime p. +1 inv ← p− f +▷ Get the additive inverse of f in Zp +2 if reg+ f < p then +3 +reg ← reg+ f +4 else +5 +reg ← reg−inv +6 end +be directly built in Tofino. To address this issue, we divide +the IDsum field into multiple counters. Rather than encod- +ing complete flow IDs, each counter only encodes specific +ID fragments. Considering that a 32-bit counter can support +at most 32-bit primes, and thus can encode at most 31-bit +ID fragment, we need four 32-bit counters to simulate the +IDsum field. Specifically, the division of the IDsum field is +shown in Figure 16. The first three 32-bit counters encode +the lower 31-bits of the source IP address, the destination IP +address, and the concatenation of the source port and desti- +nation port, respectively. The last one 32-bit counter encodes +the rest 11-bit ID fragment (1-bit source IP address + 1-bit +destination IP address + 1-bit source port + 8-bit protocol), +and the other unused 20 bits are used to encode a fingerprint +to improve decoding success rate. In summary, each bucket +of FermatSketch consists of five 32-bit counters: four coun- +ters to encode the IDsum field and the fingerprint field, and a +counter to encode the count field. Considering that there is no +dependency between the five counters in any bucket of Fer- +matSketch, a bucket array of FermatSketch can be built with +five 32-bit counter arrays, each of which is built on a register +and accessed by a SALU. During insertion, for any of the four +counter arrays encoding the IDsum field and fingerprint field, +the SALU needs to insert the specific ID fragment into its +counter through modular addition. As shown in Algorithm +3, the SALU simulates the modular addition with logic con- +sisting of a conditional judgement and two branches. Such +logic is naturally supported by SALUs. For the other counter +array encoding the count field, the SALU simply increments +its counter by one. In this way, the SALUs simulate the be- +havior of FermatSketch. By duplicating these five registers +and SALUs d times, we can easily build a d-array FermatS- +24 + +ketch. Note that we use registers consisting of 32-bit counters, +but not registers consisting of pairs of 32-bit counters that +can further save SALU resources, to simulate the buckets of +FermatSketch. This is because the logic used to simulate the +modular addition requires two 32-bit metadata ( f and inv) as +input, which is just the maximum number that a SALU can +support. However, encoding two ID fragments with a SALU +requires four 32-bit metadata as input, which is beyond the +capabilities of SALU. +Upstream flow encoder: Fifth, unless flow f is a non- +sampled LL candidate, the packet is inserted into the upstream +flow encoder, which is deployed at stage 8-11 in ingress. The +upstream flow encoder consists of three bucket arrays for +the highest memory efficiency. Each bucket array is built as +described above, and consists of mu f buckets. The left mll +buckets, the right mhh buckets, and the middle mhl buckets +in each array form the upstream LL encoder, HH encoder, +and HL encoder, respectively. Similarly, we simulate the two +upstream flow encoders by doubling the number of buckets +in each array. Based on the hierarchy of flow f, we can easily +determine the encoder that the packet should be inserted into. +We denote the number of buckets of a bucket array of that +encoder by m′. Different from the flow classifier, the base +indexes calculate by hash functions cannot be directly used +to locate the relative positions of the mapped buckets in the +encoder. This is because a base index is uniformly distributed +on [0,2t −1], while m′, which could be any of mll, mhl, and +mhh, may not be powers of two, as they are required to vary +for supporting dynamic memory allocation. To address this +issue, we decide to use the results of base indexes modulo +m′ as the relative positions of the mapped buckets. To simu- +late modulo operation in data plane, we input the hierarchy +of flow f and a base index hb to a match-action table that +uses exact matching on flow hierarchy and range matching on +index. The table first determines m′ based on the input flow +hierarchy, then outputs the largest number that is divisible by +m′ and less than hb, and finally subtracts that number from hb. +Obviously, the result equals to hb modulo m′. In this way, we +locate the relative positions of the mapped buckets at the cost +of TCAM resources, and can finally locate the mapped buck- +ets by adding offsets corresponding to the 1-bit timestamp +and the flow hierarchy to the relative positions. Considering +that the width of base index is fixed at run-time, if its width is +too long compared to the width of m′, the match-action table +will need a lot of entries to support range matching, and thus +consumes lots of TCAM resources; if its width is just a bit +longer than the width of m′, the uniformity of the calculated +relative positions will be quite poor, leading to reduction of +the decoding success rate of FermatSketch. To address this +issue, before we input the base index to the match-action table, +we bitwise-AND the base index with a mask to guarantee that +the value range of the index is between 4m′ and 8m′, so as to +make great trade-off between the uniformity of relative posi- +tions and the consumption of TCAM resources. Note that due +to the inherent features of TCAM, when TCAM is used for +range matching, different value range would require different +number of TCAM entries for supporting modulo operation. +Downstream flow encoder: Sixth, unless flow f is a non- +sampled LL candidate, the packet is inserted into the upstream +flow encoder, which is deployed at stage 4-7 in egress. The +implementation of downstream flow encoder is almost the +same as that of upstream flow encoder, except it omits the +heavy-hitter encoder. Note that the flow hierarchy and 1-bit +timestamp are obtained from the edge switch where the packet +enters the network, and carried by recording them in three +bits of the ToS field of the IPv4 protocol. +Resources Usage: As shown in Table 1, under the parameter +settings in Section 5.2, the ChameleMon data plane consumes +SALUs most, achieving 66.7%. This is because the flow clas- +sifier, the upstream flow encoder, and the downstream flow +encoder all need SALUs to access memory. For resources +other than SALUs, ChameleMon consumes no more than +25%. Overall, the resource usage of ChameleMon is moder- +ate. Although ChameleMon indeed consumes a lot of SALUs, +the consumption of SALUs will not increase when we fur- +ther enlarge the above sketches. With the advent of Tofino 2 +switches and even Tofino 3 switches, we believe the resource +usage will be much more acceptable on these more advanced +programmable switches. +Resource +Usage +Percentage +Exact Match Input xbar +353 +22.98% +Ternary Match Input xbar +33 +4.17% +VLIW Instructions +43 +11.20% +Map RAM +102 +17.71% +SRAM +130 +13.54% +TCAM +8 +2.78% +Hash Bits +809 +16.21% +Stateful ALU +32 +66.67% +Table 1: Resources used by ChameleMon in Tofino. +D.2 +Control Plane Implementation +Central controller: The central controller integrates three +modules into a DPDK [36] program: 1) a packet receiver +module responsible for collecting sketches; 2) an analyzer +module for decoding sketches, monitoring real-time network +state, and generating reconfiguration packets for reconfigur- +ing the ChameleMon data plane; 3) a packet sender module +responsible for sending reconfiguration packets to the control +plane of each edge switch. +Switch control plane: The control plane of each edge switch +runs a C++ program to load the P4 program to the Tofino +ASIC. Every time the switch control plane receives a recon- +figuration packet, it first extracts the packet to obtain the +reconfiguration. Then, based on the reconfiguration, it gen- +erates corresponding table entries and update them to the +corresponding match-action tables in the data plane to re- +configure the switch data plane. The time consumption in +25 + +this step is shown in Figure 12 in Section 5.2. To avoid the +updated entries to interfere with the monitoring of the current +epoch, those corresponding match-action tables further use +exact matching on the 1-bit timestamp. Those newly updated +entries match the 1-bit timestamp in the next epoch, so as to +function in the next epoch. +Epoch length: On our testbed, we set the epoch length to +50ms by default, and the additional time for all traffic passing +through the network is set to 10ms (described in Appendix +B). +Clock synchronization: On our testbed, we use the well- +known software time synchronization protocol NTP [105] to +synchronize the clocks between the control planes of edge +switches and the central controller. Every 10s, every edge +switch synchronizes its clock with the central controller. The +precision of synchronization is around 0.3ms∼0.5ms, and +thus NTP can already satisfy the precision requirement for +epochs of 50ms. We can further improve the precision through +more advanced software solutions, such as Huygens [106], or +hardware solutions, such as DPTP [107], to achieve us-level +or even ns-level precision. +Data plane collection: To collect sketches from data planes +of edge switches, a naive solution is to directly use the C++ +control plane APIs provided by the Tofino SDK [108]. Cur- +rently, the most efficient strategy for this solution is to first +use bulk DMA transfer to read data plane counter arrays into +control plane buffer, and then read the counter arrays from +control plane buffer [109]. However, on our testbed, such +strategy takes about 338ms to collect only the upstream flow +encoder, which seriously limits the setting of epoch length, +and thus degrades the accuracy and timeliness of measure- +ment. To address this issue, we fully exploit the capabilities +and features of programmable data plane, including SALUs, +mirror, and recirculate ports. Specifically, the switch control +plane just needs to send several tailored packets to data plane +for collecting sketches. The tailored packet is forwarded to the +recirculate port, so as to access the counters of each sketches +in turn. Every time a tailored packet accesses a counter, lever- +aging the SALU, it reads the value and inserts the value into +its payload. Every time a tailored packet reaches the maxi- +mum transmission unit (MTU, e.g., 1514 Bytes), the switch +data plane forwards it to the switch control plane, and mirrors +a new truncated packet (e.g., 64 Bytes) to read the remaining +counters. In this way, collecting the upstream flow encoder +from the switch data plane only takes 0.44ms, which is 775 +times faster than the straightforward solution. Note that if we +allocate a certain amount of buffer to the recirculate port to +store the tailored packets, they will not be lost during the col- +lection even though the switch is extremely congested. Then, +the switch control plane can transmit the collected sketches +to the central controller through persistent TCP connections +to minimize the transmission delay as well as guarantee the +reliability of transmission. On our testbed, we simplify the +transmission process, that the tailored packets reaching the +MTU will be directly forwarded to the central controller from +switch data planes. To ensure that the tailored packets will +not be lost during the transmission, we reserve idle ports in +their forwarding paths. Overall, the central controller takes +11.33ms to collect sketches from the data plane each edge +switch, which consists five parts: 1) every time the timestamp +flips, the central controller first sleeps 1ms to eliminate the +impact caused by the error in clock synchronization, ensuring +that all the edge switches have started the current epoch; 2) the +central controller takes 2.68ms to collect the flow classifier; +3) the central controller takes 0.44ms to collect the upstream +flow encoder; 4) the central controller sleeps 6.88ms to ensure +that all the packets in the previous epoch have passed through +or lost in the network; 5) the central controller takes 0.33ms +to collect the downstream flow encoder. +E +Evaluation on Different Workloads +In this section, we show that on workloads other than DCTCP, +how ChameleMon shifts measurement attention with the +change of the number of flows or ratio of victim flows. For the +measurement attention under different number of flows, we +vary the number of flows in the network from 10K to 100K, +and fix the ratio of victim flows to 10%. For the measurement +attention under different ratios of victim flows, we vary the +ratio of victim flows from 2.5% to 25%, and fix the number +of flows to 50K. +E.1 +CACHE Workload +Measurement attention vs. number of flows (Figure 17): +As the number of flows increases from 10K to 20K, Chamele- +Mon can record all flows and victim flows, and therefore sets +both Th and Tl to 1. As the number of flows increases from +30K to 70K, ChameleMon allocates more memory to HL en- +coders and raises Th higher than 1. As the number of flows +increases from 80K to 100K, the healthy network state transi- +tions to the ill network state. ChameleMon allocates memory +to LL encoders, increases Tl and decreases the sample rate, +so as to control the number of HLs and sampled LLs. Mean- +while, ChameleMon raises Th to control the number of HH +candidates. The relatively low load factor when the number +of flows is between 80K and 100K is because of the high +skewness of CACHE workload: lower thresholds will lead to +a huge increase of the number of recorded flows, thus caus- +ing decoding failure. In fact, when the number of flows is +between 80K and 100K, the Th is set to 3, and the Tl is set +to 2. ChameleMon has tried its best to select thresholds to +maximizes the load factor. +Measurement attention vs. ratio of victim flows (Fig- +ure 18): As the ratio of victim flows increases from 2.5% +to 12.5%, ChameleMon records all victim flows by allocating +more and more memory to HL encoders. Th is not adjusted +because of the high skewness of CACHE workload: setting +26 + +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HH +HL +LL +(a) Memory division. +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +20 +40 +60 +80 +100 +# Total flows (K) +10 +0 +10 +1 +Threshold +Th +Tl +(c) Threshold. +0 +20 +40 +60 +80 +100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 17: Measurement attention vs. number of flows on CACHE workload. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HH +HL +LL +(a) Memory division. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +10 +0 +10 +1 +Threshold +Th +Tl +(c) Threshold. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 18: Measurement attention vs. ratio of victim flows on CACHE workload. +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +# Total flows (K) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +20 +40 +60 +80 +100 +# Total flows (K) +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Threshold +Th +Tl +(c) Threshold. +0 +20 +40 +60 +80 +100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 19: Measurement attention vs. number of flows on VL2 workload. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Threshold +Th +Tl +(c) Threshold. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 20: Measurement attention vs. ratio of victim flows on VL2 workload. +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +# Total flows (K) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +20 +40 +60 +80 +100 +# Total flows (K) +10 +0 +10 +1 +10 +2 +10 +3 +Threshold +Th +Tl +(c) Threshold. +0 +20 +40 +60 +80 +100 +# Total flows (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 21: Measurement attention vs. number of flows on HADOOP workload. +Th to 2 already makes a fairly small portion of flows as HH +candidates, and lower Th leads to decoding failure. As the +ratio of victim flows increases from 15% to 25%, the healthy +network state transitions to the ill network state. Chamele- +Mon allocates memory to LL encoders, increases Tl to 2 and +decreases the sample rate so as to control the number of +HLs and sampled LLs. Meanwhile, because the memory of +upstream heavy-hitter encoder and the number of flows re- +main unchanged, Th also remains unchanged. The reason why +ChameleMon suffers low load factor when the ratio of victim +flows is between 15% to 25% is also due to high skewness of +CACHE workload. Both Th and Tl are set to 2, and decrease +27 + +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory division +HHE +HLE +LLE +(a) Memory division. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0 +2 +4 +6 +8 +10 +# Decoded flows (K) +HH +HL +LL +(b) Number of decoded flows. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +10 +0 +10 +1 +10 +2 +Threshold +Th +Tl +(c) Threshold. +0 +5 +10 +15 +20 +25 +Ratio of victim flows (%) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample rate +(d) Sample rate. +Figure 22: Measurement attention vs. ratio of victim flows on HADOOP workload. +of thresholds will lead to decoding failure. ChameleMon has +tried its best to select thresholds to maximize the load factor. +E.2 +VL2 Workload +Measurement attention vs. number of flows (Figure 19): +As the number of flows increases from 10K to 20K, Chamele- +Mon can record all flows and victim flows, and therefore sets +both Th and Tl to 1. As the number of flows increases from +30K to 60K, ChameleMon allocates more and more memory +to HL encoders, and increases Th to decrease the number of +HH candidates to avoid decoding failure. As the number of +flows increases from 70K to 100K, the healthy network state +transitions to the ill network state. ChameleMon allocates +memory to LL encoders, increases Tl, and decreases the sam- +ple rate, so as to to control the number of HLs and sampled +LLs. Meanwhile, ChameleMon keeps increasing Th to control +the number of HH candidates. Throughout the experiment, +ChameleMon maintains the load factor higher than 51%. The +load factor is sightly lower, and it is because the distribution +of VL2 is highly skewed. Decreasing the thresholds by 1 will +lead to huge increase in the number of recorded flows, and +thus causing decoding failure. +Measurement attention vs. ratio of victim flows (Fig- +ure 20): As the ratio of victim flows increases from 2.5% +to 12.5%, ChameleMon records all victim flows by allocat- +ing more and more memory to HL encoders, and increases +Th to decrease the number of HH candidates. As the ratio +of victim flows increases from 15% to 25%, ChameleMon +cannot record all victim flows and thus the healthy network +state transitions to the ill network state. ChameleMon allo- +cates memory to LL encoders, increases Tl, and decreases the +sample rate so as to control the number of HLs and sampled +LLs. Meanwhile, because the memory of upstream HH en- +coders and the number of flows remain unchanged, Th also +remains unchanged. Throughout the experiment, Chamele- +Mon maintains the load factor higher than 53%. The load +factor is sightly lower, and the reason is the same as the for- +mer experiment of the the number of flow. +E.3 +HADOOP Workload +Measurement attention vs. number of flows (Figure 21): +As the number of flows increases from 10K to 20K, Chamele- +Mon can record all flows and victim flows, and therefore sets +both Th and Tl to 1. As the number of flows increases from +30K to 60K, ChameleMon allocates more and more mem- +ory to HL encoders, and increases Th to decrease the number +of HH candidates to avoid decoding failure. As the number +of flows increases from 70K to 100K, the healthy network +state transitions to the ill network state. ChameleMon allo- +cates memory to LL encoders, increases Tl, and decreases +the sample rate to control the number of HLs and sampled +LLs. Meanwhile, ChameleMon keeps increasing Th to control +the number of HH candidates. Throughout the experiment, +ChameleMon maintains the load factor higher than 47%. The +load factor is sightly lower, and it is because the distribution +of HADOOP is highly skewed. Decreasing the thresholds by +1 will lead to huge increase in the number of recorded flows, +and thus causing decoding failure. +Measurement attention vs. ratio of victim flows (Fig- +ure 22): As the ratio of victim flows increases from 2.5% +to 12.5%, ChameleMon records all victim flows by allocating +more and more memory to HL encoders, and increases Th to +decrease the number of HH candidates. As the ratio of vic- +tim flows increases from 15% to 25%, ChameleMon cannot +record all victim flows and thus transitions to ill network state. +ChameleMon allocates memory to LL encoders, increase Tl, +and decreases sample rate, so as to control the number of HHs +and HLs. Throughout the experiment, ChameleMon maintains +the load factor higher than 48%. The load factor is sightly +lower, and the reason is the same as the former experiment of +the the number of flow. +28 + diff --git a/T9AyT4oBgHgl3EQfufnB/content/tmp_files/load_file.txt b/T9AyT4oBgHgl3EQfufnB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c32662ed752358737f1bfbbcb6f6c38ffd49c08 --- /dev/null +++ b/T9AyT4oBgHgl3EQfufnB/content/tmp_files/load_file.txt @@ -0,0 +1,1629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf,len=1628 +page_content='ChameleMon: Shifting Measurement Attention as Network State Changes Kaicheng Yang†, Yuhan Wu†, Ruijie Miao†, Tong Yang†, Zirui Liu†, Zicang Xu†, Rui Qiu†, Yikai Zhao†, Hanglong Lv†, Zhigang Ji§, Gaogang Xie¶ †Department of Computer Science, Peking University, China §Huawei Technologies Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', Ltd, China ¶CNIC CAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' UCAS, China Abstract Flow-level network measurement is critical to many network applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among various measurement tasks, packet loss detection and heavy-hitter detection are two most important measurement tasks, which we call the two key tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In prac- tice, the two key tasks are often required at the same time, but existing works seldom handle both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this paper, we design ChameleMon to support the two key tasks simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' One key design/novelty of ChameleMon is to shift measurement attention as network state changes, through two dimensions of dynamics: 1) dynamically allocating memory between the two key tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) dynamically monitoring the flows of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To realize the key design, we propose a key technique, leveraging Fermat’s little theorem to devise a flexible data structure, namely FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FermatSketch is dividable, additive, and subtractive, supporting the two key tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We have fully implemented a ChameleMon prototype on a testbed with a Fat-tree topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We conduct extensive experiments and the results show ChameleMon supports the two key tasks with low memory/bandwidth overhead, and more importantly, it can automatically shift measurement at- tention as network state changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1 Introduction Network measurement provides critical statistics for various network operations, such as traffic engineering [1,2], conges- tion control [3], network accounting and billing [4], anomaly detection [5–8], and failure troubleshooting [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Recently, the emerging programmable switches inject new vitality into network measurement, and designing novel sketches for flow- level measurement capabilities on programmable switches becomes a hot topic [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among all flow-level measurement tasks, packet loss de- tection and heavy-hitter detection attract the most attention, which we call two key tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, the two key tasks are seldom considered and supported simultaneously in one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' One reason behind is that solutions designed for packet loss detection and solutions designed for heavy-hitter detection usually differ significantly in data structures and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, in practice, the two key tasks are often required at the same time, and there are only limited resources for measurement in programmable switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, the first requirement for a practical measurement system is versatile to support the two key tasks with high accuracy using limited resources, where limited resources refer to sub-linear space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Based on the first requirement, the second requirement is to pay attention to different key tasks for different network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When the network state is healthy and there are only few packet losses in the network, the system should pay more attention (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', allocate more memory) to heavy-hitter detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When the network state is ill and there are lots of packet losses in the network, the system should pay more attention to packet loss detection, especially for those flows which ex- perience a great number of packet losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, it is desired to design a practical measurement system with the following requirements: [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1] versatility requirement: supporting both packet loss detection and heavy- hitter detection simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2] efficiency requirement: achieving high accuracy with sub-linear space complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' [R2] attention requirement: paying attention to different key tasks for different network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Existing solutions can be mainly classified into three cate- gories according to supported measurement tasks: 1) Solutions for packet loss detection: Typical solutions including LossRadar [17] based on Invertible Bloom fil- ter [18], NetSight [9] based on in-band network telemetry (INT), Netseer [19] based on the advanced features of pro- grammable switches, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' These solutions fail to meet [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1] because they do not handle heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) Solutions for heavy-hitter detection: These solutions are usually based on sketches, including HashPipe [20], UnivMon [13], ElasticSketch [16], BeauCoup [12], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' These solutions fail to meet [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1] because they do not handle heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='00615v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='NI] 2 Jan 2023 3) Solutions for both tasks: These solutions support both tasks by recording exact IDs of all flows, including FlowRadar [21], OmniMon [22], Counter Braids [23], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, recording exact IDs of all flows requires memory/bandwidth overhead linear with the number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, these solutions fail to meet [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, the first two categories of solutions cannot meet [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1] due to their limited measurement capabilities, and the third category of solutions cannot meet [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2] due to their linear space complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' A naive solution meeting both [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1] and [R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2] is to combine the first two categories of solutions: choosing one solution in each category to de- tect heavy-hitters (HHs for short) and packet losses, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, such a combination fails to achieve [R2] on programmable switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The reason behind is that the data structures and operations of different categories of solutions usually differ significantly1, and therefore the solutions in different categories can only utilize their resources allocated at compile time, which prohibits flexible allocation of mem- ory resources between packet loss detection and heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, the naive solution cannot pay attention to different key tasks for different network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this paper, we design ChameleMon, which meets all the above requirements simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Besides the two key tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', packet loss detection and heavy-hitter detection, ChameleMon can support five common measurement tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Compared to the state-of-the-art solutions, for packet loss detection, ChameleMon reduces the memory overhead from proportional to the number of all flows (FlowRadar) or lost packets (LossRadar), to proportional to the number of flows experiencing packet losses, which we call victim flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' for heavy-hitter detection and the other five tasks, ChameleMon achieves at least comparable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Our ChameleMon has a key design and a key technique as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The key design of ChameleMon is to shift measurement at- tention as network state changes, which is just like the process of the chameleons changing their skin coloration, through two dimensions of dynamics: 1) dynamically allocating memory between the two key tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) dynamically monitoring the flows of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, ChameleMon monitors the net- work state and allocates memory between the two key tasks accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When the network state is healthy and only a few packet losses occur in the network, ChameleMon pays most attention to and allocates most of the memory for heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the network state degrades and packet losses increase, ChameleMon gradually shifts measurement atten- tion to and allocates more and more memory for packet loss detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, ChameleMon ranks the flows according to their importance, and selects those of most importance to monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When the network state is ill and there are too many victim flows in the network, ChameleMon selects those flows 1For example, LossRadar [17] records the IDs and existences of packets using XOR operation and addition, while HashPipe [20] records the IDs and sizes of flows using comparison, substitution, and addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' experiencing many packet losses (called heavy-losses, HLs for short) to monitor, instead of monitoring all victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Overall, when the network state continuously degrades from the healthy state to the ill state, ChameleMon runs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1) As the number of victim flows increases, Chamele- Mon leverages the first dimension of dynamic: gradually shift- ing measurement attention to and allocating more and more memory for packet loss detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) When the victim flows are too many to monitor, ChameleMon leverages the second dimension of dynamic: focusing measurement attention on HLs while monitoring a small portion of other packet losses (named light-losses, LLs for short) through sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To realize the key design, ChameleMon incorporates a key technique, leveraging Fermat’s little theorem2 to devise a flex- ible data structure, namely FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The data structure of FermatSketch is made of many same units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FermatSketch is dividable, additive, and subtractive, supporting packet loss de- tection and heavy hitter detection simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' By dividing FermatSketch into three parts to detect HLs, LLs, and HHs, ChameleMon can flexibly move the division points to shift attention and allocate memory between the two key tasks as network state changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each incoming packet, We further use a flow classifier (TowerSketch [24]) to determine which of the three parts to insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For packet loss detection, owing to Fermat’s little theorem, FermatSketch only requires memory proportional to the number of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Differently, the state-of-the-art solutions require memory proportional to the number of all flows (FlowRadar) or lost packets (LossRadar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Thanks to the visibility to per-flow size provided by Tow- ersketch, ChameleMon can support five other common mea- surement tasks, including flow size estimation, heavy-change detection, flow size distribution estimation, entropy estima- tion, and cardinality estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We have fully implemented a ChameleMon prototype on a testbed with a Fat-tree topology composed of 10 Tofino switches and 8 end-hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We conduct extensive experiments and the results show that ChameleMon supports both kinds of tasks with low memory/bandwidth overhead, and more importantly, it can automatically shift measurement attention as network state changes at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We have released all related source codes at Github3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2 Overview of ChameleMon ChameleMon monitors the network in four steps (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1) Capturing flow-level statistics on edge switches: To cap- ture desired flow-level statistics, ChameleMon deploys three sketches on the data plane of each edge switch, including a flow classifier (TowerSketch), an upstream flow encoder (our FermatSketch), and a downstream flow encoder (our Fer- matSketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To detect HHs, HLs, and LLs, the upstream and downstream flow encoders are divided into multiple parts: 1) the upstream flow encoder is divided into an upstream 2Fermat’s little theorem states that if p is a prime, then for any integer a that is indivisible by p, we have ap−1 ≡ 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='com/ChameleMoncode/ChameleMon 2 Step 1: Capture flow-level statistics Sample 0 0 1 1 0 1 0 1 0 1 Step 2: Collect from edge monitor 𝑆1 Step 3: Perform analysis Step 4: Shift attention Reallocate memory Adjust thresholds UpStream DownStream Edge Analyzer UpStream DownStream Collect HL Encoder HH Encoder LL Encoder Reconfigure data plane Flow size Heavy-change Entropy Flow size distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Packet loss Heavy-hitter Cardinality Key tasks Other tasks collect 𝑆0 monitor 𝑆0 collect 𝑆1 HL Encoder LL Encoder Classifier Figure 1: Overview of ChameleMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' HH encoder, an upstream HL encoder, and an upstream LL encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) the downstream flow encoder is divided into a downstream HL encoder and a downstream LL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For every packet with flow ID f entering the network, according to the size of flow f, the flow classifier classifies flow f into one of three hierarchies: 1) HH candidate, 2) HL candidate, or 3) LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The LL candidate is further classified into sampled LL candidate or non-sampled LL candidate through sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Based on the hierarchy of flow f, the packet is then inserted into the corresponding part of the upstream flow en- coder and downstream flow encoder when it enters and exits the network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) Collecting sketches from edge switches: A central con- troller periodically collects sketches from each edge switch to support persistent measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To avoid colliding with packet insertion when collecting sketches, each edge switch divides the timeline into consecutive fixed-length time in- tervals (called epochs), and copies a group of sketches for rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time an epoch ends, the central controller col- lects the group of sketches monitoring this epoch, and the other group of sketches starts to monitor the current epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3) Performing network-wide analysis: Every epoch, the central controller performs network-wide analysis of the col- lected sketches to support seven measurement tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' By ana- lyzing the upstream and downstream flow encoders, the cen- tral controller can support packet loss detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' By analyzing the flow classifier and the upstream HH encoder, the central controller can support heavy-hitter detection and five common tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4) Shifting measurement attention as network state changes: Every epoch, the central controller monitors the real-time network state by analyzing the collected sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, the central controller reconfigures the data plane of edge switches at run-time according to the real-time network state, shifting measurement attention through two dimensions of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In the first dimension, the central controller dy- namically allocates memory between packet loss detection and heavy-hitter detection by reallocating the memory of the upstream and downstream encoders between their different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In the second dimension, the central controller dynami- cally selects the most important flows (HH/HL/sampled LL candidates) to monitor by adjusting the thresholds for flow classification and the sample rate for sampling LL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3 Design of ChameleMon Data Plane The ChameleMon data plane consists of the flow classifier, the upstream flow encoder, and the downstream flow encoder deployed on each edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this section, we detail the design of the ChameleMon data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we propose the key technique of ChameleMon, namely FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, we detail each component of the ChameleMon data plane in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 The FermatSketch Algorithm Rationale: Our primary goal is to detect packet losses with low memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Existing solutions focus on either per- packet loss (LossRadar [17]) or all-flow visibility (FlowRadar [21]), incurring unacceptable memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To reduce overhead, we hope to aggregate all the lost packets of the same flow to detect per-flow packet losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' It is very challeng- ing because existing solutions commonly use XOR operation for high memory efficiency and hardware-friendliness, but simply using XOR operation to aggregate flow IDs of lost packets causes every two lost packets of the same flow to cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' While invertible Bloom lookup table (IBLT) [25] can overcome this challenge as IBLT uses addi- tion to aggregate flow IDs, such design requires computation over large numbers, and thus complicates the implementation of IBLT on programmable switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this challenge while maintaining hardware-friendliness, we devise FermatS- ketch, which uses modular addition to aggregate flow IDs and leverages Fermat’s little theorem to extract flow IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Data structure (Figure 2): FermatSketch has d equal-sized bucket arrays B1,··· ,Bd, each of which consists of m buck- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each bucket array Bi is associated with a pairwise- independent hash function hi(·) that maps each incoming packet into one bucket (called mapped bucket) in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each bucket Bi[j] contains two fields: 1) a count field Bc i [j] record- ing the number of packets mapped into the bucket;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) an IDsum field BID i [j] recording the result of the sum of flow IDs of packets mapped into the bucket modulo a prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' At initialization, we set all fields of all buckets in FermatSketch to zero, and p to a prime that must be larger than any avail- able flow ID f and the size of any flow, so as to make use of Fermat’s little theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Encoding/Insertion operation (Figure 2): To encode an incoming packet with flow ID f, we first calculate the d hash functions to locate the d mapped buck- ets: B1[h1( f)],B2[h2(f)],··· ,Bd[hd(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each mapped bucket Bi[hi( f)], we update it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we in- crement its count field Bc i [hi( f)] by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, we up- date its IDsum field through modular addition: BID i [f)] ← ��BID i [hi(f)]+ f � mod p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The pseudo-code of encoding operation is shown in Algorithm 1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Decoding operation: The decoding operation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' which can ex- tract exact flow IDs and flow sizes from FermatSketch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' has two important suboperations: 1) pure bucket verification that verifies whether a bucket only records packets of a single 3 HH HIHHTH+ 1 8 0 1 0 0 0 4 0 0 1 0 0 0 4 0 0 0 ℬ1 𝐶 ℬ1 𝐼𝐷 ℬ1 1 ~ℬ1 4 ℬ2 1 ~ℬ2 4 Modular Addition ℬ2 𝐶 ℬ2 𝐼𝐷 0 2 0 0 0 1 0 0 1 0 1 0 4 0 8 0 ℬ1 𝐶 ℬ1 𝐼𝐷 ℬ2 𝐶 ℬ2 𝐼𝐷 + 1 8 𝑝 = 11 𝑓 Packet with 𝑓 = 8 Figure 2: An example of encoding/insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' flow (pure bucket);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) single flow extraction that extracts and deletes a single flow and its size from all its mapped buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Next, we first propose the workflow of the decoding operation, and then detail the two suboperations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The pseudo-code of decoding operation is shown in Algorithm 2 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Decoding workflow (Figure 3): The decoding operation proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1 Traverse FermatSketch and push all non-zero buckets to the decoding queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2 Pop a bucket from the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3 For the popped bucket, we perform pure bucket verifica- tion to verify whether it is a pure bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If not, we simply ignore the bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4 If so, we perform single flow extraction to extract and delete a single flow and its size from the pure bucket as well as the other mapped buckets of the single flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5 We insert the extracted single flow and its size into a hash table, namely Flowset, which is used to record all the extracted flows and their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We regard all flows recorded in the Flowset as the flows previously encoded into FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 6 Except the popped pure bucket, we push the other mapped non-zero buckets of the extracted flow into the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 7 Check whether the queue is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If so, the decoding stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Otherwise, go back to step 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' After stopping, if there are still non-zero buckets in FermatSketch, the decoding is considered as failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Otherwise, the decoding is considered as successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Pure bucket verification: The pure bucket verification re- ports whether one given bucket is pure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', only records a single flow), but it may misjudge a non-pure bucket as a pure one with a small probability 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Suppose a bucket Bi[j] only records a single flow f ′, it should satisfy that (Bc i [j] × f ′) mod p = BID i [j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Leveraging Fermat’s little theorem, we can get that f ′ = (BID i [j]×(Bc i [j])p−2) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that bucket Bi[j] should be one of the d mapped buckets of flow f ′, to verify whether Bi[j] is a pure bucket, we propose a verifica- tion method namely rehashing verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we calculate the ith hash function hi(·) to locate the ith mapped bucket of f ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', we calculate hi( f ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then we check whether hi( f ′) is equal to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If so, we consider Bi[j] as a pure bucket recording flow f ′ with size Bc i [j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that the false positive rate of pure bucket verification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', the probability of misjudging a non- pure bucket as a pure one, is 1 m, which is calculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For any non-pure bucket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' we can calculate its flow ID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' which ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐼𝐷 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝑓′ = ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐼𝐷 1 × (ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐶[1])𝑝−2 = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Pure Bucket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Verification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℎ2 𝑓′ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 1 ~ℬ1 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 1 ~ℬ2 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Modular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Subtraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐼𝐷 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐼𝐷 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝐼𝐷 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='ℬ1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='POP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='PUSH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='··· ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Single Flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Rehashing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝑝 = 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='CHECK: if non-zero bucket exists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='𝑓′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Insert into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Flowset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Initialization: push all non-zero buckets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Figure 3: An example of decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' should be considered as a random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The probability that a random ID is hashed to the same bucket is 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Single flow extraction: To extract/delete flow f ′ from Bi[j] as well as its other mapped buckets, first, we locate its other (d − 1) mapped buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, for each mapped bucket Bi′[hi′( f ′)], we decrement its count field Bc i′[hi′(f ′)] by Bc i [j], and update its IDsum field to ((BID i′ [hi′( f ′)]−BID i [j]) mod p) through modular subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Addition/Subtraction operations: Adding/Subtracting Fer- matSketch FS1 to/from FermatSketch FS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FS1 and FS2 must use the same parameters including the hash functions, number of arrays, number of buckets, and primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each bucket of FS2, we update it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we locate the bucket of FS1 that is in the same position as it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, we add/subtract the count field of the located bucket of FS1 to/from its count field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Third, we modular add/subtract the IDsum field of the located bucket of FS1 to/from its IDsum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' [Optional] fingerprint verification: To reduce the false pos- itive rate of pure bucket verification, we can perform an extra verification method, namely fingerprint verification, by ex- tending the IDsum field in each bucket by w bits and using the extra w bits as a fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For simplicity and accuracy, we recommend implementing FermatSketch without finger- prints in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Only if there is some memory that can hardly be utilized due to hardware constraints unless used as fingerprints4, we recommend implementing FermatSketch with fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Discussion: We further discuss the space and time complexity of FermatSketch in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3, where we also show how the decoding operation of FermatSketch automatically elimi- nates the false positives generated in pure bucket verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary: 1) FermatSketch achieves the highest memory efficiency when d is set to 3, that on average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='23 buckets can record a flow and its size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) when used as a hash table, its memory overhead is proportional to the number of inserted flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3) when used for packet loss detection, its memory 4An example is the implementation of FermatSketch in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Please refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4 overhead is proportional to the number of victim flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4) the time complexity of decoding is O(md2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Mathematical analysis: Let FermatSketch consists of d bucket arrays, each of which consists of m buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Let M be the number of flows inserted into that FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 to show that when FermatSketch is configured correctly and M is not too small, the decoding of FermatS- ketch has an extremely small probability of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the detailed proof, please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Suppose md > cdM + ε and M ⩾ Ω(d4dlogd(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' the decoding of FermatSketch fails with probability O( 1 Md−2 ), where both ε and cd are small constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' cd = � sup � α ���α ∈ (0,1),∀x ∈ (0,1),1−e−dαxd−1��−1 For example, c3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='222,c4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='295,c5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 ChameleMon Data Plane Components As shown in Figure 1, every packet entering the network undergoes the three components of the ChameleMon data plane in sequence: 1) the flow classifier, 2) the upstream flow encoder, and 3) the downstream flow encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Flow Classifier Rationale: To detect HHs, HLs, and LLs, ChameleMon de- ploys the flow classifier in the ingress of each edge switch, so as to classify flows into different hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' While it is easy to select HHs to monitor according to flow sizes, it is not easy to select HLs to monitor because we can hardly predict how many packets a flow will lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Our observation is that for each flow, the number of its lost packets cannot exceed its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, the sizes of HLs should have a minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon selects flows whose sizes exceed this value to monitor, so as to approximate the monitoring of HLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, the flow classifier classifies flows purely according to flow sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Thus, we choose TowerSketch [24], a simple, accurate, and hardware-friendly sketch, as the flow classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Data Structure: The flow classifier consists of l equal-sized arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The ith array Ai consists of wi δi-bit counters, where wi ×δi is a constant and δi−1 < δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Also, array Ai is associated with a pairwise-independent hash function si(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each δi- bit counter, its maximum value 2δi −1 is used to represent the state that it is overflowed, and thus be regarded as +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Insertion: To insert a packet with flow ID f, we first calculate the l hash functions to locate l counters: A1[s1( f)],A2[s2( f)],··· ,Al[sl( f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We call these counters the l mapped counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, for each of the l mapped counters, we increment it by one unless it is overflowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Online query: To query the size of flow f online, we sim- ply report the minimum value among the l mapped counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Packet processing: For a packet with flow ID f entering the network, the flow classifier processes it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we insert it into the flow classifier and query the size of flow f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, with the queried flow size, we classify flow f into the corresponding hierarchy according to two thresholds Th and Tl, where Th is used for selecting HH candidates, and Tl is used for selecting HL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In general, it satisfies that Tl <= Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the flow size is larger than or equal to Th, flow f is classified as a HH candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the flow size is less than Tl, flow f is classified as a LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the flow size is between Tl and Th, flow f is classified as a HL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The LL candidate is further classified into sampled LL candidate or non-sample LL candidate through sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Upstream Flow Encoder Rationale: To support packet loss detection, ChameleMon deploys the upstream flow encoder in the ingress of each edge switch just after the flow classifier, so as to encode the packets entering the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, the upstream flow encoder should contain two FermatSketches to encode HL candidates and sampled LL candidates individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, for better mon- itoring of the network state, ChameleMon monitors a portion of LLs to maintain an overview of all victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Besides, to support heavy-hitter detection, the upstream flow encoder should contain a FermatSketch to encode HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, the upstream flow encoder should consist of three FermatSketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Data structure: The upstream flow encoder is a d-array Fer- matSketch divided into three d-array FermatSketches: 1) an upstream HH encoder for encoding HH candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) an upstream HL encoder for encoding HL candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3) an upstream LL encoder for encoding sampled LL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We denote the number of buckets per array of the upstream flow encoder, HH encoder, HL encoder, and LL encoder by muf , mhh, mhl, and mll, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Obviously, it satisfies that muf = mhh +mhl +mll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Packet processing: For a packet with flow ID f entering the network, the upstream flow encoder processes it by encoding the packet into one of the encoders corresponding to the hier- archy of flow f unless flow f is a non-sampled LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, the hierarchy of flow f can be directly obtained because the upstream flow encoder and the flow classifier are deployed on the same edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 Downstream Flow Encoder Rationale: To support packet loss detection, ChameleMon de- ploys the downstream flow encoder in the egress of each edge switch, so as to encode the packets exiting the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the downstream flow encoder is not responsible for heavy-hitter detection, it should consist of two FermatSketches to encode HL candidates and sampled LL candidates individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Data structure: The downstream flow encoder is a d-array FermatSketch divided into two d-array FermatSketches: 1) a downstream HL encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) a downstream LL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To support packet loss detection, the number of buckets per array of the downstream HL encoder and LL encoder must also be mhl and mll, respectively, so as to support addition and subtraction operations with the corresponding upstream 5 encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We denote the number of buckets per array of the downstream flow encoder by md f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In general, it satisfies that md f < muf , and therefore satisfies that md f ⩾ mhl +mll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Packet processing: For a packet with flow ID f exiting the network, the downstream flow encoder processes it by en- coding the packet into one of the encoders corresponding to the hierarchy of flow f unless flow f is a non-sampled LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, packets of HH candidates are also encoded into the downstream HL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Different from the upstream flow encoder, the downstream flow encoder cannot directly obtain the flow hierarchy from the flow classifier, as a flow could enter and exit the network at different edge switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, first, considering that there are four flow hi- erarchies, we can use ⌈log(4)⌉ = 2 bits in the original packet header to transmit this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For example, for IPv4 protocol, we can use the unused bits in the type of service (ToS) field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If there are not enough unused bits, second, we can transmit the flow hierarchy in an INT-like [26] manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4 Design of ChameleMon Control Plane The ChameleMon control plane consists of a central con- troller, as well as the control plane of each edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this section, we detail the design of the ChameleMon con- trol plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We begin by laying out how the ChameleMon control plane collects sketches from the ChameleMon data plane, then introduce how to support seven measurement tasks with the collected sketches, and finally propose how to shift measurement attention as network state changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Collection from ChameleMon Data Plane The central controller needs to periodically collect sketches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', the flow classifier, the upstream flow encoder, and the downstream flow encoder, from the ChameleMon data plane, so as to support persistent measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, the collec- tion cannot be completed in an instant, and thus inevitably col- lide with packet insertion if there is only a group of sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, if the central controller wants to collect sketches at time t, it will inevitably collect some counters inserted by packets after t, which could result in decoding failure of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, ChameleMon takes two steps: 1) timeline division and 2) clock synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Next, we just briefly cover the two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We detail the two steps in Appendix B, where we further analyze the appropriate time for the central controller to collect sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Timeline division: Each edge switch periodically flips a 1- bit timestamp to divide the timeline into fixed-length time intervals (called epochs) with interleaved 0/1 timestamp, and copies a group of sketches for rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each group of sketches corresponds to a distinct timestamp value, and monitors the epochs with that timestamp value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Clock synchronization: The central controller also main- tains a 1-bit periodically flipping timestamp, and periodically synchronizes its clock with the control plane of each edge switch, so as to make opportunities for collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time the locally maintained 1-bit timestamp flips, an epoch ends, the central controller starts to collect the group of sketches monitoring this epoch, and the other group of sketches starts to monitor the current epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Measurement Tasks With the collected sketches, the central controller can support packet loss detection, heavy-hitter detection, and five other common measurement tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Packet loss detection: reporting each victim flow and the number of its lost packets in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The central con- troller can support packet loss detection by analyzing the upstream and downstream flow encoders collected from each edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, for each edge switch, we decode the upstream HH encoder to obtain the HH Flowset, and then reinsert each flow with its size in the HH Flowset into the upstream HL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, we add up the up- stream/downstream HL/LL encoder of each edge switch through addition operation to obtain the cumulative up- stream/downstream HL/LL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Third, we subtract the cumulative downstream HL/LL encoder from the cumula- tive upstream HL/LL encoder to obtain the delta HL/LL en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Fourth, we decode the delta HL/LL encoder to obtain the HL/LL Flowset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Finally, we report the flows in the HL Flowset as HLs, and the flows in the LL Flowset but not in the HL Flowset as LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each of these flows, its estimated number of lost packets is the sum of its size in the HL Flowset and the LL Flowset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each edge switch, the central controller can support the following six tasks by analyzing the flow classifier and upstream HH encoder collected from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, by synthesizing the results of each edge switch, the central controller can easily support these tasks in a network-wide manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We detail these six tasks from the perspective of an edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Heavy-hitter detection: reporting flows whose sizes exceed ∆h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we decode the upstream HH encoder to obtain the HH Flowset, which records flows with ID fi and size qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For any flow f j in the HH Flowset, if its estimated flow size Th +qj is larger than ∆h, we report it as a HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that Th is the threshold used for selecting HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Flow size estimation: reporting flow size of flow f j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Sim- ilarly, we obtain the HH Flowset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If flow f j is in the HH Flowset, we report its flow size as Th + qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Otherwise, we report its flow size as query result from the flow classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Heavy-change detection: reporting flows whose sizes change beyond ∆c in two adjacent epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Similarly, we ob- tain the HH Flowset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For any flow f j in the HH Flowset of either epoch, we estimate its flow size in the two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the difference between the two estimated flow sizes is larger than ∆c, we report flow f j as a heavy-change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Cardinality estimation: reporting number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We ap- ply linear-counting algorithm [27] to the counter array with the most counters in the flow classifier to estimate cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Flow size distribution estimation: reporting distribution of flow sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We apply MRAC algorithm [28] to each counter 6 array in the flow classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Array Ai provides the estimated distribution of flow size in range [2δi−1 −1,2δi −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The re- maining distribution of flow size in range [2δi −1,+∞) is es- timated from the flows larger than 2δi −2 in the HH Flowset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Entropy estimation: reporting entropy of flow sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Based on the estimated flow size distribution, we can easily com- pute the entropy as follows: −∑ � ni · i N log i N � , where ni is the number of flows of size i, and N = ∑(i·ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 Shifting Measurement Attention A practical measurement system should pay attention to dif- ferent tasks for different network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When there are only rare packet losses in network, the system should pay more attention to and allocate more memory for heavy-hitter de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In contrast, when there are lots of packet losses in network, the system should pay more attention to and allocate more memory for packet loss detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Aiming at this target, ChameleMon decides to shift mea- surement attention as network changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time all the sketches monitoring the previous epoch are collected, ChameleMon takes two phases to shift measurement attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, the central controller monitors the real-time network state, including the number and flow size distribution of flows and victim flows, by analyzing the collected sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Sec- ond, the central controller reconfigures the ChameleMon data plane according to the real-time network state, so as to support packet loss detection and heavy-hitter detection simultane- ously while maintaining high memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The central controller not only reallocates memory of the upstream and downstream encoders between their different parts, but also adjusts the thresholds for flow classification and the sample rate for sampling LL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To avoid interference with the monitoring of the current epoch, the reconfiguration will not function immediately, but in the next epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For ChameleMon, the network state could be clearly clas- sified into two levels: 1) healthy network state that Chamele- Mon can allocate sufficient memory to monitor all victim flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) ill network state that ChameleMon cannot allo- cate sufficient memory to monitor all victim flows, and thus must select HLs to monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each level of network state, ChameleMon behaves almost the same in shifting measure- ment attention, and we detail how it behaves in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Healthy Network State Suppose the previously monitored network state is healthy, and now the central controller starts to shift measurement attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Currently, the LL encoders are not allocated any memory as ChameleMon can monitor all victim flows, and Tl must be 1 as no flows should be classified into LL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The memory allocation between the upstream HH encoder and the upstream HL encoder is flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Monitoring real-time network state: The monitoring pro- ceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, for each edge switch, the central con- troller estimates the number of flows and flow size distribution as described above (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, for each edge switch, the central controller obtains the number of HH candidates by decoding the upstream HH encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' After all decoding stops, if the decoding of any upstream HH encoder fails, the central controller stops the monitoring as the decoding of the delta HL encoder requires reinserting the decoded HH candidates into the upstream HL encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Third, the central controller obtains the number of HLs (equals to victim flows for healthy network state) by decoding the delta HL encoder as described above (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding fails, the central controller estimates the number of HLs by applying linear-counting algorithm to any bucket array of the delta HL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Reconfiguring ChameleMon data plane: The core idea of reconfiguration is to first ensure the successful decoding of FermatSketches for supporting packet loss detection and heavy-hitter detection, while maintaining high memory uti- lization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The reconfiguration proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 1: We focus on the successful decoding of the upstream HH encoders as they are decoded first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each edge switch, if the decoding of the upstream HH encoder fails, the central controller turns up Th according to the number of flows and flow size distribution, controlling the expected load factor5 of the upstream HH encoder at around 70%6, so as to maintain high memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' After turning up Th, the central con- troller stops the reconfiguration as the decoding of the delta HL encoder cannot proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 2: We focus on the successful decoding and high mem- ory utilization of the delta HL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding of the delta HL encoder fails, according to the estimated number of HLs, the central controller estimates the required mem- ory for 70% load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the maximum memory that the HL encoders can be allocated to, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', all the memory of the downstream flow encoder, cannot cover the required memory, the healthy network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this case, the central controller 1) reallocates the memory inside the upstream and downstream flow encoders as the fixed allocation described in the ill network state (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2), 2) sets Tl to Th, and 3) adjusts the sample rate for 70% load factor of the delta LL encoder assuming that each HL will be a LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Otherwise, the central controller just expands the HL encoders to the required memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding of the delta HL encoder succeeds and its load factor is lower than 60%, the central controller tries to compress the HL encoders to ap- proach 70% load factor for high memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, we reserve the minimum memory for the HL encoders to handle the potential small burst of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 3: After all the memory reallocation, we focus on the suc- cessful decoding and high memory utilization of the upstream HH encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each edge switch, with the number of HH 5Load factor refers to the ratio of the number of recorded flows to the num- ber of buckets of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The maximum load factor of FermatSketch is around 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3% = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 6Here, we decide not to pursue the maximum load factor for two reasons: 1) the potential increase of HH candidates in the current epoch and 2) the inevitable estimation error in linear-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 7 candidates and the memory of the upstream HH encoder, the central controller further estimates the expected load factor of the upstream HH encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' if the expected load factor of the upstream HH encoder is lower than 60% or larger than 70%, the central controller turns down or up Th to approach 70% load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Ill Network State Suppose the previously monitored network state is ill, and now the central controller starts to shift measurement attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Currently, all the HH, HL and LL encoders are allocated fixed memory, and Tl must be larger than 1 to select HL candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, the upstream HH encoder is compressed to the minimum memory, which is the memory difference between the upstream flow encoder and the downstream flow encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Monitoring real-time network state: The monitoring pro- ceeds in a similar way to that of the healthy network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In addition, the central controller obtains the number of LLs by decoding the delta LL encoder as described above (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding fails, the central controller estimates the number of LLs by applying linear-counting algorithm to the delta LL encoder, and then stops the monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If both decod- ing of the delta HL and LL encoders succeeds, the central controller estimates the number and flow size distribution of victim flows as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, the central controller samples the HLs with the same sampling method and rate as LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, the central controller merges sampled HLs and sam- pled LLs to obtain sampled victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Third, the central controller estimates the flow size distribution of victim flows through querying the flow size of each sampled victim flow, and the number of victim flows through dividing the number of sampled victim flows by sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding of the delta HL encoder fails, the central controller regards the estimated flow size distribution of sampled LLs, which is also estimated by querying flow sizes, as the flow size distribution of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Reconfiguring ChameleMon data plane: The core idea of reconfiguration is the same as that of the healthy network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The reconfiguration proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 1: We focus on the successful decoding of the upstream HH encoders, and the reconfiguration proceeds the same as the first step of the healthy network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In addition, we focus on the successful decoding of the delta LL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding of the delta LL encoder fails, according to the estimated number of LLs, the central controller adjusts the sample rate to make the delta LL encoder approach 70% load factor, and then stops the reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 2: We focus on the successful decoding of the delta HL encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the decoding of the delta HL encoder fails, according to the estimated flow size distribution of victim flows, assuming that each victim flow larger than Tl will be a HL, the central controller turns up Tl to make the delta HL encoder approach 70% load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 3: we focus on the high memory utilization of the HL and LL encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If both the decoding of the delta HL and LL encoders succeeds, according to the estimated number of victim flows, the central controller estimates the required memory for monitoring all the victim flows with 70% load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the downstream flow encoder can cover the required memory, the ill network state transitions to the healthy net- work state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this case, the central controller 1) eliminates the LL encoders, 2) allocates the required memory (at least the reserved minimum memory) to the HL encoders, and 3) sets Tl to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the downstream flow encoder cannot cover the re- quired memory, and the load factor of the delta HL encoder or the delta LL encoder is lower than 60%, the central controller turns up Tl or the sample rate according to the estimated flow size distribution of victim flows or the estimated number of LLs, respectively, so as to approach 70% load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Step 4: After all the memory reallocation, we focus on the successful decoding and high memory utilization of the up- stream HH encoders, and the reconfiguration proceeds the same as the third step of the healthy network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5 Evaluation We conduct various experiments on CPU platform and our testbed, and focus on the following five key questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We place the experimental results that can answer the second question in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' How much memory/time can ChameleMon save in packet loss detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' (Figure 4 - 6) We implement Fer- matSketch and its competitors in C++, and use CAIDA dataset [29] to evaluate their memory and time overhead for packet loss detection on CPU platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Results show that FermatSketch can save memory in all cases and time in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' How accurately can ChameleMon support heavy-hitter detection and other five tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' (Figure 14) We imple- ment the combination of TowerSketch and FermatSketch and its competitors in C++, and use CAIDA dataset to eval- uate their accuracy for these six tasks on CPU platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Results show that the combination can achieve at least com- parable accuracy in all six tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Can ChameleMon automatically shift measurement at- tention?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' (Figure 7-8) We generate workloads according to widely used traffic distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', DCTCP [30]) for eval- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the above workloads to evaluate Chamele- Mon by generating different network states on our testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Results show that ChameleMon can always automatically shift measurement attention between packet loss detection and heavy-hitter detection at run-time, and maintains high memory utilization in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' How fast can ChameleMon shift measurement atten- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' (Figure 9) We use the above workloads to evaluate ChameleMon over a large time window, in which the net- work state changes 8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Results show that ChameleMon can shift measurement attention within at most 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' How fast can ChameleMon monitor the network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' (Fig- ure 10-12) We use the above workloads to evaluate various 8 factors that can affect the epoch length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Results show that ChameleMon can monitor the network every 50ms on our testbed, using only two CPU cores and consuming only 320Mbps bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We believe ChameleMon can easily scale to monitor a much larger network in a faster manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Evaluation on Packet Loss Detection Dataset: We use the anonymized IP traces collected in 2018 from CAIDA [29] as dataset, and use the 32-bit source IP address as the flow ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the first 100K flows containing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3M packets for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Setup: We set up a simulation with a simple topology con- sisting of only a link on CPU platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We compare Fer- matSketch with FlowRadar [21] and LossRadar [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For FermatSketch, we set its count field and ID field to 32bits, and the number of hash functions to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For FlowRadar, we allocates 10% memory to the flow filter and 90% memory to the counting table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the flow filter, which is actually a Bloom filter [31], we sets its number of hash functions to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the counting table, we set its FlowXOR field, FlowCount field, and PacketCount field to 32bits, and its number of hash functions to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For LossRadar, we set its count field to 32bits, xorSum field to 48bits, and number of hash functions to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, the xorSum field of LossRadar encodes a 32-bit flow ID as well as a 16-bit packet-specific information that represents the order of a packet in a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each solution, we deploy it upstream and downstream of the link to encode the packets entering and exiting the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Memory/Time overhead7 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of victim flows (Fig- ure 4): Experimental results show that the memory/time over- head of FermatSketch is proportional to the number of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We let the largest 10K flows pass through the link, among which a part of flows are victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The packet loss rate of victim flows is set to 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of vic- tim flows increases, the memory/time overhead of FlowRadar remains unchanged, while that of FermatSketch increases almost linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We find when the number of victim flows exceeds 6000, the decoding time of FermatSketch exceeds that of FlowRadar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because the decoding operation of FermatSketch is more complex than FlowRadar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Compared to FlowRadar/LossRadar, FermatSketch saves up to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='9/23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 times memory and up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 times decoding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Memory/Time overhead vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' packet loss rate (Figure 5): Experimental results show that the memory/time overhead of FermatSketch is independent of the number of lost pack- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We let the largest 10K flows pass through the link, among which the largest 100 flows are victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the packet loss rate of victim flows increases, the mem- ory/time overhead of FermatSketch and FlowRadar remains unchanged, while that of LossRadar increases linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Com- pared to FlowRadar/LossRadar, FermatSketch saves up to 7The memory overhead refers to the minimum memory required to achieve 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='9% decoding success rate, and the time overhead refers to the corresponding decoding time with the minimum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2 4 6 8 10 # Victim flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 Minimum memory (MB) Fermat LossRadar FlowRadar (a) Memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2 4 6 8 10 # Victim flows (K) 0 2 4 6 8 10 Decoding time (ms) Fermat LossRadar FlowRadar (b) Time overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 4: Memory/Time overhead vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 Loss rate (%) 0 2 4 6 8 Minimum memory (MB) Fermat LossRadar FlowRadar (a) Memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 Loss rate (%) 0 20 40 60 80 100 Decoding time (ms) Fermat LossRadar FlowRadar (b) Time overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 5: Memory/Time overhead vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' packet loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 3 10 4 10 5 # Total flows 10 3 10 2 10 1 10 0 10 1 Minimum memory (MB) Fermat LossRadar FlowRadar (a) Memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 3 10 4 10 5 # Total flows 10 2 10 1 10 0 10 1 10 2 Decoding time (ms) Fermat LossRadar FlowRadar (b) Time overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 6: Memory/Time overhead vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1/6411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 times memory and up to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5/1585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 times de- coding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Memory/Time overhead vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows (Figure 6): Experimental results show that the memory/time overhead of FermatSketch is independent of the number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We let a certain number of flows pass through the link, among which the largest 100 flows are victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The packet loss rate of victim flows is set to 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases, the memory/time overhead of FermatSketch and LossRadar re- mains unchanged, while that of FlowRadar increases linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Compared to FlowRadar/LossRadar, FermatSketch saves up to 1535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0/128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 times memory and up to 821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3/23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='7 times decoding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Evaluation on Testbed Testbed setup: We have fully implemented a ChameleMon prototype on a testbed with a Fat-tree topology composed of 10 Tofino switches and 8 servers, with 1400 lines of P4 [32] code and 2400 lines of C/C++ code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each server has 48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1GHz CPU cores, 256 GB RAM, and a 40Gb Mellanox Connectx-3 Pro NIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Switches and servers are interconnected with 40Gb links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We deploy the ChameleMon data plane on all four ToR/edge switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' An additional server linked with a certain edge switch works as the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For implementation details of the ChameleMon data plane and control plane, please refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Workloads: We generate workloads consisting of UDP flows according to four widely used distribution: DCTCP [30], 9 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 10 0 10 1 10 2 10 3 10 4 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 7: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 7(a) depicts the memory division of HH encoder (HHE), HL encoder (HLE), and LL encoder (LLE) inside the upstream flow encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 7(b) depicts the number of HH candidates of an edge switch, the number of HLs in the network, and the number of sampled LLs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 10 0 10 1 10 2 10 3 10 4 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 8: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' HADOOP [33], VL2 [34] and CACHE [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the 104-bit 5-tuple as the flow ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each flow, We choose its source and destination IP address uniformly, and therefore each server sends and receives almost the same number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The packet sender and packet receiver are integrated into a program written in DPDK [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To manually control packet losses, we let switches proactively drop packets whose ECN fields are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this way, we can flexibly specify any flow as a victim flow and control its packet loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To simulate the complex and changeable network state, we set the packet loss rate of each victim flow to a random value be- tween 1% and 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To avoid packet losses due to congestion, when sending packets, we set the size of every packet to 64 bytes regardless of its original size, so as to significantly re- duce the traffic load in the network and eliminate congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Such operation does not change the number of packets of each flow, and thus has no impact on the behavior of ChameleMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Parameter settings: We set the epoch length to 50ms by default8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the flow classifier, we set it to consist of an 8-bit counter array and a 16-bit counter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We set the number of 8-bit counters w1 to 32768 and the number of 16-bit counters w2 to 16384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the upstream flow encoder and downstream flow encoder, we set them to consist of 3 bucket arrays for the highest memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We set the number of buckets per array of the upstream flow encoder mu f to 4096, and that of the downstream flow encoder md f to 3072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the healthy network state, we fix the minimum memory reserved for HL encoders to a 3-array FermatSketch with 512 buckets per array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the ill network state, we fix the upstream HH, HL, LL encoders to a 3-array FermatSketch with 1024, 2560, and 512 buckets per array, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Please refer to Table 1 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 for resource usage on Tofino switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 8For some workloads that cannot run out in 50ms, we extend the epoch length appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, on DCTCP workload, we evaluate whether Chamele- Mon can automatically shift measurement attention for dif- ferent network states9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For experimental results on the other three workloads, please refer to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows (Figure 7): Experimental results show that ChameleMon can automat- ically shift measurement attention to packet loss detection while maintaining high memory utilization, as the number of flows increases and the number of victim flows increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We vary the number of flows in the network from 10K to 100K, and fix the ratio of victim flows to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' At first, the network state is healthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 10K to 20K, ChameleMon can record all flows and victim flows, and therefore sets both Th and Tl to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 30K to 70K, ChameleMon records all victim flows by allocating more and more memory to HL encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, ChameleMon cannot record all flows, and thus in- creases Th to decrease the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 80K to 100K, ChameleMon cannot record all victim flows, and thus the network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allocates fixed memory to LL encoders, increases Tl, and decreases the sample rate, so as to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, ChameleMon keeps increasing Th to control the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, ChameleMon maintains high memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The sum of decoded flows (Figure 7(b)) always exceeds 8K unless ChameleMon can record all flows and victim flows, repre- senting a load factor larger than 65% considering that the 9For each data point of Figure 7-8, we randomly initialize the configura- tion of the ChameleMon data plane, and collect the data point after Chamele- Mon successfully shifts measurement attention and the configuration of the ChameleMon data plane is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 upstream flow encoder has 12288 buckets and the maximum load factor is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows (Fig- ure 8): Experimental results show that ChameleMon can automatically shift measurement attention to packet loss de- tection while maintaining high memory utilization, as the ratio of victim flows increases and the number of victim flows increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We fix the number of flows to 50K, and vary the ratio of victim flows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' At first, the network state is healthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the ratio of victim flows increases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5%, ChameleMon records all victim flows by allo- cating more and more memory to HL encoders, and increases Th to decrease the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the ratio of victim flows increases from 15% to 25%, ChameleMon cannot record all victim flows, and thus the network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allocates fixed memory to LL encoders, increases Tl, and decreases the sample rate, so as to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, because the memory of upstream HH en- coder and the number of flows remain unchanged, Th also remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, Chamele- Mon maintains high memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The sum of decoded flows (Figure 8(b)) always exceeds 8K, representing a load factor larger than 65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Second, on DCTCP workload, we evaluate how fast can ChameleMon shift measurement attention over a large time window, in which the network state changes 8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' epoch (Figure 9): Experimen- tal results show that ChameleMon can shift measurement attention within at most 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 9 plots the shift of measurement attention in a large time window consisting of 45 consecutive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We change the network state (either the number of flows or the victim flow ratio) every 5 epochs, and the detailed settings are shown in the top sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Over- all, the network state first degrades from the healthy network state to the ill network state, and then improves back to the healthy network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the eight changes, ChameleMon shifts measurement attention within 1 (6->7), 2 (11->13), 3 (16->19), 2 (21->23), 2 (26->28), 1 (31->32), 1 (36->37), and 1 (41->42) epochs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To evaluate how fast can ChameleMon monitor the net- work, we evaluate various factors that could affect the setting of epoch length: 1) the time and bandwidth required to collect sketches from edge switches, 2) the time required to respond to different network states, and 3) the time required to recon- figure the ChameleMon data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The central controller only uses two CPU cores in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Time/Bandwidth overhead for collection (Figure 11): Ex- perimental results show that ChameleMon consumes only a small amount of time and bandwidth in collecting all the data structures deployed on edge switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon takes a total of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='33ms to collect sketches (refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As for bandwidth, when the epoch length is set to 50ms, the bandwidth overhead for collection is 317Mbps, 0 30 60 # Flows (K) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5 1 Memory division 0 5 10 # Decoded flows (K) 10 10� 10 Threshold 0 5 10 15 20 25 30 35 40 45 Epoch 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5 1 Sample rate Total flows Victim flows HHE HH HLE HL LLE LL Th Tl Figure 9: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 5 10 15 20 25 Response time (ms) DCTCP CACHE VL2 HADOOP (a) Varying number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 5 10 15 20 25 30 Response time (ms) DCTCP CACHE VL2 HADOOP (b) Varying ratio of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 10: Response time to different network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' consuming only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8% bandwidth for the central controller equipped with a 40Gb NIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Response time to different network states (Figure 10): Ex- perimental results show that ChameleMon can always respond to different network states within 30ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We count the re- sponse time of ChameleMon to each network state previously appeared in Figure 7-8, where the response time refers to the time interval between the central controller finishing the collection of sketches and the central controller generating the reconfiguration packet10 for the ChameleMon data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Al- though the response time does not seem to show a clear trend with the network state, it is mainly determined by the number of HH candidates, because the central controller needs to first extract them from the upstream HH encoders and then rein- sert them to the upstream HL encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As shown in Figure 10(b), as the ratio of victim flows increases, the response time on all the four workloads decreases because the number of HH candidates decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The response time finally stabilizes because the fixed memory allocation in the ill network state always decodes a similar number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10The central controller sends the reconfiguration packets to edge switches to reconfigure their data planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Please refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 11 0 200 400 600 800 1000 Epoch length (ms) 0 100 200 300 400 Bandwidth (Mbps) Figure 11: Bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2 3 4 5 6 7 Reconfiguration time (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 CDF Figure 12: Reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' CDF of reconfiguration time (Figure 12): Experimental re- sults show that it takes 2∼7ms to reconfigure the Chamele- Mon data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The central controller sends 10K reconfigu- ration packets with random configuration of the ChameleMon data plane to each edge switch, and we count the time for an edge switch to execute the reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We find 60% of reconfigurations take less than 5ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The difference in time consumption is mainly because different reconfigurations re- quire updating different numbers of TCAM entries to the switch data plane for supporting dynamic memory allocation (refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Adding up the above all time consumption, we find that the overall time consumption is less than 50ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This verifies that ChameleMon can monitor the network every 50ms on our testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that 1) the central controller only uses two CPU cores in experiments and 2) monitoring the network every 50ms only consumes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8% bandwidth of a 40Gb NIC, we believe ChameleMon can easily scale to monitor a much larger network with a shorter epoch length, requiring only one server as the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 6 Related Work Prior art can be classified into four categories based on sup- ported measurement tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Prior art for packet loss detection: They can be classi- fied into two kinds of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The first kind is algorithm- oriented solutions, including LossRadar [17] based on In- vertible Bloom filter (IBF) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' LossRadar can pinpoint the location of every lost packet and infer the root causes of packet losses by deploying IBF to monitor every link in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The second kind is system-oriented solutions, including Netseer [19] and PacketScope [37] that are based on programmable switches, NetSight [9] based on In-band net- work telemetry (INT), and more [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among them, NetSight mirrors desired statistics of every packet at every hop to a cen- tral controller, incurring 7% bandwidth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Both the above kinds of solutions fail to meet versatility requirement as they do not handle heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Prior art for heavy-hitter detection: They can be classi- fied into two kinds of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The first kind of solutions is sketches specifically designed for only heavy-hitter detection, including HashPipe [20], R-HHH [39], and more [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among them, HashPipe designs a multi-stage data struc- ture and kicks out small flows through comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The sec- ond kind of solutions is sketches that support many mea- surement tasks, including UnivMon [13], ElasticSketch [16], CocoSketch [11], SketchLib [43], FlyMon [44], and more [4, 12, 14, 15, 45–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among them, CocoSketch proposes a key technique, namely stochastic variance minimization technique, to provide unbiased estimation for arbitrary partial key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Both the above kinds of solutions fail to meet versatility requirement as they do not handle packet loss detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Prior art for both tasks: These solutions record the IDs and sizes of all flows in a zero-error manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Typical solutions include FlowRadar [21], OmniMon [22], Counter Braids [23], and more [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Among them, FlowRadar encodes the IDs and sizes of all flows into a variant of IBLT [25] in switches, and then executes well-designed decoding schemes to retrieve ex- act flow IDs and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' These solutions fail to meet efficiency requirement as they record the exact IDs of all flows, incurring memory/bandwidth overhead linear with the number of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Besides, INT-based solutions that carry desired statistics in packet headers can potentially support both tasks with packet- level visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Typical solutions include INT [26], PINT [54], LightGuardian [55], and more [24, 56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, INT- based solutions suffer from granularity-cost trade-off, and thus fail to meet either versatility requirement or efficiency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Other prior art: Other than the above solutions, there are still three kinds of solutions relevant to network measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Sampling-based solutions: These solutions collect desired statistics from a subset of network traffic through packet sampling, including Csamp [59], NetFlow [60], sFlow [61], EverFlow [62], and more [33,63–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' While sampling solu- tions significantly reduce the bandwidth overhead through sampling, they cannot well handle packet loss detection as only sampled packets are measured, and thus fail to meet efficiency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Programmable-switch-assisted solutions: Besides packet loss detection, these solutions leverage the advanced fea- tures and capabilities of programmable switches to monitor micro-bursts [71], perform queue measurement [72–74], and more [75–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Host-based solutions: Due to the flexibility, abundant re- sources, and high visibility to flow-level statistics of end- hosts, these solutions are typically designed for inferring the existences, locations, and root causes of specific net- work events or network failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Typical solutions either send tailored probes into the network [10,79–85] or analyze the performance of protocol stack or other I/O [86–95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Be- sides, some solutions further leverage switches to perform measurement [96,97] or record forwarding paths [98,99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon can complement these solutions as Chamele- Mon provides flow-level statistics with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Take 007 [94] as an instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Network operators can replace the TCP monitoring agent that detects TCP retransmissions in 007 with ChameleMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' After the replacement, 007 can monitor packet losses of TCP flows as well as packet losses of flows of other protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Such extra visibility can help 007 better locate the link failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 12 7 Conclusion In this paper, we present ChameleMon, which can automat- ically shift measurement attention as network state changes at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To achieve this, ChameleMon designs FermatS- ketch, a dividable, additive, and subtractive data structure that supports both packet loss detection and heavy-hitter detection simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We have fully implemented a ChameleMon prototype on a testbed consisting of 10 Tofino switches and 8 end-hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Extensive experimental results on the testbed ver- ify that 1) ChameleMon can achieve high accuracy in packet loss detection, heavy-hitter detection, and five common mea- surement tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) ChameleMon can monitor the network every 50ms and shift measurement attention within at most 3 epochs as network changes.' metadata={'source': 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of Fer- matSketch Input: Flow ID f 1 for i ∈ [1,d] do 2 j = hi(f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3 BID i [j] = (BID i [j]+ f) mod p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4 Bc i [j]++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5 end Algorithm 2: Decoding operation of FermatSketch 1 Function IsPure(i, j): 2 f = (BID i [j]×(Bc i [j])(p−2)) mod p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3 return j == hi( f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4 Function Delete(Bi′[j′],Bi[j]): 5 Bc i′[j′] = Bc i′[j′]−Bc i [j];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 6 BID i′ [j′] = (BID i′ [j′]−BID i [j]) mod p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 7 Function Decode(): 8 Queue is an empty queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 9 Flowset is an empty map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 for i ∈ [1,d], j ∈ [1,w] do 11 if Bc i [j]!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' = 0 then 12 Queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='push(Bi[j]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 13 end 14 end 15 while !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='Queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='empty() do 16 Bi[j] = Queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 17 Queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 18 if IsPure(i, j) then 19 f ′ = (BID i [j]×(Bc i [j])(p−2)) mod p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 20 Flowset[f ′] = Flowset[f ′]+Bc i [j];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 21 for i′ ∈ [1,d] do 22 Delete (Bi′[hi′( f ′)],Bi[j]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 23 if Bc i′[hi′( f ′)]!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' = 0 then 24 Queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='push(Bi′[hi′(f ′)]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 25 end 26 end 27 end 28 end 29 return Flowset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Fingerprint Verification To reduce the false positive rate of pure bucket verification, we can perform an extra verification method, namely fingerprint verification, by extending the IDsum field in each bucket by w bits and using the extra w bits as a fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For each incoming packet with flow ID f, a new hash function hf p(·) gives it a w-bit fingerprint hf p( f) for checking whether a bucket is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For encoding operation, instead of inserting flow ID f, we insert an extended ID concatenated by flow ID f and fingerprint h f p(f), and the extended IDsum field stores the result of the sum of the extended IDs modulo prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that p must be a prime larger than any available extended ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For decoding operation, obviously, we can still perform rehashing verification with the extended ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Our fingerprint verification works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Suppose a bucket is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we reuse the the extended ID of the single flow calculated in rehashing verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, we divide the extended ID to get the flow ID and its fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the divided fingerprint equals to the fingerprint of the divided flow ID, we consider the bucket passes fingerprint verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Only buckets pass both rehashing and fingerprint verification will be considered as pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The false positive rate of only fingerprint verification is obviously 1 2w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that rehashing verification and fingerprint verification are independent, the false positive rate of pure bucket verification could be reduced to 1 2wm with w-bit fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We conduct experiments to demonstrate the effect of 8-bit fingerprint on improving the decoding success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As shown in Figure 13(a), when the number of flows is 1K, with the same number of buckets, 8-bit fingerprint can improve the decoding success rate by at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, when the number of flows is 10K, the improvement falls to at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because as the number of buckets increases, m increases, and the false positive rate of pure bucket verification quickly drops, and thus further reducing the false positive rate with fingerprint yields less improvement on the decoding suc- cess rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As shown in Figure 13(b), under the same memory usage, 8-bit fingerprint actually reduces the decoding success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because fingerprint consumes additional memory, while this memory could be used as buckets to reduce the probability of 2-core of the random hypergraph and improve the decoding success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 13(a)-(b) also demonstrate that the memory overhead of FermatSketch is proportional to the number of inserted flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, for simplicity and accuracy, we recommend implementing FermatSketch without fingerprints in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Only if there is some memory that can hardly be uti- lized due to hardware constraints unless used as fingerprints, we recommend implementing FermatSketch with fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 Discussion of FermatSketch Space complexity: Suppose FermatSketch is large enough, and the pure bucket verification has negligible false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The decoding operation is almost the same as that of IBLT [25], which is exactly the procedure used to find the 2- core of a random hypergraph [100,101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, the mem- ory overhead of FermatSketch is proportional to the number of inserted flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FermatSketch also shares similar properties with IBLT: the number of hash functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', the number of the bucket arrays d, is recommended to set to 3 for the highest 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='29 # Buckets Per Flow 0 20 40 60 80 100 Decoding Success Rate (%) 10K flows w/o fp 10K flows w/ fp 1K flows w/o fp 1K flows w/ fp (a) Same number of buckets per flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 9 10 11 12 Memory Per Flow (Byte) 0 20 40 60 80 100 Decoding Success Rate (%) 10K flows w/o fp 10K flows w/ fp 1K flows w/o fp 1K flows w/ fp (b) Same memory per flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 13: Experiments on 8-bit fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the anonymized IP traces collected in 2018 from CAIDA [29] as dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the 32-bit source IP address as the flow ID, and choose the first 10K flows for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, fp represents 8-bit fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' memory efficiency, that on average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='23 buckets can record a flow and its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='. Packet loss detection: To support packet loss detection, we can deploy a group of FermatSketches on edge switches to encode the packets entering the network, and another group of FermatSketches to encode the packets exiting the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Thanks to the additivity and subtractivity of FermatSketch, for each group, we add up the FermatSketches in it to obtain a cumulative FermatSketch encoding all the packets enter- ing or exiting the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, we subtract the cumulative FermatSketch encoding all the packets exiting the network from the other one, and the FermatSketch after subtraction just encodes all the victim flows in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This Fer- matSketch just requires memory proportional to the number of victim flows for successful decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In other words, Fer- matSketch can support packet loss detection with memory overhead proportional to the number of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Eliminating false positives in rehashing verification: Due to hash collisions, the rehashing verification will inevitably misjudge some impure buckets as pure buckets with false positive rate 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Such misjudgement will lead to extraction of flows that are not inserted into FermatSketch, and finally could hinder the decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' From another point of view, ex- tracting a flow from such a misjudged impure bucket, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', false positive, equals to inserting a wrong flow with a neg- ative size into FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The decoding operation can automatically eliminate these false positives: in the decoding procedure, these inserted wrong flows could also be extracted and deleted from FermatSketch, and then the impact caused by the false positives disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Time complexity of decoding operation: Suppose FermatS- ketch is large enough and the false positive rate in pure bucket verification is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In step 1 , we traverse FermatS- ketch and push all non-zero buckets into the decoding queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The number of these buckets is at most md, and thus the time complexity of step 1 is O(md).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In the rest steps, we process all the buckets pushed into the queue, which consists of two parts: 1) the md buckets pushed into in step 1 , and 2) the mapped buckets except the popped pure bucket of each ex- tracted flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that the number of extracted flows is bounded by the number of buckets of FermatSketch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', md, the number of buckets of the second part is O(md2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' There- fore, the time complexity of the rest steps is O(md2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Adding up the time complexities of all steps, the time complexity of decoding operation is O(md2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Let FermatSketch consists of d bucket arrays, each of which consists of m buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Let M be the number of flows inserted into that FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Suppose md > cdM +ε and M ⩾ Ω(d4dlogd(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' the decoding of FermatSketch fails with probability O( 1 Md−2 ), where both ε and cd are small constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' cd = � sup � α ���α ∈ (0,1),∀x ∈ (0,1),1−e−dαxd−1��−1 For example, c3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='222,c4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='295,c5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Proof A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 This is an analysis based on the theory of the 2- core in random hypergraph [100,101] and IBLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Compared with 2-core or IBLT, we only introduce a kind of additional error, which is the false positives when we use pure bucket verification to verify the pure buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The IBLT assumes there is no error when verifying buckets because IBLT uses addi- tional hashkeySum field that can be long enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The results of 2-core and IBLT show that the failure probability without wrong verification is O( 1 Md−2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here we aim at proving that the consequences of our false positives are negligible when M is not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In the decoding procedure, the pure bucket verification runs at most O(Md) times, and the false positive rate is O( 1 m) with only rehashing verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' By Chernoff bound, when M = O(md) and δ = O( 1 Md−2 ), the number of false positives will not exceed F = O(d3log(M)) in most cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', 1 − O(δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' A false positive will incur a wrong flow ID with a wrong single flow deletion that influences d buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' There is at most Fd buckets can be influenced, called poi- soned buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The existing study [25] of poisoned bucket shows that a small number of poisoned buckets will be au- tomatically recovered, and the probability of failure due to poisoned bucket is O( � Fd M �d) = O( d4dlogd(M) M(d−1) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When set- ting M = Ω(d4dlogd(M)), the overall failure probability is δ = O( 1 Md−2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In practice, M = Ω(d4dlogd(M)) is easy to meet because M is large and d is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Here, we only use rehashing verification for pure bucket verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The theorem can also be easily extended if we further use fingerprint verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' B Collection from ChameleMon Data Plane Timeline split: For each edge switch, we maintain a 1-bit timestamp in its ingress, which is periodically flipped by the 21 switch control plane, so as to split the timeline into consecu- tive fixed-length epochs with interleaved 0/1 timestamp value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Further, we copy an additional group of sketches in the switch data plane for rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each group of sketches corresponds to a distinct timestamp value (0/1), and monitors the epochs with that timestamp value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, at each edge switch, every packet entering the network first obtains the current timestamp value, and then is inserted into the flow classifier and upstream flow encoder corresponding to the obtained timestamp value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When the packet exits the network, it is also inserted into the downstream flow encoder corresponding to the timestamp value it obtained when it entered the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To maintain the timestamp value during the packet transmis- sion, we can use one unused bit in the original packet header as discussed above (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Clock synchronization: Through maintaining a 1-bit times- tamp and copying a group of sketches, we successfully split the timeline and insert packets of different epochs to their corresponding groups of sketches, laying a solid foundation for subsequent collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, if the clocks of the con- trol planes of edge switches are out of synchronization to some extent, we still can not find opportunities to collect the sketches without colliding with packet insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Consider- ing such an extreme situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' There are three edge switches in a given network, and the transmission time between any two edge switches is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The time offset between the control planes of two of the edge switches is exactly the size of the epoch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', at any time, the flipping timestamps of the two edge switches are different (0<->1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' There are continuous packets entering the network at the above two edge switches, and exiting the network at the third switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As a result, both groups of sketches of the third switch are continuously in- serted, and thus can never be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, the central controller synchronizes the clocks of the control planes of all edge switches with itself, trying to keep only a group of sketches being inserted at any time, so as to make opportunities to collect the other group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, we further discuss the appropriate time for the central controller to collect sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Appropriate time for collection: The central controller also maintains a 1-bit timestamp, trying to collect the group of sketches monitoring the previous epoch after it ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Before collection, the central controller should ensure that all the packets in the previous epoch have been inserted into sketches or lost in the network, so as to guarantee the correctness of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' First, we analyze an ideal situation, that the clock synchronization is zero-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For ingress sketches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', the flow classifier and the upstream flow encoder, as soon as the locally maintained 1-bit timestamp flips, the central con- troller can collect the group of ingress sketches monitoring the previous epoch from each edge switch, because all the packets in that epoch have already been inserted into ingress sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For egress sketches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', the downstream flow en- coder, every time the locally maintained timestamp flips, the central controller must first wait an additional period of time, so as to ensure that all the packets in the previous epoch have either been lost in the network, or passed through the network and been inserted into egress sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, the central con- troller can collect the group of egress sketches monitoring the previous epoch from each edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Obviously, the additional period of time should be longer than the maximum time for packet transmission in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that the buffer sizes of DCN switches are at 10MB-level [102], with 100Gb link speed, the queuing delay in a single switch is at most 1ms in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, for typical data cen- ter networks that usually have at most five hops, setting the additional time to 10ms can cope with most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, in practice, the clock synchronization can never be zero-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Therefore, before collecting both ingress and egress sketches, the central controller needs to wait for another additional pe- riod of time, which should be longer than the precision of synchronization, so as to guarantee the correctness of mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In addition, the central controller should end the collection some time before its 1-bit timestamp flips again, which should also be longer than the precision of synchro- nization, in case the packets in the next epoch are inserted into the group of sketches being collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' C Heavy-hitter Detection and Other Tasks Metrics: Average Relative Error (ARE): 1 |Ω| ∑fi∈Ω |vi−ˆni| vi , where Ω is the set including all flows, vi is the true size of flow fi, and ˆvi is the estimated size of flow fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' F1 Score: 2·PR·RR PR+RR , where PR (Precision Rate) refers to the ratio of the number of the correctly reported instances to the number of all reported instances, and RR (Recall Rate) refers to the ratio of the number of the correctly reported instances to the number of all correct instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Relative Error (RE): |True−Est| True , where True and Est are the true and estimated statistics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Weighted Mean Relative Error (WMRE) [15]: ∑z i=1 |ni− ˆni| ∑z i=1 � ni+ ˆni 2 �, where z is the maximum flow size, ni and ˆni are the true and estimated numbers of the flows of size i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Dataset: We also use the IP traces from CAIDA [29] as our dataset, and use the 32-bit source IP address as the flow ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use four traces for evaluation, each of which monitors the traffic in five seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each trace contains 63K flows and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3M packets on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We report the average accuracy that each algorithm achieves on each CAIDA trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Setup: We compare the combination of TowerSketch and FermatSketch (Tower+Fermat) with 9 algorithms: CM [51], CU [4], CountHeap [52], UnivMon [13], ElasticSketch [16], FCM-sketch [103], HashPipe [20], CocoSketch [11], and MRAC [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We do not compare with FlowRadar because FlowRadar can hardly perform successful decoding with the memory sizes we used for evaluation (200KB-600KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For heavy-hitter detection and heavy-change detection, we set 22 their thresholds ∆h and ∆c to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='02% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='01% of the total packets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', 500 and 250, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We configure Tower+Fermat and its competitors as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Overall, the configurations of these competitors are recommended in liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Tower+Fermat: For Tower, we set it to consist of an 8-bit counter array and a 16-bit counter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For Fermat, We set its count field and ID field to 32bits, and allocate 2500 buckets to it for 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='9% decoding success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' we set the threshold Th for identifying heavy-hitter candi- dates to the heavy-change threshold ∆c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', 250, for detect- ing most heavy-hitters and heavy-changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' CM/CU/CountHeap: We use 3 hash functions as recom- mended in [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We set the counter size to 32bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For CountHeap, we additionally set its heap capacity to 4096 for heavy-hitter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' UnivMon: We use 14 levels and each level can record 1000 heavy hitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Elastic: We use the hardware version of Elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the heavy part, we use 4 stages and each stage has 3072 buck- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the light part, we use a one-layer CM with 8-bit counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FCM: We use the top-k version of FCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' It is almost the same as Elastic except the light part is substituted by a 16-ary FCM whose depth is set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Hashpipe: We set the number of stages to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Coco: We use the hardware version of Coco that only uses one hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Heavy-hitter detection (Figure 14(a)): Experimental results show that Tower+Fermat achieves comparable accuracy with HashPipe, and higher accuracy than other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When using only 200KB memory, the F1 score of Tower+Fermat is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8%, while that of Elastic and FCM is lower than 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Flow size estimation (Figure 14(b)): Experimental results show that Tower+Fermat achieves comparable accuracy with FCM, and higher accuracy than other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When us- ing only 200KB memory, the ARE of Tower+Fermat is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='51 times, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='19 times, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='09 times, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='59 times smaller than that of CM, CU, Elastic, and FCM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Heavy-change detection (Figure 14(c)): Experimental re- sults show that the Tower+Fermat achieves higher accuracy than other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Tower+Fermat achieves 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6% F1 score when using only 400KB memory, while that of the other algorithms is below 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Flow size distribution estimation (Figure 14(d)): Exper- imental results show that Tower+Fermat achieves higher accuracy than Elastic and FCM, and comparable accuracy with MRAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When using 600KB memory, the WMRE of Tower+Fermat is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='039, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='09 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='42 times smaller than that of Elastic and FCM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Entropy estimation (Figure 14(e)): Experimental results show that Tower+Fermat achieves higher accuracy than Univ- Mon, and comparable accuracy with Elastic and FCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When using 600KB memory, the ARE of Tower+Fermat is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='003, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 times smaller than that of UnivMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Cardinality estimation (Figure 14(f)): Experimental results show that the Tower+Fermat achieves higher accuracy than other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' When using 600KB memory, the RE of Tower+Fermat is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0016, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='125 times, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='08 times, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='57 times smaller than that of UnivMon, Elastic, and FCM, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 200 300 400 500 600 Memory (KB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 F1 score Tower + Fermat FCM UnivMon CountHeap Elastic Hashpipe Coco (a) Heavy-hitter 200 300 400 500 600 Memory (KB) 0 2 4 6 ARE Tower + Fermat FCM CM CU Elastic (b) Flow size 200 300 400 500 600 Memory (KB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 F1 score Tower + Fermat FCM UnivMon CountHeap Elastic Coco (c) Heavy-change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 200 300 400 500 600 Memory (KB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='28 WMRE Tower + Fermat FCM MRAC Elastic (d) Flow size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 200 300 400 500 600 Memory (KB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='06 RE Tower + Fermat FCM UnivMon Elastic MRAC (e) Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 200 300 400 500 600 Memory (KB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='12 RE Tower + Fermat FCM UnivMon Elastic (f) Cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 14: Accuracy comparison for six tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' D Prototype Implementation In this section, we present the important details of Chamele- Mon prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We lay out important implementation details of the ChameleMon data plane and control plane in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 Data Plane Implementation We have fully implemented the ChameleMon data plane on the switch data planes of four edge Tofino switches in P4 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this section, we detail the implementation logic of data plane along the workflow (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Hash: First, a packet with flow ID f enters the network at an edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' With its flow ID (5-tuple) as input, the packet is hashed to multiple indexes through pairwise-independent hash functions generated from different CRC polynomials, which are deployed at stage 0 in ingress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' These hash indexes are either used as base indexes for locating the mapped coun- ters/buckets in the subsequent insertions, or used for sampling LL candidates, or used as fingerprints for improving decoding success rate of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that due to the limitation of Tofino switches, each hash index is uniformly distributed on [0, 2t −1], where t is an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1-bit flipping timestamp: Second, the packet reads the cur- rent 1-bit flipping timestamp and from a match-action table, 23 Hash Stages: #0 #1 #2-3 #4 #8-11 Time Stamp Flow Classifier Sampling Blackbox Network Stages: #4-7 Downstream Flow Encoder Ingress Pipeline Egress Pipeline Upstream Flow Encoder Figure 15: Implementation logic of ChameleMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' which is deployed at stage 1 in ingress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The 1-bit timestamp is used to indicate the corresponding group of sketches for the subsequent insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Flow classifier: Third, the packet is inserted into the flow classifier, which is deployed at stage 2-3 in ingress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The flow classifier is a TowerSketch consisting of an 8-bit counter array and a 16-bit counter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The 8-bit and 16-bit counter arrays consist of w1 8-bit and w2 16-bit counters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each counter array is built on a register and accessed by a stateful arithmetic logic unit (SALU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To save SALU resources, we simulate the two flow classifiers by doubling the number of counters of the 8-bit and 16-bit counter arrays instead of build- ing additional registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The left/right w1 8-bit and w2 16-bit counters form the flow classifier corresponding to timestamp 0/1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We use the base indexes calculated by hash functions as the relative positions of the mapped counters in the flow classifier, and add offsets corresponding to the 1-bit timestamp to the base indexes, so as to locate the mapped counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, when the timestamp is 0, the offset is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' when the timestamp is 1, the offset is w1 for 8-bit counter array or w2 for 16-bit counter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' During insertion, the SALU adopts saturated addition operation for each mapped counter, which can increment the counter to its maximum value but never overflow it, and reports the value recorded in the counter, so as to simulate the behavior of TowerSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' After insertion, we take the minimum value among the re- ported values as the size of flow f, and then input the flow size to a match-action Table that uses range matching on the flow size, so as to obtain the hierarchy of flow f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Sampling: Fourth, if flow f is classified as a LL candidate, the packet reads a value from a match-action table, which is deployed at stage 4 in ingress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We then compare the read value with a 16-bit value, which is calculated by a hash function with the 5-tuple of the flow and a random seed11 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' If the read value is equal to or larger than the 16-bit value, flow f is classified as a sampled LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Otherwise, the flow is classified as a non-sampled LL candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Obviously, to simulate a sample rate R, the value should be set to ⌈65536× R⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' FermatSketch: Before detailing the implementation of up- stream and downstream flow encoders, we present the imple- mentation of FermatSketch that they are based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To encode the 104-bit flow ID (5-tuple) of each packet, an ideal bucket in FermatSketch should contain a 105-bit IDsum field and a 32-bit count field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, because each SALU can ac- cess up to a pair of 32-bit counters, the IDsum field cannot 11The random seed is used to randomize the sampled LL candidates, and we change the seed every epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' srcPort[14:0] dstPort[15:0] 32-bit register FingerPrint[19:0] Rest[10:0] 32-bit register srcIP[30:0] 32-bit register 32-bit register dstIP[30:0] Rest[10:0] : srcIP[31] + dstIP[31] + srcPort[15] + protocol[7:0] : 1-bit reserved bit Figure 16: Division of the 5-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Algorithm 3: Simulated modular addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Input: An ID fragment f, a counter reg for encoding the ID fragment, and a prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 1 inv ← p− f ▷ Get the additive inverse of f in Zp 2 if reg+ f < p then 3 reg ← reg+ f 4 else 5 reg ← reg−inv 6 end be directly built in Tofino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, we divide the IDsum field into multiple counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Rather than encod- ing complete flow IDs, each counter only encodes specific ID fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that a 32-bit counter can support at most 32-bit primes, and thus can encode at most 31-bit ID fragment, we need four 32-bit counters to simulate the IDsum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, the division of the IDsum field is shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The first three 32-bit counters encode the lower 31-bits of the source IP address, the destination IP address, and the concatenation of the source port and desti- nation port, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The last one 32-bit counter encodes the rest 11-bit ID fragment (1-bit source IP address + 1-bit destination IP address + 1-bit source port + 8-bit protocol), and the other unused 20 bits are used to encode a fingerprint to improve decoding success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In summary, each bucket of FermatSketch consists of five 32-bit counters: four coun- ters to encode the IDsum field and the fingerprint field, and a counter to encode the count field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that there is no dependency between the five counters in any bucket of Fer- matSketch, a bucket array of FermatSketch can be built with five 32-bit counter arrays, each of which is built on a register and accessed by a SALU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' During insertion, for any of the four counter arrays encoding the IDsum field and fingerprint field, the SALU needs to insert the specific ID fragment into its counter through modular addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As shown in Algorithm 3, the SALU simulates the modular addition with logic con- sisting of a conditional judgement and two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Such logic is naturally supported by SALUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the other counter array encoding the count field, the SALU simply increments its counter by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this way, the SALUs simulate the be- havior of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' By duplicating these five registers and SALUs d times, we can easily build a d-array FermatS- 24 ketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that we use registers consisting of 32-bit counters, but not registers consisting of pairs of 32-bit counters that can further save SALU resources, to simulate the buckets of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because the logic used to simulate the modular addition requires two 32-bit metadata ( f and inv) as input, which is just the maximum number that a SALU can support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, encoding two ID fragments with a SALU requires four 32-bit metadata as input, which is beyond the capabilities of SALU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Upstream flow encoder: Fifth, unless flow f is a non- sampled LL candidate, the packet is inserted into the upstream flow encoder, which is deployed at stage 8-11 in ingress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The upstream flow encoder consists of three bucket arrays for the highest memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Each bucket array is built as described above, and consists of mu f buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The left mll buckets, the right mhh buckets, and the middle mhl buckets in each array form the upstream LL encoder, HH encoder, and HL encoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Similarly, we simulate the two upstream flow encoders by doubling the number of buckets in each array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Based on the hierarchy of flow f, we can easily determine the encoder that the packet should be inserted into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We denote the number of buckets of a bucket array of that encoder by m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Different from the flow classifier, the base indexes calculate by hash functions cannot be directly used to locate the relative positions of the mapped buckets in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because a base index is uniformly distributed on [0,2t −1], while m′, which could be any of mll, mhl, and mhh, may not be powers of two, as they are required to vary for supporting dynamic memory allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, we decide to use the results of base indexes modulo m′ as the relative positions of the mapped buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To simu- late modulo operation in data plane, we input the hierarchy of flow f and a base index hb to a match-action table that uses exact matching on flow hierarchy and range matching on index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The table first determines m′ based on the input flow hierarchy, then outputs the largest number that is divisible by m′ and less than hb, and finally subtracts that number from hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Obviously, the result equals to hb modulo m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this way, we locate the relative positions of the mapped buckets at the cost of TCAM resources, and can finally locate the mapped buck- ets by adding offsets corresponding to the 1-bit timestamp and the flow hierarchy to the relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Considering that the width of base index is fixed at run-time, if its width is too long compared to the width of m′, the match-action table will need a lot of entries to support range matching, and thus consumes lots of TCAM resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' if its width is just a bit longer than the width of m′, the uniformity of the calculated relative positions will be quite poor, leading to reduction of the decoding success rate of FermatSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, before we input the base index to the match-action table, we bitwise-AND the base index with a mask to guarantee that the value range of the index is between 4m′ and 8m′, so as to make great trade-off between the uniformity of relative posi- tions and the consumption of TCAM resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that due to the inherent features of TCAM, when TCAM is used for range matching, different value range would require different number of TCAM entries for supporting modulo operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Downstream flow encoder: Sixth, unless flow f is a non- sampled LL candidate, the packet is inserted into the upstream flow encoder, which is deployed at stage 4-7 in egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The implementation of downstream flow encoder is almost the same as that of upstream flow encoder, except it omits the heavy-hitter encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that the flow hierarchy and 1-bit timestamp are obtained from the edge switch where the packet enters the network, and carried by recording them in three bits of the ToS field of the IPv4 protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Resources Usage: As shown in Table 1, under the parameter settings in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2, the ChameleMon data plane consumes SALUs most, achieving 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' This is because the flow clas- sifier, the upstream flow encoder, and the downstream flow encoder all need SALUs to access memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For resources other than SALUs, ChameleMon consumes no more than 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Overall, the resource usage of ChameleMon is moder- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Although ChameleMon indeed consumes a lot of SALUs, the consumption of SALUs will not increase when we fur- ther enlarge the above sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' With the advent of Tofino 2 switches and even Tofino 3 switches, we believe the resource usage will be much more acceptable on these more advanced programmable switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Resource Usage Percentage Exact Match Input xbar 353 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='98% Ternary Match Input xbar 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='17% VLIW Instructions 43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='20% Map RAM 102 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='71% SRAM 130 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='54% TCAM 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='78% Hash Bits 809 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='21% Stateful ALU 32 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='67% Table 1: Resources used by ChameleMon in Tofino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 Control Plane Implementation Central controller: The central controller integrates three modules into a DPDK [36] program: 1) a packet receiver module responsible for collecting sketches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) an analyzer module for decoding sketches, monitoring real-time network state, and generating reconfiguration packets for reconfigur- ing the ChameleMon data plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3) a packet sender module responsible for sending reconfiguration packets to the control plane of each edge switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Switch control plane: The control plane of each edge switch runs a C++ program to load the P4 program to the Tofino ASIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time the switch control plane receives a recon- figuration packet, it first extracts the packet to obtain the reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, based on the reconfiguration, it gen- erates corresponding table entries and update them to the corresponding match-action tables in the data plane to re- configure the switch data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The time consumption in 25 this step is shown in Figure 12 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To avoid the updated entries to interfere with the monitoring of the current epoch, those corresponding match-action tables further use exact matching on the 1-bit timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Those newly updated entries match the 1-bit timestamp in the next epoch, so as to function in the next epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Epoch length: On our testbed, we set the epoch length to 50ms by default, and the additional time for all traffic passing through the network is set to 10ms (described in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Clock synchronization: On our testbed, we use the well- known software time synchronization protocol NTP [105] to synchronize the clocks between the control planes of edge switches and the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every 10s, every edge switch synchronizes its clock with the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The precision of synchronization is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3ms∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5ms, and thus NTP can already satisfy the precision requirement for epochs of 50ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' We can further improve the precision through more advanced software solutions, such as Huygens [106], or hardware solutions, such as DPTP [107], to achieve us-level or even ns-level precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Data plane collection: To collect sketches from data planes of edge switches, a naive solution is to directly use the C++ control plane APIs provided by the Tofino SDK [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Cur- rently, the most efficient strategy for this solution is to first use bulk DMA transfer to read data plane counter arrays into control plane buffer, and then read the counter arrays from control plane buffer [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' However, on our testbed, such strategy takes about 338ms to collect only the upstream flow encoder, which seriously limits the setting of epoch length, and thus degrades the accuracy and timeliness of measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To address this issue, we fully exploit the capabilities and features of programmable data plane, including SALUs, mirror, and recirculate ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Specifically, the switch control plane just needs to send several tailored packets to data plane for collecting sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The tailored packet is forwarded to the recirculate port, so as to access the counters of each sketches in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time a tailored packet accesses a counter, lever- aging the SALU, it reads the value and inserts the value into its payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Every time a tailored packet reaches the maxi- mum transmission unit (MTU, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', 1514 Bytes), the switch data plane forwards it to the switch control plane, and mirrors a new truncated packet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=', 64 Bytes) to read the remaining counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In this way, collecting the upstream flow encoder from the switch data plane only takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='44ms, which is 775 times faster than the straightforward solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Note that if we allocate a certain amount of buffer to the recirculate port to store the tailored packets, they will not be lost during the col- lection even though the switch is extremely congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Then, the switch control plane can transmit the collected sketches to the central controller through persistent TCP connections to minimize the transmission delay as well as guarantee the reliability of transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' On our testbed, we simplify the transmission process, that the tailored packets reaching the MTU will be directly forwarded to the central controller from switch data planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' To ensure that the tailored packets will not be lost during the transmission, we reserve idle ports in their forwarding paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Overall, the central controller takes 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='33ms to collect sketches from the data plane each edge switch, which consists five parts: 1) every time the timestamp flips, the central controller first sleeps 1ms to eliminate the impact caused by the error in clock synchronization, ensuring that all the edge switches have started the current epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 2) the central controller takes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='68ms to collect the flow classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 3) the central controller takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='44ms to collect the upstream flow encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 4) the central controller sleeps 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='88ms to ensure that all the packets in the previous epoch have passed through or lost in the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 5) the central controller takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='33ms to collect the downstream flow encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' E Evaluation on Different Workloads In this section, we show that on workloads other than DCTCP, how ChameleMon shifts measurement attention with the change of the number of flows or ratio of victim flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the measurement attention under different number of flows, we vary the number of flows in the network from 10K to 100K, and fix the ratio of victim flows to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' For the measurement attention under different ratios of victim flows, we vary the ratio of victim flows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 25%, and fix the number of flows to 50K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='1 CACHE Workload Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows (Figure 17): As the number of flows increases from 10K to 20K, Chamele- Mon can record all flows and victim flows, and therefore sets both Th and Tl to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 30K to 70K, ChameleMon allocates more memory to HL en- coders and raises Th higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 80K to 100K, the healthy network state transi- tions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allocates memory to LL encoders, increases Tl and decreases the sample rate, so as to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Mean- while, ChameleMon raises Th to control the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The relatively low load factor when the number of flows is between 80K and 100K is because of the high skewness of CACHE workload: lower thresholds will lead to a huge increase of the number of recorded flows, thus caus- ing decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' In fact, when the number of flows is between 80K and 100K, the Th is set to 3, and the Tl is set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon has tried its best to select thresholds to maximizes the load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows (Fig- ure 18): As the ratio of victim flows increases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5%, ChameleMon records all victim flows by allocating more and more memory to HL encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Th is not adjusted because of the high skewness of CACHE workload: setting 26 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HH HL LL (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 10 0 10 1 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 17: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows on CACHE workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HH HL LL (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 10 0 10 1 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 18: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows on CACHE workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 10 0 10 1 10 2 10 3 10 4 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 19: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows on VL2 workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 10 0 10 1 10 2 10 3 10 4 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 20: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows on VL2 workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 # Total flows (K) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 10 0 10 1 10 2 10 3 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 20 40 60 80 100 # Total flows (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 21: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows on HADOOP workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Th to 2 already makes a fairly small portion of flows as HH candidates, and lower Th leads to decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the ratio of victim flows increases from 15% to 25%, the healthy network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Chamele- Mon allocates memory to LL encoders, increases Tl to 2 and decreases the sample rate so as to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, because the memory of upstream heavy-hitter encoder and the number of flows re- main unchanged, Th also remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The reason why ChameleMon suffers low load factor when the ratio of victim flows is between 15% to 25% is also due to high skewness of CACHE workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Both Th and Tl are set to 2, and decrease 27 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Memory division HHE HLE LLE (a) Memory division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0 2 4 6 8 10 # Decoded flows (K) HH HL LL (b) Number of decoded flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 10 0 10 1 10 2 Threshold Th Tl (c) Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 0 5 10 15 20 25 Ratio of victim flows (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='0 Sample rate (d) Sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Figure 22: Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows on HADOOP workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' of thresholds will lead to decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon has tried its best to select thresholds to maximize the load factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='2 VL2 Workload Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows (Figure 19): As the number of flows increases from 10K to 20K, Chamele- Mon can record all flows and victim flows, and therefore sets both Th and Tl to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 30K to 60K, ChameleMon allocates more and more memory to HL encoders, and increases Th to decrease the number of HH candidates to avoid decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 70K to 100K, the healthy network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allocates memory to LL encoders, increases Tl, and decreases the sam- ple rate, so as to to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, ChameleMon keeps increasing Th to control the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, ChameleMon maintains the load factor higher than 51%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The load factor is sightly lower, and it is because the distribution of VL2 is highly skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Decreasing the thresholds by 1 will lead to huge increase in the number of recorded flows, and thus causing decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows (Fig- ure 20): As the ratio of victim flows increases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5%, ChameleMon records all victim flows by allocat- ing more and more memory to HL encoders, and increases Th to decrease the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the ratio of victim flows increases from 15% to 25%, ChameleMon cannot record all victim flows and thus the healthy network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allo- cates memory to LL encoders, increases Tl, and decreases the sample rate so as to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, because the memory of upstream HH en- coders and the number of flows remain unchanged, Th also remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, Chamele- Mon maintains the load factor higher than 53%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The load factor is sightly lower, and the reason is the same as the for- mer experiment of the the number of flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='3 HADOOP Workload Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' number of flows (Figure 21): As the number of flows increases from 10K to 20K, Chamele- Mon can record all flows and victim flows, and therefore sets both Th and Tl to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 30K to 60K, ChameleMon allocates more and more mem- ory to HL encoders, and increases Th to decrease the number of HH candidates to avoid decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the number of flows increases from 70K to 100K, the healthy network state transitions to the ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allo- cates memory to LL encoders, increases Tl, and decreases the sample rate to control the number of HLs and sampled LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Meanwhile, ChameleMon keeps increasing Th to control the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, ChameleMon maintains the load factor higher than 47%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The load factor is sightly lower, and it is because the distribution of HADOOP is highly skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Decreasing the thresholds by 1 will lead to huge increase in the number of recorded flows, and thus causing decoding failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Measurement attention vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ratio of victim flows (Fig- ure 22): As the ratio of victim flows increases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content='5%, ChameleMon records all victim flows by allocating more and more memory to HL encoders, and increases Th to decrease the number of HH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' As the ratio of vic- tim flows increases from 15% to 25%, ChameleMon cannot record all victim flows and thus transitions to ill network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' ChameleMon allocates memory to LL encoders, increase Tl, and decreases sample rate, so as to control the number of HHs and HLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' Throughout the experiment, ChameleMon maintains the load factor higher than 48%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' The load factor is sightly lower, and the reason is the same as the former experiment of the the number of flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfufnB/content/2301.00615v1.pdf'} +page_content=' 28' metadata={'source': 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Comin1, ∗ +1Department of Computer Science, Federal University of S˜ao Carlos, S˜ao Carlos, SP, Brazil +2S˜ao Carlos Institute of Physics, University of S˜ao Paulo, Av. Trab. S˜ao Carlense, +400. +Parque Arnold Schmidit, S˜ao Carlos, SP, 13566-590, Brazil +3Department of Cellular and Molecular Medicine, Faculty of Medicine, University of Ottawa, Ottawa, ON, Canada +(Dated: January, 2023) +Creating a dataset for training supervised machine learning algorithms can be a demanding task. +This is especially true for medical image segmentation since this task usually requires one or more +specialists for image annotation, and creating ground truth labels for just a single image can take up +to several hours. In addition, it is paramount that the annotated samples represent well the different +conditions that might affect the imaged tissue as well as possible changes in the image acquisition +process. This can only be achieved by considering samples that are typical in the dataset as well as +atypical, or even outlier, samples. We introduce a new sampling methodology for selecting relevant +images from a larger non-annotated dataset in a way that evenly considers both prototypical as well +as atypical samples. The methodology involves the generation of a uniform grid from a feature space +representing the samples, which is then used for randomly drawing relevant images. The selected +images provide a uniform cover of the original dataset, and thus define a heterogeneous set of images +that can be annotated and used for training supervised segmentation algorithms. We provide a case +example by creating a dataset containing a representative set of blood vessel microscopy images +selected from a larger dataset containing thousands of images. +I. +INTRODUCTION +Recent developments in neural networks led to un- +precedented results in image classification [1, 2], object +detection [3] and image segmentation [4]. +Neural net- +works, and the related field of Deep Learning, shifted the +prevalent focus on feature engineering to a more data- +centric approach, where network models can approximate +highly complex functions provided the model has enough +capacity and the dataset is sufficiently representative [5]. +Due to such impressive advancements, many authors ar- +gue that neural networks have reached human-level ac- +curacy [6, 7], or even surpassed it [8]. +The performance of neural networks has dominantly +been measured using metrics such as classification or seg- +mentation accuracy, precision, recall, and the area under +the ROI curve. However, recent studies have shown the +dangers of only considering such accuracy-related met- +rics [9–11] that provide only an aggregated, summarized, +view of the performance of machine learning algorithms +on datasets with sometimes millions of images. Such an +approach may hide important biases of the model [11]. +For instance, for medical images, a 95% accuracy is usu- +ally considered a good performance. +But what about +the remaining 5%? It is unrealistic to expect models to +reach 100% accuracy, but the samples that are not cor- +rectly processed by a neural network may hide important +biases of the model. These concerns led to the definition +of new approaches and metrics that can aid the interpre- +tation of black box models [12]. +∗ Corresponding author: chcomin@gmail.com +For the task of segmentation in medical images, which +is the focus, but not the only application of our study, +the detection of the important structures is usually only +the first step of a more elaborate procedure for measur- +ing relevant properties such as size [13], regularity [14], +length [15, 16], and curvature [16, 17]. Therefore, sys- +tematic segmentation mistakes might lead to undetected +errors when characterizing samples for clinical diagnoses +[18] and research purposes [9]. An important cause of +such systematic errors can be the presence of samples +having characteristics that occur with low frequency in a +dataset. This can happen due to additional, unexpected, +noise during image acquisition, variations in tissue stain- +ing, image artifacts, or even the presence of structures +that are anatomically different than what was expected. +Assuming for illustration purposes that the data is nor- +mally distributed, a machine learning model having good +performance around the peak of the distribution will tend +to have good average accuracy measured for the whole +dataset, even if it cannot correctly classify or segment +images that are around the tail of the distribution, which +might be important for downstream analyses. +Notice +that this discussion does not necessarily only involves +outlier images, but any image occurring with a low prob- +ability according to some criteria. Nevertheless, the con- +cept of outlier has been given different definitions [19, 20], +and it sometimes is used as a synonym for low probability. +Here we argue that a machine learning model should +have good performance, or even be directly optimized, on +both prototypical and atypical samples. This focus can +lead to models that are more robust to samples located +in a sparsely populated region of the feature space of the +dataset. In addition, it might also lead to models that +generalize better to out-of-distribution samples as well +arXiv:2301.04517v1 [cs.CV] 11 Jan 2023 + +2 +as to new datasets. We develop a simple and intuitive +sampling methodology to select a subset of images from +the dataset representing, as best as possible, a uniform +coverage of the feature space of all the samples. +This +subset can be used for measuring the homogeneity of the +performance of the model on highly distinct samples. It +can also be used for identifying regions in the feature +space where the model does not perform well. +We focus on applying the method to the important +segmentation problem of selecting relevant samples for +ground truth mask annotation. +For medical images, +manually annotating masks can be very time consum- +ing and requires skilled workers or specialists. Therefore, +when many samples are available for annotation, it is in- +teresting to select an optimal subset of samples that leads +to a model that is also expected to perform well on the +whole dataset. To this end, we present a case example of +selecting an appropriate subset of samples from a blood +vessel dataset containing 2800 images. We show that the +selected samples uniformly cover different properties of +the images and thus represent a rich set of samples that +can be annotated and used for training a segmentation +algorithm for processing the whole dataset. +A concept that is similar to the developed methodol- +ogy is the so-called coreset [21]. The aim of a coreset +is to select a subset of samples that can optimally rep- +resent the whole dataset. Many different methodologies +and criteria were developed for defining relevant coresets +[21–23]. +Indeed, the subset defined by our methodol- +ogy can be associated with a coreset, but in our case, +the aim of the generated subset and the approach used +differs markedly from the usual definition of a coreset. +The aim of our methodology is not focused on accurately +representing the whole dataset, or on preserving the ac- +curacy of a machine learning algorithm, but on provid- +ing a relevant dataset for training machine learning algo- +rithms while avoiding the underrepresentation of atypical +samples. In addition, many coreset methodologies use a +surrogate neural network to estimate latent features or +to estimate a degree of uncertainty about each sample, +while our methodology is more general in the sense that +any set of features obtained from the samples can be +used. +Those can even be specific features such as im- +age contrast and average intensity. Furthermore, many +studies consider a similarity metric for selecting relevant +samples, which is a degenerate metric and therefore can- +not provide a full representation of the data distribution. +Our methodology also has some relationship with active +learning [24–27]. But in our case, the whole subset of +samples is generated in a single run and does not require +an interactive session with a specialist. +II. +METHODOLOGY +The sampling methodology proposed in this work can +be divided into three steps: (a) dataset mapping to an n- +dimensional feature space, (b) generating a discrete rep- +resentation of this feature space, and (c) drawing points +uniformly from this feature space representation. We ex- +plain each of these steps in the following subsections. +A. +Dataset mapping +Given a dataset D = {δ1, δ2, ..., δn} with n objects, and +a function f : δi → ⃗pi that maps an object δi to a vector +⃗pi with dimension d, the new dataset mapped to a feature +space can be represented as a n × d matrix. Each line of +this matrix, which we call Dmapped, therefore represents +the features of an object f(δi). Figure 1 illustrates this +procedure for a set of images. +Each image of the set +(Figure 1a) is mapped to a point in the new feature space +(Figure 1b). +(a) +(b) +FIG. 1. Representation of the mapping procedure applied to +a set of D samples – in this case, images (a). In this example, +each image of D is mapped to a 2-d position in the new feature +space (b). +There are many different approaches for defining f(δi). +For example, if D is a set of images, we can map each +image using a combination of relevant features, such as +image contrast or signal-to-noise ratio. Feature mapping +can also be achieved through a prior unsupervised or +semi-supervised segmentation of the objects in the im- +ages. In this case, features such as object area or elonga- +tion can be used. Supervised learning can also be used to +map the images to a new feature space. In this scenario, +instead of mapping D through handcrafted features, the +features learned by a supervised segmentation algorithm, +such as a neural network, can be used. In Section III we +present an example with a mapping function defined as +a combination of global image features and handcrafted +features from semi-supervised segmentation. +B. +Feature space discretization +The second step of the sampling methodology is to de- +fine a regular grid in the feature space and to translate +each data point to the nearest point in this grid. It is +useful to first normalize the values of Dmapped to remove +differences in the scale of the features. In this work, we +used z-score normalization, but other normalizations can + +3 +be used. After normalizing Dmapped, we proceed to re- +sample the mapped values to a discrete grid. This is done +by defining a scale ν that sets the size of each grid cell +and calculating +Dgrid = floor +�Dmapped +ν +� +, +(1) +where floor is the floor function. As shown in Figure +2, this operation ensures that each value of Dgrid lies +within a regular grid. +Note that, as a consequence of +undersampling, we expect multiple data points to fall in +the same grid position, this is one of the key properties +of the method that will allow a uniform sampling of the +data. +FIG. 2. +Visualization of the feature space discretization. +Light red points correspond to the original features (as shown +in Figure 1b). Equation 1 corresponds to moving the points +to a position within the regular grid. +After feature space discretization, we generate a sparse +set of points representing an estimation of the possible +values that can be obtained in the feature space. +We +call this set the sampling set of the feature space. An +illustration of the procedure is shown in Figure 3. The +procedure works as follows. An n-dimensional discrete +hypersphere S with radius r (in grid units) centered on +each data point is defined (Figure 3a). This hypersphere +is translated to each data point position. The union of +the calculated hypersphere positions of all points define +the sampling set Ddist (Figure 3b). +This operation is +described in the following algorithm: +Algorithm 1 Estimate the sampling set of the feature +space +1: Ddist ← {} +▷ Initialize Ddist as an empty set +2: for ⃗p in Dgrid do +3: +for ⃗s in S do +4: +⃗ts = ⃗s + ⃗p +5: +Ddist = Ddist ∪ ⃗ts +6: +end for +7: end for +(a) +(b) +Discrete disk centered +at origin +Translate the disk to +each data point and add +it to the distribution +FIG. 3. Estimation of the feasible values of the feature space. +In (a) we define a sparse hypersphere (depicted as a disk for +visualization) as a set of uniformly distributed points. This +disk is translated to the center of each point in Dgrid (red +points of (b)). The union of all hypersphere points defines +Ddist. +Given that Algorithm 1 operates over a discrete space, +we expect repeated values of the hypersphere points ⃗ts. +These repeated values account for important information +regarding the spatial distribution of the points in Ddist +and can be stored and used to sample the points in a +non-uniform manner. +Since the main goal of the pre- +sented methodology is to select heterogeneous subsets of +the original data, which can optimally be done using a +uniform distribution, we ignore repeated values of ⃗ts. +C. +Uniform selection of samples +The last step of the sampling methodology consists +of drawing a set of points from the sampling set Ddist. +As illustrated in Figure 4, we draw from Ddist n points +with uniform probability (green dots in Figure 4). For +each point drawn, the closest data sample is identified +using the Euclidean distance. If the same data sample +is obtained more than once, a new point is drawn from +Ddist until n unique data samples are obtained. The set +of data samples obtained (orange stars in Figure 4) is +represented as Dsampled. +As mentioned before, a uniform sampling of Ddist al- +lows us to select prototypical and atypical samples of our + +4 +FIG. 4. +Illustration of the proposed sampling protocol. +n +random points (green dots) are drawn from the set Ddist +(blue dots). +The subset of sampled data points is defined +by the data points that are closest to each point drawn (or- +ange stars). Red squares represent the remaining data points +that were not selected. +dataset with equal probability. +Nevertheless, one can- +not guarantee that a random sampling of the estimated +distribution will lead to a solution that maximizes the +heterogeneity of the sampled subset. In particular, any +given realization of the sampling may lead to distortions +such as the selection of many samples at similar regions of +the space or the creation of large regions with no samples +selected. This is due to random fluctuations in the sam- +pling. To amend this, we define a metric called Farthest +Unselected Point (FUS), that punishes sampled subsets +with large gaps between drawn points. +Being Dsampled the set of sampled data points from +Dgrid, and ¬Dsampled the set of points from Dgrid that +were not selected in the sampling subset, the FUS met- +ric, as the name implies, measures the largest euclidean +distance between any points of ¬Dsampled and Dsampled. +Sampled subsets leading to low values of the FUS metric +should be preferred, since it leads to lower gaps between +points in Dsampled. This property is illustrated in Figure +5. In Figure 5(a), the farthest unselected point evidences +a gap of sampled points in the upper right corner of the +distribution. This gap is less pronounced in Figure 5(b), +where the points are sampled more heterogeneously and +unselected points are, in general, close to the sampled +subset distribution. With that in mind, we ultimately +choose Dsampled as the set with the lowest FUS amongst +N sampled subsets. +In our experiments, we find that +N ≥ 1000 covers a good amount of subset possibilities, +but the optimal value of N may change depending on the +dataset being studied. +To illustrate the potential of the methodology, Figure 6 +shows a comparison between the usual approach for se- +lecting a subset of the data, to simply draw the points +(a) +(b) +Large FUS +Small FUS +FIG. 5. An illustration of how minimizing the FUS metric also +minimizes gaps between sampled points. In (a), a higher dis- +tance between unselected points (depicted as red squares) and +sampled points (depicted as orange starts) indicates a large +gap between sampled points. This behavior is less pronounced +in (b), where a more heterogeneous subset was sampled. +at random with uniform probability, and the selection of +points using the presented method. Using uniform sam- +pling, the points tend to be selected according to the un- +derlying probability density function of the data, which +is usually unknown. Thus, most of the selected points +will tend to be located in denser regions of the feature +space, which consequently biases the sampled subset to- +wards the regions of the dataset with prototypical exam- +ples (Figure 6a). By drawing the subset of points using +our methodology, the selected points will tend to cover +the feature space more uniformly (Figure 6b). +III. +CASE EXAMPLE – CREATING A DATASET +FOR BLOOD VESSEL SEGMENTATION +To show the potential of the methodology, in this sec- +tion we describe an application of the method on real +data. We show how the method can aid in the selection +of a heterogeneous set of samples that can then be used +for training a supervised segmentation algorithm. +A. +Blood vessel dataset +The dataset we will use contains confocal microscopy +images of mouse brain vasculature. +The dataset has +around 2800 images having sizes from 1376 × 1104 to +2499 × 2005, totaling around 3.8GB of data. The images +were acquired under different experimental conditions in +different works published in the literature [28–30]. Condi- +tions include control animals, animals that have suffered +a deletion of chromosome 16p11.12, animals that have ex- +perienced sense deprivation or sense hyperarousal, sam- +ples from stroke regions, and also from different stages of +mouse development. +This dataset is interesting because it has a considerable + +5 +(a) +(b) +FIG. 6. Illustration of two sampling strategies. Red x’s represent the sampled points. The background color represents the +underlying probability density function of the data, with yellow representing larger values. In (a), we have a typical example +of uniformly sampling the indexes of a dataset, which leads to the selection of many samples that are similar to each other. In +(b), the points are selected using the presented methodology. In this case, the sampled points are spread out and cover the +feature space. +variety of characteristics of blood vessels. In addition, the +images represent samples obtained from hundreds of dif- +ferent animals and experimental conditions. This makes +it an excellent dataset for training machine learning al- +gorithms for blood vessel segmentation. But training su- +pervised algorithms requires the manual annotation of +the blood vessels in a subset of the images. +After annotating a few samples, we estimated that each +image in the dataset takes roughly 12 hours to fully an- +notate. Therefore, it is unfeasible to annotate the whole +dataset. Thus, it is interesting to select relevant sam- +ples for annotation in order to train a machine learning +algorithm to segment the whole dataset. As mentioned +before, this subset of samples should allow the training to +occur without biases, that is, atypical samples should be +well-represented so that the accuracy of the algorithm de- +pends as little as possible on the properties of the images +or the tissues under analysis. This means that it is inter- +esting to select both prototypical and atypical samples +for annotation. The methodology presented in Section II +was used for selecting the samples. +Each image in the dataset may include illumination +inhomogeneities, changes in contrast, different levels of +noise, as well as blood vessels having distinct charac- +teristics (e.g., caliber, tortuosity, etc). Thus, from the +original dataset, we generated a new set of images, each +having a size of 256× 256. These smaller images were +generated by extracting 256× 256 regions from the orig- +inal images. As shown in Figure 7, seven regions were +extracted from each image. The seven regions were ex- +tracted in key areas of each image, with four windows in +each of the corners of the image, a central window, and +two windows at random positions. The latter two may +overlap with the other windows. The total size of the +resulting dataset is 16800 images. +FIG. 7. An example of how we extract seven windows from +a single sample. The four corners along with the central re- +gion can capture most of the illumination inhomogeneities +that may occur due to uneven illumination of the samples. +Besides these five regions, two additional random regions are +also drawn for each image. +After obtaining the various extracted images, they +were mapped to a feature space that was used for charac- +terizing the samples. As mentioned in Section II A, image +features can be calculated by processing labels obtained +from semi-supervised segmentation. +Since our dataset +was used in previous works, each image has a respective + +6 +segmentation that was obtained using a semi-supervised +methodology. This methodology is based on the adaptive +thresholding of the original images, where the threshold +was selected manually for each image. The full details on +the segmentation procedure are described in [16]. +B. +Metrics +We consider the following features to characterize the +samples: blood vessel contrast, level of gaussian noise, +blood vessel density, and medial line heterogeneity. +The blood vessel contrast is related to the difference +in intensity between the vessels and the background of +the image. The greater the contrast, the easier it is to +detect the vessels. It can be measured using the original +image of the vessels and the respective thresholded image +containing an estimation of the pixels belonging to the +vessels. The contrast is calculated as +C1 = +¯Iv +¯If +, +(2) +where ¯Iv and ¯If are the mean intensities of, respectively, +the pixels belonging to the blood vessels and the back- +ground of the image. +The signal-to-noise level of the images can be estimated +in different ways. +We investigated different definitions +and used the method that was the most compatible with +a visual inspection of the images. The method proposed +in [31] was used. It assumes a noise with normal distribu- +tion and uses wavelets to identify the most likely standard +deviation of the noise component. To prevent the method +from capturing vessel variation, only the background of +the image was used for the estimation. +Blood vessel density is defined as the total length of +blood vessels in an image divided by the image area. To +do this, we first apply a skeletonization algorithm to ex- +tract the medial lines of the vessels. In this work, we +used the Pal´agyi-Kuba algorithm [32]. The total length +of vessels is calculated as the sum of the arc-lengths of +all vessel segments. This metric is also fully explained in +[16]. +The last metric, which we call medial line heterogene- +ity, measures the illumination changes in the vessel lu- +men. To calculate this metric, we first blur the image +using a gaussian filter with unit standard deviation to +remove extreme values. The medial line heterogeneity is +calculated as the standard deviation of the pixel values +along the medial lines of this blurred image. The medial +lines considered are the same ones used for the blood +vessel density metric. +We observed that the heterogeneity tends to be cor- +related with the average intensity of the blood vessels. +In order to remove this dependency, the medial line het- +erogeneity as well as the average intensity of the medial +lines was calculated for all images in the dataset. Then, +a straight line fit hm = a ∗ m + b was applied to the +calculated values, where m is the average intensity and +hm is the expected medial line heterogeneity associated +with m. +Next, a normalized medial line heterogeneity +was defined as ˜h = h − hm, where h is the medial line +heterogeneity calculated for an image. +C. +Subset sampling +With the four metrics described in the previous section, +we mapped our dataset to a 4-d feature space by applying +the methodology detailed in Section II A. As mentioned +before, the dataset contains 16800 images. Hence, the +whole dataset was mapped to a 16800 × 4 matrix. In the +feature space discretization step (Section II B), we used +z-score normalization and a scale of ν = 10. The hyper- +sphere was generated with a radius equal to four times +the grid space resolution. In the sampling step (Section +II C), we used N = 1000 and chose the sampling solution +that minimized the FUS metric. +The sampled subset +size contained n = 100 images. To avoid data leaking, +an additional restriction that prevented the selection of +samples from the same image was used. +The generated subset contains a heterogeneous set of +samples that can be used for annotation. It is difficult +to properly measure the heterogeneity of this subset be- +cause it would involve the estimation of the probability +density function of the original data, which is not a triv- +ial task and can be influenced by the choice of parameter +values. But it is clear that the method must naturally +lead to a uniform selection of the samples. +This is so +because the set Dgrid (defined in Section II B) represents +an estimation of the domain of the probability density +function of the data, and this domain is being sampled +uniformly. +One approach to illustrate the characteristics of the +sampled images is displayed in Figure 8, which shows +histograms of the four considered features for both the +full dataset and the sampled subset. The histograms of +individual features are not expected to be uniform since +they represent a projection of the original data into one +dimension. Still, it can be seen that the histograms of the +sampled set tend to represent a slightly flattened version +of the histograms of the original data, indicating that a +larger priority is being given to atypical samples when +compared to the original distribution. For instance, in +the original data, the chances of selecting an image with +a typical contrast (values close to zero in the z-score) +are roughly 100× larger than the chances of selecting an +image with low contrast. On the other hand, the method +defined a subset where the images with contrast close to +0 are only around 10× more frequent than low-contrast +images. +A more robust way of visually checking the sampled +subset is to visualize the data using Principal Compo- +nent Analysis (PCA). Using PCA, the original 4-d data +can be projected into 2-d with optimal preservation of +the variance of the data. Figure 9 shows the PCA pro- + +7 +20 +10 +0 +10 +20 +30 +Metric value +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Frequency +Density +20 +0 +20 +40 +60 +Metric value +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Frequency +Contrast +20 +0 +20 +40 +60 +Metric value +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Frequency +Medial line heterogeneity +20 +0 +20 +40 +60 +Metric value +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Frequency +Noise +FIG. 8. Histograms of the four features that were calculated +from the blood vessel dataset. Blue bars correspond to the +distribution of the full data. Orange bars correspond to the +distribution of the sampled subset. Note that the frequencies +were normalized by their sum. +jection of the data. The four plots included in the figure +represent the same projection, but the points are colored +according to the different features used to characterize +the images. It can be noticed that the sampling method- +ology selects a subset of images that uniformly covers the +distribution of the data. Furthermore, as also suggested +by the histograms in Figure 8, the sampling was capable +of covering the full range of values of every considered +feature. +The subset of images selected by the method is shown +in Figure 10. The subset indeed contains a heterogeneous +set of images covering many different values of the consid- +ered features (e.g., low contrast, high vessel density, etc). +This subset can then be manually annotated and used for +training a machine learning algorithm to segment the full +dataset. Since atypical samples are well-represented, we +expect fewer biases in downstream analyses using the ob- +tained segmentation. For instance, some of the samples +in the dataset come from animals who suffered hemor- +rhagic strokes. These samples are very different from the +typical samples contained in the dataset, and they would +be largely underrepresented if a usual uniform sampling +of the original data, which is associated with sampling +the data according to its distribution, was performed. +40 +20 +0 +20 +40 +60 +80 +PC 1 +40 +20 +0 +20 +40 +60 +PC 2 +Density +40 +20 +0 +20 +40 +60 +80 +PC 1 +40 +20 +0 +20 +40 +60 +PC 2 +Contrast +40 +20 +0 +20 +40 +60 +80 +PC 1 +40 +20 +0 +20 +40 +60 +PC 2 +Medial line heterogeneity +40 +20 +0 +20 +40 +60 +80 +PC 1 +40 +20 +0 +20 +40 +60 +PC 2 +Noise +FIG. 9. PCA of the blood vessel dataset. Red points cor- +respond to the sampled subset obtained by the sampling +methodology. +Other points correspond to the unselected +points, with their colors representing the value of each one +of the four original metrics: vessel density, contrast, medial +line heterogeneity, and image noise. +IV. +CONCLUSION +Selecting appropriate images for training machine +learning algorithms is an important task that has been +given relatively little attention. This is because the usual +approach is to use as many images as possible. While this +approach is relevant for general classification problems, +for medical image segmentation, where image annotation +can be very costly, the images used must be carefully se- +lected in order to ensure good coverage of different tissue +appearances and imaging variations. +In addition, it is +important that the selected images do not lead to biases +in downstream tasks related to tissue characterization. +For instance, we argued that training segmentation algo- +rithms mostly on prototypical images can lead to incor- +rect measurements on samples having unusual properties +(e.g., very bright or very noisy). +We developed an intuitive sampling methodology that +evenly selects, as best as possible, both typical and atyp- +ical samples for creating a rich dataset that can then +be annotated and used for training segmentation algo- +rithms. One important property of the method is that +it provides an intuitive uniform grid in the feature space +that can be used for further analyses. For example, one +can study the accuracy of the segmentation on differ- +ent regions of the grid to identify regions where samples +are not being correctly segmented. A robust algorithm +should provide good segmentation no matter if a sample + +8 +FIG. 10. Subset obtained by our methodology. Notice that the images cover a wide range of values in the feature space defined +by our four features. Contrast variation and vessel density are the easier features to visually verify. Medial line heterogeneity +can be verified by how much the brightness changes longitudinally within a blood vessel. Gaussian noise level is harder to +verify visually, but pronounced noise can be observed on some of the brighter images. +is too noisy, bright or dark, if it has low or high con- +trast or any other variation on relevant image properties. +Likewise, expected tissue changes in the samples should +not lead to variations in accuracy. +The method has only two parameters, the resolution +and the radius of the hyperspherical structuring element. +The resolution can be set according to the scale in the +feature space where images are expected to be very simi- +lar, and thus there is no need to consider multiple images +with such a small variation in appearance. The radius of +the hypersphere can be adjusted according to the aver- +age distance between points in the dataset. For instance, +one can calculate the average distance between the near- +est neighbors of the dataset to gain intuition about the + +59 +typical distances involved and then set the radius as a +multiple of this distance. +We applied the methodology to a blood vessel dataset +and showed that it can generate a heterogeneous set of +samples representing many possible variations of image +noise and contrast as well as blood vessel density and in- +tensity variance. A segmentation algorithm that can suc- +cessfully identify the blood vessels on the selected images +should generalize well on the whole dataset. More impor- +tantly, the selected samples guarantee that the algorithm +will not focus only on the most prototypical samples, thus +avoiding segmentation biases. +While the methodology was presented focusing on sam- +ple selection for annotation, it can also be used on any +dataset to analyze possible performance biases. For in- +stance, one could define an appropriate latent space for +the ImageNet [33] and generate a subset of samples that +uniformly covers the space. The performance of a robust +classifier should be similar in all regions of this space. +Therefore, the sampling methodology presented here can +also be used for interpreting so-called black box machine +learning algorithms [34]. +FUNDING +Cesar H. Comin thanks FAPESP (grant no. 21/12354- +8) for financial support. 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Fei-Fei, “ImageNet: A Large-Scale Hierarchical Im- +age Database,” in CVPR09 (2009). + +11 +[34] Cynthia Rudin, “Stop explaining black box machine +learning models for high stakes decisions and use inter- +pretable models instead,” Nature Machine Intelligence 1, +206–215 (2019). + diff --git a/W9E3T4oBgHgl3EQfbgrt/content/tmp_files/load_file.txt b/W9E3T4oBgHgl3EQfbgrt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..acf9d1d0d2bdd6e3429e4890845f735f621acb21 --- /dev/null +++ b/W9E3T4oBgHgl3EQfbgrt/content/tmp_files/load_file.txt @@ -0,0 +1,447 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf,len=446 +page_content='A new sampling methodology for creating rich, heterogeneous, subsets of samples for training image segmentation algorithms Matheus Viana da Silva,1 Nat´alia de Carvalho Santos,2 Baptiste Lacoste,3 and Cesar H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Comin1, ∗ 1Department of Computer Science, Federal University of S˜ao Carlos, S˜ao Carlos, SP, Brazil 2S˜ao Carlos Institute of Physics, University of S˜ao Paulo, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Trab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' S˜ao Carlense, 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Parque Arnold Schmidit, S˜ao Carlos, SP, 13566-590, Brazil 3Department of Cellular and Molecular Medicine, Faculty of Medicine, University of Ottawa, Ottawa, ON, Canada (Dated: January, 2023) Creating a dataset for training supervised machine learning algorithms can be a demanding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This is especially true for medical image segmentation since this task usually requires one or more specialists for image annotation, and creating ground truth labels for just a single image can take up to several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In addition, it is paramount that the annotated samples represent well the different conditions that might affect the imaged tissue as well as possible changes in the image acquisition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This can only be achieved by considering samples that are typical in the dataset as well as atypical, or even outlier, samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We introduce a new sampling methodology for selecting relevant images from a larger non-annotated dataset in a way that evenly considers both prototypical as well as atypical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The methodology involves the generation of a uniform grid from a feature space representing the samples, which is then used for randomly drawing relevant images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The selected images provide a uniform cover of the original dataset, and thus define a heterogeneous set of images that can be annotated and used for training supervised segmentation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We provide a case example by creating a dataset containing a representative set of blood vessel microscopy images selected from a larger dataset containing thousands of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' INTRODUCTION Recent developments in neural networks led to un- precedented results in image classification [1, 2], object detection [3] and image segmentation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Neural net- works, and the related field of Deep Learning, shifted the prevalent focus on feature engineering to a more data- centric approach, where network models can approximate highly complex functions provided the model has enough capacity and the dataset is sufficiently representative [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Due to such impressive advancements, many authors ar- gue that neural networks have reached human-level ac- curacy [6, 7], or even surpassed it [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The performance of neural networks has dominantly been measured using metrics such as classification or seg- mentation accuracy, precision, recall, and the area under the ROI curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' However, recent studies have shown the dangers of only considering such accuracy-related met- rics [9–11] that provide only an aggregated, summarized, view of the performance of machine learning algorithms on datasets with sometimes millions of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Such an approach may hide important biases of the model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For instance, for medical images, a 95% accuracy is usu- ally considered a good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' But what about the remaining 5%?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It is unrealistic to expect models to reach 100% accuracy, but the samples that are not cor- rectly processed by a neural network may hide important biases of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' These concerns led to the definition of new approaches and metrics that can aid the interpre- tation of black box models [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' ∗ Corresponding author: chcomin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='com For the task of segmentation in medical images, which is the focus, but not the only application of our study, the detection of the important structures is usually only the first step of a more elaborate procedure for measur- ing relevant properties such as size [13], regularity [14], length [15, 16], and curvature [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Therefore, sys- tematic segmentation mistakes might lead to undetected errors when characterizing samples for clinical diagnoses [18] and research purposes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' An important cause of such systematic errors can be the presence of samples having characteristics that occur with low frequency in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This can happen due to additional, unexpected, noise during image acquisition, variations in tissue stain- ing, image artifacts, or even the presence of structures that are anatomically different than what was expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Assuming for illustration purposes that the data is nor- mally distributed, a machine learning model having good performance around the peak of the distribution will tend to have good average accuracy measured for the whole dataset, even if it cannot correctly classify or segment images that are around the tail of the distribution, which might be important for downstream analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Notice that this discussion does not necessarily only involves outlier images, but any image occurring with a low prob- ability according to some criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Nevertheless, the con- cept of outlier has been given different definitions [19, 20], and it sometimes is used as a synonym for low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Here we argue that a machine learning model should have good performance, or even be directly optimized, on both prototypical and atypical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This focus can lead to models that are more robust to samples located in a sparsely populated region of the feature space of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In addition, it might also lead to models that generalize better to out-of-distribution samples as well arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='04517v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='CV] 11 Jan 2023 2 as to new datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We develop a simple and intuitive sampling methodology to select a subset of images from the dataset representing, as best as possible, a uniform coverage of the feature space of all the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This subset can be used for measuring the homogeneity of the performance of the model on highly distinct samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It can also be used for identifying regions in the feature space where the model does not perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We focus on applying the method to the important segmentation problem of selecting relevant samples for ground truth mask annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For medical images, manually annotating masks can be very time consum- ing and requires skilled workers or specialists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Therefore, when many samples are available for annotation, it is in- teresting to select an optimal subset of samples that leads to a model that is also expected to perform well on the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To this end, we present a case example of selecting an appropriate subset of samples from a blood vessel dataset containing 2800 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We show that the selected samples uniformly cover different properties of the images and thus represent a rich set of samples that can be annotated and used for training a segmentation algorithm for processing the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A concept that is similar to the developed methodol- ogy is the so-called coreset [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The aim of a coreset is to select a subset of samples that can optimally rep- resent the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Many different methodologies and criteria were developed for defining relevant coresets [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Indeed, the subset defined by our methodol- ogy can be associated with a coreset, but in our case, the aim of the generated subset and the approach used differs markedly from the usual definition of a coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The aim of our methodology is not focused on accurately representing the whole dataset, or on preserving the ac- curacy of a machine learning algorithm, but on provid- ing a relevant dataset for training machine learning algo- rithms while avoiding the underrepresentation of atypical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In addition, many coreset methodologies use a surrogate neural network to estimate latent features or to estimate a degree of uncertainty about each sample, while our methodology is more general in the sense that any set of features obtained from the samples can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Those can even be specific features such as im- age contrast and average intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Furthermore, many studies consider a similarity metric for selecting relevant samples, which is a degenerate metric and therefore can- not provide a full representation of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Our methodology also has some relationship with active learning [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' But in our case, the whole subset of samples is generated in a single run and does not require an interactive session with a specialist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' METHODOLOGY The sampling methodology proposed in this work can be divided into three steps: (a) dataset mapping to an n- dimensional feature space, (b) generating a discrete rep- resentation of this feature space, and (c) drawing points uniformly from this feature space representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We ex- plain each of these steps in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Dataset mapping Given a dataset D = {δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=', δn} with n objects, and a function f : δi → ⃗pi that maps an object δi to a vector ⃗pi with dimension d, the new dataset mapped to a feature space can be represented as a n × d matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Each line of this matrix, which we call Dmapped, therefore represents the features of an object f(δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Figure 1 illustrates this procedure for a set of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Each image of the set (Figure 1a) is mapped to a point in the new feature space (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Representation of the mapping procedure applied to a set of D samples – in this case, images (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this example, each image of D is mapped to a 2-d position in the new feature space (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' There are many different approaches for defining f(δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For example, if D is a set of images, we can map each image using a combination of relevant features, such as image contrast or signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Feature mapping can also be achieved through a prior unsupervised or semi-supervised segmentation of the objects in the im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this case, features such as object area or elonga- tion can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Supervised learning can also be used to map the images to a new feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this scenario, instead of mapping D through handcrafted features, the features learned by a supervised segmentation algorithm, such as a neural network, can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In Section III we present an example with a mapping function defined as a combination of global image features and handcrafted features from semi-supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Feature space discretization The second step of the sampling methodology is to de- fine a regular grid in the feature space and to translate each data point to the nearest point in this grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It is useful to first normalize the values of Dmapped to remove differences in the scale of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this work, we used z-score normalization, but other normalizations can 3 be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' After normalizing Dmapped, we proceed to re- sample the mapped values to a discrete grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This is done by defining a scale ν that sets the size of each grid cell and calculating Dgrid = floor �Dmapped ν � , (1) where floor is the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As shown in Figure 2, this operation ensures that each value of Dgrid lies within a regular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Note that, as a consequence of undersampling, we expect multiple data points to fall in the same grid position, this is one of the key properties of the method that will allow a uniform sampling of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Visualization of the feature space discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Light red points correspond to the original features (as shown in Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Equation 1 corresponds to moving the points to a position within the regular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' After feature space discretization, we generate a sparse set of points representing an estimation of the possible values that can be obtained in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We call this set the sampling set of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' An illustration of the procedure is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The procedure works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' An n-dimensional discrete hypersphere S with radius r (in grid units) centered on each data point is defined (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This hypersphere is translated to each data point position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The union of the calculated hypersphere positions of all points define the sampling set Ddist (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This operation is described in the following algorithm: Algorithm 1 Estimate the sampling set of the feature space 1: Ddist ← {} ▷ Initialize Ddist as an empty set 2: for ⃗p in Dgrid do 3: for ⃗s in S do 4: ⃗ts = ⃗s + ⃗p 5: Ddist = Ddist ∪ ⃗ts 6: end for 7: end for (a) (b) Discrete disk centered at origin Translate the disk to each data point and add it to the distribution FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Estimation of the feasible values of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In (a) we define a sparse hypersphere (depicted as a disk for visualization) as a set of uniformly distributed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This disk is translated to the center of each point in Dgrid (red points of (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The union of all hypersphere points defines Ddist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Given that Algorithm 1 operates over a discrete space, we expect repeated values of the hypersphere points ⃗ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' These repeated values account for important information regarding the spatial distribution of the points in Ddist and can be stored and used to sample the points in a non-uniform manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Since the main goal of the pre- sented methodology is to select heterogeneous subsets of the original data, which can optimally be done using a uniform distribution, we ignore repeated values of ⃗ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Uniform selection of samples The last step of the sampling methodology consists of drawing a set of points from the sampling set Ddist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As illustrated in Figure 4, we draw from Ddist n points with uniform probability (green dots in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For each point drawn, the closest data sample is identified using the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' If the same data sample is obtained more than once, a new point is drawn from Ddist until n unique data samples are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The set of data samples obtained (orange stars in Figure 4) is represented as Dsampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As mentioned before, a uniform sampling of Ddist al- lows us to select prototypical and atypical samples of our 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Illustration of the proposed sampling protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' n random points (green dots) are drawn from the set Ddist (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The subset of sampled data points is defined by the data points that are closest to each point drawn (or- ange stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Red squares represent the remaining data points that were not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' dataset with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Nevertheless, one can- not guarantee that a random sampling of the estimated distribution will lead to a solution that maximizes the heterogeneity of the sampled subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In particular, any given realization of the sampling may lead to distortions such as the selection of many samples at similar regions of the space or the creation of large regions with no samples selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This is due to random fluctuations in the sam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To amend this, we define a metric called Farthest Unselected Point (FUS), that punishes sampled subsets with large gaps between drawn points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Being Dsampled the set of sampled data points from Dgrid, and ¬Dsampled the set of points from Dgrid that were not selected in the sampling subset, the FUS met- ric, as the name implies, measures the largest euclidean distance between any points of ¬Dsampled and Dsampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Sampled subsets leading to low values of the FUS metric should be preferred, since it leads to lower gaps between points in Dsampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This property is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In Figure 5(a), the farthest unselected point evidences a gap of sampled points in the upper right corner of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This gap is less pronounced in Figure 5(b), where the points are sampled more heterogeneously and unselected points are, in general, close to the sampled subset distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' With that in mind, we ultimately choose Dsampled as the set with the lowest FUS amongst N sampled subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In our experiments, we find that N ≥ 1000 covers a good amount of subset possibilities, but the optimal value of N may change depending on the dataset being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To illustrate the potential of the methodology, Figure 6 shows a comparison between the usual approach for se- lecting a subset of the data, to simply draw the points (a) (b) Large FUS Small FUS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' An illustration of how minimizing the FUS metric also minimizes gaps between sampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In (a), a higher dis- tance between unselected points (depicted as red squares) and sampled points (depicted as orange starts) indicates a large gap between sampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This behavior is less pronounced in (b), where a more heterogeneous subset was sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' at random with uniform probability, and the selection of points using the presented method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Using uniform sam- pling, the points tend to be selected according to the un- derlying probability density function of the data, which is usually unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Thus, most of the selected points will tend to be located in denser regions of the feature space, which consequently biases the sampled subset to- wards the regions of the dataset with prototypical exam- ples (Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' By drawing the subset of points using our methodology, the selected points will tend to cover the feature space more uniformly (Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' CASE EXAMPLE – CREATING A DATASET FOR BLOOD VESSEL SEGMENTATION To show the potential of the methodology, in this sec- tion we describe an application of the method on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We show how the method can aid in the selection of a heterogeneous set of samples that can then be used for training a supervised segmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Blood vessel dataset The dataset we will use contains confocal microscopy images of mouse brain vasculature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The dataset has around 2800 images having sizes from 1376 × 1104 to 2499 × 2005, totaling around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='8GB of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The images were acquired under different experimental conditions in different works published in the literature [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Condi- tions include control animals, animals that have suffered a deletion of chromosome 16p11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='12, animals that have ex- perienced sense deprivation or sense hyperarousal, sam- ples from stroke regions, and also from different stages of mouse development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This dataset is interesting because it has a considerable 5 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Illustration of two sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Red x’s represent the sampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The background color represents the underlying probability density function of the data, with yellow representing larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In (a), we have a typical example of uniformly sampling the indexes of a dataset, which leads to the selection of many samples that are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In (b), the points are selected using the presented methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this case, the sampled points are spread out and cover the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' variety of characteristics of blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In addition, the images represent samples obtained from hundreds of dif- ferent animals and experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This makes it an excellent dataset for training machine learning al- gorithms for blood vessel segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' But training su- pervised algorithms requires the manual annotation of the blood vessels in a subset of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' After annotating a few samples, we estimated that each image in the dataset takes roughly 12 hours to fully an- notate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Therefore, it is unfeasible to annotate the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Thus, it is interesting to select relevant sam- ples for annotation in order to train a machine learning algorithm to segment the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As mentioned before, this subset of samples should allow the training to occur without biases, that is, atypical samples should be well-represented so that the accuracy of the algorithm de- pends as little as possible on the properties of the images or the tissues under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This means that it is inter- esting to select both prototypical and atypical samples for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The methodology presented in Section II was used for selecting the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Each image in the dataset may include illumination inhomogeneities, changes in contrast, different levels of noise, as well as blood vessels having distinct charac- teristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=', caliber, tortuosity, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Thus, from the original dataset, we generated a new set of images, each having a size of 256× 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' These smaller images were generated by extracting 256× 256 regions from the orig- inal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As shown in Figure 7, seven regions were extracted from each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The seven regions were ex- tracted in key areas of each image, with four windows in each of the corners of the image, a central window, and two windows at random positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The latter two may overlap with the other windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The total size of the resulting dataset is 16800 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' An example of how we extract seven windows from a single sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The four corners along with the central re- gion can capture most of the illumination inhomogeneities that may occur due to uneven illumination of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Besides these five regions, two additional random regions are also drawn for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' After obtaining the various extracted images, they were mapped to a feature space that was used for charac- terizing the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As mentioned in Section II A, image features can be calculated by processing labels obtained from semi-supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Since our dataset was used in previous works, each image has a respective 6 segmentation that was obtained using a semi-supervised methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This methodology is based on the adaptive thresholding of the original images, where the threshold was selected manually for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The full details on the segmentation procedure are described in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Metrics We consider the following features to characterize the samples: blood vessel contrast, level of gaussian noise, blood vessel density, and medial line heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The blood vessel contrast is related to the difference in intensity between the vessels and the background of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The greater the contrast, the easier it is to detect the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It can be measured using the original image of the vessels and the respective thresholded image containing an estimation of the pixels belonging to the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The contrast is calculated as C1 = ¯Iv ¯If , (2) where ¯Iv and ¯If are the mean intensities of, respectively, the pixels belonging to the blood vessels and the back- ground of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The signal-to-noise level of the images can be estimated in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We investigated different definitions and used the method that was the most compatible with a visual inspection of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The method proposed in [31] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It assumes a noise with normal distribu- tion and uses wavelets to identify the most likely standard deviation of the noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To prevent the method from capturing vessel variation, only the background of the image was used for the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Blood vessel density is defined as the total length of blood vessels in an image divided by the image area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To do this, we first apply a skeletonization algorithm to ex- tract the medial lines of the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In this work, we used the Pal´agyi-Kuba algorithm [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The total length of vessels is calculated as the sum of the arc-lengths of all vessel segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This metric is also fully explained in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The last metric, which we call medial line heterogene- ity, measures the illumination changes in the vessel lu- men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To calculate this metric, we first blur the image using a gaussian filter with unit standard deviation to remove extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The medial line heterogeneity is calculated as the standard deviation of the pixel values along the medial lines of this blurred image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The medial lines considered are the same ones used for the blood vessel density metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We observed that the heterogeneity tends to be cor- related with the average intensity of the blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In order to remove this dependency, the medial line het- erogeneity as well as the average intensity of the medial lines was calculated for all images in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Then, a straight line fit hm = a ∗ m + b was applied to the calculated values, where m is the average intensity and hm is the expected medial line heterogeneity associated with m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Next, a normalized medial line heterogeneity was defined as ˜h = h − hm, where h is the medial line heterogeneity calculated for an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Subset sampling With the four metrics described in the previous section, we mapped our dataset to a 4-d feature space by applying the methodology detailed in Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' As mentioned before, the dataset contains 16800 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Hence, the whole dataset was mapped to a 16800 × 4 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In the feature space discretization step (Section II B), we used z-score normalization and a scale of ν = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The hyper- sphere was generated with a radius equal to four times the grid space resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In the sampling step (Section II C), we used N = 1000 and chose the sampling solution that minimized the FUS metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The sampled subset size contained n = 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' To avoid data leaking, an additional restriction that prevented the selection of samples from the same image was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The generated subset contains a heterogeneous set of samples that can be used for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It is difficult to properly measure the heterogeneity of this subset be- cause it would involve the estimation of the probability density function of the original data, which is not a triv- ial task and can be influenced by the choice of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' But it is clear that the method must naturally lead to a uniform selection of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This is so because the set Dgrid (defined in Section II B) represents an estimation of the domain of the probability density function of the data, and this domain is being sampled uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' One approach to illustrate the characteristics of the sampled images is displayed in Figure 8, which shows histograms of the four considered features for both the full dataset and the sampled subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The histograms of individual features are not expected to be uniform since they represent a projection of the original data into one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Still, it can be seen that the histograms of the sampled set tend to represent a slightly flattened version of the histograms of the original data, indicating that a larger priority is being given to atypical samples when compared to the original distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For instance, in the original data, the chances of selecting an image with a typical contrast (values close to zero in the z-score) are roughly 100× larger than the chances of selecting an image with low contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' On the other hand, the method defined a subset where the images with contrast close to 0 are only around 10× more frequent than low-contrast images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A more robust way of visually checking the sampled subset is to visualize the data using Principal Compo- nent Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Using PCA, the original 4-d data can be projected into 2-d with optimal preservation of the variance of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Figure 9 shows the PCA pro- 7 20 10 0 10 20 30 Metric value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='16 Frequency Density 20 0 20 40 60 Metric value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='16 Frequency Contrast 20 0 20 40 60 Metric value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='16 Frequency Medial line heterogeneity 20 0 20 40 60 Metric value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='16 Frequency Noise FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Histograms of the four features that were calculated from the blood vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Blue bars correspond to the distribution of the full data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Orange bars correspond to the distribution of the sampled subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Note that the frequencies were normalized by their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' jection of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The four plots included in the figure represent the same projection, but the points are colored according to the different features used to characterize the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' It can be noticed that the sampling method- ology selects a subset of images that uniformly covers the distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Furthermore, as also suggested by the histograms in Figure 8, the sampling was capable of covering the full range of values of every considered feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The subset of images selected by the method is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The subset indeed contains a heterogeneous set of images covering many different values of the consid- ered features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=', low contrast, high vessel density, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This subset can then be manually annotated and used for training a machine learning algorithm to segment the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Since atypical samples are well-represented, we expect fewer biases in downstream analyses using the ob- tained segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For instance, some of the samples in the dataset come from animals who suffered hemor- rhagic strokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' These samples are very different from the typical samples contained in the dataset, and they would be largely underrepresented if a usual uniform sampling of the original data, which is associated with sampling the data according to its distribution, was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 40 20 0 20 40 60 80 PC 1 40 20 0 20 40 60 PC 2 Density 40 20 0 20 40 60 80 PC 1 40 20 0 20 40 60 PC 2 Contrast 40 20 0 20 40 60 80 PC 1 40 20 0 20 40 60 PC 2 Medial line heterogeneity 40 20 0 20 40 60 80 PC 1 40 20 0 20 40 60 PC 2 Noise FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' PCA of the blood vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Red points cor- respond to the sampled subset obtained by the sampling methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Other points correspond to the unselected points, with their colors representing the value of each one of the four original metrics: vessel density, contrast, medial line heterogeneity, and image noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' CONCLUSION Selecting appropriate images for training machine learning algorithms is an important task that has been given relatively little attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' This is because the usual approach is to use as many images as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' While this approach is relevant for general classification problems, for medical image segmentation, where image annotation can be very costly, the images used must be carefully se- lected in order to ensure good coverage of different tissue appearances and imaging variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' In addition, it is important that the selected images do not lead to biases in downstream tasks related to tissue characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For instance, we argued that training segmentation algo- rithms mostly on prototypical images can lead to incor- rect measurements on samples having unusual properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=', very bright or very noisy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We developed an intuitive sampling methodology that evenly selects, as best as possible, both typical and atyp- ical samples for creating a rich dataset that can then be annotated and used for training segmentation algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' One important property of the method is that it provides an intuitive uniform grid in the feature space that can be used for further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For example, one can study the accuracy of the segmentation on differ- ent regions of the grid to identify regions where samples are not being correctly segmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A robust algorithm should provide good segmentation no matter if a sample 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Subset obtained by our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Notice that the images cover a wide range of values in the feature space defined by our four features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Contrast variation and vessel density are the easier features to visually verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Medial line heterogeneity can be verified by how much the brightness changes longitudinally within a blood vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Gaussian noise level is harder to verify visually, but pronounced noise can be observed on some of the brighter images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' is too noisy, bright or dark, if it has low or high con- trast or any other variation on relevant image properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Likewise, expected tissue changes in the samples should not lead to variations in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The method has only two parameters, the resolution and the radius of the hyperspherical structuring element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The resolution can be set according to the scale in the feature space where images are expected to be very simi- lar, and thus there is no need to consider multiple images with such a small variation in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The radius of the hypersphere can be adjusted according to the aver- age distance between points in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For instance, one can calculate the average distance between the near- est neighbors of the dataset to gain intuition about the 59 typical distances involved and then set the radius as a multiple of this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' We applied the methodology to a blood vessel dataset and showed that it can generate a heterogeneous set of samples representing many possible variations of image noise and contrast as well as blood vessel density and in- tensity variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' A segmentation algorithm that can suc- cessfully identify the blood vessels on the selected images should generalize well on the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' More impor- tantly, the selected samples guarantee that the algorithm will not focus only on the most prototypical samples, thus avoiding segmentation biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' While the methodology was presented focusing on sam- ple selection for annotation, it can also be used on any dataset to analyze possible performance biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' For in- stance, one could define an appropriate latent space for the ImageNet [33] and generate a subset of samples that uniformly covers the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The performance of a robust classifier should be similar in all regions of this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Therefore, the sampling methodology presented here can also be used for interpreting so-called black box machine learning algorithms [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' FUNDING Cesar H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' Comin thanks FAPESP (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' 21/12354- 8) for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' The authors acknowledge the support of the Government of Canada’s New Frontiers in Research Fund (NFRF) (NFRFE-2019-00641) and Google’s Latin America Research Awards (LARA 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} +page_content=' [1] Shutao 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high stakes decisions and use inter- pretable models instead,” Nature Machine Intelligence 1, 206–215 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfbgrt/content/2301.04517v1.pdf'} diff --git a/XNFIT4oBgHgl3EQfiisy/content/tmp_files/2301.11292v1.pdf.txt b/XNFIT4oBgHgl3EQfiisy/content/tmp_files/2301.11292v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d280d48f993c60251a220b942029b620519e1ab2 --- /dev/null +++ b/XNFIT4oBgHgl3EQfiisy/content/tmp_files/2301.11292v1.pdf.txt @@ -0,0 +1,783 @@ +The Quantum Alternating Operator Ansatz for Satisfiability +Problems +John Golden1, Andreas B¨artschi1, Daniel O’Malley2, and Stephan Eidenbenz1 +1Information Sciences (CCS-3), Los Alamos National Laboratory, Los Alamos, NM 87545 +2Computational Earth Sciences (EES-16), Los Alamos National Laboratory, Los Alamos, NM 87545 +We comparatively study, through large- +scale numerical simulation, the performance +across a large set of Quantum Alternating +Operator Ansatz (QAOA) implementations +for finding approximate and optimum so- +lutions to unconstrained combinatorial op- +timization problems. +Our survey includes +over 100 different mixing unitaries, and we +combine each mixer with both the standard +phase separator unitary representing the ob- +jective function and a thresholded version. +Our numerical tests for randomly chosen in- +stances of the unconstrained optimization +problems Max 2-SAT and Max 3-SAT reveal +that the traditional transverse-field mixer +with the standard phase separator performs +best for problem sizes of 8 through 14 vari- +ables, while the recently introduced Grover +mixer with thresholding wins at problems of +size 6. This result (i) corrects earlier work +suggesting that the Grover mixer is a supe- +rior mixer based only on results from prob- +lems of size 6, thus illustrating the need to +push numerical simulation to larger prob- +lem sizes to more accurately predict perfor- +mance; and (ii) it suggests that more com- +plicated mixers and phase separators may +not improve QAOA performance. +1 +Introduction +Combinatorial optimization is widely viewed to be +a promising application domain for quantum com- +puting, and the Quantum Approximate Optimiza- +tion Algorithm [1] and its generalization the Quan- +tum Alternate Operator Ansatz [2] – both abbre- +viated as QAOA – are the most prominent algo- +rithms for this application. +As theoretical guar- +antees of QAOA performance are few and far be- +tween, we approach QAOA as a heuristic optimiza- +tion algorithm that may perform very well in prac- +tice despite the lack of formal performance guar- +antees. The study of QAOA performance is in its +infancy and – until the arrival of large-scale error +corrected quantum computers – we are limited to +numerical experimentation on classical computers. +QAOA follows a simple iterative approach, +evolving an initial state with alternating phase +separating and mixing unitaries. The amount of +time (also referred to as the angle) for the phasing +and mixing at each iteration (or round) is tuned +to increase constructive interference amongst ba- +sis states which correspond to high-quality solu- +tions to a target optimization problem, and simi- +larly create destructive interference amongst poor- +quality solutions. Other studies have shown that +QAOA performance depends on the choice of mix- +ing operators, both for unconstrained [3] and con- +strained [4] optimization; moreover, mixing opera- +tors may work better with some phase separators +(operators that model the objective function) than +others. +Overall, the connection between mixers, +phase separators, and QAOA performance is not +well understood. +In this paper, we choose Max +2-SAT and Max 3-SAT as our problems of study, +which are both unconstrained optimization prob- +lems that ask for a truth assignment to variables +that maximize the number of clauses in the prob- +lem instance; these problems are unconstrained as +every possible (binary) assignment of truth values +to the values is a feasible solution. +We select a +subset of mixing unitaries that spans a wide range +of possible higher order mixing terms, with the +well-known transverse-field mixer [1] and the re- +1 +arXiv:2301.11292v1 [quant-ph] 26 Jan 2023 + +cently introduced Grover mixer [5] occupying op- +posite ends of the spectrum. These mixers are then +tested on problems of up to 14 variables. Previous +work [3] tested QAOA only with transverse-field +and Grover mixers, solving Max 2-SAT and 3-SAT +problems with only six variables, and found no- +table differences in performance with Grover mixer +emerging as the winner. In contrast, our analysis +includes a much larger array of mixers and phase +separators, and tests k-SAT problems up to size +n = 14. +Our main finding is that mixer performance de- +pends on problem size, and that the transverse- +field mixer is a poor choice for small problems, +but outperforms all other mixers on problems with +≥ 10 variables. We also find that relative perfor- +mance depends on the exact choice of metric for +performance. In particular, we compare approxi- +mation ratio vs. ground state probability, and find +that certain QAOA variations can have a relatively +high approximation ratio while having low ground +state probability. +2 +QAOA Review +Using QAOA to solve an optimization problem, +defined here by a cost function C(x) on binary +strings, requires several choices: an initial state +|ψ0⟩, a mixer Hamiltonian HM, a phase separa- +tor Hamiltonian HP , and a set of 2p parameters +(commonly referred to as angles) β = {β1, . . . , βp} +and γ = {γ1, . . . , γp}. One then uses a quantum +computer to apply the mixing and phase separat- +ing unitaries over p rounds to prepare the state +|ψp⟩ = e−iγpHP e−iβpHM · · · e−iγ1HP e−iβ1HM |ψ0⟩ . +(1) +The goal is to choose the above parameters so that +sampling from |ψp⟩ is likely to return a state which +is a good (or ideally optimal) solution to the opti- +mization problem. This is generally accomplished +by fixing |ψ0⟩ , HM, HP and using classical opti- +mization techniques to modify β and γ in order to +maximize (or minimize) ⟨ψp|HC|ψp⟩, where HC, or +cost Hamiltonian, encodes the optimization prob- +lem at hand, HC |x⟩ = C(x) |x⟩. +In this work we study unconstrained problems +involving n binary variables, and we use the stan- +dard initial state |ψ0⟩ = |+⟩⊗n. Recent work has +argued in favor of using classical algorithms to gen- +erate initial states which are weighted in favor of +likely good solutions [6, 7], and the effect of this +“warm-start” approach in conjunction with differ- +ent mixers and phase separators, particularly for +small number of rounds, is worth future study. +Mixers for unconstrained optimization can take +many forms. +Two well-known examples are the +transverse-field mixer, �n +i=1 Xi, originally intro- +duced in [1], and the Grover mixer, |ψ0⟩ ⟨ψ0|, orig- +inally introduced in the context of k-SAT in [3] +and generalized in [5]. +For unconstrained prob- +lems, where all n-bit strings represent feasible so- +lutions, the space of possible mixer Hamiltonians +is very large. +In this work we restrict ourselves +to mixers composed of products of Pauli X oper- +ators which are symmetric across qubits and have +constant coefficient across terms. This choice en- +compasses many of the mixers previously studied +in the literature, e.g. both the transverse-field and +Grover mixers, however the vast majority of this +subspace remains unexplored. +This subspace of +mixers can be parameterized by sets of integers, +where each integer indicates the different degrees +of X products to include in the sum. For exam- +ple, the set {1} is equivalent to the transverse- +field mixer defined above, and the set {1, 2} gives +�n +i=1 Xi + �n +i 6 +We study QAOA performance in solving conjunc- +tive normal form satisfiability optimization prob- +lems. +Such problems are generally phrased in +terms of n Boolean variables organized into m +clauses of length k, and are referred to as k-SAT. +In our context we are specifically interested on the +Max k-SAT variation, that is, the goal is to find +the variable assignment(s) which satisfy the maxi- +mal number of clauses (and will simply be referred +to as k-SAT from here on). The computational re- +sources necessary to solve this problem are known +to depend on the clause density, α = m/n, for both +classical [13] and quantum [3] solvers. In general, +finding the maximum satisfiability for a random +k-SAT instance with low clause density requires +less computational resources than one with a high +clause density. Furthermore, there exists a phase +transition, where the difficulty ramps up sharply. +In the case of QAOA case, difficulty here is mea- +sured as the number of QAOA rounds necessary to +reach a state composed only of optimal solutions. +This phase transition was shown in [3], but only at +n = 6. +Here we extend this analysis to higher n. +In +Fig. 1 we show how the difficulty scales with clause +density for both 2- and 3-SAT. These results show +that the clause densities associated with the hard +regime gradually increase. +However, the rate of +increase appears quite small, and for problems up +to n = 14 it is sufficient to set the clause density +at 3 for 2-SAT and 6 for 3-SAT to be in the hard +regime. +3.2 +Broad mixer survey +Having now ensured we are studying difficult ran- +dom 2- and 3-SAT instances, we can now compare +the performance of a wide variety of QAOA im- +plementations at different n. This will answer two +questions: which QAOA implementation performs +the best at a given n, and does relative perfor- +mance depend on n? For this section we will focus +solely on the performance metric of approximation +ratio as a function of number of QAOA rounds p, +covering up to p = 10. +3 + +0 +1 +2 +3 +4 +5 +clause density +100 +101 +num. rounds to reach ground state +n= +6 +8 +10 +12 +14 +2-SAT +0 +2 +4 +6 +8 +10 +clause density +100 +101 +num. rounds to reach ground state +n= +6 +8 +10 +12 +14 +3-SAT +Figure 1: Number of rounds with Grover-Th necessary to achieve an approximation ratio of 1 as a function of clause +density. Data represents mean results over 100 random k-SAT instances. Setting the clause density at 3,6 ensures “hard” +problems for 2-SAT, 3-SAT (respectively). +The subspace of mixers as described in Sec. 1 +contains 2n−1 different mixers (ignoring W = {}), +so testing the performance of every mixer is compu- +tationally infeasible for n > 6. Instead, we focused +on the following collection of 3(n − 1) mixers: +W ∈ +� +� +� +� +� +� +� +{1, . . . , i} for i = 2, . . . , n, +{i, . . . , n} for i = 2, . . . , n − 1, +{i} for i = 1, . . . , n. +(3) +These mixers capture a wide range of different mix- +ing approaches. For example, the transverse-field +mixer W = {1} mixes solution vectors which are +Hamming distance 1 apart, while the mixer W = +{n} mixes a solution with its inverse (Hamming +distance n), and the Grover mixer W = {1, . . . , n} +mixes all states equally. The choice of mixers in +Eq. 3 interpolates smoothly between these differ- +ent extremes. Furthermore, for each mixer, we test +both the -Obj and -Th phase separators. +Thus +for each n we study 6(n − 1) different QAOA im- +plementations. We tested these implementations +via highly optimized QAOA statevector simulation +code on a cluster of 12 NVIDIA RTX A6000s, each +with 48GB RAM. A single n = 10 problem instance +took O(1) day to simulate 10 rounds of the 54 dif- +ferent implementations. +Fig. 2 shows the relative performance of these +QAOA implementations for n = 6, 8, 10, as mea- +sured by approximation ratio for p ≤ 10. Due to +the large number of implementations, it is difficult +to precisely yet succinctly summarize the differ- +ences in performance for all of the mixers, how- +ever some broad trends do emerge. Most apparent +from visual inspection is that QAOA implementa- +tions with the threshold-based phase separator (i.e. +green lines) tend to achieve higher approximation +ratios than those with the objective-based phase +separator (i.e. blue lines) for n ≤ 10. The best +performing -Th implementations all tended to per- +form very similarly, this can be seen in the thick +green lines (which represent many overlapping -Th +results). These high-performing -Th implementa- +tions generally involved a large number of X’s, in- +cluding Grover-Th as well as e.g. {n/2, . . . , n} and +{1, . . . , n/2}. +“Individual” mixers, e.g. +{1}-Th, +performed the worst amongst the -Th implementa- +tions. The inverse is true for the -Obj implementa- +tions, that is, “small” mixers such as {1} and {1, 2} +tended to perform the best, while mixers that in- +cluded many products (notably including Grover- +Obj) performed the worst. +3.3 +Transverse vs. Grover up to n = 14 +The most surprising result of Fig. 2, however, is +that as n increases a small number of objective- +based separators steadily improve, and by n = 10 +the best performing implementations are {1}-Obj +4 + +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +2-SAT +n=6 +n=8 +n=10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +3-SAT +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +rounds +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +approx. ratio +Objective +Threshold +Figure 2: Approximation ratio as a function of rounds for the 6(n−1) mixers described in Eq. 3, with both -Obj (blue) and +-Th (green) phase separators. Data averaged over 10 random problem instances per n. For n = 6, the threshold QAOA +implementations tend to perform better than those with objective value-based phased separators. However, certain -Obj +implementations improve steadily with n, and the highest performing implementations at n = 10 are -Obj for both 2- +and 3-SAT. +and {1, 2}-Obj (for both 2- and 3-SAT). Threshold- +based implementations still tend to perform better. +For example, in the n = 10 3-SAT case, 23 out +of the top 26 implementations are threshold-based +(for 2-SAT the proportion is 22/26). However, the +{1}-Obj and {1, 2}-Obj implementations (as well +as the {2}-Obj implementation in the case of 2- +SAT) are clearly achieving higher approximation +ratios. In Fig. 3 we specifically compare the rel- +ative performance of {1}-Obj, i.e. the transverse- +field mixer, against both Grover-Th and Grover- +Obj up to n = 14. Grover-Th is included in the +comparison as it is representative of the best-case +-Th performance, and Grover-Obj is included as it +is the primary mixer of interest in [3]. Here we see +even more clearly how the superior performance +of Transverse-Obj is only apparent as n increases. +This result stands in contrast to the conclusions +drawn in [3], where they observed that Grover-Obj +outperformed Transverse-Obj at n = 6 and implied +that this result held for more general n. +3.4 +Approximation Ratio vs. +Ground State +Probability +We +now +study +the +relative +performance +of +Transverse-Obj, Grover-Th, and Grover-Obj when +measured by ground state probability (GSP), see +Fig. 4. This particular choice of regime (p ≤ 6 up +to n = 14) and performance metric is meant to +highlight the practical differences between QAOA +implementations in an approximate optimization +context. It is important to emphasize here that the +angles β, γ are still chosen to optimize ⟨HC⟩, as op- +posed to GSP, since GSP is only measurable once +one has already solved the optimization problem +of interest. So it is quite striking that the different +QAOA implementations isolate the optimal states +to vastly differing degrees. At n = 6, Grover-Th +quickly creates a |ψp⟩ dominated by optimal states. +For these small problems, setting the threshold suf- +ficiently high to only filter out the optimal states +is the best strategy for both improving ⟨HC⟩ as +well as GSP. In this case, Grover-Th is equiva- +5 + +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +2-SAT +n=6 +n=10 +n=14 +1 +2 +3 +4 +5 +6 +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +3-SAT +1 +2 +3 +4 +5 +6 +rounds +1 +2 +3 +4 +5 +6 +approx. ratio +Transverse-Obj +Grover-Th +Grover-Obj +Figure 3: Ground state probability as a function of rounds for Transverse-Obj, Grover-Th, and Grover-Obj for rounds ≤ 6 +across a range of n for 2- and 3-SAT. Data averaged over 10 random problem instances. Transverse-Obj significantly +outperforms the others for n ≥ 10. +lent to a direct Grover search for optimal states. +Meanwhile, Transverse-Obj and Grover-Obj both +improve steadily with each round. +However, by +n = 10, both of the Grover implementations fea- +ture drastically lower GSPs, while Transverse-Obj +has only decreased slightly, and the story contin- +ues at n = 14. Comparing the results of Figs. 3 +and 4, which are obtained from the same prob- +lem instances and β, γ, we can infer that for +n ≥ 10 the Grover mixer, with both -Obj and - +Th phase separators, improves the approximation +ratio by increasing the amplitude of a large number +of states with sub-optimal objective values. Mean- +while, the Transverse-Obj implementation is much +more effective at increasing the amplitude of opti- +mal states. +4 +Conclusions and Future Directions +In this work we have conducted the first large- +scale comparison of QAOA mixers and phase sep- +arators for unconstrained optimization problems. +The central conclusion of this work is that the tra- +ditional transverse-field mixer with objective value +phase separator is the most performant QAOA im- +plementation as problem size grows. This stands +in contrast to prior work [3], which conjectured +that mixing Hamiltonians that mixed across a +wide range of states were more effective. Instead, +we argue the opposite: +the effectiveness of the +transverse-field mixer is due to the fact that it pre- +dominantly mixes states which are close in Ham- +ming distance. This is useful because states which +are nearby in Hamming space often have simi- +lar objective values for a random k-SAT instance. +Used in conjunction with the objective value phase +separator, Transverse-Obj is thus more able to +mix states with similar objective value and create +the constructive/destructive interference necessary +for QAOA performance. By contrast, the Grover +mixer mixes all states equally. +This would be a +useful feature for a completely random optimiza- +tion problem, that is, one where similar states have +no correlation in objective value. In such a case, +where there is no structure to the problem, Grover- +Th represents an optimal approach as it is effec- +tively just searching via Grover’s algorithm, which +6 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2-SAT +n=6 +n=10 +n=14 +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +3-SAT +1 +2 +3 +4 +5 +6 +rounds +1 +2 +3 +4 +5 +6 +ground state prob. +Transverse-Obj +Grover-Th +Grover-Obj +Figure 4: Ground state probability as a function of rounds for Transverse-Obj, Grover-Th, and Grover-Obj for rounds +≤ 6 across a range of n for 2- and 3-SAT. Data taken from the same 10 random instances as Fig. 3. Transverse-Obj +significantly outperforms the others for n ≥ 10. +is known to be optimal for unstructured search. +However, in the case of highly structured optimiza- +tion problems such as k-SAT, the fine-tuned mix- +ing capabilities of the transverse-field mixer, com- +bined with the additional information captured by +the objective phase separator (as opposed to the +the more brute-force thresholded version) gives the +best performance. Our results in effect show how +QAOA exploits problem structure to move beyond +Grover’s unstructured search performance. +Future work [14] is needed to precisely quantify +the different effects and capabilities of these mix- +ers. Furthermore, it is intriguing to consider mix- +ers custom designed for the specific energy spectra +of different optimization problems, e.g. +MaxCut +vs. +Max 2-SAT vs. +Max 3-SAT. The results of +this study suggest that brute-force, or maximally +mixing QAOA is unlikely to lead to optimal perfor- +mance for large problem instances. Instead, mix- +ers and phase separators that are highly tailored +to specific problem classes may unlock the fullest +of QAOA’s potential. +References +[1] Edward Farhi, Jeffrey Goldstone, and Sam +Gutmann. “A Quantum Approximate Opti- +mization Algorithm” (2014). arXiv:1411.4028. +[2] Stuart +Hadfield, +Zhihui +Wang, +Bryan +O’Gorman, +Eleanor +G +Rieffel, +Davide +Venturelli, and Rupak Biswas. +“From the +quantum approximate optimization algorithm +to a quantum alternating operator ansatz”. +Algorithms 12, 34 (2019). arXiv:1709.03489. +[3] V. Akshay, H. Philathong, M. E. S. Morales, +and J. D. Biamonte. “Reachability Deficits in +Quantum Approximate Optimization”. Phys- +ical Review Letters 124, +090504 (2020). +arXiv:1906.11259. +[4] John Golden, +Andreas B¨artschi, +Stephan +Eidenbenz, +and +Daniel +O’Malley. +“Ev- +idence +for +super-polynomial +advantage +of +QAOA over unstructured search” (2022). +arXiv:2202.00648. +[5] Andreas B¨artschi and Stephan Eidenbenz. +“Grover Mixers for QAOA: Shifting Com- +7 + +plexity from Mixer Design to State Prepara- +tion”. In IEEE International Conference on +Quantum Computing & Engineering QCE’20. +Pages 72–82. (2020). arXiv:2006.00354. +[6] Daniel J Egger, Jakub Mareˇcek, and Stefan +Woerner. “Warm-starting quantum optimiza- +tion”. Quantum 5, 479 (2021). +[7] Reuben Tate, Majid Farhadi, Creston Herold, +Greg Mohler, and Swati Gupta. +“Bridging +classical and quantum with sdp initialized +warm-starts for qaoa” (2020). +[8] John +Golden, +Andreas +B¨artschi, +Daniel +O’Malley, +and +Stephan +Eidenbenz. +“Threshold-Based +Quantum +Optimiza- +tion”. In IEEE International Conference on +Quantum Computing & Engineering QCE’21. +Pages 137–147. (2021). arXiv:2106.13860. +[9] Jonathan Wurtz and Danylo Lykov. +“The +fixed angle conjecture for qaoa on regular +maxcut graphs” (2021). +[10] Jeremy Cook, Stephan Eidenbenz, and An- +dreas B¨artschi. +“The Quantum Alternat- +ing Operator Ansatz on Maximum k-Vertex +Cover”. In IEEE International Conference on +Quantum Computing & Engineering QCE’20. +Pages 83–92. (2020). arxiv:1910.13483. +[11] Jonathan Wurtz and Peter J. Love. “Coun- +terdiabaticity +and +the +quantum +approxi- +mate optimization algorithm”. +Quantum 6, +635 (2022). +[12] Charles Moussa, Hao Wang, Thomas B¨ack, +and Vedran Dunjko. “Unsupervised strategies +for identifying optimal parameters in quantum +approximate optimization algorithm”. +EPJ +Quantum Technology9 (2022). +[13] Weixiong Zhang. “Phase transitions and back- +bones of 3-sat and maximum 3-sat”. In Princi- +ples and Practice of Constraint Programming +— CP 2001. Pages 153–167. Berlin, Heidel- +berg (2001). Springer Berlin Heidelberg. +[14] John Golden, Yigit Subasi, Andreas B¨artschi, +and Stephan Eidenbenz. “Optimal mixing in +qaoa” (2023). +8 + diff --git a/XNFIT4oBgHgl3EQfiisy/content/tmp_files/load_file.txt b/XNFIT4oBgHgl3EQfiisy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7baf263186aca84b1b3f63fae5b5ea99a04931e --- /dev/null +++ b/XNFIT4oBgHgl3EQfiisy/content/tmp_files/load_file.txt @@ -0,0 +1,341 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf,len=340 +page_content='The Quantum Alternating Operator Ansatz for Satisfiability Problems John Golden1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Andreas B¨artschi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Daniel O’Malley2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' and Stephan Eidenbenz1 1Information Sciences (CCS-3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' NM 87545 2Computational Earth Sciences (EES-16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' NM 87545 We comparatively study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' through large- scale numerical simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' the performance across a large set of Quantum Alternating Operator Ansatz (QAOA) implementations for finding approximate and optimum so- lutions to unconstrained combinatorial op- timization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Our survey includes over 100 different mixing unitaries, and we combine each mixer with both the standard phase separator unitary representing the ob- jective function and a thresholded version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Our numerical tests for randomly chosen in- stances of the unconstrained optimization problems Max 2-SAT and Max 3-SAT reveal that the traditional transverse-field mixer with the standard phase separator performs best for problem sizes of 8 through 14 vari- ables, while the recently introduced Grover mixer with thresholding wins at problems of size 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' This result (i) corrects earlier work suggesting that the Grover mixer is a supe- rior mixer based only on results from prob- lems of size 6, thus illustrating the need to push numerical simulation to larger prob- lem sizes to more accurately predict perfor- mance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' and (ii) it suggests that more com- plicated mixers and phase separators may not improve QAOA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' 1 Introduction Combinatorial optimization is widely viewed to be a promising application domain for quantum com- puting, and the Quantum Approximate Optimiza- tion Algorithm [1] and its generalization the Quan- tum Alternate Operator Ansatz [2] – both abbre- viated as QAOA – are the most prominent algo- rithms for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' As theoretical guar- antees of QAOA performance are few and far be- tween, we approach QAOA as a heuristic optimiza- tion algorithm that may perform very well in prac- tice despite the lack of formal performance guar- antees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' The study of QAOA performance is in its infancy and – until the arrival of large-scale error corrected quantum computers – we are limited to numerical experimentation on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' QAOA follows a simple iterative approach, evolving an initial state with alternating phase separating and mixing unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' The amount of time (also referred to as the angle) for the phasing and mixing at each iteration (or round) is tuned to increase constructive interference amongst ba- sis states which correspond to high-quality solu- tions to a target optimization problem, and simi- larly create destructive interference amongst poor- quality solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Other studies have shown that QAOA performance depends on the choice of mix- ing operators, both for unconstrained [3] and con- strained [4] optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' moreover, mixing opera- tors may work better with some phase separators (operators that model the objective function) than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Overall, the connection between mixers, phase separators, and QAOA performance is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' In this paper, we choose Max 2-SAT and Max 3-SAT as our problems of study, which are both unconstrained optimization prob- lems that ask for a truth assignment to variables that maximize the number of clauses in the prob- lem instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' these problems are unconstrained as every possible (binary) assignment of truth values to the values is a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' We select a subset of mixing unitaries that spans a wide range of possible higher order mixing terms, with the well-known transverse-field mixer [1] and the re- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content='11292v1 [quant-ph] 26 Jan 2023 cently introduced Grover mixer [5] occupying op- posite ends of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' These mixers are then tested on problems of up to 14 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Previous work [3] tested QAOA only with transverse-field and Grover mixers, solving Max 2-SAT and 3-SAT problems with only six variables, and found no- table differences in performance with Grover mixer emerging as the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' In contrast, our analysis includes a much larger array of mixers and phase separators, and tests k-SAT problems up to size n = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Our main finding is that mixer performance de- pends on problem size, and that the transverse- field mixer is a poor choice for small problems, but outperforms all other mixers on problems with ≥ 10 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' We also find that relative perfor- mance depends on the exact choice of metric for performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' In particular, we compare approxi- mation ratio vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' ground state probability, and find that certain QAOA variations can have a relatively high approximation ratio while having low ground state probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' 2 QAOA Review Using QAOA to solve an optimization problem, defined here by a cost function C(x) on binary strings, requires several choices: an initial state |ψ0⟩, a mixer Hamiltonian HM, a phase separa- tor Hamiltonian HP , and a set of 2p parameters (commonly referred to as angles) β = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' , βp} and γ = {γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' , γp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' One then uses a quantum computer to apply the mixing and phase separat- ing unitaries over p rounds to prepare the state |ψp⟩ = e−iγpHP e−iβpHM · · · e−iγ1HP e−iβ1HM |ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' (1) The goal is to choose the above parameters so that sampling from |ψp⟩ is likely to return a state which is a good (or ideally optimal) solution to the opti- mization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' This is generally accomplished by fixing |ψ0⟩ , HM, HP and using classical opti- mization techniques to modify β and γ in order to maximize (or minimize) ⟨ψp|HC|ψp⟩, where HC, or cost Hamiltonian, encodes the optimization prob- lem at hand, HC |x⟩ = C(x) |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' In this work we study unconstrained problems involving n binary variables, and we use the stan- dard initial state |ψ0⟩ = |+⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Recent work has argued in favor of using classical algorithms to gen- erate initial states which are weighted in favor of likely good solutions [6, 7], and the effect of this “warm-start” approach in conjunction with differ- ent mixers and phase separators, particularly for small number of rounds, is worth future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Mixers for unconstrained optimization can take many forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' Two well-known examples are the transverse-field mixer, �n i=1 Xi, originally intro- duced in [1], and the Grover mixer, |ψ0⟩ ⟨ψ0|, orig- inally introduced in the context of k-SAT in [3] and generalized in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' For unconstrained prob- lems, where all n-bit strings represent feasible so- lutions, the space of possible mixer Hamiltonians is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' In this work we restrict ourselves to mixers composed of products of Pauli X oper- ators which are symmetric across qubits and have constant coefficient across terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' This choice en- compasses many of the mixers previously studied in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' both the transverse-field and Grover mixers, however the vast majority of this subspace remains unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' This subspace of mixers can be parameterized by sets of integers, where each integer indicates the different degrees of X products to include in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFIT4oBgHgl3EQfiisy/content/2301.11292v1.pdf'} +page_content=' For exam- ple, the set {1} is equivalent to the transverse- field mixer defined above, and the set {1, 2} gives �n i=1 Xi + �n i 1), where m is odd, 4ℓ2 ≡ ±1 (mod k), +and either m > 1 or 2ℓ ̸≡ ±1 (mod k). +(1.6) Theorem (cf. Theorem 4.1). Tr(m, n, r) is nontrivially unstable if and only if it is: +(1) Tr(2, 4k, 4) ∼= Tr(2, 4k, −2) ∼= Tr(4, 2k, 2), or +(2) Tr(2, 4k, 2k + 1), or +(3) Tr(2, 4k, 2k) ∼= Tr(2, 4k, 2k + 2) ∼= Tr(2k, 4, 2), or +(4) Tr(4, 2k, 0) ∼= Tr(4, 2k, 4) ∼= Tr(2k, 4, 0), or +(5) Tr(2k, 4, 1), with k > 1, or Tr(2k, 4, −1), or +(6) Tr(4, 2, 1), or Tr(4, 3, −1). +(If k is odd, then the graphs in (3) are isomorphic to the graphs in (4). If k is even, then the two +graphs in (5) are isomorphic to each other.) + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +3 +(1.7) Example. By searching Theorem 1.5 for cases where r = 0, we see that, for n ≥ m ≥ 2, the +Cartesian product Cn □ Cm is nontrivially unstable if and only if n = 2m and m is odd. +(1.8) Remark. Theorems 1.5 and 1.6 only list graphs that are nontrivially unstable. However, it +is easy to check whether a toroidal grid is trivially unstable. First, note that they are all connected. +Also: +(a) Qd(m, n, r) is bipartite if and only if n and m + r are even. +(b) Qd(m, n, r) has twin vertices if and only if m = 2 and r = ±2 (cf. Remark 3.2)). +(c) Tr(m, n, r) is never bipartite (because it has triangles). +(d) Tr(m, n, r) has twin vertices if and only if (m, n, r) ∈ {(2, 4, 1), (3, 3, 0)} (cf. Lemma 4.2). +(1.9) Remarks. +(1) The assumption that m, n ≥ 2 is not stated explicitly in [16]. Wilson’s conjectures also seem +to implicitly assume that gcd(n, r) ̸= 1. We do not make this assumption, so Theorems 1.5 +and 1.6 include include infinite families of graphs that are not listed in Wilson’s conjectures. +(2) There are other differences between Wilson’s conjectures [16, pp. 380 and 381] and our +statements of the results. In particular: +(a) Wilson omits Qd(m, 4k, −k) and Qd(2m, km, ±4ℓ), and usually omits Tr(m, n, m − r) +when Tr(m, n, r) is listed. Corollary 2.4 explains that they are alternate representations +of other graphs in the list. +(b) Wilson requires k to be odd in 1.5(2), but we omit this redundant condition: it is a +consequence of the equation 4ℓ2 ≡ ±1 (mod k). +(c) Wilson uses 4k in 1.6(4), instead of 2k. That eliminates the overlap with 1.6(3). +(3) The two occurrences of “±” in 1.5(2) cause redundancy (and could therefore be omitted), +because −ℓ satisfies the congruence 4ℓ2 ≡ ±1 (mod k) whenever ℓ does. +(1.10) Remark. Following a section of preliminaries, Theorem 3.1 is proved in Section 3, and +Theorem 4.1 is provided in Section 4. +Theorems 3.1 and 4.1 are slightly more general than +Theorems 1.5 and 1.6. For example, their statements in the language of abelian Cayley graphs +allow for the case where m is equal to 1. See the well-known Lemma 2.3 for the translation between +the two languages. + +2. Preliminaries +All graphs in this paper are simple (no loops or multiple edges). +2A. Abelian Cayley graphs +(2.1) Definition (cf. [3, p. 34]). Let S be a subset of an additive abelian group G, such that +S = −S and 0 /∈ S. The corresponding abelian Cayley graph Cay(G; S) is the graph whose vertices +are the elements of G, and with an edge joining the vertices g and h if and only if g = h + s for +some s ∈ S. +(2.2) Remark. The adjective “abelian” in “abelian Cayley graph” is to emphasize the assumption +that G is abelian, so we will sometimes omit it when it is not relevant. (The usual definition of +Cay(G; S) does not require G to be abelian, but we have no need for the nonabelian case in this +paper.) +The following simple (and well known) observation notes that the toroidal grids Qd(m, n, r) and +Tr(m, n, r) are isomorphic to abelian Cayley graphs. The minus sign in Tr(m, n, −r) is because +the definition of Tr(m, n, −r) would naturally identify it with the Cayley graph having a − b as the +third generator, but, for our purposes, it is more convenient to use a + b. +(2.3) Lemma. Given m, n, r ∈ Z (with m, n ≥ 2), let +G = ⟨ a, b | ma = rb, nb = 0, a + b = b + a ⟩, +so G is an abelian group of order mn. Then +Qd(m, n, r) ∼= Cay(G; ±a, ±b) +and +Tr(m, n, −r) ∼= Cay +� +G; ±a, ±b, ±(a + b) +� +. +(2.4) Corollary. Qd(m, n, r) ∼= Qd(m, n, −r) and Tr(m, n, r) ∼= Tr(m, n, m − r). +Proof. (Qd) We have ma = −r(−b) and n(−b) = 0, so using −b in the place of b yields a +representation of Cay(G; ±a, ±b) as Qd(m, n, −r). +(Tr) Let c = −(a + b), so {±a, ±b, ±(a + b)} = {±c, ±b, ±(c + b)}. Then +mc = −ma − mb = −rb − mb = −(r + m)b, +so using c in the place of a yields a representation of Cay(G; ±a, ±b, ±(a + b)) as Tr(m, n, r + m). +Therefore Tr(m, n, −r) ∼= Tr(m, n, r + m). +□ +(2.5) Remark. By replacing a with −a, the proof of Corollary 2.4 shows that Qd(m, n, r) ∼= +Qd(m, n, −r). However, the same trick does not work for Tr(m, n, r): if a is replaced with −a, +then the equation a + b + c = 0 forces b and c to also be replaced with their negatives. Since +m(−a) = −ma = −rb = r(−b), this does not result in any change in the parameter r. +Here is an abelian Cayley graph that appears in the statement of Theorem 3.1(4): +(2.6) Notation. M2n = Cay(Z2n; ±1, n) is the Moebius ladder with 2n vertices. +4 + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +5 +2B. Some classes of stable/unstable abelian Cayley graphs +(2.7) Theorem (Morris [11, Thm. 1.1]). There are no nontrivially unstable abelian Cayley graphs +of odd order. +Recall that if G is cyclic, then Cay(G; S) is a circulant graph. The following result is stated only +for circulant graphs in [7], but exactly the same proof applies to abelian Cayley graphs. +(2.8) Proposition (Hujdurovi´c-Mitrovi´c-Morris, cf. [7, Prop. 4.2]). There are no nontrivially +unstable abelian Cayley graphs of valency ≤ 3. +(2.9) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Thm. 4.3]). A circulant graph Cay(Zn, {±a, ±b}) +of valency 4 is unstable if and only if either it is trivially unstable, or one of the following conditions +is satisfied (perhaps after interchanging a and b): +(1) n is divisible by 8 and gcd +� +|a|, |b| +� += 4, or +(2) n ≡ 2 (mod 4), gcd(b, n) = 1, and a ≡ ℓb + (n/2) (mod n), for some ℓ ∈ Z, such that +ℓ2 ≡ ±1 (mod n). +(2.10) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Thm. 5.1]). A circulant graph Cay(Zn; S) of +valency 5 is unstable if and only if either it is trivially unstable, or it is either: +(1) Cay(Z12k; ±s, ±2k, 6k) with s odd, or +(2) Cay(Z8; ±1, ±3, 4) +(2.11) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Cor. 6.8]). A circulant graph +X = Cay(Zn, {±a, ±b, ±c}) +of valency 6 is unstable if and only if either it is trivially unstable, or it is one of the following: +(1) Cay(Z8k, {±a, ±b, ±2k}), where a and b are odd, +(2) Cay(Z4k, {±a, ±b, ±b + 2k}), where a is odd and b is even, +(3) Cay +� +Z4k, +� +±a, ±(a + k), ±(a − k) +�� +, where a ≡ 0 (mod 4) and k is odd, +(4) Cay(Z8k, {±a, ±b, 4k ± b}), where a is even and |a| is divisible by 4, +(5) Cay(Z8k, {±a, ±k, ±3k}), where a ≡ 0 (mod 4) and k is odd, +(6) Cay(Z4k, {±a, ±b, ±mb + 2k}), where +gcd(m, 4k) = 1, +(m − 1)a ≡ 2k (mod 4k), +and +either m2 ≡ 1 (mod 4k) or (m2 + 1)b ≡ 0 (mod 4k), +(7) Cay(Z8k, {±a, ±b, ±c}), where there exists m ∈ Z, such that +gcd(m, 8k) = 1, +m2 ≡ 1 (mod 8k), +and +(m − 1)a ≡ (m + 1)b ≡ (m + 1)c ≡ 4k (mod 8k). +2C. Criteria for stability or instability +(2.12) Lemma (cf. [2, Lem. 2.4]). A connected, abelian Cayley graph X = Cay(G; S) is unstable if +and only if there exists α ∈ Aut BX, such that α(0, 0) = (0, 0), but α(0, 1) ̸= (0, 1). +(2.13) Remark. It is easy to see (and well known) that an abelian Cayley graph Cay(G; S) has +twin vertices if and only if S + z = S, for some nonzero z ∈ G. In other words, S is a union of +cosets of ⟨z⟩. By passing to a subgroup of ⟨z⟩, there is no harm in assuming that |z| is prime. +The following result was stated only for circulant graphs in [6, Prop. 3.7] (which is a slight +generalization of [16, Thm. C.4]), but the same proof applies more generally. (In fact, the proof +even applies without the assumption that z has order 2, if S + z = −(S + z) is symmetric. And +there is no need for G to be abelian.) + +6 +DAVE WITTE MORRIS +(2.14) Lemma ([6, Prop. 3.7], cf. [16, Thm. C.4]). An abelian Cayley graph Cay(G; S) is unstable +if Cay(G; S) ∼= Cay(G; S + z), for some element z of order 2 in G. +(2.15) Lemma (Wilson [16, §2.2]). A graph X is unstable if it has an automorphism α, such that +the subgraph induced by the set of un-fixed vertices is disconnected and has a component C, such +that C is bipartite, and either α(C) ̸= C or each of the two bipartition sets of C is α-invariant. +(2.16) Proposition (Hujdurovi´c-Mitrovi´c [5, Prop. 3.2]). Let X be a connected graph with more +than one vertex, and assume that X satisfies the following conditions: +(1) Every edge of X lies on a triangle. +(2) For every x ∈ V (X), it holds that: +(a) every vertex at distance 2 from x has a neighbour at distance 3 from x, and +(b) every vertex at distance 3 from x has a neighbour at distance 4 from x. +Then X is stable. +2D. Other results on automorphisms and isomorphisms +(2.17) Proposition (Baik-Feng-Sim-Xu [1, Thm. 1.1]). Let S be a generating set of an abelian +group G, such that S = −S, 0 /∈ S, and, for all s, t, u, v ∈ S: +s + t = u + v ̸= 0 =⇒ {s, t} = {u, v}. +If α is any automorphism of the graph Cay(G; S), such that α(0) = 0, then α is an automorphism +of the group G (i.e., α(g + h) = α(g) + α(h), for all g, h ∈ G). +(2.18) Definition ([4, p. 35]). Recall that the Cartesian product X □ Y of two graphs X and Y +has vertex set V (X) × V (Y ), and two vertices (x1, y1) and (x2, y2) are adjacent if and only if either +• x1 = x2 and y1y2 ∈ E(Y ), or +• y1 = y2 and x1x2 ∈ E(X). +(2.19) Proposition (cf. [4, Thm. 6.10, p. 69]). Let X be a connected graph. If there does not exist +a graph Y, such that X ∼= Y □ K2, then +Aut(X □ K2) = Aut X × S2 and Aut(X □ C4) = Aut X × Aut C4. +We will use the following elementary observation in part (4) of the proof of Lemma 3.3. +(2.20) Lemma. Let X and Y be graphs. If Y is bipartite, then B(X □ Y ) ∼= (BX) □ Y . +2E. Stability of a few specific graphs +(2.21) Example ([12, Example 2.2]). If n ≥ 3, then the complete graph Kn is stable. (But K2 is +bipartite, and is therefore unstable.) +(2.22) Lemma. For 2 ≤ n ≤ 7, the abelian Cayley graph +Cay +� +Zn × Z2; ±(1, 0), ±(1, 1), (0, 1) +� +is stable, unless n = 4, in which case it is unstable. +Proof. This can be checked very quickly by computer. +For example, the sagemath program in +Figure 1 can be executed on https://cocalc.com. (The program also verifies Lemma 2.24.) +□ +(2.23) Remark. Most cases of Lemma 2.22 can be settled quite easily without a computer: +• If n is odd, then Zn × Z2 is cyclic, so Theorem 2.10 can be applied. +• If n = 2, then the Cayley graph is K4, which is stable by Example 2.21. + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +7 +• If n = 4, then part (1) of the proof of Lemma 4.3 explains why the Cayley graph is unstable. +Therefore, n = 6 is the only case that requires effort (or a computer). +(2.24) Lemma. For 3 ≤ n ≤ 12, the abelian Cayley graph +Cay +� +Zn × Z3; ±(1, 0), ±(1, 1), ±(0, 1) +� +is stable, unless n = 3, in which case it is unstable (and is listed in Theorem 4.1(5)). +Proof. As mentioned in the proof of Lemma 2.22, the stability/instability of these graphs is +calculated by the sagemath program in Figure 1. +For n = 3, the elements a = (1, 0) and b = (0, 1) have order 3. Also, if we let c = −(1, 1), then +a + b + c = (0, 0). Therefore, the Cayley graph is described in Theorem 4.1(5). +□ +for n in range(2, 13): +for k in [2, 3]: +G = direct_product_permgroups( +[CyclicPermutationGroup(n), +CyclicPermutationGroup(k)]) +a, b = G.gens() +assert {a.order(), b.order()} == {n, k} +X = Graph(G.cayley_graph(generators=[a, b, a*b])) +AutX = X.automorphism_group() +K2 = graphs.CompleteGraph(2) +BX = X.categorical_product(K2) +AutBX = BX.automorphism_group() +if 2 * AutX.order() != AutBX.order(): +print(n, k, "unstable") +Figure 1. A sagemath [14] program to verify Lemmas 2.22 and 2.24. + +3. Unstable abelian Cayley graphs of valency 4 +This section proves the following theorem, which implies Theorem 1.5. +It also generalizes +Theorem 2.9, which handles the case where G is cyclic; however, our argument relies on Theorem 2.9, +so we are not providing an independent proof of that result. +(3.1) Theorem. A connected abelian Cayley graph Cay(G; S) of valency 4 is unstable if and only +if either it is bipartite, or it is in the following list (up to a group isomorphism ): +(1) Cay(G; ±a, ±b), where |⟨a⟩ ∩ ⟨b⟩| = 4. +(2) Cay(G; ±a, ±b), where |G : ⟨b⟩| = m, ma = 2ℓmb, |b| = 2km, and 4ℓ2 ≡ ±1 (mod k). +(3) Cay(G; ±a, ±b), where 2a = 2b. +(4) Cay +� +Z2n × Z2; ±(1, 0), (n, 0), (0, 1) +� ∼= M2n □ K2. +(a) If n = 2, this is isomorphic to K4 □ K2, and can also be realized as +Cay +� +Z2 × Z2 × Z2; (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1) +� +. +Before proving this theorem, let us show that it implies Theorem 1.5. +Proof of Theorem 1.5. The graphs in Theorem 1.5 are required to be nontrivially unstable. +Therefore, we see from Proposition 2.8 that they must have valency 4. Hence, it suffices to show +that the graphs in Theorem 1.5 are precisely the graphs that arise from Theorem 3.1 by applying +Lemma 2.3 (and are not trivially unstable), and satisfy the additional assumption that m, n ≥ +2 (where n = |b| and m = |G : ⟨b⟩|). +To do this, we consider each part of the statement of +Theorem 3.1 individually. We also find the toroidal grids that are obtained by applying Lemma 2.3 +after interchanging a and b. (And we know from Corollary 2.4 that Qd(m, n, r) ∼= Qd(m, n, −r).) +(1) The conditions in 1.5(1) that n = 4k and r = ±k are a direct translation of the fact that +|⟨a⟩ ∩ ⟨b⟩| = 4. The additional condition that m + k is odd ensures that the grid is not +trivially unstable (see Remark 1.8). +Since the condition in 3.1(1) is symmetric in a and b, no additional examples are obtained +by interchanging a and b. +(2) The grid Qd(m, 2km, 2ℓm) of 1.5(2) is obtained from a direct translation of the conditions +in 3.1(2). The condition that m is odd ensures that the grid is not trivially unstable (see +Remark 1.8). However, the definition of Qd(m, n, r) requires m > 1. +Now, we let a play the role of b in Lemma 2.3. Note that gcd(k, ℓ) = 1, because 4ℓ2 ≡ +1 (mod k). Therefore, we have +|a| = |G: ⟨b⟩| · |⟨a⟩ ∩ ⟨b⟩| = m · +2km +gcd(2km, 2ℓm) = m · 2km +2m = km. +Hence, |G : ⟨a⟩| = |G|/|a| = m(2km)/(km) = 2m. Also, since ma = 2ℓmb and 4ℓ2 ≡ +±1 (mod k) (and |b| = 2km), we have +4ℓma = (4ℓ)(2ℓmb) = 4ℓ2(2mb) = ±2mb. +So this yields the graph Qd(2m, km, ±4ℓm) of 1.5(2) (even if m = 1). +However, this graph has twin vertices (and is therefore trivially stable) if (and only if) +2m = 2 and 4ℓm ≡ ±2 (mod km) (see Remark 1.8). This situation is ruled out by assuming +(at the end of 1.5(2)) that either m > 1 or 2ℓ ̸≡ ±1 (mod k). +(3) This is trivially unstable (see Remark 3.2). +(4) The generating sets arising here are not of the form {±a, ±b}, so Lemma 2.3 cannot be +applied. These Cayley graphs are therefore not needed to find all of the toroidal grids. +□ +8 + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +9 +(3.2) Remark. It is easy to determine whether a particular Cayley graph listed in Theorem 3.1 is +trivially unstable. First, note that the graph is assumed to be connected (i.e., it is assumed that S +generates G). Case 1 of the proof shows that the examples with twin vertices are precisely those +in (3). So all that remains is to determine which of them are bipartite (which is usually answered +by Remark 1.8(a)). +3.1(1) This is bipartite if and only if |a|/4 + |b|/4 is even. +It has twin vertices if and only if +|a| = |b| = 8 (in which case, it is also bipartite). +3.1(2) This is bipartite if and only if m is even. It has twin vertices if and only if m = 1 and +2ℓ ≡ ±1 (mod k). +3.1(3) As mentioned above, this graph has twin vertices, and is therefore trivially unstable. (For +completeness, we observe that it is bipartite if and only if |a| and |b| are even.) +3.1(4) This is bipartite if and only if n is odd. (It never has twin vertices.) +To avoid cluttering the main part of the proof of Theorem 3.1, we present one direction of the +argument in the following lemma. It is mostly (or entirely?) known: the instability of the graphs +in (1) and (2) was proved by S. Wilson [16, §A.4.1], and the rest is very easy. However, Wilson gave +only a one-sentence sketch of his proofs, so we will provide a fairly complete argument for every +case. +(3.3) Lemma (cf. S. Wilson [16, §A.4.1]). All of the graphs listed in Theorem 3.1 are unstable. +Proof. We consider each part of the statement of the theorem individually. We may assume each +Cayley graph is not bipartite (for otherwise it is trivially unstable). +(1) (S. Wilson [16, Thm. Q.1, p. 380]) Let m = |G : ⟨b⟩| and n = |b|, and choose r ∈ Z, such that +ma = rb. Also, let z be the element of order 2 in ⟨a⟩ ∩ ⟨b⟩. Since |b| is divisible by 4, we know that +|b + z| = |b| = n, so |G : ⟨b + z⟩| = |G : ⟨b⟩| = m. Also note that, since |⟨a⟩ ∩ ⟨b⟩| = 4, we have +|rb| = 4, so −rb = rb + z = rb + (m + r)z (since m + r is odd, because the Cayley graph is not +bipartite). Therefore +m(−a + z) = −ma + mz = −rb + mz = rb + (m + r)z + mz = r(b + z). +Therefore, there is an automorphism ϕ of G, such that ϕ(a) = −a + z and ϕ(b) = b + z. Then ϕ +is an isomorphism from Cay(G; S) to Cay(G; S + z). This implies that Cay(G; S) is unstable (see +Lemma 2.14). +(2) (S. Wilson [16, Thm. Q.2, p. 381]) Let n = 2km = |b| and r = 2ℓm, so ma = rb. Also, let z +be the element of order 2 in ⟨b⟩. Note that: +• m is odd, because X is bipartite and r = 2ℓm is even, and +• gcd(2ℓ, k) = 1, because 4ℓ2 ≡ 1 (mod k) (so k is odd). +Then, since |b| = 2km and km is odd, we see that |b + z| = km. +Also (using the fact that +gcd(ℓ, k) = 1), we have +|a| = m · +|b| +gcd(r, |b|) = m · +2km +gcd(2ℓm, 2km) = m · k. +Since mk is odd, this implies |a + z| = 2mk = |b|. +Also note that, since 4ℓ2 = pk ± 1 for some p ∈ Z (and p must be odd), we have +4ℓ2b = (pk ± 1)b = pkb ± b = pz ± b = z ± b. +Therefore +r(a + z) = 2ℓm(a + z) = 2ℓrb + 0 = 2ℓ(2ℓmb) = 4ℓ2mb = m(z ± b) = ±m(b + z), +Hence, there is an automorphism ϕ of G, such that ϕ(b) = a + z and either ϕ(a) = b + z or +ϕ(a) = −b + z. In either case, ϕ is an isomorphism from Cay(G; S) to Cay(G; S + z). This implies +that Cay(G; S) is unstable (see Lemma 2.14). + +10 +DAVE WITTE MORRIS +(3) Cay(G; S) has twin vertices (see Remark 3.2), so it is trivially unstable. +(4) Since the Cayley graph is not bipartite, we know that n is even. By Lemma 2.20, we have +BX = B(M2n □ K2) ∼= (BM2n) □ K2 ∼= (C2n □ K2) □ K2 ∼= C2n □ C4. +So |Aut BX| ≥ |Aut C2n| · |Aut C4| = 4n · 8 = 32n. +If n ≥ 4, then +|Aut X| = |Aut(M2n □ K2)| = 2|Aut M2n| = 8n < 1 +2|Aut BX|, +so X is unstable. +For the special case where n = 2, we have X ∼= K4 □ K2, so . +|Aut X| = |Aut K4| · |Aut K2| = 4! · 2. +However, +BX ∼= (BK4) □ K2 ∼= (K2 □ K2 □ K2) □ K2, +so |Aut BX| = 4! · 24 ≫ 2 |Aut X|. Therefore X is unstable. +□ +Proof of Theorem 3.1. (⇐) See Lemma 3.3. +(⇒) Let X = Cay(G; S), and assume that X is connected and unstable, but not bipartite. We +will show that X is in the list. +Case 1. +Assume X is trivially unstable. Since X is assumed to be connected and nonbipartite, +it must have twin vertices. Therefore, S is a union of cosets of some subgroup ⟨z⟩ of prime order +(see Remark 2.13). Since X has valency 4, we know that |S| = 4, so we must have |z| = 2 (since |z| +is a prime number that divides |S|). +Subcase 1.1. Assume S = {±a, ±b}, where |a|, |b| > 2. We may assume a + z ∈ {−a, b} +(perhaps after replacing b with its negative). +• If a+z = −a, then 2a = z (and −a+z = a). Also, b+z /∈ {±a}, so we must have b+z = −b, +which implies 2b = z. Therefore 2a = z = 2b, so (3) is satisfied. +• If a + z = b, then 2b = 2(a + z) = 2a + 2z = 2a + 0 = 2a, so (3) is satisfied. +Subcase 1.2. Assume S = {±a, b, c}, where |a| > 2 and |b| = |c| = 2. Since b + z ∈ S and +2(b + z) = 2b + 2z = 0 + 0 = 0, we must have b + z = c (and hence c + z = b). So a + z = −a, +which implies z = 2a (and |a| = 4). Now, since X is not bipartite, there exist p, q, r ∈ Z, such that +pa + qb + rc = 0 and p + q + r is odd. Then +0 = pa + qb + rc = pa + qb + r(b + 2a) ≡ (q + r)b (mod a). +If q + r is odd, this implies b ∈ ⟨a⟩, so b = z (since z is the unique element of order 2 in ⟨a⟩. But +then c = b + z = z + z = 0, which contradicts the fact that |c| = 2. +So q + r is even. Therefore p is odd, so pa = ±a (since |a| = 4). Then ±a = −(qb + rc) ∈ ⟨b, c⟩ ∼= +Z2 × Z2. This is impossible, since |a| = 4. +Subcase 1.3. Assume S = {a, b, c, d}, where |a| = |b| = |c| = |d| = 2. We may assume, +without loss of generality, that a + z = b and c + z = d, and also, since X is not bipartite, that +a + b + c = 0. But then +c = a + b = a + (a + z) = 0 + z = z, +so +d = c + z = c + c = 0, +which contradicts the fact that |d| = 2. +Assumption A. In the remaining cases of the proof, we assume that X is nontrivially unstable. + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +11 +Case 2. Assume G is cyclic. We see from Theorem 2.9 that X is listed in either (1) or (2) (with +m = 1). +Case 3. Assume that S contains at least one element of order 2. If every element of S has +order 2, then (since X is not bipartite) it is not difficult to see that X is the Cayley graph K4 □ K2 +that is listed in (4a). +Therefore, we may assume that S contains precisely two elements of order 2, so we may write +S = {a, b, ±c}, where |a| = |b| = 2 and |c| ≥ 3. Since ⟨c⟩ has at most one element of order 2, we +have |⟨a, b⟩ ∩ ⟨c⟩| ∈ {1, 2}. Let n = |c|. +Subcase 3.1. Assume |⟨a, b⟩ ∩ ⟨c⟩| = 1. Then +X ∼= Cn □ C4 ∼= Cay +� +Zn × Z4; ±(1, 0), ±(0, 1) +� +. +This Cayley graph has no elements of order 2 in the generating set, so it is considered in a later +case. Also note that n must be odd, since X is not bipartite. Then it is not difficult to see that +this Cayley graph is not listed in any of the parts of the statement of the theorem, so it is stable. +Subcase 3.2. Assume |⟨a, b⟩ ∩ ⟨c⟩| = 2, but ⟨c⟩ ∩ {a, b} = ∅. Then X is a prism with 2n +vertices, plus an edge from each vertex to its antipodal vertex. (Note that n must be even, since +⟨c⟩ has a subgroup of order 2.) Therefore, it is not difficult to see that X ∼= Cay(G′; ±x, ±y), where +G′ = ⟨ x, y | 4x = ny = 0, 2x = (n/2)y, x + y = y + x ⟩. +The generating set of this abelian Cayley graph has no elements of order 2, so the Cayley graph is +considered in a later case. Since X is not bipartite, n/2 must be odd. Therefore, it is not difficult +to see that this Cayley graph is not listed in any of the parts of the statement of the theorem, so +it is stable. +Subcase 3.3. Assume ⟨a, b⟩ ∩ ⟨c⟩ = ⟨a⟩. Then X ∼= Mn □ K2 is listed in (4). +Assumption B. In the remaining cases of the proof, we assume that S has no elements of order 2. +Therefore, we may write +S = {±a, ±b}, where |a|, |b| > 2. +Case 4. +Assume there is a group automorphism α of G × Z2, such that α is an automorphism +of BX, and α(0, 1) ̸= (0, 1). Since G × {0} and G × {1} are the bipartition sets of BX, we know +that each of these sets is α-invariant. Therefore +• α(g, 0) = +� +ϕ(g), 0 +� +, for some automorphism ϕ of G, and +• α(0, 1) = (z, 1), for some element z of order 2. +Since α is an automorphism of BX, we must have α +� +S×{1} +� += S×{1}, so ϕ(S) = S+z. Therefore +ϕ is an isomorphism from Cay(G; S) to Cay(G; S + z). +We may assume (by interchanging a and b, if necessary) that |b| is divisible by (at least) the +largest power of 2 that divides |a|. By Theorem 2.7 (and Assumption A), this implies that +|b| is even. +Let +m = |G : ⟨b⟩|. +Then ma ∈ ⟨b⟩, so we may choose r ∈ {0, 1, . . . , |b| − 1}, such that +ma + rb = 0. +Since X is not bipartite, we know that +m + r is odd. +Subcase 4.1. Assume ϕ(b) ∈ {±b + z}. Since ϕ is a homomorphism (and |z| = 2), this implies +that ϕ +� +{±b} +� += {±b+z}. Then, since ϕ is a bijection from S to S +z, we must have ϕ(a) = ǫa+z, + +12 +DAVE WITTE MORRIS +for some ǫ ∈ {±1}. We may assume, without loss of generality, that ϕ(b) = b + z (by composing +with the automorphism x �→ −x, if necessary). Then +0 = ϕ(ma + rb) +(ma + rb = 0) += m ϕ(a) + r ϕ(b) +(ϕ is a group automorphism) += m (ǫa + z) + r (b + z) += ǫma + rb + z +(|z| = 2 and m + r is odd). +If ǫ = 1, then ǫma + rb = ma + rb = 0, so z = 0, which contradicts the fact that |z| = 2. +Therefore, we must have ǫ = −1, so −ma+rb = z. Subtracting this from the equation ma+rb = +0, we conclude that 2ma = z has order 2, so ma has order 4. Thus, the Cayley graph is listed +in (1). +Subcase 4.2. Assume ϕ(b) ∈ {±a + z}. We may assume, without loss of generality, that +ϕ(b) = a + z (by composing with the automorphism x �→ −x, if necessary). We have ϕ(a) = ǫb + z, +for some ǫ ∈ {±1}. +Note that |b| = |ϕ(b)| = |a + z|. Therefore, either |b| = 2|a| (and |a| is odd) or |b| = |a|. Hence, +gcd +� +|b|, r +� +∈ {m, 2m}. If m is even, this implies that r is also even, which contradicts the fact that +m + r is odd. Therefore m is odd, so r is even. Hence, we must have gcd +� +|b|, r +� += 2m, so we may +write +|b| = 2km and r = 2ℓm, +for some k, ℓ ∈ Z. +We have +0 = ϕ(ma + rb) +(ma + rb = 0) += m ϕ(a) + r ϕ(b) +(ϕ is a group automorphism) += m (ǫb + z) + r (a + z) += ra + ǫmb + z +(|z| = 2 and m + r is odd) += 2ℓma + ǫmb + z +(definition of ℓ). +We also have +2ℓma + 4ℓ2mb = 2ℓ(ma + rb) = 2ℓ(0) = 0, +so, by subtracting these two equations, we conclude that (4ℓ2 − ǫ)mb = z. Since |b| = 2km, and +|z| = 2, then 4ℓ2 ≡ ǫ (mod k). Thus, the Cayley graph is listed in (2). +Remark. The remaining cases are copied almost verbatim from the analogous cases in [7, proof +of Thm. 4.3]. +Case 5. +Assume 2s ̸= 2t, for all s, t ∈ S, such that s ̸= t. +By Lemma 2.12, there is an +automorphism α of BX that fixes (0, 0), but does not fix (0, 1). +We may assume α is not a +group automorphism, for otherwise Case 4 applies. Therefore, Proposition 2.17 implies there exist +s, t, u, v ∈ S such that s + t = u + v ̸= 0 and {s, t} ̸= {u, v}. From the assumption of this case, we +see that this implies 3b = ±a (perhaps after interchanging a with b). Then +G = ⟨a, b⟩ = ⟨3b, b⟩ = ⟨b⟩ is cyclic, +so Case 2 applies. +Case 6. +The remaining case. Since Case 5 does not apply, we have 2s = 2t, for some s, t ∈ S, +such that s ̸= t. +Subcase 6.1. Assume that t = −s. Then |s| = 4. Therefore, if we assume, without loss of +generality, that s = a, then we have i := |⟨a⟩ ∩ ⟨b⟩| ∈ {1, 2, 4}. In all cases, we will show that G is +cyclic, so Case 2 applies. + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +13 +If i = 1, then G = ⟨a⟩ × ⟨b⟩. Since X is not bipartite, this implies |b| is odd, so gcd +� +|a|, |b| +� += 1. +Therefore G is cyclic. +If i = 2, then there is some k ∈ Z, such that a2 = bk. Since X is not bipartite, we know k is odd. +So ⟨a2⟩ has odd index in ⟨b⟩. This implies that ⟨a⟩ has odd index in G and is therefore a Sylow +2-subgroup. So the Sylow 2-subgroup of G is cyclic. All of the other Sylow subgroups of G are +contained in ⟨b⟩, and are therefore also cyclic. So G is an abelian group whose Sylow subgroups +are cyclic. Therefore G is cyclic. +If i = 4, then a ∈ ⟨b⟩, so G = ⟨a, b⟩ = ⟨b⟩ is cyclic. +Subcase 6.2. Assume that t ̸= −s. Therefore, we may assume s = a and t = b, so 2a = 2b. +If we let z = b − a, this implies that 2z = 0, so z = −z. Then a = b + z and b = a − z = a + z, so +S = S +z, which contradicts the assumption that Cay(G : S) is nontrivially unstable (and therefore +has no twin vertices). +□ + +4. Some unstable abelian Cayley graphs of valency 6 +In this section, we prove the following theorem, which implies Theorem 1.6. (Although the title +of this section specifies “valency 6,” the theorem also applies to some graphs of smaller valency, +because a, b, and/or c may have order 2.) +(4.1) Theorem. Let {a, b, c} be a generating set of a finite abelian group G, such that +a + b + c = 0 +and +the sets {±a}, {±b}, {±c} are distinct. +The Cayley graph X = Cay(G; ±a, ±b, ±c) is unstable if and only if one of the following conditions +is satisfied (perhaps after permuting a, b, and c): +(1) |a| = 4 and |G| is divisible by 8. +(2) 2a = 2b and |G| is divisible by 8. +(3) |a| = 8 and b = 3a. +(4) |a| = 12 and b = 4a. +(5) |a| = |b| = 3. +Before proving this theorem, let us show that it implies Theorem 1.6. +Proof of Theorem 1.6. As in the proof of Theorem 1.5, we use Lemma 2.3. We will show that +the graphs in Theorem 1.6 are precisely those that arise from Theorem 4.1 (and are not the two +trivially unstable graphs in Remark 1.8(d)), and satisfy the additional assumption that m, n ≥ 2 +(where n = |b| and m = |G : ⟨b⟩|). To do this, we consider each part of the statement of Theorem 4.1 +individually. We also consider appropriate permutations of a, b, and c. This is made easier by the +observation that Corollary 2.4 determines the result of interchanging a with c. +(1) Since |a| = 4, we must have m ∈ {1, 2, 4}. Since |G| is divisible by 8, but |a| is not, we know +that n and r are even. +• If m = 4, then r = 0 (because ma = 4a = 0). This yields the graph Tr(4, 2k, 0) of 1.6(4). +By Corollary 2.4, this is isomorphic to Tr(4, 2k, 4). +By using a in the role of b in Lemma 2.3, we obtain Tr(2k, 4, 0), which is also listed +in 1.6(4). If k is even, then applying Corollary 2.4 does not give anything new. However, +if k is odd, then this graph is isomorphic to Tr(2k, 4, 2), as mentioned at the end of the +statement of Theorem 1.6. +• If m = 2, then r = n/2 (because ma = 2a has order 2). Since r is even, this yields the +graph Tr(2, 4k, 2k) of 1.6(3). By Corollary 2.4, this is isomorphic to Tr(2, 4k, 2 − 2k) = +Tr(2, 4k, 2k + 2). +By using a in the role of b in Lemma 2.3, we obtain Tr(2k, 4, 2), which is also listed +in 1.6(3). If k is even, then, as above, applying Corollary 2.4 does not give anything new. +However, if k is odd, then this graph is isomorphic to Tr(2k, 4, 0), as mentioned at the end +of the statement of Theorem 1.6. +• If m = 1, then we cannot directly apply Lemma 2.3, because the definition of Tr(m, n, r) +requires m > 1. +However, we may use a in the role of b. This yields Tr(2k, 4, r), and we have r ∈ {±1} +because ⟨b⟩ = G (since m = 1). Therefore, the graph is listed in 1.6(5). If k is even, then +the two graphs are isomorphic (by Corollary 2.4), as mentioned at the end of the statement +of Theorem 1.6. The graph Tr(2, 4, 1) is trivially unstable (see Remark 1.8), so we require +k > 1 in Tr(2k, 4, 1). +(2) Write |G| = 8k. +14 + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +15 +• Suppose, for the moment, that ⟨b⟩ = G. Since 2a = 2b, but a ̸= b, we must have a = +(4k + 1)b. Then a and b both generate G, so neither can play the role of b in Lemma 2.3. +However, we may let c play this role. Note that c = −(a + b) = −(4k + 2)b = (4k − 2)b, so +|G : ⟨c⟩| = 2 and +2a = 2b = (2k − 1)(4k − 2)b = (2k − 1)c, +so this yields the graph Tr +� +2, 4k, −(2k − 1) +� += Tr(2, 4k, 2k + 1), which is listed in 1.6(2). +Applying Corollary 2.4 to this graph does not yield anything new. +• We may now assume ⟨b⟩ ̸= G. Then, since 2a = 2b, we have m = r = 2 and n = |G|/m = +8k/2 = 4k. +So this yields the graph Tr(2, 4k, −2) of 1.6(1). +By Corollary 2.4, this is +isomorphic to Tr(2, 4k, 4). +Note that 4a = 2a + 2a = 2a + 2b = 2(a + b) = −2c. Therefore, by using c in the role +of b, we obtain the graph Tr(4, 2k, 2) of 1.6(1). Applying Corollary 2.4 to this graph does +not give us anything new. +(3) Since ⟨a⟩ = ⟨b⟩ = G, neither a nor b can play the role of b in Lemma 2.3. Letting c play +the role of b yields the graph Tr(4, 2, 1) (because c = −(a + b) = −4a has order 2). This is listed +in 1.6(6). Applying Corollary 2.4 to this graph does not yield anything new. +(4) Since G = ⟨a⟩, we have |G| = |a| = 12. Then n = |b| = 3 and m = |G|/|b| = 4. Also, +ma = 4a = b, so r = 1. Therefore, we have the graph Tr(4, 3, −1) of 1.6(6). Applying Corollary 2.4 +to this graph does not yield anything new. +Since ⟨a⟩ = ⟨c⟩ = G, neither a nor c can play the role of b in Lemma 2.3. +(5) This graph is trivially unstable (see Lemma 4.2(2)). +□ +To shorten the main argument, we establish three minor results that deal with parts of the proof +of Theorem 4.1. +(4.2) Lemma. Let {a, b, c} be a generating set of a finite abelian group G, such that +a + b + c = 0 +and +the sets {±a}, {±b}, {±c} are distinct. +The Cayley graph X = Cay(G; ±a, ±b, ±c) has twin vertices (or, equivalently, is trivially unstable) +if and only if (perhaps after permuting a, b, and c) either: +(1) |a| = 8 and b = 2a, or +(2) |a| = |b| = 3. +In each case, the Cayley graph is listed in Theorem 4.1. +Proof. (⇐) Up to a group isomorphism, X is either Cay(Z8; S8) or Cay(Z3 × Z3; S3), where +S8 = {±1, ±2, ±3} +and +S3 = {±(1, 0), ±(0, 1), ±(1, 1)}. +We have S8 = S8 + 4 and S3 = S3 + (−1, 1), so both Cayley graphs have twin vertices, and are +therefore trivially unstable. +Since 2 is an element of order 4 in Z8, the first Cayley graph is listed in 4.1(1). The second is +listed in 4.1(5). +(⇒) Let S = {±a, ±b, ±c}. Since X has twin vertices, we see from Remark 2.13 that S is a +union of cosets of some subgroup ⟨z⟩ of prime order. Since |S| ≤ 6, we must have |z| ∈ {2, 3, 5}. +Case 1. Assume |z| = 2. Then −s + z = −(s + z) for all s ∈ S, so the permutation x �→ x + z +induces a well-defined action on +� +{±a}, {±b}, {±c} +� +. +Since the permutation has order 2, this +implies s + z = −s for some s ∈ S. Assume without loss of generality that s = a, so |a| = 4 and +z = 2a. +If b + z = −b, then we also have c + z = −c, so +0 = −(a + b + c) = (a + z) + (b + z) + (c + z) = (a + b + c) + z = 0 + z = z, +which contradicts the fact that |z| = 2. + +16 +DAVE WITTE MORRIS +Therefore, we must have b + z = ±c, so a + b ± (b + z) = 0. +• For the minus sign, we have 0 = a + b − (b + z) = a + z. This contradicts the fact that +|a| = 4 ̸= |z|. +• For the plus sign, we have 0 = a + 2b + z = −a + 2b, so a = 2b. Since |a| = 4, this implies +|b| = 8, so X is the Cayley graph in (1). +Case 2. Assume |z| = 3. This implies |S| = 6, so no element of S has order 2. Therefore, if C +is any coset of ⟨z⟩, then C ̸= −C (since |C| is odd). Since the cosets form a partition, we conclude +that C ∩ −C = ∅. Hence, we may assume that C contains a, b, and ±c. Then +0 = a + (a + z) ± (a + 2z), +so either 3a = 0 or a = z. However, a ̸= z, since 0 /∈ S. Therefore 3a = 0, which means |a| = 3. +Since z also has order 3, X is the Cayley graph in (2). +Case 3. Assume |z| = 5. Then |S| = 5, so some element s of S has order 2. Since G = ⟨S⟩ = ⟨s, z⟩, +this implies |G| = 10. More precisely, up to a group isomorphism, we have X = Cay(Z10; ±1, ±3, 5). +However, it is not possible to choose representatives of {±1}, {±3}, and {5} whose sum is 0, so +this case is not possible. +□ +(4.3) Lemma. All of the Cayley graphs listed in the statement of Theorem 4.1 are unstable. +Proof. (1) (Wilson [16, Thms. T.2 and T.3, p. 381]) Let z = 2a, and define ϕ: G → G by ϕ(pa+qb) = +pa + q(b + z). Since |a| = 4, we have |z| = 2, so ϕ is well-defined. (Since |G| is divisible by 8, but +|a| is not, and ⟨a, b⟩ = G, we know that |⟨b⟩ : ⟨a⟩ ∩ ⟨b⟩| is even. Therefore, if p1a + q1b = p2a + q2b, +then q1 ≡ q2 (mod 2), so p1a + q1(b + z) = p2a + q2(b + z).) Then it is easy to see that ϕ is an +automorphism of G. Also, we have +ϕ(a) = a = −3a = −a + 2a = −a + z, +so ϕ(S) = S + z. Hence, ϕ is an isomorphism from Cay(G; S) to Cay(G; S + z). This implies +Cay(G; S) is unstable (see Lemma 2.14). +(2) (Wilson [16, Thm. T.1, p. 381]) There is an automorphism ϕ of G that interchanges a and b +(and fixes c). Then ϕ(S) = S, so ϕ is an automorphism of Cay(G; S). Also note that ϕ fixes each +element of the index-2 subgroup ⟨2a, c⟩. +• If ⟨a⟩ ̸= G, then the subgraph induced by the set of un-fixed vertices consists of two cycles +of length |c|, and these two cycles are interchanged by ϕ. Since |c| = |a|/2 = |G|/4 is even, +these cycles are bipartite. Therefore Cay(G; S) is unstable by Lemma 2.15. +• If ⟨a⟩ = G, then (since |G| is divisible by 8) we may assume G = Z8k and a = 1. Since +2a = 2b, we know that b − a has order 2, and is therefore equal to 4k. +Hence S = +{±1, 4k ± 1, 4k ± 2}, so we see that X is unstable by letting a = 4k + 2 and b = 1 in +Theorem 2.11(4). +(3) We have X ∼= Cay(Z8; ±1, ±3, 4), which is unstable by Theorem 2.10(2). +(4) Since X ∼= Cay(Z12; ±1, ±4, ±5) = Cay(Z12; ±1, ±4, ±7), we see that it is unstable from +Theorem 4.1(4) with k = 3 and a = 4. +(5) X is trivially unstable by Lemma 4.2(2). +□ +(4.4) Lemma. Let S = {±1, ±b, ±(b + 1)}, for some b ∈ Zn, such that b /∈ {0, ±1, −2} (so the sets +{±1}, {±b}, {±(b + 1)} are distinct). If Cay(Zn; S) is unstable, then it is listed in Theorem 4.1. +Proof. Let X = Cay(Zn; S). We may assume X is nontrivially unstable, for otherwise Lemma 4.2 +applies. Also, since cyclic groups have no more than one element of order 2, we know that the + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +17 +valency of X is either 5 or 6. Therefore, the Cayley graph X is listed in either Theorem 2.10 or +Theorem 2.11. +Case 1. Assume X is listed in Theorem 2.10. In 2.10(1), one generator is odd, but the other two +generators are even, so a + b + c ̸≡ 0 (mod 2). Hence, the equation a + b + c = 0 is not satisfied. +So X must the graph in 2.10(2), which is listed in 4.1(3). +Case 2. +Assume X is listed in Theorem 2.11. We consider each of the seven lists of graphs +individually. +2.11(1): The element 2k has order 4, so X is listed in 4.1(1). +2.11(2): The generator ±a is odd, but ±b and ±b + 2k are even, so the equation a + b + c = 0 +is not satisfied. +2.11(3): For some choice of the signs, we must have ±a±(a+k)±(a−k) = 0, so 3a ≡ 0 (mod k). +Since k is odd and we also have a ≡ 0 (mod 4), this implies 3a = 0 in Z4k. But since 1 is in the +generating set, we must also have a − k = ±1 (perhaps after replacing a with −a). So a ∈ {k ± 1}. +Therefore 3k ± 3 = 3a ≡ 0 (mod 4k). From this (and the fact that a ̸= 0), we conclude that k = 3. +Since k ± 1 = a ≡ 0 (mod 4), we see that a = 4, so X is listed in 4.1(4). +2.11(4): For some c ∈ {4k ±b}, we have ±a+b+c = 0. However, if c = 4k −b, then this implies +a = 4k, which contradicts the fact that |a| is divisible by 4. Therefore, we must have c = 4k + b. +Then 2b = 2c, so this Cayley graph is listed in 4.1(2). +2.11(5): The condition a + b + c = 0 implies a ≡ 0 (mod k), so all of the elements of S are +divisible by k. Since 1 ∈ S, we conclude that k = 1, so n = 8. Since a ≡ 0 (mod 4), this implies +a = 4. So X is the Cayley graph in 4.1(3). (Alternatively, we have |a| = 2, which contradicts the +fact that the valency of X is assumed to be 6 in the current case.) +2.11(6): Since m is odd, we know that b and mb+2k have the same parity. Hence, the equation +a + b + c = 0 implies that a is even, so a ̸= ±1. Therefore, we may assume 1 ∈ {b, mb + 2k}. Since +the assumptions imply m2b ≡ ±b (and m is odd), we have ±b = m(mb + 2k) + 2k, so there is no +harm in assuming b = 1. +Since m2 ≡ 1 (mod 4), we know that m2 + 1 ̸≡ 0 (mod 4), so (m2 + 1)b = m2 + 1 ̸≡ 0 (mod 4k). +Therefore, we must have m2 ≡ 1 (mod 4k). +For an appropriate choice of the sign (and perhaps replacing a with its negative), we have +−a ± b + (mb + 2k) = 0, so +a = mb + 2k ± b = m + 2k ± 1. +However, +(m − 1)(m + 2k + 1) = m2 − 1 + (m − 1)(2k) ≡ 1 − 1 + 0 = 0 ̸≡ 2k ≡ (m − 1)a (mod 4k). +So a ̸= m + 2k + 1. Hence, we must have +a = m + 2k − 1. +Then, modulo 4k, we have +2k ≡ (m − 1)a = (m − 1)(m + 2k − 1) = m2 − 2m + 1 + (m − 1)(2k) ≡ 1 − 2m + 1 + 0 = −2(m − 1). +This means m ≡ k + 1 (mod 2k), so m is either k + 1 or −k + 1. +Since m is odd, this implies that k is even, so |Z4k| is divisible by 8. It also implies that a is +either 3k or k. In either case, a has order 4. So X is listed in 4.1(1). +2.11(7): If a = 1, then m = 4k + 1, so b and c must be even, which contradicts the equation +a + b + c = 0. +Thus, we may assume b = 1. Then m = 4k − 1, so a ≡ 2k (mod 4k), which means a ∈ {±2k}. +Therefore a has order 4, so the Cayley graph is listed in 4.1(1). +□ +As final preparation for proof of Theorem 4.1, let us recall some useful notation. + +18 +DAVE WITTE MORRIS +(4.5) Notation (cf. [6, Defn. 2.3], [9, Notn. 2.5]). Assume X = Cay(G; S) is an abelian Cayley +graph. +(1) For g ∈ G, let �g = (g, 1), so BX = Cay(G × Z2; �S ). +(2) For s1, . . . , sℓ ∈ S, and a starting point v ∈ V (BX), we use ( �s1, �s2, . . . , �sℓ) to denote +the walk v, v + �s1, v + �s1 + �s2, . . . , v + �s1 + �s2 + · · · + �sℓ in BX. +(3) ( �s1 +m1, �s2 +m2, . . . , �sℓ +mℓ) denotes the sequence consisting of m1 copies of �s1, followed by m2 +copies of �s2, followed by . . . , followed by mℓ copies of �sℓ. +Proof of Theorem 4.1. (⇐) See Lemma 4.3. +(⇒) Let S = {±a, ±b, ±c}, so X = Cay(G; S). Assuming that X is unstable, we will show that +(at least) one of the listed conditions is satisfied. Note that, since the equation a + b + c = 0 is +completely symmetric, we are free to permute the elements of {a, b, c} arbitrarily. +By Lemma 2.12, we may let α be an automorphism of BX that fixes (0, 0), but does not fix +(0, 1). Since G × {0} and G × {1} are the bipartition sets of BX, we know that each of these sets +is α-invariant. +Case 1. +Assume |G| is odd. By Theorem 2.7, we know that X is trivially unstable. Therefore, +Lemma 4.2 applies. +Case 2. +Assume G = ⟨s⟩ for some s ∈ S. See Lemma 4.4. +Case 3. +Assume |c| = 2. We may assume Case 2 does not apply; therefore c /∈ ⟨a⟩. If we let +n = |a|, then +X ∼= Cay +� +Zn × Z2; ±(1, 0), ±(1, 1), (0, 1) +� +. +• If n ≥ 8, then Proposition 2.16 tells us that X is stable. +• If n ≤ 7, and n ̸= 4, then Lemma 2.22 tells us that X is stable. +• If n = 4, then |a| = 4 and |G| = 8, so condition (1) is satisfied. +Assumption C. Henceforth, we assume |s| ≥ 3 for all s ∈ S (so X has valency 6), and |G| is +even. +Case 4. +Assume |s| = 4, for some s ∈ S. We may assume |G| = 4k, where k is odd, for +otherwise condition (1) is satisfied. Then G is cyclic (since |a| = 4 and G/⟨a⟩ is generated by b). +We may also assume that X is nontrivially unstable, for otherwise Lemma 4.2 applies. Then X +must be listed in Theorem 2.11. More precisely, since |G| is not divisible by 8, it must be listed in +2.11(2), 2.11(3), or 2.11(6). We will look at each of these possibilities separately (similarly to how +these cases were considered in the proof of Lemma 4.4). +2.11(2): Since the generator ±a is odd, but the other two are even, the equation a + b + c = 0 +cannot be satisfied. +2.11(3): Since S must contain the element k of order 4, but a ≡ 0 (mod 4), we must have +a + k = k or a + k = −k or a − k = k or a − k = −k, +so a ∈ {0, 2k}. However, we know a ̸= 0. And a ̸= 2k, because 2k ̸≡ 0 (mod 4) (or because we are +assuming that no element of S has order 2). +2.11(6): Since m is odd, we know that b and mb + 2k have the same parity. Therefore, the +equation a + b + c = 0 implies that a is even. So a ̸= ±k. Then {±b, ±mb + 2k} contains k, and +is therefore contained in ⟨k⟩. This is impossible, because ⟨k⟩ does not contain 4 distinct nonzero +elements. +Case 5. +Assume α is a group automorphism. Then +• α(g, 0) = +� +ϕ(g), 0 +� +, for some automorphism ϕ of G, and +• α(0, 1) = (z, 1), for some element z of order 2. + +AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID +19 +Then, since α +� +S × {1} +� += S × {1}, we have ϕ(S) = S + z, so ϕ is an isomorphism from Cay(G; S) +to Cay(G; S + z). +We have ϕ(s) ∈ ±{a, b, c} + z for all s ∈ {a, b, c}. At least two elements of S must use the +same sign, which implies there exists ǫ ∈ {0, 1}, such that ϕ(S) contains at least two elements of +{ǫa + z, ǫb + z, ǫc + z}. We may assume ǫ = 1 (by composing with the automorphism x �→ −x if +necessary). Then, by permuting {a, b, c}, we may assume that a + z and b + z are in ϕ({a, b, c}). +Therefore +0 = ϕ(a + b + c) = ϕ(a) + ϕ(b) + ϕ(c) = (a + z) + (b + z) ± (c + z) = (a + b) ± c + z = −c ± c + z. +Since z ̸= 0, we conclude that 2c = z, so c has order 4. Therefore, Case 4 applies. +Case 6. +Assume 2s ̸= 2t, for all s, t ∈ S, such that s ̸= t. +We may assume α is not a +group automorphism, for otherwise Case 5 applies. Therefore, Proposition 2.17 implies there exist +s, t, u, v ∈ S such that s + t = u + v ̸= 0 and {s, t} ̸= {u, v}. From the assumption of this case, we +see that this implies either 3a = ±c or 2a = ±b ± c (perhaps after permuting a, b, and c). +• If 3a = ±c, then c = ±3a ∈ ⟨a⟩, so G = ⟨a⟩. Therefore Case 2 applies. +• If 2a = ±b − c = ±b + (a + b), then a = ±b + b ∈ ⟨b⟩, so G = ⟨b⟩. Therefore Case 2 applies. +So we may assume +2a = ±b + c = ±b − (a + b) ∈ {−a, −a − 2b}, +so 3a ∈ {0, −2b}. +• If 3a = −2b, then a = −2a − 2b = −2(a + b) = 2c ∈ ⟨c⟩, so G = ⟨c⟩. Therefore Case 2 +applies. +Hence, we may assume +3a = 0, so |a| = 3. +We may also assume a /∈ ⟨b⟩, for otherwise Case 2 applies. +So ⟨a⟩ ∩ ⟨b⟩ = {0}. +This implies +G = ⟨a⟩ × ⟨b⟩, so, letting n = |b|, we have +X ∼= Cay +� +Z3 × Zn; ±(1, 0), ±(0, 1), ±(1, 1) +� +. +Note that: +• (�a3) and (�a−3) are paths of length 3 in BX from (0, 0) to �0 = (0, 1), +• (�b3) and (�c−3) are paths of length 3 in BX from (0, 0) to 3�b, and +• (�b−3) and (�c3) are paths of length 3 in BX from (0, 0) to 3�c. +Also note that if s1, s2, and s3 are not all equal to each other, then the number of permutations +of the list ( �s1, �s2, �s3) is divisible by 3. Therefore, �0, 3�b, and 3�c are the only vertices of BX for +which the number of paths of length 3 from (0, 0) to the vertex is not divisible by 3. Hence, the +set {�0, 3�b, 3�c} is α-invariant. +We may assume |b| > 12, for otherwise Lemma 2.24 applies. +Then the number of paths of +length 6 from 3�c = −3�b to 3�b is +�6 +3 +� ++ 2: +• +�6 +3 +� +permutations of the path (�b3, �c−3), and +• the paths (�b6) and (�c−6). +This is much smaller than the number of paths of length 6 from �0 to either 3�b or 3�c. For example, +paths from �0 to 3�b include: +• +�6 +3 +� +permutations of the path (�a3,�b3), +• +�6 +3 +� +permutations of the path (�a3, �c−3), and +• others. + +20 +DAVE WITTE MORRIS +Thus, the vertex (0, 1) = �0 is uniquely determined as an element of the α-invariant set {�0, 3�b, 3�c}, +so it must be fixed by α. This contradicts the choice of α. +Case 7. +The remaining case. Since Case 6 does not apply, we have 2s = 2t, for some s, t ∈ S, +such that s ̸= t. Since Case 4 does not apply, we know s ̸= −t. Therefore, we may assume s = a +and t ∈ {±b}, so 2a ∈ {±2b}. However, if 2a = −2b, then +2c = −2(a + b) = −(2a + 2b) = −(−2b + 2b) = −0 = 0, +so Case 3 applies. Thus, we must have 2a = 2b. So b = a + z for some element z of order 2. +We may assume G ̸= ⟨a⟩ and G ̸= ⟨b⟩ (otherwise, Case 2 applies), so b /∈ ⟨a⟩ and a /∈ ⟨b⟩. Hence +z /∈ ⟨a⟩ and z /∈ ⟨b⟩. So +X ∼= Cay +� +Zn × Z2; ±(1, 0), ±(1, 1), ±(2, 1) +� +, +where n = |a| = |b| is even and, up to a group isomorphism, a = (1, 0), b = (1, 1), and c = −(2, 1). +We may assume X does not satisfy condition (2), so +n/2 is odd. +Also, since Case 4 (and Case 3) does not apply, we have n > 4. In addition, since Case 6 does not +apply, we know 2a ̸= −2c, so n ̸= 6. Therefore (since 8/2 is not odd), we have +n > 8. +Note that, for every v ∈ V (BX), the fact that 2a = 2b implies that v + 2 �c and v − 2 �c are the +only vertices of BX that are joined to v by a unique path of length 2. It follows from this that +α(v + k�c) = α(v) ± k�c, for all k ∈ Z and all v ∈ V (BX). +(4.6) +Since α fixes 0, we conclude that {±k �c} is α-invariant, for all k ∈ Z. +Let Y be the spanning subgraph of BX that is obtained by removing all edges of the form +(v, v + �c). By (4.6), we know that Y is α-invariant, so α is an automorphism of Y . +Now, � +(0, 0) and � +(0, 1) are the only vertices that are at distance 2 from both �c and −�c in Y . Since +{±�c} is α-invariant, this implies that +� � +(0, 0), � +(0, 1) +� +is also α-invariant. +Since |�c| = n, we have (n/2)�c = −(n/2)�c. Therefore, we see from (4.6) that (n/2)�c is fixed by α. +However, since n/2 is odd, we have +� +(0, 1) = n +2 +� +(2, 1) = n +2 �c. +Hence, � +(0, 1) is fixed by α. Therefore, since � +(0, 0) is the only other element of the invariant set +� � +(0, 0), � +(0, 1) +� +, we conclude that � +(0, 0) is also fixed by α. This contradicts the choice of α. +□ + +References +[1] Y.-G. Baik, Y.-Q. Feng, H.-S. 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B 98 (2008) 359–383. +MR 2389604, doi:10.1016/j.jctb.2007.08.001 +Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, +T1K 3M4, Canada +Email address: dmorris@deductivepress.ca, https://deductivepress.ca/dmorris +21 + diff --git a/XtE5T4oBgHgl3EQfCg6t/content/tmp_files/load_file.txt b/XtE5T4oBgHgl3EQfCg6t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b934a6fc6c4d6eb5bc85c746a4bd6bb15300a86 --- /dev/null +++ b/XtE5T4oBgHgl3EQfCg6t/content/tmp_files/load_file.txt @@ -0,0 +1,1281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf,len=1280 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='05396v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='CO] 13 Jan 2023 AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID DAVE WITTE MORRIS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The Cartesian product of two cycles (Cn□Cm) has a natural embedding on the torus, such that each face of the embedding is a 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The toroidal grid Qd(m, n, r) is a generalization of this in which there is a shift by r when traversing the meridian of length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In 2008, Steve Wilson found two interesting infinite families of (nonbipartite) toroidal grids that are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (By definition, this means that the canonical bipartite double cover of the grid has more than twice as many automorphisms as the grid has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') It is easy to see that bipartite grids are also unstable, because the canonical double cover is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Furthermore, there are degenerate cases in which there exist two different vertices that have the same neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This paper proves Wilson’s conjecture that Qd(m, n, r) is stable for all other values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In addition, we prove an analogous conjecture of Wilson for the triangular grids Tr(m, n, r) that are obtained by adding a diagonal to each face of Qd(m, n, r) (with all of the added diagonals parallel to each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Introduction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1) Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The canonical bipartite double cover [15] of a graph X is the bipartite graph BX with V (BX) = V (X) × {0, 1}, where (v, 0) is adjacent to (w, 1) in BX ⇐⇒ v is adjacent to w in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Letting S2 be the symmetric group on the 2-element set {0, 1}, it is clear that Aut X × S2 is a subgroup of Aut BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If this subgroup happens to be all of Aut BX, then we say that X is stable [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Understanding unstable graphs is a fundamental problem in the study of automorphisms of direct products (see [11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6]), and also arises in other contexts (see the introductions of [12] and [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In the appendix of [16], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson stated conjectures about exactly which graphs in certain families are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Here is the current status of each of these conjectures: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1) Wilson’s conjecture about circulant graphs is known to be false: a counterexample was published in [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 156], and infinite families of additional counterexamples can be found in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We still do not know which circulant graphs are unstable, but progress was made in [2, 6, 7, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2) Wilson’s conjecture on generalized Petersen graphs is correct [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 1 2 DAVE WITTE MORRIS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3) There does not seem to have been any progress on Wilson’s conjecture about rose window graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4) This paper proves (slight generalizations of) Wilson’s two conjectures about toroidal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We now state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2) Definition ([16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 380 and 381]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Given m, n ∈ Z (with m, n ≥ 2), and r ∈ Zn, we can number the vertices of the cycle Cn with the elements of Zn, and number the vertices of the path Pm+1 with the elements of {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (In the special case where m = 2, we let C2 = K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') Then (1) Qd(m, n, r) is the graph that is obtained from the Cartesian product Cn□Pm+1 by identifying the vertex (x, m) with (x + r, 0), for each x ∈ Zn, and (2) Tr(m, n, r) is the graph that is obtained from Qd(m, n, r) by adding an edge from (x, y) to (x + 1, y − 1) for each x ∈ Zn and y ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (See Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 for reformulations of these definitions in the language of Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Qd(m, n, r) has a natural embedding on the torus, such that each face of the embedding is a 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (In the special case where r = 0, the graph Qd(m, n, 0) is isomorphic to the Cartesian product Cn □ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') So Qd(m, n, r) is often called a toroidal grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The graph Tr(m, n, r) is obtained by adding a diagonal in each face of Qd(m, n, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, the faces of its natural toroidal embedding are triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' There are some trivial reasons for a graph to be unstable [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 360]: (1) Every disconnected graph is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) Every bipartite graph with a nontrivial automorphism is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) If two different vertices of a graph have the same neighbours, then the graph is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (These are called “twin vertices” [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') This motivates the following definition: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4) Definition ([16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 360]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' An unstable graph is nontrivially unstable if it is connected and nonbipartite, and has no twin vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Otherwise, it is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') The following two results were conjectured by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson [16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 380 and 381], who proved the direction (⇐) of each theorem (except parts (5) and (6) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (We write “±” in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 because Qd(m, n, r) is always isomorphic to Qd(m, n, −r), as explained in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also recall that the parameter r in Qd(m, n, r) and Tr(m, n, r) is taken modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5) Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Qd(m, n, r) is nontrivially unstable if and only if it is: (1) Qd(m, 4k, ±k), where m + k is odd, or (2) Qd(2m, km, ±4ℓm) (∼= Qd(m, 2km, ±2ℓm) if m > 1), where m is odd, 4ℓ2 ≡ ±1 (mod k), and either m > 1 or 2ℓ ̸≡ ±1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6) Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Tr(m, n, r) is nontrivially unstable if and only if it is: (1) Tr(2, 4k, 4) ∼= Tr(2, 4k, −2) ∼= Tr(4, 2k, 2), or (2) Tr(2, 4k, 2k + 1), or (3) Tr(2, 4k, 2k) ∼= Tr(2, 4k, 2k + 2) ∼= Tr(2k, 4, 2), or (4) Tr(4, 2k, 0) ∼= Tr(4, 2k, 4) ∼= Tr(2k, 4, 0), or (5) Tr(2k, 4, 1), with k > 1, or Tr(2k, 4, −1), or (6) Tr(4, 2, 1), or Tr(4, 3, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (If k is odd, then the graphs in (3) are isomorphic to the graphs in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If k is even, then the two graphs in (5) are isomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7) Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By searching Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 for cases where r = 0, we see that, for n ≥ m ≥ 2, the Cartesian product Cn □ Cm is nontrivially unstable if and only if n = 2m and m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6 only list graphs that are nontrivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, it is easy to check whether a toroidal grid is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' First, note that they are all connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also: (a) Qd(m, n, r) is bipartite if and only if n and m + r are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (b) Qd(m, n, r) has twin vertices if and only if m = 2 and r = ±2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (c) Tr(m, n, r) is never bipartite (because it has triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (d) Tr(m, n, r) has twin vertices if and only if (m, n, r) ∈ {(2, 4, 1), (3, 3, 0)} (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='9) Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1) The assumption that m, n ≥ 2 is not stated explicitly in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson’s conjectures also seem to implicitly assume that gcd(n, r) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We do not make this assumption, so Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6 include include infinite families of graphs that are not listed in Wilson’s conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) There are other differences between Wilson’s conjectures [16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 380 and 381] and our statements of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In particular: (a) Wilson omits Qd(m, 4k, −k) and Qd(2m, km, ±4ℓ), and usually omits Tr(m, n, m − r) when Tr(m, n, r) is listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 explains that they are alternate representations of other graphs in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (b) Wilson requires k to be odd in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(2), but we omit this redundant condition: it is a consequence of the equation 4ℓ2 ≡ ±1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (c) Wilson uses 4k in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(4), instead of 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' That eliminates the overlap with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) The two occurrences of “±” in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(2) cause redundancy (and could therefore be omitted), because −ℓ satisfies the congruence 4ℓ2 ≡ ±1 (mod k) whenever ℓ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Following a section of preliminaries, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 is proved in Section 3, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 is provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 are slightly more general than Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For example, their statements in the language of abelian Cayley graphs allow for the case where m is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' See the well-known Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 for the translation between the two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Preliminaries All graphs in this paper are simple (no loops or multiple edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Abelian Cayley graphs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1) Definition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let S be a subset of an additive abelian group G, such that S = −S and 0 /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The corresponding abelian Cayley graph Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is the graph whose vertices are the elements of G, and with an edge joining the vertices g and h if and only if g = h + s for some s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The adjective “abelian” in “abelian Cayley graph” is to emphasize the assumption that G is abelian, so we will sometimes omit it when it is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (The usual definition of Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) does not require G to be abelian, but we have no need for the nonabelian case in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') The following simple (and well known) observation notes that the toroidal grids Qd(m, n, r) and Tr(m, n, r) are isomorphic to abelian Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The minus sign in Tr(m, n, −r) is because the definition of Tr(m, n, −r) would naturally identify it with the Cayley graph having a − b as the third generator, but, for our purposes, it is more convenient to use a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Given m, n, r ∈ Z (with m, n ≥ 2), let G = ⟨ a, b | ma = rb, nb = 0, a + b = b + a ⟩, so G is an abelian group of order mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then Qd(m, n, r) ∼= Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b) and Tr(m, n, −r) ∼= Cay � G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b, ±(a + b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4) Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Qd(m, n, r) ∼= Qd(m, n, −r) and Tr(m, n, r) ∼= Tr(m, n, m − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Qd) We have ma = −r(−b) and n(−b) = 0, so using −b in the place of b yields a representation of Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b) as Qd(m, n, −r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Tr) Let c = −(a + b), so {±a, ±b, ±(a + b)} = {±c, ±b, ±(c + b)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then mc = −ma − mb = −rb − mb = −(r + m)b, so using c in the place of a yields a representation of Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b, ±(a + b)) as Tr(m, n, r + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore Tr(m, n, −r) ∼= Tr(m, n, r + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By replacing a with −a, the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 shows that Qd(m, n, r) ∼= Qd(m, n, −r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, the same trick does not work for Tr(m, n, r): if a is replaced with −a, then the equation a + b + c = 0 forces b and c to also be replaced with their negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since m(−a) = −ma = −rb = r(−b), this does not result in any change in the parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Here is an abelian Cayley graph that appears in the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(4): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6) Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' M2n = Cay(Z2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, n) is the Moebius ladder with 2n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4 AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 5 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Some classes of stable/unstable abelian Cayley graphs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7) Theorem (Morris [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' There are no nontrivially unstable abelian Cayley graphs of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Recall that if G is cyclic, then Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is a circulant graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The following result is stated only for circulant graphs in [7], but exactly the same proof applies to abelian Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8) Proposition (Hujdurovi´c-Mitrovi´c-Morris, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' There are no nontrivially unstable abelian Cayley graphs of valency ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='9) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A circulant graph Cay(Zn, {±a, ±b}) of valency 4 is unstable if and only if either it is trivially unstable, or one of the following conditions is satisfied (perhaps after interchanging a and b): (1) n is divisible by 8 and gcd � |a|, |b| � = 4, or (2) n ≡ 2 (mod 4), gcd(b, n) = 1, and a ≡ ℓb + (n/2) (mod n), for some ℓ ∈ Z, such that ℓ2 ≡ ±1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A circulant graph Cay(Zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) of valency 5 is unstable if and only if either it is trivially unstable, or it is either: (1) Cay(Z12k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±s, ±2k, 6k) with s odd, or (2) Cay(Z8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, ±3, 4) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11) Theorem (Hujdurovi´c-Mitrovi´c-Morris [7, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A circulant graph X = Cay(Zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±c}) of valency 6 is unstable if and only if either it is trivially unstable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' or it is one of the following: (1) Cay(Z8k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±2k}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where a and b are odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) Cay(Z4k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b + 2k}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where a is odd and b is even,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) Cay � Z4k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' � ±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(a + k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(a − k) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where a ≡ 0 (mod 4) and k is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) Cay(Z8k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4k ± b}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where a is even and |a| is divisible by 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (5) Cay(Z8k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±3k}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where a ≡ 0 (mod 4) and k is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (6) Cay(Z4k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±mb + 2k}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where gcd(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4k) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (m − 1)a ≡ 2k (mod 4k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' and either m2 ≡ 1 (mod 4k) or (m2 + 1)b ≡ 0 (mod 4k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (7) Cay(Z8k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' {±a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±c}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' where there exists m ∈ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' such that gcd(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 8k) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' m2 ≡ 1 (mod 8k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' and (m − 1)a ≡ (m + 1)b ≡ (m + 1)c ≡ 4k (mod 8k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Criteria for stability or instability (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='12) Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [2, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A connected, abelian Cayley graph X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable if and only if there exists α ∈ Aut BX, such that α(0, 0) = (0, 0), but α(0, 1) ̸= (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='13) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It is easy to see (and well known) that an abelian Cayley graph Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) has twin vertices if and only if S + z = S, for some nonzero z ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In other words, S is a union of cosets of ⟨z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By passing to a subgroup of ⟨z⟩, there is no harm in assuming that |z| is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The following result was stated only for circulant graphs in [6, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7] (which is a slight generalization of [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4]), but the same proof applies more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (In fact, the proof even applies without the assumption that z has order 2, if S + z = −(S + z) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' And there is no need for G to be abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') 6 DAVE WITTE MORRIS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='14) Lemma ([6, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' An abelian Cayley graph Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable if Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) ∼= Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z), for some element z of order 2 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='15) Lemma (Wilson [16, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A graph X is unstable if it has an automorphism α, such that the subgraph induced by the set of un-fixed vertices is disconnected and has a component C, such that C is bipartite, and either α(C) ̸= C or each of the two bipartition sets of C is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='16) Proposition (Hujdurovi´c-Mitrovi´c [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let X be a connected graph with more than one vertex, and assume that X satisfies the following conditions: (1) Every edge of X lies on a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) For every x ∈ V (X), it holds that: (a) every vertex at distance 2 from x has a neighbour at distance 3 from x, and (b) every vertex at distance 3 from x has a neighbour at distance 4 from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then X is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Other results on automorphisms and isomorphisms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='17) Proposition (Baik-Feng-Sim-Xu [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let S be a generating set of an abelian group G, such that S = −S, 0 /∈ S, and, for all s, t, u, v ∈ S: s + t = u + v ̸= 0 =⇒ {s, t} = {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If α is any automorphism of the graph Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S), such that α(0) = 0, then α is an automorphism of the group G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=', α(g + h) = α(g) + α(h), for all g, h ∈ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='18) Definition ([4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Recall that the Cartesian product X □ Y of two graphs X and Y has vertex set V (X) × V (Y ), and two vertices (x1, y1) and (x2, y2) are adjacent if and only if either x1 = x2 and y1y2 ∈ E(Y ), or y1 = y2 and x1x2 ∈ E(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='19) Proposition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let X be a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If there does not exist a graph Y, such that X ∼= Y □ K2, then Aut(X □ K2) = Aut X × S2 and Aut(X □ C4) = Aut X × Aut C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We will use the following elementary observation in part (4) of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='20) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let X and Y be graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If Y is bipartite, then B(X □ Y ) ∼= (BX) □ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Stability of a few specific graphs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='21) Example ([12, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n ≥ 3, then the complete graph Kn is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (But K2 is bipartite, and is therefore unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='22) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For 2 ≤ n ≤ 7, the abelian Cayley graph Cay � Zn × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(1, 1), (0, 1) � is stable, unless n = 4, in which case it is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This can be checked very quickly by computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For example, the sagemath program in Figure 1 can be executed on https://cocalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (The program also verifies Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') □ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='23) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Most cases of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='22 can be settled quite easily without a computer: If n is odd, then Zn × Z2 is cyclic, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n = 2, then the Cayley graph is K4, which is stable by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 7 If n = 4, then part (1) of the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 explains why the Cayley graph is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, n = 6 is the only case that requires effort (or a computer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='24) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For 3 ≤ n ≤ 12, the abelian Cayley graph Cay � Zn × Z3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(1, 1), ±(0, 1) � is stable, unless n = 3, in which case it is unstable (and is listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' As mentioned in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='22, the stability/instability of these graphs is calculated by the sagemath program in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For n = 3, the elements a = (1, 0) and b = (0, 1) have order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, if we let c = −(1, 1), then a + b + c = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, the Cayley graph is described in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ for n in range(2, 13): for k in [2, 3]: G = direct_product_permgroups( [CyclicPermutationGroup(n), CyclicPermutationGroup(k)]) a, b = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='gens() assert {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='order(), b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='order()} == {n, k} X = Graph(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='cayley_graph(generators=[a, b, a*b])) AutX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='automorphism_group() K2 = graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='CompleteGraph(2) BX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='categorical_product(K2) AutBX = BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='automorphism_group() if 2 * AutX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='order() !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='= AutBX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='order(): print(n, k, "unstable") Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A sagemath [14] program to verify Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Unstable abelian Cayley graphs of valency 4 This section proves the following theorem, which implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It also generalizes Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='9, which handles the case where G is cyclic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' however, our argument relies on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='9, so we are not providing an independent proof of that result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1) Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' A connected abelian Cayley graph Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) of valency 4 is unstable if and only if either it is bipartite, or it is in the following list (up to a group isomorphism ): (1) Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b), where |⟨a⟩ ∩ ⟨b⟩| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b), where |G : ⟨b⟩| = m, ma = 2ℓmb, |b| = 2km, and 4ℓ2 ≡ ±1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b), where 2a = 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) Cay � Z2n × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), (n, 0), (0, 1) � ∼= M2n □ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (a) If n = 2, this is isomorphic to K4 □ K2, and can also be realized as Cay � Z2 × Z2 × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Before proving this theorem, let us show that it implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The graphs in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 are required to be nontrivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we see from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8 that they must have valency 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, it suffices to show that the graphs in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5 are precisely the graphs that arise from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 (and are not trivially unstable), and satisfy the additional assumption that m, n ≥ 2 (where n = |b| and m = |G : ⟨b⟩|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' To do this, we consider each part of the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We also find the toroidal grids that are obtained by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 after interchanging a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (And we know from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 that Qd(m, n, r) ∼= Qd(m, n, −r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') (1) The conditions in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(1) that n = 4k and r = ±k are a direct translation of the fact that |⟨a⟩ ∩ ⟨b⟩| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The additional condition that m + k is odd ensures that the grid is not trivially unstable (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since the condition in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1) is symmetric in a and b, no additional examples are obtained by interchanging a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) The grid Qd(m, 2km, 2ℓm) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(2) is obtained from a direct translation of the conditions in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The condition that m is odd ensures that the grid is not trivially unstable (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, the definition of Qd(m, n, r) requires m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Now, we let a play the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that gcd(k, ℓ) = 1, because 4ℓ2 ≡ 1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we have |a| = |G: ⟨b⟩| · |⟨a⟩ ∩ ⟨b⟩| = m · 2km gcd(2km, 2ℓm) = m · 2km 2m = km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, |G : ⟨a⟩| = |G|/|a| = m(2km)/(km) = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, since ma = 2ℓmb and 4ℓ2 ≡ ±1 (mod k) (and |b| = 2km), we have 4ℓma = (4ℓ)(2ℓmb) = 4ℓ2(2mb) = ±2mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So this yields the graph Qd(2m, km, ±4ℓm) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(2) (even if m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, this graph has twin vertices (and is therefore trivially stable) if (and only if) 2m = 2 and 4ℓm ≡ ±2 (mod km) (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This situation is ruled out by assuming (at the end of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5(2)) that either m > 1 or 2ℓ ̸≡ ±1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) This is trivially unstable (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) The generating sets arising here are not of the form {±a, ±b}, so Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3 cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' These Cayley graphs are therefore not needed to find all of the toroidal grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ 8 AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 9 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It is easy to determine whether a particular Cayley graph listed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' First, note that the graph is assumed to be connected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=', it is assumed that S generates G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 1 of the proof shows that the examples with twin vertices are precisely those in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So all that remains is to determine which of them are bipartite (which is usually answered by Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1) This is bipartite if and only if |a|/4 + |b|/4 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It has twin vertices if and only if |a| = |b| = 8 (in which case, it is also bipartite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(2) This is bipartite if and only if m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It has twin vertices if and only if m = 1 and 2ℓ ≡ ±1 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(3) As mentioned above, this graph has twin vertices, and is therefore trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (For completeness, we observe that it is bipartite if and only if |a| and |b| are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(4) This is bipartite if and only if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (It never has twin vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') To avoid cluttering the main part of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1, we present one direction of the argument in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It is mostly (or entirely?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') known: the instability of the graphs in (1) and (2) was proved by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson [16, §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1], and the rest is very easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, Wilson gave only a one-sentence sketch of his proofs, so we will provide a fairly complete argument for every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3) Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson [16, §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' All of the graphs listed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We consider each part of the statement of the theorem individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume each Cayley graph is not bipartite (for otherwise it is trivially unstable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 380]) Let m = |G : ⟨b⟩| and n = |b|, and choose r ∈ Z, such that ma = rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, let z be the element of order 2 in ⟨a⟩ ∩ ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |b| is divisible by 4, we know that |b + z| = |b| = n, so |G : ⟨b + z⟩| = |G : ⟨b⟩| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also note that, since |⟨a⟩ ∩ ⟨b⟩| = 4, we have |rb| = 4, so −rb = rb + z = rb + (m + r)z (since m + r is odd, because the Cayley graph is not bipartite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore m(−a + z) = −ma + mz = −rb + mz = rb + (m + r)z + mz = r(b + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, there is an automorphism ϕ of G, such that ϕ(a) = −a + z and ϕ(b) = b + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then ϕ is an isomorphism from Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) to Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies that Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Wilson [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 381]) Let n = 2km = |b| and r = 2ℓm, so ma = rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, let z be the element of order 2 in ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that: m is odd, because X is bipartite and r = 2ℓm is even, and gcd(2ℓ, k) = 1, because 4ℓ2 ≡ 1 (mod k) (so k is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then, since |b| = 2km and km is odd, we see that |b + z| = km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also (using the fact that gcd(ℓ, k) = 1), we have |a| = m · |b| gcd(r, |b|) = m · 2km gcd(2ℓm, 2km) = m · k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since mk is odd, this implies |a + z| = 2mk = |b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also note that, since 4ℓ2 = pk ± 1 for some p ∈ Z (and p must be odd), we have 4ℓ2b = (pk ± 1)b = pkb ± b = pz ± b = z ± b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore r(a + z) = 2ℓm(a + z) = 2ℓrb + 0 = 2ℓ(2ℓmb) = 4ℓ2mb = m(z ± b) = ±m(b + z), Hence, there is an automorphism ϕ of G, such that ϕ(b) = a + z and either ϕ(a) = b + z or ϕ(a) = −b + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In either case, ϕ is an isomorphism from Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) to Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies that Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 10 DAVE WITTE MORRIS (3) Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) has twin vertices (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2), so it is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) Since the Cayley graph is not bipartite, we know that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='20, we have BX = B(M2n □ K2) ∼= (BM2n) □ K2 ∼= (C2n □ K2) □ K2 ∼= C2n □ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So |Aut BX| ≥ |Aut C2n| · |Aut C4| = 4n · 8 = 32n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n ≥ 4, then |Aut X| = |Aut(M2n □ K2)| = 2|Aut M2n| = 8n < 1 2|Aut BX|, so X is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For the special case where n = 2, we have X ∼= K4 □ K2, so .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' |Aut X| = |Aut K4| · |Aut K2| = 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, BX ∼= (BK4) □ K2 ∼= (K2 □ K2 □ K2) □ K2, so |Aut BX| = 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' · 24 ≫ 2 |Aut X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore X is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇐) See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇒) Let X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S), and assume that X is connected and unstable, but not bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We will show that X is in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume X is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X is assumed to be connected and nonbipartite, it must have twin vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, S is a union of cosets of some subgroup ⟨z⟩ of prime order (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X has valency 4, we know that |S| = 4, so we must have |z| = 2 (since |z| is a prime number that divides |S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume S = {±a, ±b}, where |a|, |b| > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume a + z ∈ {−a, b} (perhaps after replacing b with its negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If a+z = −a, then 2a = z (and −a+z = a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, b+z /∈ {±a}, so we must have b+z = −b, which implies 2b = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore 2a = z = 2b, so (3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If a + z = b, then 2b = 2(a + z) = 2a + 2z = 2a + 0 = 2a, so (3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume S = {±a, b, c}, where |a| > 2 and |b| = |c| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since b + z ∈ S and 2(b + z) = 2b + 2z = 0 + 0 = 0, we must have b + z = c (and hence c + z = b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So a + z = −a, which implies z = 2a (and |a| = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Now, since X is not bipartite, there exist p, q, r ∈ Z, such that pa + qb + rc = 0 and p + q + r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then 0 = pa + qb + rc = pa + qb + r(b + 2a) ≡ (q + r)b (mod a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If q + r is odd, this implies b ∈ ⟨a⟩, so b = z (since z is the unique element of order 2 in ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' But then c = b + z = z + z = 0, which contradicts the fact that |c| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So q + r is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore p is odd, so pa = ±a (since |a| = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then ±a = −(qb + rc) ∈ ⟨b, c⟩ ∼= Z2 × Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This is impossible, since |a| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume S = {a, b, c, d}, where |a| = |b| = |c| = |d| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume, without loss of generality, that a + z = b and c + z = d, and also, since X is not bipartite, that a + b + c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' But then c = a + b = a + (a + z) = 0 + z = z, so d = c + z = c + c = 0, which contradicts the fact that |d| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In the remaining cases of the proof, we assume that X is nontrivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 11 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume G is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We see from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='9 that X is listed in either (1) or (2) (with m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume that S contains at least one element of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If every element of S has order 2, then (since X is not bipartite) it is not difficult to see that X is the Cayley graph K4 □ K2 that is listed in (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we may assume that S contains precisely two elements of order 2, so we may write S = {a, b, ±c}, where |a| = |b| = 2 and |c| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since ⟨c⟩ has at most one element of order 2, we have |⟨a, b⟩ ∩ ⟨c⟩| ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let n = |c|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |⟨a, b⟩ ∩ ⟨c⟩| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then X ∼= Cn □ C4 ∼= Cay � Zn × Z4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(0, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This Cayley graph has no elements of order 2 in the generating set, so it is considered in a later case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also note that n must be odd, since X is not bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then it is not difficult to see that this Cayley graph is not listed in any of the parts of the statement of the theorem, so it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |⟨a, b⟩ ∩ ⟨c⟩| = 2, but ⟨c⟩ ∩ {a, b} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then X is a prism with 2n vertices, plus an edge from each vertex to its antipodal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Note that n must be even, since ⟨c⟩ has a subgroup of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') Therefore, it is not difficult to see that X ∼= Cay(G′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±x, ±y), where G′ = ⟨ x, y | 4x = ny = 0, 2x = (n/2)y, x + y = y + x ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The generating set of this abelian Cayley graph has no elements of order 2, so the Cayley graph is considered in a later case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X is not bipartite, n/2 must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, it is not difficult to see that this Cayley graph is not listed in any of the parts of the statement of the theorem, so it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume ⟨a, b⟩ ∩ ⟨c⟩ = ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then X ∼= Mn □ K2 is listed in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assumption B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In the remaining cases of the proof, we assume that S has no elements of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we may write S = {±a, ±b}, where |a|, |b| > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume there is a group automorphism α of G × Z2, such that α is an automorphism of BX, and α(0, 1) ̸= (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since G × {0} and G × {1} are the bipartition sets of BX, we know that each of these sets is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore α(g, 0) = � ϕ(g), 0 � , for some automorphism ϕ of G, and α(0, 1) = (z, 1), for some element z of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since α is an automorphism of BX, we must have α � S×{1} � = S×{1}, so ϕ(S) = S+z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore ϕ is an isomorphism from Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) to Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume (by interchanging a and b, if necessary) that |b| is divisible by (at least) the largest power of 2 that divides |a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7 (and Assumption A), this implies that |b| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let m = |G : ⟨b⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then ma ∈ ⟨b⟩, so we may choose r ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , |b| − 1}, such that ma + rb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X is not bipartite, we know that m + r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume ϕ(b) ∈ {±b + z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since ϕ is a homomorphism (and |z| = 2), this implies that ϕ � {±b} � = {±b+z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then, since ϕ is a bijection from S to S +z, we must have ϕ(a) = ǫa+z, 12 DAVE WITTE MORRIS for some ǫ ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume, without loss of generality, that ϕ(b) = b + z (by composing with the automorphism x �→ −x, if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then 0 = ϕ(ma + rb) (ma + rb = 0) = m ϕ(a) + r ϕ(b) (ϕ is a group automorphism) = m (ǫa + z) + r (b + z) = ǫma + rb + z (|z| = 2 and m + r is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If ǫ = 1, then ǫma + rb = ma + rb = 0, so z = 0, which contradicts the fact that |z| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we must have ǫ = −1, so −ma+rb = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subtracting this from the equation ma+rb = 0, we conclude that 2ma = z has order 2, so ma has order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Thus, the Cayley graph is listed in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume ϕ(b) ∈ {±a + z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume, without loss of generality, that ϕ(b) = a + z (by composing with the automorphism x �→ −x, if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We have ϕ(a) = ǫb + z, for some ǫ ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that |b| = |ϕ(b)| = |a + z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, either |b| = 2|a| (and |a| is odd) or |b| = |a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, gcd � |b|, r � ∈ {m, 2m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If m is even, this implies that r is also even, which contradicts the fact that m + r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore m is odd, so r is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, we must have gcd � |b|, r � = 2m, so we may write |b| = 2km and r = 2ℓm, for some k, ℓ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We have 0 = ϕ(ma + rb) (ma + rb = 0) = m ϕ(a) + r ϕ(b) (ϕ is a group automorphism) = m (ǫb + z) + r (a + z) = ra + ǫmb + z (|z| = 2 and m + r is odd) = 2ℓma + ǫmb + z (definition of ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We also have 2ℓma + 4ℓ2mb = 2ℓ(ma + rb) = 2ℓ(0) = 0, so, by subtracting these two equations, we conclude that (4ℓ2 − ǫ)mb = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |b| = 2km, and |z| = 2, then 4ℓ2 ≡ ǫ (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Thus, the Cayley graph is listed in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The remaining cases are copied almost verbatim from the analogous cases in [7, proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume 2s ̸= 2t, for all s, t ∈ S, such that s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='12, there is an automorphism α of BX that fixes (0, 0), but does not fix (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume α is not a group automorphism, for otherwise Case 4 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='17 implies there exist s, t, u, v ∈ S such that s + t = u + v ̸= 0 and {s, t} ̸= {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' From the assumption of this case, we see that this implies 3b = ±a (perhaps after interchanging a with b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then G = ⟨a, b⟩ = ⟨3b, b⟩ = ⟨b⟩ is cyclic, so Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The remaining case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since Case 5 does not apply, we have 2s = 2t, for some s, t ∈ S, such that s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume that t = −s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then |s| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, if we assume, without loss of generality, that s = a, then we have i := |⟨a⟩ ∩ ⟨b⟩| ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In all cases, we will show that G is cyclic, so Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 13 If i = 1, then G = ⟨a⟩ × ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X is not bipartite, this implies |b| is odd, so gcd � |a|, |b| � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore G is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If i = 2, then there is some k ∈ Z, such that a2 = bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X is not bipartite, we know k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So ⟨a2⟩ has odd index in ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies that ⟨a⟩ has odd index in G and is therefore a Sylow 2-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So the Sylow 2-subgroup of G is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' All of the other Sylow subgroups of G are contained in ⟨b⟩, and are therefore also cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So G is an abelian group whose Sylow subgroups are cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore G is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If i = 4, then a ∈ ⟨b⟩, so G = ⟨a, b⟩ = ⟨b⟩ is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Subcase 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume that t ̸= −s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we may assume s = a and t = b, so 2a = 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If we let z = b − a, this implies that 2z = 0, so z = −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then a = b + z and b = a − z = a + z, so S = S +z, which contradicts the assumption that Cay(G : S) is nontrivially unstable (and therefore has no twin vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Some unstable abelian Cayley graphs of valency 6 In this section, we prove the following theorem, which implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Although the title of this section specifies “valency 6,” the theorem also applies to some graphs of smaller valency, because a, b, and/or c may have order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1) Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let {a, b, c} be a generating set of a finite abelian group G, such that a + b + c = 0 and the sets {±a}, {±b}, {±c} are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The Cayley graph X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b, ±c) is unstable if and only if one of the following conditions is satisfied (perhaps after permuting a, b, and c): (1) |a| = 4 and |G| is divisible by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) 2a = 2b and |G| is divisible by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) |a| = 8 and b = 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) |a| = 12 and b = 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (5) |a| = |b| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Before proving this theorem, let us show that it implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' As in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We will show that the graphs in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6 are precisely those that arise from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 (and are not the two trivially unstable graphs in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8(d)), and satisfy the additional assumption that m, n ≥ 2 (where n = |b| and m = |G : ⟨b⟩|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' To do this, we consider each part of the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We also consider appropriate permutations of a, b, and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This is made easier by the observation that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 determines the result of interchanging a with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1) Since |a| = 4, we must have m ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |G| is divisible by 8, but |a| is not, we know that n and r are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If m = 4, then r = 0 (because ma = 4a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This yields the graph Tr(4, 2k, 0) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4, this is isomorphic to Tr(4, 2k, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By using a in the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3, we obtain Tr(2k, 4, 0), which is also listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If k is even, then applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 does not give anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, if k is odd, then this graph is isomorphic to Tr(2k, 4, 2), as mentioned at the end of the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If m = 2, then r = n/2 (because ma = 2a has order 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since r is even, this yields the graph Tr(2, 4k, 2k) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4, this is isomorphic to Tr(2, 4k, 2 − 2k) = Tr(2, 4k, 2k + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By using a in the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3, we obtain Tr(2k, 4, 2), which is also listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If k is even, then, as above, applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 does not give anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, if k is odd, then this graph is isomorphic to Tr(2k, 4, 0), as mentioned at the end of the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If m = 1, then we cannot directly apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3, because the definition of Tr(m, n, r) requires m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, we may use a in the role of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This yields Tr(2k, 4, r), and we have r ∈ {±1} because ⟨b⟩ = G (since m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, the graph is listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If k is even, then the two graphs are isomorphic (by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4), as mentioned at the end of the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The graph Tr(2, 4, 1) is trivially unstable (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='8), so we require k > 1 in Tr(2k, 4, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) Write |G| = 8k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 14 AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 15 Suppose, for the moment, that ⟨b⟩ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since 2a = 2b, but a ̸= b, we must have a = (4k + 1)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then a and b both generate G, so neither can play the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, we may let c play this role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that c = −(a + b) = −(4k + 2)b = (4k − 2)b, so |G : ⟨c⟩| = 2 and 2a = 2b = (2k − 1)(4k − 2)b = (2k − 1)c, so this yields the graph Tr � 2, 4k, −(2k − 1) � = Tr(2, 4k, 2k + 1), which is listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 to this graph does not yield anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may now assume ⟨b⟩ ̸= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then, since 2a = 2b, we have m = r = 2 and n = |G|/m = 8k/2 = 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So this yields the graph Tr(2, 4k, −2) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4, this is isomorphic to Tr(2, 4k, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that 4a = 2a + 2a = 2a + 2b = 2(a + b) = −2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, by using c in the role of b, we obtain the graph Tr(4, 2k, 2) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 to this graph does not give us anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) Since ⟨a⟩ = ⟨b⟩ = G, neither a nor b can play the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Letting c play the role of b yields the graph Tr(4, 2, 1) (because c = −(a + b) = −4a has order 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This is listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 to this graph does not yield anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) Since G = ⟨a⟩, we have |G| = |a| = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then n = |b| = 3 and m = |G|/|b| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, ma = 4a = b, so r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we have the graph Tr(4, 3, −1) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4 to this graph does not yield anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since ⟨a⟩ = ⟨c⟩ = G, neither a nor c can play the role of b in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (5) This graph is trivially unstable (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ To shorten the main argument, we establish three minor results that deal with parts of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let {a, b, c} be a generating set of a finite abelian group G, such that a + b + c = 0 and the sets {±a}, {±b}, {±c} are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The Cayley graph X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±a, ±b, ±c) has twin vertices (or, equivalently, is trivially unstable) if and only if (perhaps after permuting a, b, and c) either: (1) |a| = 8 and b = 2a, or (2) |a| = |b| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In each case, the Cayley graph is listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇐) Up to a group isomorphism, X is either Cay(Z8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S8) or Cay(Z3 × Z3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S3), where S8 = {±1, ±2, ±3} and S3 = {±(1, 0), ±(0, 1), ±(1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We have S8 = S8 + 4 and S3 = S3 + (−1, 1), so both Cayley graphs have twin vertices, and are therefore trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since 2 is an element of order 4 in Z8, the first Cayley graph is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The second is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇒) Let S = {±a, ±b, ±c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since X has twin vertices, we see from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='13 that S is a union of cosets of some subgroup ⟨z⟩ of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |S| ≤ 6, we must have |z| ∈ {2, 3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |z| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then −s + z = −(s + z) for all s ∈ S, so the permutation x �→ x + z induces a well-defined action on � {±a}, {±b}, {±c} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since the permutation has order 2, this implies s + z = −s for some s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume without loss of generality that s = a, so |a| = 4 and z = 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If b + z = −b, then we also have c + z = −c, so 0 = −(a + b + c) = (a + z) + (b + z) + (c + z) = (a + b + c) + z = 0 + z = z, which contradicts the fact that |z| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 16 DAVE WITTE MORRIS Therefore, we must have b + z = ±c, so a + b ± (b + z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For the minus sign, we have 0 = a + b − (b + z) = a + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This contradicts the fact that |a| = 4 ̸= |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For the plus sign, we have 0 = a + 2b + z = −a + 2b, so a = 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |a| = 4, this implies |b| = 8, so X is the Cayley graph in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |z| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies |S| = 6, so no element of S has order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, if C is any coset of ⟨z⟩, then C ̸= −C (since |C| is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since the cosets form a partition, we conclude that C ∩ −C = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, we may assume that C contains a, b, and ±c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then 0 = a + (a + z) ± (a + 2z), so either 3a = 0 or a = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, a ̸= z, since 0 /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore 3a = 0, which means |a| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since z also has order 3, X is the Cayley graph in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |z| = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then |S| = 5, so some element s of S has order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since G = ⟨S⟩ = ⟨s, z⟩, this implies |G| = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' More precisely, up to a group isomorphism, we have X = Cay(Z10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, ±3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, it is not possible to choose representatives of {±1}, {±3}, and {5} whose sum is 0, so this case is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' All of the Cayley graphs listed in the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1 are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1) (Wilson [16, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2 and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 381]) Let z = 2a, and define ϕ: G → G by ϕ(pa+qb) = pa + q(b + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |a| = 4, we have |z| = 2, so ϕ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Since |G| is divisible by 8, but |a| is not, and ⟨a, b⟩ = G, we know that |⟨b⟩ : ⟨a⟩ ∩ ⟨b⟩| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, if p1a + q1b = p2a + q2b, then q1 ≡ q2 (mod 2), so p1a + q1(b + z) = p2a + q2(b + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') Then it is easy to see that ϕ is an automorphism of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, we have ϕ(a) = a = −3a = −a + 2a = −a + z, so ϕ(S) = S + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, ϕ is an isomorphism from Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) to Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) (Wilson [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 381]) There is an automorphism ϕ of G that interchanges a and b (and fixes c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then ϕ(S) = S, so ϕ is an automorphism of Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also note that ϕ fixes each element of the index-2 subgroup ⟨2a, c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If ⟨a⟩ ̸= G, then the subgraph induced by the set of un-fixed vertices consists of two cycles of length |c|, and these two cycles are interchanged by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |c| = |a|/2 = |G|/4 is even, these cycles are bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If ⟨a⟩ = G, then (since |G| is divisible by 8) we may assume G = Z8k and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since 2a = 2b, we know that b − a has order 2, and is therefore equal to 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence S = {±1, 4k ± 1, 4k ± 2}, so we see that X is unstable by letting a = 4k + 2 and b = 1 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) We have X ∼= Cay(Z8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, ±3, 4), which is unstable by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4) Since X ∼= Cay(Z12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, ±4, ±5) = Cay(Z12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±1, ±4, ±7), we see that it is unstable from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(4) with k = 3 and a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (5) X is trivially unstable by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4) Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let S = {±1, ±b, ±(b + 1)}, for some b ∈ Zn, such that b /∈ {0, ±1, −2} (so the sets {±1}, {±b}, {±(b + 1)} are distinct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If Cay(Zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is unstable, then it is listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let X = Cay(Zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume X is nontrivially unstable, for otherwise Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, since cyclic groups have no more than one element of order 2, we know that the AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 17 valency of X is either 5 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, the Cayley graph X is listed in either Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10 or Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume X is listed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10(1), one generator is odd, but the other two generators are even, so a + b + c ̸≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, the equation a + b + c = 0 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So X must the graph in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='10(2), which is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume X is listed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We consider each of the seven lists of graphs individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(1): The element 2k has order 4, so X is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(2): The generator ±a is odd, but ±b and ±b + 2k are even, so the equation a + b + c = 0 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(3): For some choice of the signs, we must have ±a±(a+k)±(a−k) = 0, so 3a ≡ 0 (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since k is odd and we also have a ≡ 0 (mod 4), this implies 3a = 0 in Z4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' But since 1 is in the generating set, we must also have a − k = ±1 (perhaps after replacing a with −a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So a ∈ {k ± 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore 3k ± 3 = 3a ≡ 0 (mod 4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' From this (and the fact that a ̸= 0), we conclude that k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since k ± 1 = a ≡ 0 (mod 4), we see that a = 4, so X is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(4): For some c ∈ {4k ±b}, we have ±a+b+c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, if c = 4k −b, then this implies a = 4k, which contradicts the fact that |a| is divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we must have c = 4k + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then 2b = 2c, so this Cayley graph is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(5): The condition a + b + c = 0 implies a ≡ 0 (mod k), so all of the elements of S are divisible by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since 1 ∈ S, we conclude that k = 1, so n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since a ≡ 0 (mod 4), this implies a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So X is the Cayley graph in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (Alternatively, we have |a| = 2, which contradicts the fact that the valency of X is assumed to be 6 in the current case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(6): Since m is odd, we know that b and mb+2k have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, the equation a + b + c = 0 implies that a is even, so a ̸= ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we may assume 1 ∈ {b, mb + 2k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since the assumptions imply m2b ≡ ±b (and m is odd), we have ±b = m(mb + 2k) + 2k, so there is no harm in assuming b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since m2 ≡ 1 (mod 4), we know that m2 + 1 ̸≡ 0 (mod 4), so (m2 + 1)b = m2 + 1 ̸≡ 0 (mod 4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we must have m2 ≡ 1 (mod 4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For an appropriate choice of the sign (and perhaps replacing a with its negative), we have −a ± b + (mb + 2k) = 0, so a = mb + 2k ± b = m + 2k ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, (m − 1)(m + 2k + 1) = m2 − 1 + (m − 1)(2k) ≡ 1 − 1 + 0 = 0 ̸≡ 2k ≡ (m − 1)a (mod 4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So a ̸= m + 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, we must have a = m + 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then, modulo 4k, we have 2k ≡ (m − 1)a = (m − 1)(m + 2k − 1) = m2 − 2m + 1 + (m − 1)(2k) ≡ 1 − 2m + 1 + 0 = −2(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This means m ≡ k + 1 (mod 2k), so m is either k + 1 or −k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since m is odd, this implies that k is even, so |Z4k| is divisible by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It also implies that a is either 3k or k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In either case, a has order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So X is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(7): If a = 1, then m = 4k + 1, so b and c must be even, which contradicts the equation a + b + c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Thus, we may assume b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then m = 4k − 1, so a ≡ 2k (mod 4k), which means a ∈ {±2k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore a has order 4, so the Cayley graph is listed in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ As final preparation for proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1, let us recall some useful notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 18 DAVE WITTE MORRIS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5) Notation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' [6, Defn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3], [9, Notn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) is an abelian Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (1) For g ∈ G, let �g = (g, 1), so BX = Cay(G × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' �S ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (2) For s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , sℓ ∈ S, and a starting point v ∈ V (BX), we use ( �s1, �s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , �sℓ) to denote the walk v, v + �s1, v + �s1 + �s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , v + �s1 + �s2 + · · · + �sℓ in BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (3) ( �s1 m1, �s2 m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , �sℓ mℓ) denotes the sequence consisting of m1 copies of �s1, followed by m2 copies of �s2, followed by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' , followed by mℓ copies of �sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇐) See Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (⇒) Let S = {±a, ±b, ±c}, so X = Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assuming that X is unstable, we will show that (at least) one of the listed conditions is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that, since the equation a + b + c = 0 is completely symmetric, we are free to permute the elements of {a, b, c} arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='12, we may let α be an automorphism of BX that fixes (0, 0), but does not fix (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since G × {0} and G × {1} are the bipartition sets of BX, we know that each of these sets is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |G| is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='7, we know that X is trivially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume G = ⟨s⟩ for some s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' See Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |c| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume Case 2 does not apply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' therefore c /∈ ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If we let n = |a|, then X ∼= Cay � Zn × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(1, 1), (0, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n ≥ 8, then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='16 tells us that X is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n ≤ 7, and n ̸= 4, then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='22 tells us that X is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If n = 4, then |a| = 4 and |G| = 8, so condition (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assumption C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Henceforth, we assume |s| ≥ 3 for all s ∈ S (so X has valency 6), and |G| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume |s| = 4, for some s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume |G| = 4k, where k is odd, for otherwise condition (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then G is cyclic (since |a| = 4 and G/⟨a⟩ is generated by b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may also assume that X is nontrivially unstable, for otherwise Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then X must be listed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' More precisely, since |G| is not divisible by 8, it must be listed in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(2), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(3), or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We will look at each of these possibilities separately (similarly to how these cases were considered in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(2): Since the generator ±a is odd, but the other two are even, the equation a + b + c = 0 cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(3): Since S must contain the element k of order 4, but a ≡ 0 (mod 4), we must have a + k = k or a + k = −k or a − k = k or a − k = −k, so a ∈ {0, 2k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, we know a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' And a ̸= 2k, because 2k ̸≡ 0 (mod 4) (or because we are assuming that no element of S has order 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='11(6): Since m is odd, we know that b and mb + 2k have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, the equation a + b + c = 0 implies that a is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So a ̸= ±k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then {±b, ±mb + 2k} contains k, and is therefore contained in ⟨k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This is impossible, because ⟨k⟩ does not contain 4 distinct nonzero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume α is a group automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then α(g, 0) = � ϕ(g), 0 � , for some automorphism ϕ of G, and α(0, 1) = (z, 1), for some element z of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' AUTOMORPHISMS OF THE CANONICAL DOUBLE COVER OF A TOROIDAL GRID 19 Then, since α � S × {1} � = S × {1}, we have ϕ(S) = S + z, so ϕ is an isomorphism from Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S) to Cay(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' S + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We have ϕ(s) ∈ ±{a, b, c} + z for all s ∈ {a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' At least two elements of S must use the same sign, which implies there exists ǫ ∈ {0, 1}, such that ϕ(S) contains at least two elements of {ǫa + z, ǫb + z, ǫc + z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume ǫ = 1 (by composing with the automorphism x �→ −x if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then, by permuting {a, b, c}, we may assume that a + z and b + z are in ϕ({a, b, c}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore 0 = ϕ(a + b + c) = ϕ(a) + ϕ(b) + ϕ(c) = (a + z) + (b + z) ± (c + z) = (a + b) ± c + z = −c ± c + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since z ̸= 0, we conclude that 2c = z, so c has order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, Case 4 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Assume 2s ̸= 2t, for all s, t ∈ S, such that s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume α is not a group automorphism, for otherwise Case 5 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='17 implies there exist s, t, u, v ∈ S such that s + t = u + v ̸= 0 and {s, t} ̸= {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' From the assumption of this case, we see that this implies either 3a = ±c or 2a = ±b ± c (perhaps after permuting a, b, and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If 3a = ±c, then c = ±3a ∈ ⟨a⟩, so G = ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If 2a = ±b − c = ±b + (a + b), then a = ±b + b ∈ ⟨b⟩, so G = ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So we may assume 2a = ±b + c = ±b − (a + b) ∈ {−a, −a − 2b}, so 3a ∈ {0, −2b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' If 3a = −2b, then a = −2a − 2b = −2(a + b) = 2c ∈ ⟨c⟩, so G = ⟨c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, we may assume 3a = 0, so |a| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may also assume a /∈ ⟨b⟩, for otherwise Case 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So ⟨a⟩ ∩ ⟨b⟩ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This implies G = ⟨a⟩ × ⟨b⟩, so, letting n = |b|, we have X ∼= Cay � Z3 × Zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(0, 1), ±(1, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that: (�a3) and (�a−3) are paths of length 3 in BX from (0, 0) to �0 = (0, 1), (�b3) and (�c−3) are paths of length 3 in BX from (0, 0) to 3�b, and (�b−3) and (�c3) are paths of length 3 in BX from (0, 0) to 3�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also note that if s1, s2, and s3 are not all equal to each other, then the number of permutations of the list ( �s1, �s2, �s3) is divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, �0, 3�b, and 3�c are the only vertices of BX for which the number of paths of length 3 from (0, 0) to the vertex is not divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, the set {�0, 3�b, 3�c} is α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume |b| > 12, for otherwise Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='24 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Then the number of paths of length 6 from 3�c = −3�b to 3�b is �6 3 � + 2: �6 3 � permutations of the path (�b3, �c−3), and the paths (�b6) and (�c−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This is much smaller than the number of paths of length 6 from �0 to either 3�b or 3�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' For example, paths from �0 to 3�b include: �6 3 � permutations of the path (�a3,�b3), �6 3 � permutations of the path (�a3, �c−3), and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' 20 DAVE WITTE MORRIS Thus, the vertex (0, 1) = �0 is uniquely determined as an element of the α-invariant set {�0, 3�b, 3�c}, so it must be fixed by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This contradicts the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Case 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' The remaining case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since Case 6 does not apply, we have 2s = 2t, for some s, t ∈ S, such that s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since Case 4 does not apply, we know s ̸= −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we may assume s = a and t ∈ {±b}, so 2a ∈ {±2b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, if 2a = −2b, then 2c = −2(a + b) = −(2a + 2b) = −(−2b + 2b) = −0 = 0, so Case 3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Thus, we must have 2a = 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So b = a + z for some element z of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume G ̸= ⟨a⟩ and G ̸= ⟨b⟩ (otherwise, Case 2 applies), so b /∈ ⟨a⟩ and a /∈ ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence z /∈ ⟨a⟩ and z /∈ ⟨b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' So X ∼= Cay � Zn × Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' ±(1, 0), ±(1, 1), ±(2, 1) � , where n = |a| = |b| is even and, up to a group isomorphism, a = (1, 0), b = (1, 1), and c = −(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' We may assume X does not satisfy condition (2), so n/2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Also, since Case 4 (and Case 3) does not apply, we have n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' In addition, since Case 6 does not apply, we know 2a ̸= −2c, so n ̸= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore (since 8/2 is not odd), we have n > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Note that, for every v ∈ V (BX), the fact that 2a = 2b implies that v + 2 �c and v − 2 �c are the only vertices of BX that are joined to v by a unique path of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' It follows from this that α(v + k�c) = α(v) ± k�c, for all k ∈ Z and all v ∈ V (BX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6) Since α fixes 0, we conclude that {±k �c} is α-invariant, for all k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Let Y be the spanning subgraph of BX that is obtained by removing all edges of the form (v, v + �c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6), we know that Y is α-invariant, so α is an automorphism of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Now, � (0, 0) and � (0, 1) are the only vertices that are at distance 2 from both �c and −�c in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since {±�c} is α-invariant, this implies that � � (0, 0), � (0, 1) � is also α-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Since |�c| = n, we have (n/2)�c = −(n/2)�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, we see from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='6) that (n/2)�c is fixed by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' However, since n/2 is odd, we have � (0, 1) = n 2 � (2, 1) = n 2 �c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Hence, � (0, 1) is fixed by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Therefore, since � (0, 0) is the only other element of the invariant set � � (0, 0), � (0, 1) � , we conclude that � (0, 0) is also fixed by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' This contradicts the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' □ References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Baik, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content=' Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfCg6t/content/2301.05396v1.pdf'} +page_content='-S.' 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index 0000000000000000000000000000000000000000..4f687114c66aa5b82cf5eed5fabeaa213d1be5d5 --- /dev/null +++ b/_dE4T4oBgHgl3EQfEQsW/content/tmp_files/2301.04875v1.pdf.txt @@ -0,0 +1,362 @@ +COLOR-NEURACRYPT: PRIVACY-PRESERVING COLOR-IMAGE CLASSIFICATION +USING EXTENDED RANDOM NEURAL NETWORKS +Zheng Qi, AprilPyone MaungMaung and Hitoshi Kiya +Tokyo Metropolitan University +6-6, Asahigaoka, Hino-shi, Tokyo, 191–0065, Japan +Phone/FAX:+81-042-585-8454 +E-mail: {qi-zheng@ed., apmaung@, kiya@}tmu.ac.jp +Abstract +In recent years, with the development of cloud computing +platforms, privacy-preserving methods for deep learning have +become an urgent problem. NeuraCrypt is a private random +neural network for privacy-preserving that allows data own- +ers to encrypt the medical data before the data uploading, and +data owners can train and then test their models in a cloud +server with the encrypted data directly. However, we point +out that the performance of NeuraCrypt is heavily degraded +when using color images. In this paper, we propose a Color- +NeuraCrypt to solve this problem. Experiment results show +that our proposed Color-NeuraCrypt can achieve a better clas- +sification accuracy than the original one and other privacy- +preserving methods. +1. INTRODUCTION +In recent years, the spread of deep neural networks (DNNs) [1] +has greatly contributed to solving complex tasks for many ap- +plications, and it has been very popular for data owners to +train DNNs on large amounts of data in cloud servers. How- +ever, data privacy such as personal medical records, may be +compromised in that process, because a third party can access +the uploaded data illegally, so it is necessary to protect data +privacy in cloud environments, and privacy-preserving meth- +ods for deep learning have become an urgent challange [2]. +One of the most efficient solutions is to encrypt data before +the data uploading, so that data owners can train and then +test their DNNs in a cloud server with the encrypted data +directly [3–6]. +NeuraCrypt [7] is a private random neural network that al- +lows us to encrypt data before uploading. Vision Transforma- +tion (ViT) [8] models have been demonstrated to maintain a +high classification performance for medical images (with one +channel) under the use of NeuraCrypt, but we point out that +the performance of NeuraCrypt is heavily degraded when us- +ing color images. In this paper, we extend NeuraCrypt from +one channel to three channels, called Color-NeuraCrypt, to +Client +User +Cloud Server +(Untrusted) +DNN +Data storage +Training Set +Testing +Random Neural Network +Trained +Testing Set +Random Neural Network +DNN +Training +Fig. 1: Framework of proposed method. +avoid performance degradation. Experiment results show that +our proposed Color-NeuraCrypt achieved a better classifica- +tion accuracy than the original one and outperformed other +privacy-preserving methods on the CIFAR-10 dataset. +2. RELATED WORK +Lightweight privacy-preserving methods, called learnable +encryption, have almost the same usage scenario as the ran- +dom neural network. +Generally, Privacy-preserving image +classification methods have to satisfy two requirements: high +classification accuracy and strong robustness against various +attacks. Tanaka first introduced a block-wise learnable image +encryption (LE) method with an adaptation layer [9], which +is used prior to a classifier to reduce the influence of im- +age encryption. Another encryption method is a pixel-wise +encryption (PE) method in which negative-positive transfor- +mation and color component shuffling are applied without +using any adaptation layer [10]. However, both encryption +methods are not robust enough against ciphertext-only at- +tacks as in [11]. To enhance the security of encryption, LE +was extended by adding a block scrambling step and a pixel +encryption operation with multiple keys (hereinafter denoted +as ELE) [12]. However, ELE still has a lower accuracy than +arXiv:2301.04875v1 [cs.CR] 12 Jan 2023 + +(a) +(b) +Fig. 2: Architecture of two random neural networks (a) Neu- +raCrypt (b) Color-NeuraCrypt (proposed) +that of using plain images. Recently, block-wise learnable +encryption methods with an isotropic network have been pro- +posed to reduce the influence of image encryption [13, 14]. +Meanwhile, NeuraCrypt was proposed with ViT and achieved +a good performance on grayscale medical images, but its per- +formance degraded heavily for color images. In addition, it +cannot be directly applied to a standard pre-trained ViT. Ac- +cordingly, we propose a novel random neural network called +Color-NeuraCypt to improve these issues that the conven- +tional methods have. +3. PROPOSED METHOD +3.1. Overview +Figure 1 depicts the framework of the proposed scheme. A +user encrypts training images by using a random neural net- +work and sending the encrypted images to a cloud provider. +Next, the cloud provider trains a ViT model with the uploaded +encrypted images without perceiving any visual information. +After training, the user also encrypts the testing images us- +ing the same random neural network as in training, and sends +that to the cloud server. Data privacy can be protected in both +training and testing processes in this framework. +3.2. Color-NeuraCrypt +NeuraCrypt is a randomly constructed neural network to en- +code input data [7] as shown in Fig. 2a. It consists of patch +embedding, several blocks of a 1 × 1 convolutional layer, po- +sition embedding, and linear projection. It can achieve a high +classification accuracy for grayscale medical images, but its +performance significantly drops for color images (see Sec- +tion 4). +To avoid performance degradation for color images, +we propose a novel random neural network called Color- +NeuraCrypt. +Figure 2b shows the architecture of Color- +NeuraCrypt. There are two major differences between the +two random neural networks: +• The output of NeuraCrypt is a patch representation. In +contrast, the output of Color-NeuraCrypt is an image +because we add a pixel shuffling layer at the end of +the Color-NeuraCrypt to reshape a patch representa- +tion. Figure 3 shows an example of plain and encrypted +images. +• NeuraCrypt randomly permutes patches at the output +independently for each image in patch shuffling. +In +contrast, to align with a standard ViT, we remove the +patch shuffling step but still retain the random position +embedding to hide the spatial information of plain im- +ages. +Furthermore, we utilize a standard pre-trained ViT, which +has trainable patch embedding and position embedding. We +fine-tune ViT with encrypted images for training and testing. +(a) Plain +(b) Encrypted +Fig. 3: Example of plain and encrypted images. +4. EXPERIMENTS +We conducted image classification experiments on the MNIST [17] +and CIFAR-10 [18] datasets. The MNIST dataset consists of +70,000 grayscale images (dimension of 1 × 28 × 28) of hand- +written digits with ten classes, where 60,000 images are for +training and 10,000 for testing. The CIFAR-10 dataset con- +sists of 60,000 color images (dimension of 1 × 28 × 28), +where 50,000 images are for training and 10,000 for testing. + +NeuraCrypt +Classifier +Input +Patch Embedding +Label +× depth +个 +1×1, Conv +Activation +MLP +不 +Activation +Linear Projection +Transformer Encoder +BatchNorm +Patch Shuffling +....... +Position EmbeddingColor-NeuraCrypt +Classifier +Input +Label +depth +Standard ViT +Patch Embedding +1x1, Conv +MLP +Activation +个 +个 +Activation +Transformer Encoder +BatchNorm +个 +BatchNorm ++ +Position Embedding +Position Embedding +Pixel Shuffle Layer +Patch EmbeddingTable 1: Comparison of accuracy (%) of image classification. EtC and ELE are block-wise learnable image encryption methods +that use a Shakedrop network as a classifier [12,15], and an adaptation network (denoted as AdaptNet) is applied to reduce the +influence of image encryption. +Encryption +Classifier +Image size +Accuracy +MINST +CIFAR-10 +EtC [12,16] +AdaptNet+ShakeDrop +32 × 32 +– +89.09 +ELE [12] +AdaptNet+ShakeDrop +32 × 32 +– +83.06 +NeuraCrypt [7] +ViT-B 16 +224 × 224 +97.93 +70.60 +Color-NeuraCrypt(proposed) +ViT-B 16 +224 × 224 +– +96.20 +Plain +ShakeDrop +32 × 32 +– +96.70 +Plain +ViT-B 16 +224 × 224 +99.71 +99.10 +We used a PyTorch implementation of ViT1 and fine- +tuned the ViT-B 16 model which was pre-trained with the +ImageNet21k dataset. To maximize the classification perfor- +mance, we followed the training settings from [8] except for +the learning rate. The parameters of the stochastic gradient +descent (SGD) optimizer for encrypted images that we used +were: a momentum of 0.9, a weight decay of 0.0005, and +a learning rate value of 0.03-0.1. In addition, the depth of +NeuraCrypt and Color-NeuraCrypt was set to 4. +As shown in Table 1, ViT models with NeuraCrypt per- +formed with 97.93% accuracy on the MNIST dataset, which +was similar to medical images. However, it achieved unsat- +isfactory accuracy on the CIFAR-10 dataset. These results +confirmed that NeuraCrypt is very effective on grayscale +images but difficult to be applied to color images. Table 1 +also shows the classification performance of other privacy- +preserving methods. +Our Color-NeuraCrypt outperformed +not only NeuraCrypt but also the two block-wise encryption +methods (ELE and EtC [16]) on the CIFAR-10 dataset, so the +proposed method was confirmed to be more suitable for color +images. +5. CONCLUSION AND FUTURE WORK +In this research, we proposed a random neural network, called +Color-NeuraCrypt for privacy-preserving. Color images en- +crypted by Color-NeuraCrypt can be applied to ViT mod- +els for both training and testing directly. Experiment results +showed that our Color-NeuraCrypt achieved a better accu- +racy than NeuraCrypt and other privacy-preserving methods +on color images. +As a random neural network is considered as an encryp- +tion method for privacy-preserving, its security needs to be +evaluated. For example, we can correctly match a plain and +encrypted sample using the algorithm in [19]. Furthermore, +a random neural network can hide the visual information of +plain images, but it is hard to secrete some transparent in- +1https://github.com/jeonsworld/ViT-pytorch +formation, such as the distribution of the dataset and the en- +cryption scheme. An attacker may perform a ciphertext-only +attack via that information to reconstruct visual information +from encrypted images. +Acknowledgment +This study was partially supported by JSPS KAKENHI +(Grant Number JP21H01327). +6. REFERENCES +[1] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” +nature, vol. 521, no. 7553, pp. 436–444, 2015. +[2] H. Kiya, M. AprilPyone, Y. Kinoshita, S. Imaizumi, and +S. Shiota, “An overview of compressible and learnable +image transformation with secret key and its applica- +tions,” APSIPA Transactions on Signal and Information +Processing, vol. 11, no. 1, e11, 2022. +[3] H. Kiya, R. Iijima, A. Maungmaung, and Y. Kinoshita, +“Image and model transformation with secret key for vi- +sion transformer,” IEICE TRANSACTIONS on Informa- +tion and Systems, vol. 106, no. 1, pp. 2–11, 2023. +[4] H. Kiya, T. Nagamori, S. Imaizumi, and S. 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Heigold, S. Gelly, J. Uszkoreit, and +N. Houlsby, “An image is worth 16x16 words: Trans- +formers for image recognition at scale,” in Proceedings +of International Conference on Learning Representa- +tions (ICLR), 2021. +[9] M. Tanaka, “Learnable image encryption,” in Pro- +ceddings of International Conference on Consumer +Electronics-Taiwan (ICCE-TW), 2018, pp. 1–2. +[10] W. Sirichotedumrong, Y. Kinoshita, and H. Kiya, +“Pixel-based image encryption without key manage- +ment for privacy-preserving deep neural networks,” +IEEE Access, vol. 7, pp. 177 844–177 855, 2019. +[11] H. Ito, Y. Kinoshita, M. AprilPyone, and H. Kiya, “Im- +age to perturbation: An image transformation network +for generating visually protected images for privacy- +preserving deep neural networks,” IEEE Access, vol. 9, +pp. 64 629–64 638, 2021. +[12] K. Madono, M. Tanaka, M. Onishi, and T. Ogawa, +“Block-wise scrambled image recognition using adapta- +tion network,” in Proceedings of Workshop on Artificial +Intelligence of Things (AAAI-WS), 2020. +[13] Z. Qi, A. MaungMaung, Y. Kinoshita, and H. Kiya, +“Privacy-preserving image classification using vision +transformer,” in Proceedings of 2022 30th European +Signal Processing Conference (EUSIPCO), 2022, pp. +543–547. +[14] M. AprilPyone and H. Kiya, “Privacy-preserving image +classification using an isotropic network,” IEEE Multi- +Media, vol. 29, no. 2, pp. 23–33, 2022. +[15] Y. Yamada, M. Iwamura, T. Akiba, and K. Kise, “Shake- +drop regularization for deep residual learning,” IEEE +Access, vol. 7, pp. 186 126–186 136, 2019. +[16] T. Chuman, +W. Sirichotedumrong, +and H. Kiya, +“Encryption-then-compression +systems +using +grayscale-based image encryption for jpeg images,” +IEEE Transactions on Information Forensics and +security, vol. 14, no. 6, pp. 1515–1525, 2018. +[17] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, +“Gradient-based learning applied to document recog- +nition,” Proceedings of the IEEE, vol. 86, no. 11, pp. +2278–2324, 1998. +[18] A. Krizhevsky and G. Hinton, “Learning multiple lay- +ers of features from tiny images,” University of Toronto, +Tech. Rep., 2009. +[19] N. Carlini, S. Garg, S. Jha, S. Mahloujifar, M. Mah- +moody, and F. Tramer, “Neuracrypt is not private,” 2021. +[Online]. Available: https://arxiv.org/abs/2108.07256 + diff --git a/_dE4T4oBgHgl3EQfEQsW/content/tmp_files/load_file.txt b/_dE4T4oBgHgl3EQfEQsW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..87c6d82c3167b40e1ccce2d2aed24fbd6c56be30 --- /dev/null +++ b/_dE4T4oBgHgl3EQfEQsW/content/tmp_files/load_file.txt @@ -0,0 +1,246 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf,len=245 +page_content='COLOR-NEURACRYPT: PRIVACY-PRESERVING COLOR-IMAGE CLASSIFICATION USING EXTENDED RANDOM NEURAL NETWORKS Zheng Qi, AprilPyone MaungMaung and Hitoshi Kiya Tokyo Metropolitan University 6-6, Asahigaoka, Hino-shi, Tokyo, 191–0065, Japan Phone/FAX:+81-042-585-8454 E-mail: {qi-zheng@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=', apmaung@, kiya@}tmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='jp Abstract In recent years, with the development of cloud computing platforms, privacy-preserving methods for deep learning have become an urgent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' NeuraCrypt is a private random neural network for privacy-preserving that allows data own- ers to encrypt the medical data before the data uploading, and data owners can train and then test their models in a cloud server with the encrypted data directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' However, we point out that the performance of NeuraCrypt is heavily degraded when using color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In this paper, we propose a Color- NeuraCrypt to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Experiment results show that our proposed Color-NeuraCrypt can achieve a better clas- sification accuracy than the original one and other privacy- preserving methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' INTRODUCTION In recent years, the spread of deep neural networks (DNNs) [1] has greatly contributed to solving complex tasks for many ap- plications, and it has been very popular for data owners to train DNNs on large amounts of data in cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' How- ever, data privacy such as personal medical records, may be compromised in that process, because a third party can access the uploaded data illegally, so it is necessary to protect data privacy in cloud environments, and privacy-preserving meth- ods for deep learning have become an urgent challange [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' One of the most efficient solutions is to encrypt data before the data uploading, so that data owners can train and then test their DNNs in a cloud server with the encrypted data directly [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' NeuraCrypt [7] is a private random neural network that al- lows us to encrypt data before uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Vision Transforma- tion (ViT) [8] models have been demonstrated to maintain a high classification performance for medical images (with one channel) under the use of NeuraCrypt, but we point out that the performance of NeuraCrypt is heavily degraded when us- ing color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In this paper, we extend NeuraCrypt from one channel to three channels, called Color-NeuraCrypt, to Client User Cloud Server (Untrusted) DNN Data storage Training Set Testing Random Neural Network Trained Testing Set Random Neural Network DNN Training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 1: Framework of proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' avoid performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Experiment results show that our proposed Color-NeuraCrypt achieved a better classifica- tion accuracy than the original one and outperformed other privacy-preserving methods on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' RELATED WORK Lightweight privacy-preserving methods, called learnable encryption, have almost the same usage scenario as the ran- dom neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Generally, Privacy-preserving image classification methods have to satisfy two requirements: high classification accuracy and strong robustness against various attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Tanaka first introduced a block-wise learnable image encryption (LE) method with an adaptation layer [9], which is used prior to a classifier to reduce the influence of im- age encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Another encryption method is a pixel-wise encryption (PE) method in which negative-positive transfor- mation and color component shuffling are applied without using any adaptation layer [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' However, both encryption methods are not robust enough against ciphertext-only at- tacks as in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' To enhance the security of encryption, LE was extended by adding a block scrambling step and a pixel encryption operation with multiple keys (hereinafter denoted as ELE) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' However, ELE still has a lower accuracy than arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='04875v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='CR] 12 Jan 2023 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 2: Architecture of two random neural networks (a) Neu- raCrypt (b) Color-NeuraCrypt (proposed) that of using plain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Recently, block-wise learnable encryption methods with an isotropic network have been pro- posed to reduce the influence of image encryption [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Meanwhile, NeuraCrypt was proposed with ViT and achieved a good performance on grayscale medical images, but its per- formance degraded heavily for color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In addition, it cannot be directly applied to a standard pre-trained ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Ac- cordingly, we propose a novel random neural network called Color-NeuraCypt to improve these issues that the conven- tional methods have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' PROPOSED METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Overview Figure 1 depicts the framework of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' A user encrypts training images by using a random neural net- work and sending the encrypted images to a cloud provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Next, the cloud provider trains a ViT model with the uploaded encrypted images without perceiving any visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' After training, the user also encrypts the testing images us- ing the same random neural network as in training, and sends that to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Data privacy can be protected in both training and testing processes in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Color-NeuraCrypt NeuraCrypt is a randomly constructed neural network to en- code input data [7] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' It consists of patch embedding, several blocks of a 1 × 1 convolutional layer, po- sition embedding, and linear projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' It can achieve a high classification accuracy for grayscale medical images, but its performance significantly drops for color images (see Sec- tion 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' To avoid performance degradation for color images, we propose a novel random neural network called Color- NeuraCrypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Figure 2b shows the architecture of Color- NeuraCrypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' There are two major differences between the two random neural networks: The output of NeuraCrypt is a patch representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In contrast, the output of Color-NeuraCrypt is an image because we add a pixel shuffling layer at the end of the Color-NeuraCrypt to reshape a patch representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Figure 3 shows an example of plain and encrypted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' NeuraCrypt randomly permutes patches at the output independently for each image in patch shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In contrast, to align with a standard ViT, we remove the patch shuffling step but still retain the random position embedding to hide the spatial information of plain im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Furthermore, we utilize a standard pre-trained ViT, which has trainable patch embedding and position embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' We fine-tune ViT with encrypted images for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' (a) Plain (b) Encrypted Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 3: Example of plain and encrypted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' EXPERIMENTS We conducted image classification experiments on the MNIST [17] and CIFAR-10 [18] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' The MNIST dataset consists of 70,000 grayscale images (dimension of 1 × 28 × 28) of hand- written digits with ten classes, where 60,000 images are for training and 10,000 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' The CIFAR-10 dataset con- sists of 60,000 color images (dimension of 1 × 28 × 28), where 50,000 images are for training and 10,000 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' NeuraCrypt Classifier Input Patch Embedding Label × depth 个 1×1, Conv Activation MLP 不 Activation Linear Projection Transformer Encoder BatchNorm Patch Shuffling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Position EmbeddingColor-NeuraCrypt Classifier Input Label depth Standard ViT Patch Embedding 1x1, Conv MLP Activation 个 个 Activation Transformer Encoder BatchNorm 个 BatchNorm + Position Embedding Position Embedding Pixel Shuffle Layer Patch EmbeddingTable 1: Comparison of accuracy (%) of image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' EtC and ELE are block-wise learnable image encryption methods that use a Shakedrop network as a classifier [12,15], and an adaptation network (denoted as AdaptNet) is applied to reduce the influence of image encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Encryption Classifier Image size Accuracy MINST CIFAR-10 EtC [12,16] AdaptNet+ShakeDrop 32 × 32 – 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='09 ELE [12] AdaptNet+ShakeDrop 32 × 32 – 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='06 NeuraCrypt [7] ViT-B 16 224 × 224 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='93 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='60 Color-NeuraCrypt(proposed) ViT-B 16 224 × 224 – 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='20 Plain ShakeDrop 32 × 32 – 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='70 Plain ViT-B 16 224 × 224 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='71 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='10 We used a PyTorch implementation of ViT1 and fine- tuned the ViT-B 16 model which was pre-trained with the ImageNet21k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' To maximize the classification perfor- mance, we followed the training settings from [8] except for the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' The parameters of the stochastic gradient descent (SGD) optimizer for encrypted images that we used were: a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='9, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='0005, and a learning rate value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='03-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' In addition, the depth of NeuraCrypt and Color-NeuraCrypt was set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' As shown in Table 1, ViT models with NeuraCrypt per- formed with 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='93% accuracy on the MNIST dataset, which was similar to medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' However, it achieved unsat- isfactory accuracy on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' These results confirmed that NeuraCrypt is very effective on grayscale images but difficult to be applied to color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Table 1 also shows the classification performance of other privacy- preserving methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Our Color-NeuraCrypt outperformed not only NeuraCrypt but also the two block-wise encryption methods (ELE and EtC [16]) on the CIFAR-10 dataset, so the proposed method was confirmed to be more suitable for color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this research, we proposed a random neural network, called Color-NeuraCrypt for privacy-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Color images en- crypted by Color-NeuraCrypt can be applied to ViT mod- els for both training and testing directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Experiment results showed that our Color-NeuraCrypt achieved a better accu- racy than NeuraCrypt and other privacy-preserving methods on color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' As a random neural network is considered as an encryp- tion method for privacy-preserving, its security needs to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' For example, we can correctly match a plain and encrypted sample using the algorithm in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' Furthermore, a random neural network can hide the visual information of plain images, but it is hard to secrete some transparent in- 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content='com/jeonsworld/ViT-pytorch formation, such as the distribution of the dataset and the en- cryption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE4T4oBgHgl3EQfEQsW/content/2301.04875v1.pdf'} +page_content=' An attacker may perform a ciphertext-only attack via that 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Modified gravity and massive neutrinos. +Rodrigo Calder´on +,1, 2, ∗ Benjamin L’Huillier +,3, † David Polarski +,2, ‡ +Arman Shafieloo +,1, 4, § and Alexei A. Starobinsky +5, 6, ¶ +1Korea Astronomy and Space Science Institute, Daejeon 34055, South Korea +2Laboratoire Charles Coulomb, Universit´e de Montpellier & CNRS, 34095 Montpellier, France +3Department of Physics and Astronomy, Sejong University, Seoul 05006, South Korea +4University of Science and Technology, Daejeon 34113, South Korea +5L. D. Landau Institute for Theoretical Physics RAS, Chernogolovka, Moscow region 142432, Russia +6Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia +(Dated: January 3, 2023) +Based on a formalism introduced in our previous work, we reconstruct the phenomenological +function Geff(z) describing deviations from General Relativity (GR) in a model-independent manner. +In this alternative approach, we model µ ≡ Geff/G as a Gaussian process and use forecasted growth- +rate measurements from a stage-IV survey to reconstruct its shape for two different toy-models. +We follow a two-step procedure: (i) we first reconstruct the background expansion history from +Supernovae (SNe) and Baryon Acoustic Oscillation (BAO) measurements; (ii) we then use it to +obtain the growth history fσ8, that we fit to redshift-space distortions (RSD) measurements to +reconstruct Geff. We find that upcoming surveys such as the Dark Energy Spectroscopic Instrument +(DESI) might be capable of detecting deviations from GR, provided the dark energy behavior is +accurately determined. We might even be able to constrain the transition redshift from G → Geff for +some particular models. We further assess the impact of massive neutrinos on the reconstructions +of Geff (or µ) assuming the expansion history is given, and only the neutrino mass is free to vary. +Given the tight constraints on the neutrino mass, and for the profiles we considered in this work, we +recover numerically that the effect of such massive neutrinos do not alter our conclusions. Finally, +we stress that incorrectly assuming a ΛCDM expansion history leads to a degraded reconstruction +of µ, and/or a non-negligible bias in the (Ωm,0,σ8,0)-plane. +I. +INTRODUCTION +Addressing the late-time accelerated phase of expan- +sion of the Universe remains a major challenge for funda- +mental physics [1, 2]. Though most observations to date +are in agreement with the standard (Concordance) model +of cosmology (ΛCDM), alternative explanations for Dark +Energy (DE)—other than a cosmological constant Λ— +are still up for debate. In particular, modifying the laws +of gravity (beyond Einstein’s GR) at large-scales remains +a tantalizing possibility [3, 4]. Besides the exact nature +of the dark energy (DE) component and its (effective) +equation of state, additional modifications come with the +properties of the relativistic degrees of freedom, notably +the neutrino sector. Interestingly, despite the wide class +of modified-gravity (MG) scenarios explored in the last +decades, observations seem to suggest that GR remains +our best description of gravitational interactions, where +dark energy is in the form of a cosmological constant in +the Einstein field equations. For example, the detection +of GW 170817, together with its electromagnetic counter- +part GRB 170817A [5], implies that gravitational waves +∗ calderon@kasi.re.kr +† benjamin@sejong.ac.kr +‡ david.polarski@umontpellier.fr +§ shafieloo@kasi.re.kr +¶ alstar@landau.ac.ru +travel at the speed of light—ruling out a large subclass of +Horndeski models predicting a tensor speed cT ̸= c at the +present epoch [6]. Hence the detection of gravitational +waves (GW) have added stringent constraints on modi- +fied gravity models in addition to local constraints. Note +that a viable cosmic expansion history can give addi- +tional strong constraints, for example on f(R) models [7] +1. At the phenomenological level, most modified theories +of gravity predict a time (and possibly scale) dependent +effective gravitational coupling Geff(z) [11, 12] entering +the equation for the growth of perturbations. Thus, de- +tecting a deviation from Newton’s constant would be a +smoking gun for physics beyond ΛCDM and even beyond +GR. +Let us present now the basic formalism of our +approach, starting with the background. +We con- +sider here spatially flat Friedmann-Lemaˆıtre-Robertson- +Walker universes with +h2(z) ≡ H2/H2 +0 = Ωm,0(1+z)3 +(1−Ωm,0)fDE(z) , (1) +where fDE = ρDE(z)/ρDE(z = 0). While the second term +in (1) becomes generically subdominant in the past for +viable cosmologies, this has to be enforced explicitly at +1 Viable cosmological models of the present Universe in f(R) grav- +ity satisfying these constraints were independently constructed +soon after that paper in [8–10]. +arXiv:2301.00640v1 [astro-ph.CO] 2 Jan 2023 + +2 +high redshifts (where no data are available) once we use +Gaussian Processes in order to reconstruct h(z) [13]. We +stress further that the parameter Ωm,0 refers to clustered +dust-like matter only. The second term of (1) is more +general than the compact notation suggests, see the dis- +cussion given in [13]. We turn now to the perturbations. +We use the following conventions and notations [11] (see +also e.g. [14]) in the conformal Newtonian gauge, where +the perturbed FLRW metric is described by (c = 1) +ds2 = −(1 + 2φ)dt2 + (1 − 2ψ)a2dx2, +(2) +where φ and ψ are the Bardeen potentials. Phenomeno- +logically, on subhorizon scales, in many modified gravity +models the departure from the standard perburbations +growth in GR is encoded in the modified Poisson equa- +tion [11] (see also e.g. [14–16]) +∇2φ = 4πGeff(a, k) ρ ≡ 4πGµ(a, k) ρ . +(3) +GR corresponds obviously to µ ≡ 1. The relation be- +tween the Bardeen potentials is expressed as follows +φ ≡ η(a, k) ψ , +(4) +the two potentials are generically unequal in these mod- +els. The subhorizon modes are essentially affected by µ as +is explicit from Eq. (5) given below, while super horizon +modes are affected by both µ and η [15]. In this work, +given the datasets considered, we restrict our attention to +µ (see e.g. [17, 18] for constraints on η). In what follows, +we will use Geff and µ interchangeably, since µ is just +Geff in units of G. The growth of dust-like subhorizon +matter perturbations in the Quasi-Static Approximation +(QSA) is then governed by [11] +¨δ + 2H ˙δ = 4πG ρ δ µ(z, k), +(5) +where δ ≡ δρ/ρ is the density contrast of dust-like +matter. +For modes of cosmological interest, the k- +dependence of µ is often mild and can be neglected in +a first approach [19–21] – see e.g. [22–25] for current +and future constraints on the scale-dependence of µ. +Note that this is certainly the case for the unscreened +scalar-tensor model considered in [11]. We will restrict +ourselves here to phenomenological models where µ or +Geff is scale independent. +The above equation can be re-written in terms of the +growth factor f ≡ δ′/δ, to give +f ′ + +� +f + 2 + h′ +h +� +f − 3 +2Ωm(z)µ(z) = 0 , +(6) +where a prime stands for derivative with respect to +N ≡ ln a. +From an observational standpoint, redshift +space distortions (RSD) provides us with growth rate +measurements of the quantity +fσ8 ≡ σ8,0 +δ0 +fδ = σ8,0 +δ0 +δ′, +with +δ0 = δ(z = 0) . +(7) +We remind that the quantities Ωi appearing in (1) and +(6) are defined in the standard way as in GR with the +help of Newton’s constant G. +In this work, we will use the synergy between geomet- +rical background probes (Type Ia Supernovae [SN] and +Baryon Acoustic Oscillations [BAO]) and growth mea- +surements from RSD to constrain the phenomenological +function µ(z) describing the departures from GR. While +current analysis pipelines rely on various assumptions +(namely, Λ+GR) when extracting the cosmological infor- +mation from large-scale structure, in particular the BAO +and RSD measurements, we expect that our results will +remain essentially unaffected when such effects are taken +into account. +The paper is organized as follows. +We start by de- +scribing in detail the methodology and the data used in +Section II. In Section III, we apply the method to simu- +lated RSD data generated with µ ̸= 1 in both idealistic +and realistic scenarios and further discuss the implica- +tions of the results. We also comment on the effects of +incorrectly assuming a ΛCDM expansion history on the +reconstructions in Section III C. In Section IV, we con- +sider separately the inclusion of massive neutrinos. +II. +METHOD AND DATA +A. +Models & Mock Data +For the data, we generate mock fσ8 measurements for +a (stage-IV) DESI-like survey following Tables 2.3-2.7 in +[26] (covering 14K deg2) and for different behaviours of +Geff that we aim to reconstruct. Namely, we consider an +f(R)-inspired bump-like profile (which we refer to simply +as “Bump”) and a smooth step-like transition (“Dip” +hereafter) in the recent past towards the weak gravity +regime (Geff < G), see e.g. [27, 28] 2. These two profiles +are treated purely phenomenologically here, indeed vi- +able f(R) theories are actually screened and allow Geff,0 +to deviate from G today. +Nonetheless, due to the k- +dependence of µ which we do not discuss here, cosmic +scales smaller than some critical scale would experience +a boost in their growth in the recent past. +In the case of the dip, we consider it mainly to assess +whether such profiles can be accurately reconstructed us- +ing our model-independent approach. Note in this con- +text that a decreasing µ is impossible in massless scalar- +tensor models [31]. To summarize, these hybrid profiles +allow to test our reconstruction independently of any the- +oretical prior. +The behaviors of the phenomenological functions +µfid(z) used to generate the data are depicted by the +2 Indeed, both such profiles can occur in viable cosmological mod- +els in f(R) gravity, see [29] in particular, especially in the case +of oscillations around phantom divide [30]. + +3 +dashed-lines in the upper panel of Fig. 1, while the corre- +sponding growth fσ8(z) evolutions are shown in the lower +panel. We also make use of stage-IV SN+BAO data to +determine the background expansion history h(z) with- +out relying on a specific parametric model, as explained +in §III B. The fiducial background used to generate the +data is a Chevallier-Polarski-Linder (CPL) model [32], +extensively discussed in [13] with +θfid = {Ωfid +m,0 = 0.28, wfid +0 += −0.95, wfid +a += 0.3, +hfid = 0.7, σfid +8,0 = 0.81}, +(8) +where H0 = 100 h km s−1 Mpc−1. +More details on the +background-only (SN+BAO) mock data can also be +found in [13]. +Already at this stage, let us note that +modified theories of gravity can lead to a modified Chan- +drasekhar mass (with mch ∼ G−3/2 +eff +[33]), relevant for +SNeIa analyses, which can affect the absolute magnitude +(e.g. +∆M = +15 +4 µ(z) in scalar-tensor theories [34, 35]) +and hence the distance measurements obtained from such +standard candles [36–38]. This effect has even been pro- +posed as a possible explanation for the mismatch between +early and late-time measurements of the Hubble constant +H0, see e.g. [39–45]. However, for our purposes, we ne- +glect these effects and assume the h(z) measurements ob- +tained from SNe are independent of µ in the current anal- +ysis. The inclusion of these effects for a specific model +might be the subject of future works. +B. +The Method +To explore possible modifications of gravity at late- +times, we model Geff(z) as a Gaussian Process3 (GP) +centered around Newton’s constant G, such that +µ(z; σf, ℓf, zc) = +� +GP( ¯f = 1, k(σf, ℓf)), +for z < zc +1, +for z ≥ zc +(9) +so that we recover GR at large-z. We further impose the +conditions µ(z = 0) = 1 ± 0.05 and µ′(z = 0) = µ′(zc) = +0, where ′ ≡ d/dz—see Appendix A for details. +The +second condition allows us to smoothly recover Geff = G +above a certain zc and at z = 0, while exploring possible +departures from GR at intermediate redshifts 0.1 < z < +10 (see e.g. [18, 43, 47–50] for other approaches). The +first condition is not necessary (see our discussion at the +beginning of this Section), but from a technical point of +view it can help guiding our reconstructions at very low +z where we are volume-limited and uncertainties become +quite large. Furthermore, when dealing with real data, +3 We do not delve into the details of Gaussian Process modeling +here, instead we refer the reader to our previous work [13] and +the excellent review [46] for more. +Parameter +σ8,0 +log10 σf +log10 ℓf log10 zc +Prior +[0.5, 1.2] [−3, 0.5] [−1, 0.2] [−1, 1] +TABLE I. Uniform priors for the parameters used in the +MCMC analyses. +we do not know the true behaviour of µ, and whether +the underlying model is screened or not, hence the two +representative behaviours at z = 0 chosen for our profiles. +It is comforting to find that the first condition does not +alter the reconstruction of the second profile around z = +0 as illustrated by the blue curves in Fig. 1. +We use a squared exponential kernel given by +k(x, x′; σf, ℓf) = σ2 +f e−(x−x′)2/2ℓ2 +f , +(10) +where σf and ℓf determines the amplitude and typical +length-scale of the correlations, respectively [46]. +In a Bayesian spirit, we give flat priors to the cosmolog- +ical and (hyper)parameters, listed in Table I. We sample +the parameter space using Markov Chain Monte Carlo +(MCMC) methods, as implemented in emcee [51, 52]. At +each step in the MCMC, we draw a sample of µ(N = +ln a) ≡ Geff/G ∼ GP(1, K), characterized by (σf, ℓf, zc), +and solve the growth equation, with a given σ8,0, to ob- +tain a solution fσ8(z) that we confront to RSD data. +Those samples of µ(z) retracing a similar shape to µfid +will yield a better fit to growth data, and thus will be sta- +tistically favored in the long run. Averaging over a large +number of realizations gives the median shape of µ(z) and +95% (2σ) confidence intervals around it. This is along the +lines of what was done in [13] to reconstruct fDE, but this +time we also include conditions on the derivatives of the +GP, to smoothly recover the form in Eq. (9), following +the formalism described in Appendix A. +III. +RESULTS AND DISCUSSIONS +A. +Ideal case: Background is perfectly known +We first consider the idealistic case where the back- +ground expansion history is perfectly known. In other +words, we fix Ωm,0 and σ8,0 to their fiducial values, and +further assume that the dark energy evolution is known +fDE(z) = f fid +DE. Although this is far from being a realistic +scenario, it allows us to test our method and quantify +the uncertainties purely coming from the modifications +of gravity, encoded in Geff. +The posterior distributions for µ(z) assuming perfect +knowledge of h(z) and σ8,0 are shown in Fig. 1. If the +background (and the amplitude of fluctuations σ8,0) are +perfectly known, the RSD data alone is enough to per- +form an accurate (within 2σ) reconstruction of the un- +derlying theory of gravity, i.e. Geff(z). In the next sub- +section, we take a more realistic approach, where only + +4 +z +0.8 +1.0 +1.2 +1.4 +Geff/G +Bump +Dip +µfid +0 +1 +2 +3 +4 +z +0.2 +0.3 +0.4 +0.5 +fσ8 +FIG. 1. Reconstructions of Geff in the idealistic case where +the background h(z) and amplitude of fluctuations σ8,0 are +perfectly known. Solid lines and shaded regions correspond +to the median, 68 and 95% confidence intervals around it, +respectively. Dashed lines correspond to the fiducial cosmolo- +gies generating the DESI-like (RSD) data. The redshift zc +of the transition to GR, as well as the hyperparameters σf +and ℓf appearing in (9) are nonetheless free parameters to be +determined by the data. Both of these reconstructions detect +deviations from GR (µ = 1) at more than 2σ for z ∼ 1. +minimal assumptions on the background are made4 and +h(z) is purely determined from the data. +B. +Realistic case: σ8,0 free - Ωm,0 and fDE(z) +determined by SN+BAO +In this section, instead of assuming a parametric form +for h(z), we use the reconstructed expansion history as +determined by SN+BAO data. In practice, this amounts +to obtaining an expansion history h(z) from the samples +of fDE and calculating angular and luminosity distances +which are then fitted to the data, as explained in [13]. +The degeneracies between σ8,0, Ωm,0 and Geff makes it +very hard to say something about the underlying theory +of gravity, given the quality of the data and in particu- +lar, when all parameters are free to vary. To circumvent +this issue, we assume a single expansion history, as deter- +mined solely by the data. More specifically, the expan- +sion history h(z), along with the value of Ωm,0—needed +for solving the growth equation (6)—are the median of +4 We only assume a flat FLRW universe, and that the Hubble rate +is a sum of a matter term and an “effective” DE component [13] +0 +5 +10 +15 +fDE +Fid +ΛCDM +10−1 +100 +101 +102 +z +−2 +0 +2 +∆H/H [%] +FIG. 2. Top : Reconstruction of the DE evolution fDE(z). +Bottom : Relative (percentage) errors in the background re- +constructions from forecasted SN+BAO measurements. The +orange line correspond to the true fiducial background in (8), +while gray lines depict the reconstructed median, 68 and 95% +confidence levels around it. Dashed-black line correspond to +ΛCDM’s best-fit (fDE = 1, Ωm,0 = 0.3103) to SN+BAO data. +all the realizations drawn from the Markov SN+BAO +chains5, obtained in [13]. Indeed, it was shown in [13] +that our method is able to capture a large class of DE +models, even those where the contribution from DE is not +negligible at high-z. Our reconstruction of h(z) is accu- +rate to ≲ 1% across the entire redshift range of interest— +see Fig. 2. The amplitude of the fluctuations, σ8,0, now +becomes a free parameter, and we sample the full pa- +rameter space θ = {σ8,0, log10 σf, log10 ℓf, log10 zc} in the +range given by Table I. In Fig. 3, we show the reconstruc- +tions when using the median of h(z) and median Ωm,0 +from the SN+BAO chains. As expected, the uncertain- +ties in the reconstructions increase with respect to those +in Fig.1, as σ8,0 is now a free parameter which is some- +what degenerate with Geff, allowing for more flexibility +in the samples of Geff drawn at each step in MCMC. +The advantage of taking this approach, is that we do +not make any assumption on the evolution of DE, and we +are able to effectively reconstruct any expansion history +directly from the data, by reconstructing fDE(z). More- +over, this disentangles the uncertainties coming from the +growth evolution fσ8(z) and those coming from the back- +ground expansion h(z). This also allows us to point down +a value for Ωm,0, which is of course anti-correlated with +σ8,0, which is in turn anti-correlated with Geff. Thus, +allowing for more constraining power on the quantity of +interest µ(z) from RSD alone. The two-dimensional pos- +teriors of the quantity µ at two different redshifts z = 0 +5 The posterior distributions correspond to the blue contours +shown in Fig. 6 of Calder´on et al. [13]. + +5 +z +0.8 +1.0 +1.2 +1.4 +1.6 +Geff/G +Bump +Dip +1 +2 +3 +4 +z +0.2 +0.3 +0.4 +0.5 +fσ8 +σ8,0 = 0.814+0.022 +−0.032 +σ8,0 = 0.806+0.017 +−0.013 +FIG. 3. Realistic case where σ8,0 is allowed to vary, and the +background h(z) is determined by SN+BAO (gray lines in +Fig. 2). The fiducial cosmologies used to generate the fσ8(z) +measurements are shown by the dashed-lines. Despite having +larger error confidence intervals with respect to the idealistic +case in Fig. 1, both of these reconstructions are still able to +rule out GR at more than 2σ at z ∼ 1. +and z = 1.4 are shown on Fig. 4. At z = 1.4, where +most of the constraining power of RSD measurements +lies, the bump-like posteriors in red exclude GR (µ = 1, +in dashed) at > 2σ, while the posteriors for the dip-like +profile in blue are marginally consistent with GR at the +2σ level. At low redshift, because of the large uncertain- +ties in fσ8, the posteriors are much broader and provide +a ∼ 20% constraints on µ(z = 0). +We note that the +study of peculiar velocities using SNIa from ZTF and +LSST can potentially improve the measurements of the +growth at very low-z by a factor of 2 with respect to DESI +[53]—see also [54] for other interesting constraints using +gravitational waves and galaxies’ peculiar velocities. In- +terestingly, because the redshift zc in (9) of the transition +from G → Geff is a free parameter, our method allows us +to constrain when the departures from GR start taking +place (see Fig. 7 and the discussions in Appendix B) +C. +Incorrectly assuming a ΛCDM background +Cosmological observations suggest that dark energy +is in the form of a cosmological constant Λ. +Because +of its simplicity and agreement with observations, it +remains the standard model of cosmology today. Thus, +most cosmological analyses are done within the ΛCDM +framework, which might lead to biased reconstructions + +Bump +Dip +0.75 +0.90 +1.05 +1.20 +(z = 0) +0.70 +0.75 +0.80 +0.85 +8, 0 +0.9 +1.2 +1.5 +1.8 +(z = 1.4) +0.75 +0.90 +1.05 +1.20 +(z = 0) +0.9 +1.2 +1.5 +1.8 +(z = 1.4) +FIG. 4. Marginalized posterior distributions of the relevant +cosmological parameters, when using our model-independent +reconstructions of h(z), shown in gray in Fig.2 (where the +unknown function fDE(z) is reconstructed in a fully model +independent way and Ωm,0 is fixed to the median of all pos- +sible values obtained from the SN+BAO chains—c.f. Section +III B). +z +0.8 +1.0 +1.2 +1.4 +1.6 +Geff/G +ΛCDM +Bump +Dip +1 +2 +3 +4 +z +0.2 +0.3 +0.4 +0.5 +fσ8 +σ8,0 = 0.812+0.013 +−0.021 +σ8,0 = 0.784+0.023 +−0.015 +FIG. 5. +Reconstructions of µ(z) when assuming the best- +fit ΛCDM’s expansion history, with Ωm,0 = 0.31. Incorrectly +assuming a ΛCDM background leads to biased determinations +of σ8,0 and a degraded reconstruction of µ(z), despite being +perfectly consistent with fσ8(z), as can be seen from the lower +panel. + +6 +if DE is not constant, as for the fiducial cosmology +considered here. In this section, we explore the effects +of incorrectly assuming a ΛCDM background expansion +history in the reconstructions of µ(z). In other words, +we fit a ΛCDM model to the SN+BAO mock data +described before and find the corresponding best-fit +value for Ωm,0 (and thus ΩΛ,0 = 1 − Ωm,0). We remind +the reader that the mock data are generated from a +time-evolving CPL dark energy model, given by Eq. +(8). +We then use this expansion history to solve for +the perturbations and reconstruct µ(z), as explained +in the previous sections. +The black dashed-lines in +Fig. 2 show the best-fit ΛCDM expansion history (with +ΩΛCDM,bf +m,0 += 0.3103+0.0025 +−0.0024), compared to the fiducial one +with Ωfid +m,0 = 0.28 in orange (hence, representing a ∼ 12σ +bias in the fractional matter density). +Despite having +almost identical H(z), the differences in the DE evolu- +tion fDE(z) and biased Ωm,0 translate into a degraded +reconstruction of Geff, shown in Fig. 5—to be compared +with Fig. +3. +We also find that the inferred value of +σ8,0 can be biased σ8,0 ∼ 0.78 vs. σfid +8,0 = 0.81 (which +corresponds to a ∼ 1.2σ bias in the inferred amplitude +of fluctuations) for the case of the dip (in blue)—see +Table II. As understood from our previous work [13], +from the background-only (SN+BAO) stand-point, the +lack of DE at high-z is compensated by higher values of +Ωm,0, which translates into lower values of σ8,0 (or lower +Geff < Gfid +eff) to maintain the agreement with growth-rate +measurements of fσ8(z). This is a perfect example of +what might happen if one incorrectly assumes DE is +constant, the background expansion history might be +consistent with the geometrical probes (SN+BAO), but +a tension might appear in the amplitude of fluctuations +σ8,0 inferred from LSS observables. +Despite the bias +in the cosmological parameters Ωm,0 and σ8,0—and +for the specific cases of µ(z) considered here—the +reconstructions are still able to capture the main trends +in µ(z). +Finally, let us note that for the step-like transition in +blue, the reason why the reconstructions deviate some- +how from the fiducial µfid(z) (in dashed) at very low-z is +because of our theoretical prior Geff(z = 0) ≃ G, which +tends to draw our GP samples back to 1. We stress that +this prior does not need to be imposed, as we do not +necessarily have Geff(z = 0) ≃ G in most MG theories. +We have in mind here theories without screening mech- +anisms that do require Geff ≃ G today to satisfy local +constraints, e.g. [55]. Despite this prior, because of the +large uncertainties in RSD measurements at z ∼ 0, our +reconstructions are still able to capture (within 2σ) the +true fiducial µfid +Dip. +IV. +EFFECT OF MASSIVE NEUTRINOS +In this section, we consider universes containing mas- +sive neutrinos. +We want to investigate how well our +reconstruction of µ fares in their presence. +It is well +known that free-streaming species with non-zero mass +(here massive neutrinos) lead to a suppression of gravi- +tational clustering on scales below a characteristic scale, +corresponding to their free-streaming length. +Hence, +while massive neutrinos contribute to the universe ex- +pansion in the same way as usual dust-like matter (cor- +responding to Ωm), they are absent from the driving term +in the matter perturbations growth. Hence we have in +front of us a situation where the parameter Ωm,0 does +not represent all dust-like components at low redshifts. +Indeed, one cannot distinguish massive neutrinos from +dust-like matter purely from geometric probes at low z. +In this case, the splitting in (1), while sensible theoreti- +cally, is somewhat ambiguous regarding expansion data if +we have no additional information on Ωm,0 or Ων,0. This +ambiguity however gets broken once we consider the per- +turbations growth. In a first step, we assume the presence +of massive neutrinos and we work with equation (12) be- +low (instead of (1)). +So, while we reconstruct µ as a +Gaussian process, we assume the background expansion +is known up to the two parameters Ωm,0 and mν. Here +however, we have only one free parameter left. Indeed, +in this Section we fix the present relative energy density +Ωtot +m,0 of all components which behave like dust at low z, +namely +Ωtot +m,0 ≡ Ωm,0 + Ων,0 = Ωcdm,0 + Ωb,0 + Ων,0 , +(11) +where Ωcdm,0, Ωb,0, and Ων,0 are the present relative +densities of cold dark matter, baryons, and massive neu- +trinos respectively. Note that the couple of parameters +(Ωm,0, mν) and +� +Ωtot +m,0, mν +� +carry the same information. +We assume now that h2(z) is given by: +h2(z) = Ωm,0 (1 + z)3 + ΩΛ,0 +(12) ++ Ωγ,0 (1 + z)4 +� +1 + 0.2271 Neff +3 +� +i +fν +�mνi +Tν +�� +, +where fν(y) ≃ (1 + (Ay)p)1/p is a fit provided in Ref. +[56], with A = 180ζ(3) +7π4 +and p = 1.83. This fitting function +fν describes the evolution from the relativistic behav- +ior when mν ≪ Tν (Tν ∼ a−1) to the non-relativistic +regime when we have eventually mν ≫ Tν. Like in (1), +the first term appearing in (12) corresponds to the frac- +tional amount of matter that clusters. In order to test +our reconstruction in the presence of massive neutrinos, +it is more relevant to consider universes sharing identical +Ωtot +m,0 rather than identical Ωm,0, but with different Ωm,0, +or equivalently different neutrino masses mν. +Clearly, +the parameters Ωtot +m,0 and mν, completely define the back- +ground expansion (12). +The driving term in the perturbations growth equation +depends on the combination µ Ωm. Hence for modified +gravity and in the presence of massive neutrinos, this + +7 +combination is modified at low redshifts as follows +GΩtot +m → GeffΩm = GΩtot +m µ +� +1 − Ων +Ωtot +m +� +≈ 0.965 mν +0.5 eVh−2 +70 µ GΩtot +m , +(13) +where we evidently have Ωtot +m += Ωm in the absence of +massive neutrinos, and h70 = H0/70 km s−1 Mpc−1. For +the values we take here, the change comes essentially +from modified gravity. +Here, we forecast the future surveys’ potential to re- +construct the coupling strength µ(z) in the presence of +massive neutrinos and purely from RSD measurements +of fσ8(z). +As before, we generate mock data from a +fiducial model; this time we choose a (ΛCDMν) cosmol- +ogy containing 2 massless and 1 massive neutrinos, with +mfid +ν += 0.5 eV. Although this mass is larger than what is +currently allowed by cosmological observations6 [57, 58], +it is still within the allowed mass range probed by terres- +trial experiments, which constrain m2 +ν ≡ Σi |Uei|2 m2 +i = +0.26+0.34 +−0.34 eV2, yielding an upper bound on the electron +(anti)-neutrino mass mν < 0.8 eV at 90% CL [59]7. The +rest of the cosmological parameters are fixed to Planck’s +best-fit values. +Due to the growth suppression from +such a massive neutrino, the normalization of the mat- +ter power spectrum Pm(k, z = 0), characterized by σ8,0, +is now σfid +8,0 ≃ 0.73, lower than in the previous sections +(where σ8,0 was fixed to σfid +8,0 = 0.81). +In what follows, we assume that this normalization +(σfid +8,0 = 0.73, as obtained for µ = 1) is the same for +all profiles of Geff. Although the actual normalization +of the Pm(k, z = 0) would indeed depend on the theory +of gravity, we generate mock data for different profiles +of µ from the same value of σ8,0. +We stress that this +choice is arbitrary, as we are dealing with simulated data +and we are interested in assessing whether the theory +of gravity µ(z) and σ8,0 are accurately recovered by our +model-independent reconstructions, which do not know +anything about the underlying theory that generates the +data. +We +then +sample +the +parameters +θ += +{σ8,0, mν, log10 σf, log10 ℓf, log10 zc}, with mν ∈ [0, 1] to +see the impact of a varying neutrino mass on the recon- +structions of µ(z). +The posterior distributions for the +relevant cosmological parameters are shown on Fig. 6. +The value of σ8,0 is anti-correlated with the reconstruc- +tions of µ, mainly seen in the (σ8,0, µ(z = 1.4))-plane. +Large deviations from GR, up to µ(z = 1.4) ∼ 1.8 can +be achieved, provided that the amplitude of fluctuations +σ8,0 is low (σ8,0 ∼ 0.65). A slight (negative) correlation +between Ωm,0 and σ8,0 is also obtained, as expected. +6 Cosmological constraints are indirect and somewhat model- +dependent, unlike ground-based experiments. +7 Note that masses of usual and sterile neutrinos mν ∼ 1 eV are +well possible in viable f(R) cosmological models [60, 61]. + +Bump +Dip +0.300 +0.305 +0.310 +0.315 +0.320 +m, 0 +0.75 +0.90 +1.05 +1.20 +(z = 0) +0.64 +0.68 +0.72 +0.76 +0.80 +8, 0 +0.75 +1.00 +1.25 +1.50 +1.75 +(z = 1.4) +0.300 +0.305 +0.310 +0.315 +0.320 +m, 0 +0.75 +0.90 +1.05 +1.20 +(z = 0) +0.75 +1.00 +1.25 +1.50 +1.75 +(z = 1.4) +FIG. 6. Marginalized posterior distributions for the parame- +ters in the presence of massive neutrinos. This figure is the +same as Fig. +4, but this time assuming the background is +known h(z) (up to 1 free parameter mν) given by Eq. (12), +including relativistic species and when the neutrino mass is +free to vary (c.f. Section IV). +The enhanced suppression of growth (due to larger +mass mν, hence smaller Ωm,0 = Ωtot +m,0 − Ων,0) needs to +be compensated by larger values of σ8,0, to maintain +the agreement with fσ8 measurements. +Despite these +correlations, the reconstructions of µ(z) remain accurate, +and does not seem to be affected by a varying neutrino +mass (other than increasing the uncertainties in the +reconstructions, due to an additional free parameter). +The fiducial value for σ8,0, shown as a dashed vertical +line in Fig. 6, is also accurately recovered. +Finally, let us note that we separately tested our recon- +structions in the presence of massive neutrinos without +assuming the functional form of h(z), given by (12) but +using instead the (reconstructed) effective fDE(z) in Eq. +(1), which captures the effect of relativistic species [13]. +Our conclusions remain unaltered, but no information on +the neutrino mass can be obtained. +V. +CONCLUSIONS +In a companion paper Calder´on et al. [13], we jointly +reconstructed the growth and expansion histories inside +GR directly from the data and using minimal assump- +tions. We showed that our framework is able to capture +a wide variety of behaviors in the DE component. +In +this work, we extend our methodology to include pos- + +8 +sible modifications of gravity at late-times, as encoded +by the function Geff(z) appearing in the (modified) Pois- +son equation. We illustrate the efficiency of our method +in reconstructing different theories of gravity by recon- +structing two phenomenological shapes of µ(z) ≡ Geff/G. +As an example, we consider a “bump” and a smooth +transition (“dip”) towards the weak gravity regime in +the recent past. We used the reconstructed h(z) from +background-only data, as obtained in [13] in order to fit +fσ8(z) to RSD mock data, thereby constraining µ(z) us- +ing minimal assumptions. +We also explore the effects +of incorrectly assuming a ΛCDM background. In both +cases, the fiducial µ(z) is within the 1σ confidence in- +tervals of our reconstructions, if the background is accu- +rately determined, and within 2σ if we incorrectly assume +the ΛCDM’s best-fit h(z). Finally, we explored the im- +pact of massive neutrinos on the reconstructions of µ(z). +To summarize, let us list a few important results. +• If the background is given (Fig. 1), or accurately +reconstructed from SN+BAO (Fig. 2), our recon- +structions of Geff(z) are able to distinguish both +fiducial µ-profiles from GR at ≳ 2σ (see Figs. 1 +and 3). +• Incorrectly assuming a ΛCDM expansion (with the +best-fit Ωm,0 to background probes) can lead to bi- +ased/degraded reconstructions (red-shaded regions +in Fig. +5) and/or biased estimations of the am- +plitude of fluctuations σ8,0 (see Table II). This is +despite the perfect agreement with fσ8(z) measure- +ments, as shown in the lower panel of Fig. 5. +• The posterior distributions for the hyperparame- +ters clearly show the need for a deviation from the +mean ¯f = 1, i.e. GR is not a good description of +the data. This is understood because the marginal- +ized contours in Fig. 7 suggest σf ̸= 0. Interest- +ingly, the redshift of the transition zc is also not +compatible with 0, and we have a “detection” on +when this transition from G → Geff happens. +In this work, we used forecasted (stage-IV) SN+BAO +data to reconstruct the DE evolution fDE(z)—which de- +termines the expansion history h(z)—and separately re- +constructed µ(z) using DESI-like fσ8(z) measurements +for two different toy models of Geff. +We expect our +methodology to hold for essentially any (viable) form of +Geff. We showed that for both profiles considered in this +work, the reconstructions are able to detect the devia- +tions from GR at ≳ 2σ in the redshift range 0.5 ≲ z ≲ 1.5 +where DESI’s (RSD) constraining power lies. The inclu- +sion of external data sets, such as the (modified) lumi- +nosity distance of gravitational waves dGW +L +(z) [62] or the +Integrated Sachs-Wolfe effect (ISW) seen in the temper- +ature anisotropies of the Cosmic Microwave Background +(CMB) in cross-correlation with LSS surveys would pro- +vide interesting (model-independent) constraints on the +allowed deviations from GR [63]. Moreover, we note that +the effect of massive neutrinos would be tracked more + +Bump +Dip +2.4 +1.6 +0.8 +0.0 +log10 +f +0.75 +0.50 +0.25 +0.00 +log10 +f +0.70 +0.75 +0.80 +0.85 +8, 0 +0.80.40.00.40.8 +log10zc +2.4 +1.6 +0.8 +0.0 +log10 +f +0.75 +0.50 +0.25 +0.00 +log10 +f +0.80.4 +0.00.40.8 +log10zc +FIG. 7. Marginalized posterior distributions for the relevant +parameters from the RSD chains. The background expansion +history used in the analysis is fixed to the median h(z) ob- +tained from the SN+BAO chains, shown as a gray line in Fig. +2. +accurately if we allow for a scale-dependent growth. We +leave such extensions for future work. +ACKNOWLEDGEMENTS +BL acknowledges the support of the National Research +Foundation of Korea (NRF-2019R1I1A1A01063740 and +NRF-2022R1F1A1076338) and the support of the Korea +Institute for Advanced Study (KIAS) grant funded by +the government of Korea. AS would like to acknowledge +the support by National Research Foundation of Korea +NRF2021M3F7A1082053, and the support of the Korea +Institute for Advanced Study (KIAS) grant funded by the +government of Korea. AAS was partly supported by the +project number 0033-2019-0005 of the Russian Ministry +of Science and Higher Education. +Appendix A: Gaussian process with observations on +the derivatives +In this section, we describe a less common use of Gaus- +sian process when we also observe the derivative of the +function f to be reconstructed [46, 64]. We note that in +this section, f denotes a general function, not the growth +rate. In our case, f = µ(z). In addition to observations +of y, we also “observe” y′ = f ′(x) + ε, where +ε ∼ N(0, Cy′) +(A1) + +9 +Model +σ8,0 +log10 zc +ln L +Bump +0.814+0.022 +−0.032 0.524+0.342 +−0.092 −10.15+0.78 +−1.48 +Dip +0.806+0.017 +−0.013 +0.79+0.15 +−0.40 +−10.36+0.96 +−1.46 +Bump (ΛCDM) 0.812+0.013 +−0.021 +0.86+0.10 +−0.42 +−10.35+0.94 +−1.97 +Dip (ΛCDM) +0.784+0.023 +−0.015 −0.17+0.20 +−0.34 +−10.5+1.0 +−1.3 +TABLE II. Marginalized constraints (median and 68% con- +fidence levels) on the relevant parameters. We compare the +effects of incorrectly assuming ΛCDM (bottom-two rows) vs +the reconstructed h(z) from SN+BAO in the upper-two rows. +is a Gaussian noise and Cy′ is the covariance of the deriva- +tives. We further assume that y and y′ are uncorrelated. +Therefore, the vector +� +�� +y +y′ +f +f ′ +� +�� +(A2) +is jointly gaussian, and the posterior predictive distribu- +tion can be calculated using +� +f +f ′ +� +|y, y′, X, X∗ ∼ N +�� ¯f +¯f ′ +� +; +� +A − CB−1CT �� +, (A3) +where the mean is +� ¯f +¯f ′ +� += CB−1 +� +y − µy +y′ − µy′ +� +, +(A4) +and the covariance matrix is given by +A = +� +K∗∗ K01 +∗∗ +K10 +∗∗ K11 +∗∗ +� +∈ M2n∗,2n∗, +(A5a) +B = +� +K + Cy +K01 +K10 +K11 + Cy′ +� +∈ Mn+n′, +(A5b) +CT = +� +K∗ +K01 +∗ +K10 +∗ +K11 +∗ +� +∈ Mn+n′,2n∗, +(A5c) +where +K = k(X, X), +(A6a) +K∗ = k(X, X∗), +(A6b) +K∗∗ = k(X∗, X∗), +(A6c) +and for any matrix X, +Xi,j = +∂i+jX +∂Xi∂Xj . +(A7) +This formalism allows us to impose theoretical priors on +the samples of µ(z) and its derivative µ′(z) to smoothly +recover the expected GR behaviour at early times (see +Eq. 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D 85, 123530 (2012), +arXiv:1204.2272. + diff --git a/aNAyT4oBgHgl3EQfv_ls/content/tmp_files/load_file.txt b/aNAyT4oBgHgl3EQfv_ls/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f94f2dc942a3a3cedcc3d0c5ea7dccfcca5f4de6 --- /dev/null +++ b/aNAyT4oBgHgl3EQfv_ls/content/tmp_files/load_file.txt @@ -0,0 +1,1078 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf,len=1077 +page_content='Joint reconstructions of growth and expansion histories from stage-IV surveys with minimal assumptions II: Modified gravity and massive neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Rodrigo Calder´on ,1, 2, ∗ Benjamin L’Huillier ,3, † David Polarski ,2, ‡ Arman Shafieloo ,1, 4, § and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Starobinsky 5, 6, ¶ 1Korea Astronomy and Space Science Institute, Daejeon 34055, South Korea 2Laboratoire Charles Coulomb, Universit´e de Montpellier & CNRS, 34095 Montpellier, France 3Department of Physics and Astronomy, Sejong University, Seoul 05006, South Korea 4University of Science and Technology, Daejeon 34113, South Korea 5L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Landau Institute for Theoretical Physics RAS, Chernogolovka, Moscow region 142432, Russia 6Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia (Dated: January 3, 2023) Based on a formalism introduced in our previous work, we reconstruct the phenomenological function Geff(z) describing deviations from General Relativity (GR) in a model-independent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this alternative approach, we model µ ≡ Geff/G as a Gaussian process and use forecasted growth- rate measurements from a stage-IV survey to reconstruct its shape for two different toy-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We follow a two-step procedure: (i) we first reconstruct the background expansion history from Supernovae (SNe) and Baryon Acoustic Oscillation (BAO) measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (ii) we then use it to obtain the growth history fσ8, that we fit to redshift-space distortions (RSD) measurements to reconstruct Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We find that upcoming surveys such as the Dark Energy Spectroscopic Instrument (DESI) might be capable of detecting deviations from GR, provided the dark energy behavior is accurately determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We might even be able to constrain the transition redshift from G → Geff for some particular models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We further assess the impact of massive neutrinos on the reconstructions of Geff (or µ) assuming the expansion history is given, and only the neutrino mass is free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Given the tight constraints on the neutrino mass, and for the profiles we considered in this work, we recover numerically that the effect of such massive neutrinos do not alter our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Finally, we stress that incorrectly assuming a ΛCDM expansion history leads to a degraded reconstruction of µ, and/or a non-negligible bias in the (Ωm,0,σ8,0)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' INTRODUCTION Addressing the late-time accelerated phase of expan- sion of the Universe remains a major challenge for funda- mental physics [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Though most observations to date are in agreement with the standard (Concordance) model of cosmology (ΛCDM), alternative explanations for Dark Energy (DE)—other than a cosmological constant Λ— are still up for debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In particular, modifying the laws of gravity (beyond Einstein’s GR) at large-scales remains a tantalizing possibility [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Besides the exact nature of the dark energy (DE) component and its (effective) equation of state, additional modifications come with the properties of the relativistic degrees of freedom, notably the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Interestingly, despite the wide class of modified-gravity (MG) scenarios explored in the last decades, observations seem to suggest that GR remains our best description of gravitational interactions, where dark energy is in the form of a cosmological constant in the Einstein field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' For example, the detection of GW 170817, together with its electromagnetic counter- part GRB 170817A [5], implies that gravitational waves ∗ calderon@kasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='kr † benjamin@sejong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='kr ‡ david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='polarski@umontpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='fr § shafieloo@kasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='kr ¶ alstar@landau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='ru travel at the speed of light—ruling out a large subclass of Horndeski models predicting a tensor speed cT ̸= c at the present epoch [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Hence the detection of gravitational waves (GW) have added stringent constraints on modi- fied gravity models in addition to local constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Note that a viable cosmic expansion history can give addi- tional strong constraints, for example on f(R) models [7] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' At the phenomenological level, most modified theories of gravity predict a time (and possibly scale) dependent effective gravitational coupling Geff(z) [11, 12] entering the equation for the growth of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Thus, de- tecting a deviation from Newton’s constant would be a smoking gun for physics beyond ΛCDM and even beyond GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Let us present now the basic formalism of our approach, starting with the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We con- sider here spatially flat Friedmann-Lemaˆıtre-Robertson- Walker universes with h2(z) ≡ H2/H2 0 = Ωm,0(1+z)3 +(1−Ωm,0)fDE(z) , (1) where fDE = ρDE(z)/ρDE(z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' While the second term in (1) becomes generically subdominant in the past for viable cosmologies, this has to be enforced explicitly at 1 Viable cosmological models of the present Universe in f(R) grav- ity satisfying these constraints were independently constructed soon after that paper in [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00640v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='CO] 2 Jan 2023 2 high redshifts (where no data are available) once we use Gaussian Processes in order to reconstruct h(z) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We stress further that the parameter Ωm,0 refers to clustered dust-like matter only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The second term of (1) is more general than the compact notation suggests, see the dis- cussion given in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We turn now to the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We use the following conventions and notations [11] (see also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [14]) in the conformal Newtonian gauge, where the perturbed FLRW metric is described by (c = 1) ds2 = −(1 + 2φ)dt2 + (1 − 2ψ)a2dx2, (2) where φ and ψ are the Bardeen potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Phenomeno- logically, on subhorizon scales, in many modified gravity models the departure from the standard perburbations growth in GR is encoded in the modified Poisson equa- tion [11] (see also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [14–16]) ∇2φ = 4πGeff(a, k) ρ ≡ 4πGµ(a, k) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (3) GR corresponds obviously to µ ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The relation be- tween the Bardeen potentials is expressed as follows φ ≡ η(a, k) ψ , (4) the two potentials are generically unequal in these mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The subhorizon modes are essentially affected by µ as is explicit from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (5) given below, while super horizon modes are affected by both µ and η [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this work, given the datasets considered, we restrict our attention to µ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [17, 18] for constraints on η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In what follows, we will use Geff and µ interchangeably, since µ is just Geff in units of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The growth of dust-like subhorizon matter perturbations in the Quasi-Static Approximation (QSA) is then governed by [11] ¨δ + 2H ˙δ = 4πG ρ δ µ(z, k), (5) where δ ≡ δρ/ρ is the density contrast of dust-like matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' For modes of cosmological interest, the k- dependence of µ is often mild and can be neglected in a first approach [19–21] – see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [22–25] for current and future constraints on the scale-dependence of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Note that this is certainly the case for the unscreened scalar-tensor model considered in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We will restrict ourselves here to phenomenological models where µ or Geff is scale independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The above equation can be re-written in terms of the growth factor f ≡ δ′/δ, to give f ′ + � f + 2 + h′ h � f − 3 2Ωm(z)µ(z) = 0 , (6) where a prime stands for derivative with respect to N ≡ ln a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' From an observational standpoint, redshift space distortions (RSD) provides us with growth rate measurements of the quantity fσ8 ≡ σ8,0 δ0 fδ = σ8,0 δ0 δ′, with δ0 = δ(z = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (7) We remind that the quantities Ωi appearing in (1) and (6) are defined in the standard way as in GR with the help of Newton’s constant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this work, we will use the synergy between geomet- rical background probes (Type Ia Supernovae [SN] and Baryon Acoustic Oscillations [BAO]) and growth mea- surements from RSD to constrain the phenomenological function µ(z) describing the departures from GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' While current analysis pipelines rely on various assumptions (namely, Λ+GR) when extracting the cosmological infor- mation from large-scale structure, in particular the BAO and RSD measurements, we expect that our results will remain essentially unaffected when such effects are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We start by de- scribing in detail the methodology and the data used in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In Section III, we apply the method to simu- lated RSD data generated with µ ̸= 1 in both idealistic and realistic scenarios and further discuss the implica- tions of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We also comment on the effects of incorrectly assuming a ΛCDM expansion history on the reconstructions in Section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In Section IV, we con- sider separately the inclusion of massive neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' METHOD AND DATA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Models & Mock Data For the data, we generate mock fσ8 measurements for a (stage-IV) DESI-like survey following Tables 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='7 in [26] (covering 14K deg2) and for different behaviours of Geff that we aim to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Namely, we consider an f(R)-inspired bump-like profile (which we refer to simply as “Bump”) and a smooth step-like transition (“Dip” hereafter) in the recent past towards the weak gravity regime (Geff < G), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [27, 28] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' These two profiles are treated purely phenomenologically here, indeed vi- able f(R) theories are actually screened and allow Geff,0 to deviate from G today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Nonetheless, due to the k- dependence of µ which we do not discuss here, cosmic scales smaller than some critical scale would experience a boost in their growth in the recent past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In the case of the dip, we consider it mainly to assess whether such profiles can be accurately reconstructed us- ing our model-independent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Note in this con- text that a decreasing µ is impossible in massless scalar- tensor models [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' To summarize, these hybrid profiles allow to test our reconstruction independently of any the- oretical prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The behaviors of the phenomenological functions µfid(z) used to generate the data are depicted by the 2 Indeed, both such profiles can occur in viable cosmological mod- els in f(R) gravity, see [29] in particular, especially in the case of oscillations around phantom divide [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 3 dashed-lines in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1, while the corre- sponding growth fσ8(z) evolutions are shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We also make use of stage-IV SN+BAO data to determine the background expansion history h(z) with- out relying on a specific parametric model, as explained in §III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The fiducial background used to generate the data is a Chevallier-Polarski-Linder (CPL) model [32], extensively discussed in [13] with θfid = {Ωfid m,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='28, wfid 0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='95, wfid a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3, hfid = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='7, σfid 8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='81}, (8) where H0 = 100 h km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' More details on the background-only (SN+BAO) mock data can also be found in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Already at this stage, let us note that modified theories of gravity can lead to a modified Chan- drasekhar mass (with mch ∼ G−3/2 eff [33]), relevant for SNeIa analyses, which can affect the absolute magnitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' ∆M = 15 4 µ(z) in scalar-tensor theories [34, 35]) and hence the distance measurements obtained from such standard candles [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This effect has even been pro- posed as a possible explanation for the mismatch between early and late-time measurements of the Hubble constant H0, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [39–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' However, for our purposes, we ne- glect these effects and assume the h(z) measurements ob- tained from SNe are independent of µ in the current anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The inclusion of these effects for a specific model might be the subject of future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The Method To explore possible modifications of gravity at late- times, we model Geff(z) as a Gaussian Process3 (GP) centered around Newton’s constant G, such that µ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' σf, ℓf, zc) = � GP( ¯f = 1, k(σf, ℓf)), for z < zc 1, for z ≥ zc (9) so that we recover GR at large-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We further impose the conditions µ(z = 0) = 1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='05 and µ′(z = 0) = µ′(zc) = 0, where ′ ≡ d/dz—see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The second condition allows us to smoothly recover Geff = G above a certain zc and at z = 0, while exploring possible departures from GR at intermediate redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='1 < z < 10 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [18, 43, 47–50] for other approaches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The first condition is not necessary (see our discussion at the beginning of this Section), but from a technical point of view it can help guiding our reconstructions at very low z where we are volume-limited and uncertainties become quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Furthermore, when dealing with real data, 3 We do not delve into the details of Gaussian Process modeling here, instead we refer the reader to our previous work [13] and the excellent review [46] for more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Parameter σ8,0 log10 σf log10 ℓf log10 zc Prior [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2] [−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5] [−1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2] [−1, 1] TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Uniform priors for the parameters used in the MCMC analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' we do not know the true behaviour of µ, and whether the underlying model is screened or not, hence the two representative behaviours at z = 0 chosen for our profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' It is comforting to find that the first condition does not alter the reconstruction of the second profile around z = 0 as illustrated by the blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We use a squared exponential kernel given by k(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' σf, ℓf) = σ2 f e−(x−x′)2/2ℓ2 f , (10) where σf and ℓf determines the amplitude and typical length-scale of the correlations, respectively [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In a Bayesian spirit, we give flat priors to the cosmolog- ical and (hyper)parameters, listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We sample the parameter space using Markov Chain Monte Carlo (MCMC) methods, as implemented in emcee [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' At each step in the MCMC, we draw a sample of µ(N = ln a) ≡ Geff/G ∼ GP(1, K), characterized by (σf, ℓf, zc), and solve the growth equation, with a given σ8,0, to ob- tain a solution fσ8(z) that we confront to RSD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Those samples of µ(z) retracing a similar shape to µfid will yield a better fit to growth data, and thus will be sta- tistically favored in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Averaging over a large number of realizations gives the median shape of µ(z) and 95% (2σ) confidence intervals around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This is along the lines of what was done in [13] to reconstruct fDE, but this time we also include conditions on the derivatives of the GP, to smoothly recover the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (9), following the formalism described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Ideal case: Background is perfectly known We first consider the idealistic case where the back- ground expansion history is perfectly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In other words, we fix Ωm,0 and σ8,0 to their fiducial values, and further assume that the dark energy evolution is known fDE(z) = f fid DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Although this is far from being a realistic scenario, it allows us to test our method and quantify the uncertainties purely coming from the modifications of gravity, encoded in Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The posterior distributions for µ(z) assuming perfect knowledge of h(z) and σ8,0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' If the background (and the amplitude of fluctuations σ8,0) are perfectly known, the RSD data alone is enough to per- form an accurate (within 2σ) reconstruction of the un- derlying theory of gravity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Geff(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In the next sub- section, we take a more realistic approach, where only 4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 Geff/G Bump Dip µfid 0 1 2 3 4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 fσ8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Reconstructions of Geff in the idealistic case where the background h(z) and amplitude of fluctuations σ8,0 are perfectly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Solid lines and shaded regions correspond to the median, 68 and 95% confidence intervals around it, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Dashed lines correspond to the fiducial cosmolo- gies generating the DESI-like (RSD) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The redshift zc of the transition to GR, as well as the hyperparameters σf and ℓf appearing in (9) are nonetheless free parameters to be determined by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Both of these reconstructions detect deviations from GR (µ = 1) at more than 2σ for z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' minimal assumptions on the background are made4 and h(z) is purely determined from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Realistic case: σ8,0 free - Ωm,0 and fDE(z) determined by SN+BAO In this section, instead of assuming a parametric form for h(z), we use the reconstructed expansion history as determined by SN+BAO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In practice, this amounts to obtaining an expansion history h(z) from the samples of fDE and calculating angular and luminosity distances which are then fitted to the data, as explained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The degeneracies between σ8,0, Ωm,0 and Geff makes it very hard to say something about the underlying theory of gravity, given the quality of the data and in particu- lar, when all parameters are free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' To circumvent this issue, we assume a single expansion history, as deter- mined solely by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' More specifically, the expan- sion history h(z), along with the value of Ωm,0—needed for solving the growth equation (6)—are the median of 4 We only assume a flat FLRW universe, and that the Hubble rate is a sum of a matter term and an “effective” DE component [13] 0 5 10 15 fDE Fid ΛCDM 10−1 100 101 102 z −2 0 2 ∆H/H [%] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Top : Reconstruction of the DE evolution fDE(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Bottom : Relative (percentage) errors in the background re- constructions from forecasted SN+BAO measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The orange line correspond to the true fiducial background in (8), while gray lines depict the reconstructed median, 68 and 95% confidence levels around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Dashed-black line correspond to ΛCDM’s best-fit (fDE = 1, Ωm,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3103) to SN+BAO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' all the realizations drawn from the Markov SN+BAO chains5, obtained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Indeed, it was shown in [13] that our method is able to capture a large class of DE models, even those where the contribution from DE is not negligible at high-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Our reconstruction of h(z) is accu- rate to ≲ 1% across the entire redshift range of interest— see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The amplitude of the fluctuations, σ8,0, now becomes a free parameter, and we sample the full pa- rameter space θ = {σ8,0, log10 σf, log10 ℓf, log10 zc} in the range given by Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 3, we show the reconstruc- tions when using the median of h(z) and median Ωm,0 from the SN+BAO chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' As expected, the uncertain- ties in the reconstructions increase with respect to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='1, as σ8,0 is now a free parameter which is some- what degenerate with Geff, allowing for more flexibility in the samples of Geff drawn at each step in MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The advantage of taking this approach, is that we do not make any assumption on the evolution of DE, and we are able to effectively reconstruct any expansion history directly from the data, by reconstructing fDE(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' More- over, this disentangles the uncertainties coming from the growth evolution fσ8(z) and those coming from the back- ground expansion h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This also allows us to point down a value for Ωm,0, which is of course anti-correlated with σ8,0, which is in turn anti-correlated with Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Thus, allowing for more constraining power on the quantity of interest µ(z) from RSD alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The two-dimensional pos- teriors of the quantity µ at two different redshifts z = 0 5 The posterior distributions correspond to the blue contours shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6 of Calder´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='6 Geff/G Bump Dip 1 2 3 4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 fσ8 σ8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='814+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='032 σ8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='806+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='013 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Realistic case where σ8,0 is allowed to vary, and the background h(z) is determined by SN+BAO (gray lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The fiducial cosmologies used to generate the fσ8(z) measurements are shown by the dashed-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Despite having larger error confidence intervals with respect to the idealistic case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1, both of these reconstructions are still able to rule out GR at more than 2σ at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' and z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' At z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4, where most of the constraining power of RSD measurements lies, the bump-like posteriors in red exclude GR (µ = 1, in dashed) at > 2σ, while the posteriors for the dip-like profile in blue are marginally consistent with GR at the 2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' At low redshift, because of the large uncertain- ties in fσ8, the posteriors are much broader and provide a ∼ 20% constraints on µ(z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We note that the study of peculiar velocities using SNIa from ZTF and LSST can potentially improve the measurements of the growth at very low-z by a factor of 2 with respect to DESI [53]—see also [54] for other interesting constraints using gravitational waves and galaxies’ peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In- terestingly, because the redshift zc in (9) of the transition from G → Geff is a free parameter, our method allows us to constrain when the departures from GR start taking place (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7 and the discussions in Appendix B) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Incorrectly assuming a ΛCDM background Cosmological observations suggest that dark energy is in the form of a cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Because of its simplicity and agreement with observations, it remains the standard model of cosmology today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Thus, most cosmological analyses are done within the ΛCDM framework, which might lead to biased reconstructions Bump Dip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='20 (z = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='85 8, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='20 (z = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Marginalized posterior distributions of the relevant cosmological parameters, when using our model-independent reconstructions of h(z), shown in gray in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 (where the unknown function fDE(z) is reconstructed in a fully model independent way and Ωm,0 is fixed to the median of all pos- sible values obtained from the SN+BAO chains—c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Section III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='6 Geff/G ΛCDM Bump Dip 1 2 3 4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 fσ8 σ8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='812+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='021 σ8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='784+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='015 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Reconstructions of µ(z) when assuming the best- fit ΛCDM’s expansion history, with Ωm,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Incorrectly assuming a ΛCDM background leads to biased determinations of σ8,0 and a degraded reconstruction of µ(z), despite being perfectly consistent with fσ8(z), as can be seen from the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6 if DE is not constant, as for the fiducial cosmology considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this section, we explore the effects of incorrectly assuming a ΛCDM background expansion history in the reconstructions of µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In other words, we fit a ΛCDM model to the SN+BAO mock data described before and find the corresponding best-fit value for Ωm,0 (and thus ΩΛ,0 = 1 − Ωm,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We remind the reader that the mock data are generated from a time-evolving CPL dark energy model, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We then use this expansion history to solve for the perturbations and reconstruct µ(z), as explained in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The black dashed-lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2 show the best-fit ΛCDM expansion history (with ΩΛCDM,bf m,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3103+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0025 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0024), compared to the fiducial one with Ωfid m,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='28 in orange (hence, representing a ∼ 12σ bias in the fractional matter density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Despite having almost identical H(z), the differences in the DE evolu- tion fDE(z) and biased Ωm,0 translate into a degraded reconstruction of Geff, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 5—to be compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We also find that the inferred value of σ8,0 can be biased σ8,0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='78 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' σfid 8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='81 (which corresponds to a ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2σ bias in the inferred amplitude of fluctuations) for the case of the dip (in blue)—see Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' As understood from our previous work [13], from the background-only (SN+BAO) stand-point, the lack of DE at high-z is compensated by higher values of Ωm,0, which translates into lower values of σ8,0 (or lower Geff < Gfid eff) to maintain the agreement with growth-rate measurements of fσ8(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This is a perfect example of what might happen if one incorrectly assumes DE is constant, the background expansion history might be consistent with the geometrical probes (SN+BAO), but a tension might appear in the amplitude of fluctuations σ8,0 inferred from LSS observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Despite the bias in the cosmological parameters Ωm,0 and σ8,0—and for the specific cases of µ(z) considered here—the reconstructions are still able to capture the main trends in µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Finally, let us note that for the step-like transition in blue, the reason why the reconstructions deviate some- how from the fiducial µfid(z) (in dashed) at very low-z is because of our theoretical prior Geff(z = 0) ≃ G, which tends to draw our GP samples back to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We stress that this prior does not need to be imposed, as we do not necessarily have Geff(z = 0) ≃ G in most MG theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We have in mind here theories without screening mech- anisms that do require Geff ≃ G today to satisfy local constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Despite this prior, because of the large uncertainties in RSD measurements at z ∼ 0, our reconstructions are still able to capture (within 2σ) the true fiducial µfid Dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' EFFECT OF MASSIVE NEUTRINOS In this section, we consider universes containing mas- sive neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We want to investigate how well our reconstruction of µ fares in their presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' It is well known that free-streaming species with non-zero mass (here massive neutrinos) lead to a suppression of gravi- tational clustering on scales below a characteristic scale, corresponding to their free-streaming length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Hence, while massive neutrinos contribute to the universe ex- pansion in the same way as usual dust-like matter (cor- responding to Ωm), they are absent from the driving term in the matter perturbations growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Hence we have in front of us a situation where the parameter Ωm,0 does not represent all dust-like components at low redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Indeed, one cannot distinguish massive neutrinos from dust-like matter purely from geometric probes at low z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this case, the splitting in (1), while sensible theoreti- cally, is somewhat ambiguous regarding expansion data if we have no additional information on Ωm,0 or Ων,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This ambiguity however gets broken once we consider the per- turbations growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In a first step, we assume the presence of massive neutrinos and we work with equation (12) be- low (instead of (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' So, while we reconstruct µ as a Gaussian process, we assume the background expansion is known up to the two parameters Ωm,0 and mν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Here however, we have only one free parameter left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Indeed, in this Section we fix the present relative energy density Ωtot m,0 of all components which behave like dust at low z, namely Ωtot m,0 ≡ Ωm,0 + Ων,0 = Ωcdm,0 + Ωb,0 + Ων,0 , (11) where Ωcdm,0, Ωb,0, and Ων,0 are the present relative densities of cold dark matter, baryons, and massive neu- trinos respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Note that the couple of parameters (Ωm,0, mν) and � Ωtot m,0, mν � carry the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We assume now that h2(z) is given by: h2(z) = Ωm,0 (1 + z)3 + ΩΛ,0 (12) + Ωγ,0 (1 + z)4 � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='2271 Neff 3 � i fν �mνi Tν �� , where fν(y) ≃ (1 + (Ay)p)1/p is a fit provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [56], with A = 180ζ(3) 7π4 and p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This fitting function fν describes the evolution from the relativistic behav- ior when mν ≪ Tν (Tν ∼ a−1) to the non-relativistic regime when we have eventually mν ≫ Tν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Like in (1), the first term appearing in (12) corresponds to the frac- tional amount of matter that clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In order to test our reconstruction in the presence of massive neutrinos, it is more relevant to consider universes sharing identical Ωtot m,0 rather than identical Ωm,0, but with different Ωm,0, or equivalently different neutrino masses mν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Clearly, the parameters Ωtot m,0 and mν, completely define the back- ground expansion (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The driving term in the perturbations growth equation depends on the combination µ Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Hence for modified gravity and in the presence of massive neutrinos, this 7 combination is modified at low redshifts as follows GΩtot m → GeffΩm = GΩtot m µ � 1 − Ων Ωtot m � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='965 mν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 eVh−2 70 µ GΩtot m , (13) where we evidently have Ωtot m = Ωm in the absence of massive neutrinos, and h70 = H0/70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' For the values we take here, the change comes essentially from modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Here, we forecast the future surveys’ potential to re- construct the coupling strength µ(z) in the presence of massive neutrinos and purely from RSD measurements of fσ8(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' As before, we generate mock data from a fiducial model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' this time we choose a (ΛCDMν) cosmol- ogy containing 2 massless and 1 massive neutrinos, with mfid ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Although this mass is larger than what is currently allowed by cosmological observations6 [57, 58], it is still within the allowed mass range probed by terres- trial experiments, which constrain m2 ν ≡ Σi |Uei|2 m2 i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='34 eV2, yielding an upper bound on the electron (anti)-neutrino mass mν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 eV at 90% CL [59]7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The rest of the cosmological parameters are fixed to Planck’s best-fit values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Due to the growth suppression from such a massive neutrino, the normalization of the mat- ter power spectrum Pm(k, z = 0), characterized by σ8,0, is now σfid 8,0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='73, lower than in the previous sections (where σ8,0 was fixed to σfid 8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In what follows, we assume that this normalization (σfid 8,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='73, as obtained for µ = 1) is the same for all profiles of Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Although the actual normalization of the Pm(k, z = 0) would indeed depend on the theory of gravity, we generate mock data for different profiles of µ from the same value of σ8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We stress that this choice is arbitrary, as we are dealing with simulated data and we are interested in assessing whether the theory of gravity µ(z) and σ8,0 are accurately recovered by our model-independent reconstructions, which do not know anything about the underlying theory that generates the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We then sample the parameters θ = {σ8,0, mν, log10 σf, log10 ℓf, log10 zc}, with mν ∈ [0, 1] to see the impact of a varying neutrino mass on the recon- structions of µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The posterior distributions for the relevant cosmological parameters are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The value of σ8,0 is anti-correlated with the reconstruc- tions of µ, mainly seen in the (σ8,0, µ(z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4))-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Large deviations from GR, up to µ(z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 can be achieved, provided that the amplitude of fluctuations σ8,0 is low (σ8,0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' A slight (negative) correlation between Ωm,0 and σ8,0 is also obtained, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6 Cosmological constraints are indirect and somewhat model- dependent, unlike ground-based experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7 Note that masses of usual and sterile neutrinos mν ∼ 1 eV are well possible in viable f(R) cosmological models [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Bump Dip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='320 m, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='20 (z = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='80 8, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='320 m, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='20 (z = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Marginalized posterior distributions for the parame- ters in the presence of massive neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This figure is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 4, but this time assuming the background is known h(z) (up to 1 free parameter mν) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (12), including relativistic species and when the neutrino mass is free to vary (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The enhanced suppression of growth (due to larger mass mν, hence smaller Ωm,0 = Ωtot m,0 − Ων,0) needs to be compensated by larger values of σ8,0, to maintain the agreement with fσ8 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Despite these correlations, the reconstructions of µ(z) remain accurate, and does not seem to be affected by a varying neutrino mass (other than increasing the uncertainties in the reconstructions, due to an additional free parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The fiducial value for σ8,0, shown as a dashed vertical line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 6, is also accurately recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Finally, let us note that we separately tested our recon- structions in the presence of massive neutrinos without assuming the functional form of h(z), given by (12) but using instead the (reconstructed) effective fDE(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (1), which captures the effect of relativistic species [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Our conclusions remain unaltered, but no information on the neutrino mass can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' CONCLUSIONS In a companion paper Calder´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [13], we jointly reconstructed the growth and expansion histories inside GR directly from the data and using minimal assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We showed that our framework is able to capture a wide variety of behaviors in the DE component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this work, we extend our methodology to include pos- 8 sible modifications of gravity at late-times, as encoded by the function Geff(z) appearing in the (modified) Pois- son equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We illustrate the efficiency of our method in reconstructing different theories of gravity by recon- structing two phenomenological shapes of µ(z) ≡ Geff/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' As an example, we consider a “bump” and a smooth transition (“dip”) towards the weak gravity regime in the recent past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We used the reconstructed h(z) from background-only data, as obtained in [13] in order to fit fσ8(z) to RSD mock data, thereby constraining µ(z) us- ing minimal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We also explore the effects of incorrectly assuming a ΛCDM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In both cases, the fiducial µ(z) is within the 1σ confidence in- tervals of our reconstructions, if the background is accu- rately determined, and within 2σ if we incorrectly assume the ΛCDM’s best-fit h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Finally, we explored the im- pact of massive neutrinos on the reconstructions of µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' To summarize, let us list a few important results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' If the background is given (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1), or accurately reconstructed from SN+BAO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2), our recon- structions of Geff(z) are able to distinguish both fiducial µ-profiles from GR at ≳ 2σ (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Incorrectly assuming a ΛCDM expansion (with the best-fit Ωm,0 to background probes) can lead to bi- ased/degraded reconstructions (red-shaded regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 5) and/or biased estimations of the am- plitude of fluctuations σ8,0 (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This is despite the perfect agreement with fσ8(z) measure- ments, as shown in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The posterior distributions for the hyperparame- ters clearly show the need for a deviation from the mean ¯f = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' GR is not a good description of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This is understood because the marginal- ized contours in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7 suggest σf ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Interest- ingly, the redshift of the transition zc is also not compatible with 0, and we have a “detection” on when this transition from G → Geff happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In this work, we used forecasted (stage-IV) SN+BAO data to reconstruct the DE evolution fDE(z)—which de- termines the expansion history h(z)—and separately re- constructed µ(z) using DESI-like fσ8(z) measurements for two different toy models of Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We expect our methodology to hold for essentially any (viable) form of Geff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We showed that for both profiles considered in this work, the reconstructions are able to detect the devia- tions from GR at ≳ 2σ in the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 ≲ z ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5 where DESI’s (RSD) constraining power lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The inclu- sion of external data sets, such as the (modified) lumi- nosity distance of gravitational waves dGW L (z) [62] or the Integrated Sachs-Wolfe effect (ISW) seen in the temper- ature anisotropies of the Cosmic Microwave Background (CMB) in cross-correlation with LSS surveys would pro- vide interesting (model-independent) constraints on the allowed deviations from GR [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Moreover, we note that the effect of massive neutrinos would be tracked more Bump Dip 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 log10 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00 log10 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='85 8, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 log10zc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 log10 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00 log10 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='8 log10zc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Marginalized posterior distributions for the relevant parameters from the RSD chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' The background expansion history used in the analysis is fixed to the median h(z) ob- tained from the SN+BAO chains, shown as a gray line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' accurately if we allow for a scale-dependent growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We leave such extensions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' ACKNOWLEDGEMENTS BL acknowledges the support of the National Research Foundation of Korea (NRF-2019R1I1A1A01063740 and NRF-2022R1F1A1076338) and the support of the Korea Institute for Advanced Study (KIAS) grant funded by the government of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' AS would like to acknowledge the support by National Research Foundation of Korea NRF2021M3F7A1082053, and the support of the Korea Institute for Advanced Study (KIAS) grant funded by the government of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' AAS was partly supported by the project number 0033-2019-0005 of the Russian Ministry of Science and Higher Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Appendix A: Gaussian process with observations on the derivatives In this section, we describe a less common use of Gaus- sian process when we also observe the derivative of the function f to be reconstructed [46, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We note that in this section, f denotes a general function, not the growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In our case, f = µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' In addition to observations of y, we also “observe” y′ = f ′(x) + ε, where ε ∼ N(0, Cy′) (A1) 9 Model σ8,0 log10 zc ln L Bump 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='814+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='524+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='342 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='092 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='78 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='48 Dip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='806+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='40 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='96 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='46 Bump (ΛCDM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='812+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='42 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='94 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='97 Dip (ΛCDM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='784+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='34 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='3 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Marginalized constraints (median and 68% con- fidence levels) on the relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We compare the effects of incorrectly assuming ΛCDM (bottom-two rows) vs the reconstructed h(z) from SN+BAO in the upper-two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' is a Gaussian noise and Cy′ is the covariance of the deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' We further assume that y and y′ are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Therefore, the vector � �� y y′ f f ′ � �� (A2) is jointly gaussian, and the posterior predictive distribu- tion can be calculated using � f f ′ � |y, y′, X, X∗ ∼ N �� ¯f ¯f ′ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' � A − CB−1CT �� , (A3) where the mean is � ¯f ¯f ′ � = CB−1 � y − µy y′ − µy′ � , (A4) and the covariance matrix is given by A = � K∗∗ K01 ∗∗ K10 ∗∗ K11 ∗∗ � ∈ M2n∗,2n∗, (A5a) B = � K + Cy K01 K10 K11 + Cy′ � ∈ Mn+n′, (A5b) CT = � K∗ K01 ∗ K10 ∗ K11 ∗ � ∈ Mn+n′,2n∗, (A5c) where K = k(X, X), (A6a) K∗ = k(X, X∗), (A6b) K∗∗ = k(X∗, X∗), (A6c) and for any matrix X, Xi,j = ∂i+jX ∂Xi∂Xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (A7) This formalism allows us to impose theoretical priors on the samples of µ(z) and its derivative µ′(z) to smoothly recover the expected GR behaviour at early times (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Appendix B: Supplementary Material Inspecting the posterior distributions of the hyperpa- rameters, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7, can yield additional infor- mation on the Geff reconstructions and put interesting constraints on the departures from GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' First, let us note that the inferred value of σ8,0 is unbiased in both cases, when the evolution of DE fDE(z) is reconstructed us- ing our model independent approach [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This is not the case when one (incorrectly) assumes a ΛCDM ex- pansion history (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Second, both the “bump” and “dip” reconstructions seem to require a deviation from the mean function ¯f = µ = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' GR), as the pos- teriors of log10 σf are not compatible with σf → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' This suggest that GR is not a good description of the growth fσ8(z) history and that the data requires extra flexibil- ity, as encoded by the GP kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Lastly, the posteriors of log10 zc seem to peak at the redshift zc ∼ 3 where the departures from GR actually takes place (de- picted by the vertical dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Weinberg, The Cosmological Constant Problem, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 61, 1 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Sahni and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Starobinsky, The Case for a positive cosmological Lambda term, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' D 9, 373 (2000), arXiv:astro-ph/9904398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Clifton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Ferreira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Padilla, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Skordis, Modified gravity and cosmology, Physics Reports 513, 1–189 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Tsujikawa, Modified gravity models of dark energy, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' 800, 99 (2010), arXiv:1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content='0191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Abbott, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Acernese, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} +page_content=' Ackley, C.' metadata={'source': 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+page_content='2272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfv_ls/content/2301.00640v1.pdf'} diff --git a/adAzT4oBgHgl3EQfLPsZ/vector_store/index.faiss b/adAzT4oBgHgl3EQfLPsZ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7ae381d65bd01d83ce84fcb4f13bd6e790d7e99d --- /dev/null +++ b/adAzT4oBgHgl3EQfLPsZ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:54987cbf3f4ceead82a1c267cdc2a4c9bf9e476f570de6ac7ac89dd389d62cc8 +size 4390957 diff --git a/b9E4T4oBgHgl3EQfow0v/vector_store/index.faiss b/b9E4T4oBgHgl3EQfow0v/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2c87ea0591ac48c731db273c12b400a037ca86ec --- /dev/null +++ b/b9E4T4oBgHgl3EQfow0v/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b4d82b8e9f584f8cebc58a948f0ce007379ba93aab5bd0d834e2d8f83e3a5447 +size 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Research at Meta +3FAIR +Abstract +We introduce the novel-view acoustic synthesis (NVAS) +task: given the sight and sound observed at a source view- +point, can we synthesize the sound of that scene from an +unseen target viewpoint? We propose a neural rendering +approach: Visually-Guided Acoustic Synthesis (ViGAS) net- +work that learns to synthesize the sound of an arbitrary +point in space by analyzing the input audio-visual cues. +To benchmark this task, we collect two first-of-their-kind +large-scale multi-view audio-visual datasets, one synthetic +and one real. We show that our model successfully reasons +about the spatial cues and synthesizes faithful audio on both +datasets. To our knowledge, this work represents the very +first formulation, dataset, and approach to solve the novel- +view acoustic synthesis task, which has exciting potential +applications ranging from AR/VR to art and design. Un- +locked by this work, we believe that the future of novel-view +synthesis is in multi-modal learning from videos. +1. Introduction +Replaying a video recording from a new viewpoint1 has +many applications in cinematography, video enhancement, +and virtual reality. For example, it can be used to edit a +video, simulate a virtual camera, or, given a video of a per- +sonal memory, even enable users to experience a treasured +moment again—not just on a 2D screen, but in 3D in a vir- +tual or augmented reality, thus ‘reliving’ the moment. +While the applications are exciting, there are still many +unsolved technical challenges. Recent advances in 3D re- +construction and novel-view synthesis (NVS) address the +problem of synthesizing new images of a given scene [31, +33, 43]. However, thus far, the view synthesis problem is +concerned with creating visuals alone; the output is silent or +at best naively adopts the sounds of the original video (from +the “wrong” viewpoint). Without sound, the emotional and +cognitive significance of the replay is severely diminished. +In this work, we address this gap and introduce the new +task of novel-view acoustic synthesis (NVAS). 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Novel-view acoustic synthesis task. Given audio-visual +observations from one viewpoint and the relative target viewpoint +pose, render the sound received at the target viewpoint. Note that +the target is expressed as the desired pose of the microphones; the +image at that pose (right) is neither observed nor synthesized. +this task is to synthesize the sound in a scene from a new +acoustic viewpoint, given only the visual and acoustic input +from another source viewpoint in the same scene (Fig. 1). +NVAS is very different from the existing NVS task, +where the goal is to reconstruct images instead of sounds, +and these differences present new challenges. First, the 3D +geometry of most real-life scenes changes in a limited man- +ner during the recording. On the contrary, sound changes +substantially over time, so the reconstruction target is highly +dynamic. Secondly, visual and audio sensors are very dif- +ferent. +A camera matrix captures the light in a highly- +directional manner, and a single image comprises a large 2D +array of pixels. In contrast, sounds are recorded with one or +two microphones which are at best weakly-directional, pro- +viding only a coarse sampling of the sound field. Thirdly, +the frequency of light waves is much higher than that of +sound waves; the length of audio waves is thus larger to the +point of being comparable to the size of geometric features +of the scene, meaning that effects such as diffraction are +often dominant, and spatial resolution is low. As a result, +techniques that require spatial precision, such as triangula- +tion and segmentation, are not applicable to audio. Lastly, +sounds mix together, making it difficult to segment them, +arXiv:2301.08730v1 [cs.CV] 20 Jan 2023 + +and they are affected by environmental effects such as re- +verberation that are distributed and largely unobservable. +While the NVS and NVAS tasks are indeed very dif- +ferent, we hypothesize that NVAS is an inherently multi- +modal task. In fact, vision can play an important role in +achieving accurate sound synthesis. First, establishing cor- +respondences between sounds and their sources as they ap- +pear in images can provide essential cues for resynthesiz- +ing the sounds realistically. For instance, human speech is +highly directional and sounds very differently if one faces +the speaker or their back, which can only be inferred from +visual cues. In addition, the environment acoustics also af- +fect the sound one hears as a function of the scene geom- +etry, materials, and emitter/receiver locations. The same +source sounds very differently if it is located in the center of +a room, at the corner, or in a corridor, for example. In short, +vision provides cues about space and geometry that affect +sound, and are difficult to estimate from the sound alone. +In order to validate our hypothesis, we propose a novel +visually-guided acoustic synthesis network that analyzes au- +dio and visual features and synthesizes the audio at a target +location. More specifically, the network first takes as in- +put the image observed at the source viewpoint in order to +infer global acoustic and geometric properties of the envi- +ronment along with the bounding box of the active speaker. +The network then reasons how the speaker and scene geom- +etry change in 3D based on the relative target pose with a +fusion network. We inject the fused features into audio with +a gated multi-modal fusion network and model the acoustic +changes between viewpoints with a time-domain model. +In order to conduct our experiments on the new NVAS +task, we require suitable training and benchmarking data, +of which currently there is none available. To address that, +we contribute two new datasets: one real (Replay-NVAS) +and one synthetic (SoundSpaces-NVAS). The key feature +of these datasets is to record the sight and sound of dif- +ferent scenes from multiple cameras/viewpoints. Replay- +NVAS contains video recordings of groups of people per- +forming social activities (e.g., chatting, watching TV, doing +yoga, playing instruments) from 8 surrounding viewpoints +simultaneously. It contains 72 hours of highly realistic ev- +eryday conversation and social interactions in one home- +like environment. To our knowledge, Replay-NVAS repre- +sents the first large-scale real-world dataset enabling NVAS. +This dataset would also greatly benefit many other exist- +ing tasks including NVS, active speaker localization, etc. +For SoundSpaces-NVAS, we render 1.3K hours of audio- +visual data based on the SoundSpaces [7] platform. Using +this simulator, one can easily change the scene geometry +and the positions of speakers, cameras, and microphones. +This data serves as a powerful test bed with clean ground +truth for a large collection of home environments, offer- +ing a good complement to Replay-NVAS. For both datasets, +we capture binaural audio, which is what humans perceive +with two ears. Together the datasets contain 1,372 hours +of audio-visual capture, with 1,032 speakers across 121 3D +scenes. We will release both datasets. +We show that our model outperforms traditional signal +processing approaches as well as learning-based baselines, +often by a substantial margin, in a quantitative evaluation +and a human study. We show qualitative examples where +the model predicts acoustic changes according to the view- +point changes, e.g., left channel becomes louder when the +viewpoint changes from left to right. +In a nutshell, we +present the first work that deals with novel-view acoustic +synthesis, and contribute two large-scale datasets along with +a novel neural rendering approach for solving the task. +2. Related Work +Novel-view synthesis (NVS). +Kickstarted by advances in +neural rendering [33,51], many recent works consider vari- +ants of the NVS problem. Most approaches assume dozens +of calibrated images for reconstructing a single static scene. +Closer to monocular video NVS, authors have considered +reducing the number of input views [19, 24, 37, 45, 60] and +modelling dynamic scenes [26,27,41,42,53,55]. However, +none of these works tackle audio. +Acoustic matching and spatialization. +NVAS requires +accounting for (1) the environmental acoustics and (2) the +geometric configuration of the target microphone(s) (e.g., +monaural vs binaural). +Modelling environmental acous- +tics has been addressed extensively by the audio commu- +nity [4, 25]. Room impulse response (RIR) functions char- +acterize the environment acoustics as a transfer function be- +tween the emitter and receiver, accounting for the scene +geometry, materials, and emitter/receiver locations. Esti- +mating the direct-to-reverberant ratio and the reverberation +time, is sufficient to synthesize simple RIRs that match au- +dio in a plausible manner [11,15,21,29,36,59]. These meth- +ods do not synthesize for a target viewpoint, rather they +resynthesize to match an audio sample. In [46, 47] sound +from a moving emitter is spatialized towards a receiver con- +ditioned on the tracked 3D location of the emitter. +Recently, the vision community explores using visual in- +formation to estimate environmental acoustics [6,50]. How- +ever, these works only synthesize acoustics for a given +viewpoint rather than a novel viewpoint. In addition, they +have only addressed monaural audio, which is more forgiv- +ing than binaural because humans do not perceive subtle ab- +solute acoustic properties, but can detect easily inconsisten- +cies in the sounds perceived by the two ears. Recent work +spatializes monaural sounds by upmixing them to multiple +channels conditioned on the video, where the sound emit- +ters are static [16, 35]. Because the environment, emitter +and receiver are static, so are the acoustics. Other work +predicts impulse responses in simulation either for a single + +environment [28], or by using few-shot egocentric obser- +vations [30], or by using the 3D scene mesh [44]. While +simulated results are satisfying, those models’ impact on +real-world data is unknown, especially for scenarios where +human speakers move and interact with each other. Unlike +any of the above, we introduce and tackle the NVAS prob- +lem, accounting for both acoustics and spatialization, and +we propose a model that addresses the problem effectively +on both synthetic and real-world data. +Audio-visual learning. +Recent advances in multi-modal +video understanding enable new forms of self-supervised +cross-modal feature learning from video [2, 23, 34], sound +source localization [18,20,54], and audio-visual speech en- +hancement and source separation [1, 13, 32, 39, 61]. +All +of these existing tasks and datasets only deal with a single +viewpoint. We introduce the first audio-visual learning task +and dataset that deals with multi-view audio-visual data. +3. The Novel-view Acoustic Synthesis Task +We introduce a new task, novel-view acoustic synthesis +(NVAS). Assuming there are N sound emitters in the scene +(emitter i emits Ci from location Li), given the audio AS +and video VS observed at the source viewpoint S, the goal +is to synthesize the audio AT at the target viewpoint T, as +it would sound from the target location, specified by the +relative pose PT of the target microphone (translation and +orientation) with respect to the source view (Fig. 1). Fur- +thermore, we assume that the active sound emitters in the +environment are visible in the source camera, but we make +no assumptions about the camera at the target location. +The sound received at any point R can be expressed as: +AR = F(L1,...,N, C1,...,N, R | E), +(1) +where R is the receiver location (S or T) and E is the +environment. The emitted sounds Ci are not restricted to +speech but can be ambient noise, sounding objects, etc. +Our goal here is to learn a transfer function T (·) defined +as AT = T (AS, VS, PT ), where S, T, L1,...,N, C1,...,N, E +are not directly given and need to be inferred from VS and +PT , which makes the task inherently multi-modal. +This task is challenging because the goal is to model the +sound field of a dynamic scene and capture acoustic changes +between viewpoints given one pair of audio-visual measure- +ments. While traditional signal processing methods can be +applied, we show in Sec. 6 that they perform poorly. In this +work, we present a neural rendering approach. +4. Datasets +We introduce two datasets for the NVAS task: +live +recordings (Sec. 4.1), and simulated audio in scanned real- +world environments (Sec. 4.2) (see Fig. 2). The former is +real and covers various social scenarios, but offers limited +Figure 2. Example source and target views for the two introduced +datasets: Replay-NVAS (left) and SoundSpaces-NVAS (right). +diversity of sound sources, viewpoints and environments, +and is noisy. The latter has a realism gap, but allows perfect +control over these aforementioned elements. +Both datasets focus on human speech given its relevance +in applications. However, our model design is not specific +to speech. For both datasets, we capture binaural audio, +which best aligns with human perception. Note that for both +datasets, we collect multiple multi-modal views for train- +ing and evaluation; during inference the target viewpoint(s) +(and in some cases target environment) are withheld. We +will release both datasets to assist future research. +4.1. The Replay-NVAS Dataset +Replay-NVAS contains multi-view captures of acted +scenes in apartments. We capture 109 different scenarios +(e.g., having a conversation, having dinner, or doing yoga) +from 8 different viewpoints. In total, we collect 72 hours of +video data, involving 32 participants across all scenarios. +In each scenario, we invite 2–4 participants to act on a +given topic. +Each participant wears a near-range micro- +phone, providing a clean recording of their own speech. The +scene is captured by 8 DLSR cameras, each augmented with +a 3Dio binaural microphone. In this way, the data captures +video and audio simultaneously from multiple cameras, re- +sulting in 56 possible source/target viewpoint combinations +for each scene. The videos are recorded at 30 FPS and the +audio is recorded with a 48k sampling rate. We use a clap- +per at the beginning of the recording for temporal synchro- +nization. Each scenario lasts 3–8 min. We use off-the-shelf +software for multi-view camera calibration (see Supp.). +To construct the dataset, we extract one-second long +clips from each video with overlapping windows. We auto- +matically remove silent and noisy clips based on the energy +of near-range microphones, which results in 77K/12K/2K +clips in total for train/val/test (details in Supp.) +During +training, for one sample, we randomly select two out of +eight viewpoints, one as the source and one as the target. +This dataset is very challenging. It covers a wide range +of social activities. It is harrowed by ambient sound, room + +reverberation, overlapping speech and non-verbal sounds +such as clapping and instruments. Participants can move +freely in the environment. We believe that this data will be +useful to the community beyond the NVAS task as it can +be used for benchmarking many other problems, including +active speaker localization, source separation, and NVS. +4.2. The SoundSpaces-NVAS Dataset +In this dataset, we synthesize multi-view audio-visual +data of two people having conversations in 3D scenes. In +total, we construct 1.3K hours of audio-visual data for a to- +tal of 1,000 speakers, 120 3D scenes and 200K viewpoints. +Our goal is to construct audio-visual data with strong +spatial and acoustic correspondences across multiple view- +points, meaning that the visual information should indicate +what the audio should sound like, e.g., observing speaker +on the left should indicate the left ear is louder and ob- +serving speaker at a distance should indicate there is higher +reverberation. We use the SoundSpaces 2.0 platform [7], +which allows highly realistic audio and visual rendering for +arbitrary camera and microphone locations in 3D scans of +real-world environments [5, 52, 58]. It accounts for all ma- +jor real-world acoustics phenomena: direct sounds, early +specular/diffuse reflections, reverberation, binaural spatial- +ization, and effects from materials and air absorption. +We use the Gibson dataset [58] for scene meshes and +LibriSpeech [40] for speech samples. As we are simulating +two people having conversations, for a given environment, +we randomly sample two speaker locations within 3 m and +insert two copyright-free mannequins (one male and one fe- +male) at these two locations.2 We then randomly sample +four nearby viewpoints facing the center of the two speak- +ers at a height of 1.5 m (Fig. 2, right). For each speaker, +we select a speech sample from LibriSpeech with matching +gender. We render images at all locations as well as binau- +ral impulse response for all pairs of points between speakers +and viewpoints. The received sound is obtained by convolv- +ing the binaural impulse response with the speech sample. +During training, for one sample, we randomly sample +two out of four rendered viewpoints, one as the source and +one as the target. We also randomly choose one speaker to +be active, simulating what we observe on the real data (i.e., +usually only one person speaks at a time). +5. Visually-Guided Acoustic Synthesis +We introduce a new method, Visually-Guided Acoustic +Synthesis (ViGAS), to address the NVAS problem, taking +as input sound and an image and outputting the sound from +a different target microphone pose. +ViGAS consists of five components: ambient sound sep- +aration, active speaker localization, visual acoustic network, +acoustic synthesis, and temporal alignment. The high-level +2https://renderpeople.com/free-3d-people +idea is to separate the observed sound into primary and am- +bient, extract useful visual information (active speaker and +acoustic features), and use this information to guide acous- +tic synthesis for the primary sound. Temporal alignment is +performed during training for better optimization. ViGAS +is discussed in detail next and summarised in Fig. 3. +5.1. Ambient Sound Separation +ViGAS starts by decomposing the input sound into pri- +mary and ambient (traffic, electric noise from a fridge or the +A/C, etc.). Ambient sound is important for realism, but it +also interferes with learning the model because it can carry +significant energy, making the model focus on it rather than +on the primary sounds, and its spatial distribution is very +different from the primary sounds. +By explicitly separating primary and ambient sounds, +ViGAS: (1) accounts for the fact that the transfer func- +tions of primary and ambient sounds are very different and +thus difficult to model together; (2) avoids wasting repre- +sentational power on modelling ambient sounds that might +be difficult to reconstruct accurately and depend less on +the viewpoint; and (3) prevents ambient sounds, which are +noise-like and high-energy, from dominating learning and +reconstruction. In practice, as we show in Sec. 6, without +the ambient sound separation, the model performs poorly. +The goal of ambient sound separation is thus to con- +struct a function (AC, AN) = P(AS) that separates the +input sound AS into primary sound AC and ambient sound +AS. Existing approaches to this problem are based on sig- +nal processing [3,12] or learning [10,14]. We find that pre- +trained speech enhancement models such as Denoiser [10] +tend to aggressively remove the noise including the primary +sound, which hinders re-synthesis. We thus opt for band- +pass filtering, passing frequencies within a certain range +and rejecting/attenuating frequencies outside of it, which +we found to work well. We cut frequencies below 80 Hz +for SoundSpaces-NVAS and 150 Hz for Replay-NVAS. +5.2. Active Speaker Localization +Knowing where the emitters of different primary sounds +are located in the environment can help to solve the NVAS +task. In this paper, we focus on localizing the active speaker, +although there can be other important primary sound events +like instruments playing, speakers interacting with objects, +etc. The goal of active speaker localization is to predict +the bounding box of the active speaker in each frame of +the video (examples in Fig. 4). The bounding box is in the +format of (ymin, ymax, xmin, xmax) and x, y are normalized +to [0, 1] by the image width and height, respectively. +On SoundSpaces-NVAS, this task is relatively easy be- +cause of the strong correspondence between the appearance +of the speaker and the gender of the speech sample, which +enables to easily train a classifier for active speakers. 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Visually Guided Acoustic Synthesis (ViGAS). Given the input audio AS, we first separate out the ambient sound to focus on +the sound of interest. We take the source audio and source visual to localize the active speaker on the 2D image. We also extract the visual +acoustic features of the environment by running an encoder on the source visual. We concatenate the active speaker feature, source visual +features, and the target pose, and fuse these features with a MLP. We feed both the audio stream AC and fused visual feature VC into the +acoustic synthesis network, which has M stacked audio-visual fusion blocks. In each block, the audio sequence is processed by dilated +conv1d layers and the visual features are processed by conv1d layers. Lastly, the previously separated ambient sound is added back to the +waveform. During training, our temporal alignment module shifts the prediction by the amount of delay estimated between the source and +the target audio to align the prediction well with the target. +ever, this is much harder on Replay-NVAS because cameras +record speakers from a distance and from diverse angles, +meaning that lip motion, the main cue used by speaker lo- +calization methods [20,48,54], is often not visible. Hence, +the model has to rely on other cues to identify the speaker +(such as body motion, gender or identity). Furthermore, +sometimes people speak or laugh over each other. +Since our focus is not speaker localization, for the +Replay-NVAS we assume that this problem is solved by +an external module that does audio-visual active speaker +localization. +To approximate the output of such a mod- +ule automatically, we rely on the near-range audio record- +ings. Specifically, we first run an off-the-shelf detection and +tracker [9] on the video at 5 FPS and obtain, with some man- +ual refinement, bounding boxes Bi +t for i = 1, . . . , N at each +frame t. We manually assign the near-range microphone au- +dio Ai +N to each tracked person. We select the active speaker +D based on the maximum energy of each near-range micro- +phone, i.e., D = argmaxi +� � Ai +N[t : t+∆t]2� +, where ∆t +is the time interval we use to calculate the audio energy. We +output bounding box BD as the localization feature VL. +5.3. Visual Acoustic Network and Fusion +The active speaker bounding box BD only disam- +biguates the active speaker from all visible humans on 2D, +which is not enough to indicate where the speaker is in 3D. +To infer that, the visual information is also needed. Since +there is usually not much movement in one second (the +length of the input video clip), the video clip does not pro- +vide much extra information compared to a single frame. +Thus, we choose the middle frame to represent the clip and +extract the visual acoustic features VE from the input RGB +image with a pretrained ResNet18 [17] before the average +pooling layer to preserve spatial information. To reduce the +feature size, we feed VE into a 1D convolution with kernel +size 1 and output channel size 8. We then flatten the visual +features to obtain feature VF . +The target pose is specified as the translation along +x, y, z axes plus difference between orientations of the +source “view” and the target “view” expressed via rotation +angles: +y (roll), +x (pitch) and +z (yaw). We encode +each angle α as its sinusoidal value: (sin(α), cos(α)). +Similarly, the target pose is not enough by itself to in- +dicate where the target viewpoint T is in the 3D space; to +infer that, the source view VS is again needed. For example, +in top row of Fig 4, for target viewpoint 3, “two meters to +the right and one meter forward” is not enough to indicate +the target location is in the corridor, while the model can +reason that based on the source view. +We use a fusion network to predict a latent representa- +tion of the scene variables S, T, LD, E (cf. Sec. 3) by first +concatenating [VL, PT , VF ] and then feeding it through a +multilayer perceptron (MLP). See Fig. 3 for the network. +5.4. Acoustic Synthesis +With the separated primary sound AC and the visual +acoustic feature VC as input, the goal of the acoustic syn- +thesis module is to transform AC guided by VC. We design +the acoustic synthesis network to learn a non-linear trans- +fer function (implicitly) that captures these major acoustic +phenomena, including the attenuation of sound in space, the +directivity of sound sources (human speech is directional), +the reverberation level, the head-related transfer function, +as well as the frequency-dependent acoustic phenomena. +Training end-to-end makes it possible to capture these sub- +tle and complicated changes in the audio. + +Inspired by recent advances in time-domain signal mod- +eling [38,47], we design the network as M stacked synthe- +sis blocks, where each block consists of multiple conv1D +layers. We first encode the input audio AC into a latent +space, which is then fed into the synthesis block. The key +of the synthesis block is a gated multimodal fusion network +that injects the visual information into the audio as follows: +z =tanh(pk +A(Ak +F )+pk +V (VC))⊙σ(qk +A(Ak +F )+qk +V (VC)), (2) +where ⊙ indicates element-wise multiplication, σ is a logis- +tic sigmoid function, k = 1, . . . , M is the layer index and +p, q are both learnable 1D convolutions. +After passing z through a sinusoidal activation function, +the network uses two separate conv1D layers to process the +feature, one producing the residual connection Ak+1 +F +and +one producing the skip connection Ak+1 +P +. All skip connec- +tions Ak+1 +P +are mean pooled and fed into a decoder to pro- +duce the output AO. We add back the separated ambient +sound AN as the target audio estimate: ˆAT = AO + AN. +See Supp. for more details on the architecture. +5.5. Temporal Alignment +In order for the model to learn well, it is important that +input and output sounds are temporally aligned. While the +Replay-NVAS data is already synchronised based on the +clapper sound, due to the finite speed of sound, the sounds +emitted from different locations may still arrive at micro- +phones with a delay slightly different from the one of the +clapper, causing misalignments that affect training. +To align source and target audio for training, we find the +delay τ that maximizes the generalized cross-correlation: +RAS,AT (τ) = Et[hS(t) · hT (t − τ)], +(3) +where hS and hT are the feature embedding for AS and +AT respectively at time t. +We use the feature extractor +h from the generalized cross-correlation phase transform +(GCC-PHAT) algorithm [22], which whitens the audio by +dividing by the magnitude of the cross-power spectral den- +sity. After computing τ, we shift the prediction AO by τ +samples to align with the AT and obtain AL. Note that +alignment is already exact for SoundSpaces-NVAS. +5.6. Loss +To compute the loss, we first encode the audio with +the short-time Fourier transform (STFT), a complex-valued +matrix representation of the audio where the y axis repre- +sents frequency and the x axis is time. We then compute the +magnitude of the STFT, and optimize the L1 loss between +the the predicted and ground truth magnitudes as follows: +L = +��||STFT(AL)||2 − ||STFT(A′ +T )||2 +��, +(4) +where A′ +T is the primary sound separated from AT with +P(·). By taking the magnitude, we do not model the ex- +act phase values, which we find hinders learning if being +included in the loss. See implementation details in Supp. +6. Experiments +We compare with several traditional and learning-based +baselines and show that ViGAS outperforms them in both a +quantitative evaluation and a human subject study. +Evaluation. +We measure performance from three aspects: +1. closeness to GT as measured by the magnitude spectro- +gram distance (Mag). 2. correctness of the spatial sound as +measured by the left-right energy ratio error (LRE), i.e., +the difference of ratio of energy between left and right chan- +nels and 3. correctness of the acoustic properties measured +by RT60 error (RTE) [6, 50], i.e., the error in reverbera- +tion time decaying by 60dB (RT60). We use a pretrained +model [6] to estimate RT60 directly from speech. +Baselines. +We consider the following baselines: 1. In- +put audio. Copying the input to the output. 2. TF Es- +timator [56] + Nearest Neighbor, i.e. storing the transfer +function estimated during training and retrieving the nearest +neighbor during test time. We estimate transfer functions +with a Wiener filter [56] and index them with the ground- +truth locations of the speaker, source viewpoint, and target +viewpoint for the single environment setup and their rel- +ative pose for the novel environment setup. At test time, +this method searches the database to find the nearest trans- +fer function and applies it on the input audio. 3. Digital +Signal Processing (DSP) [8] approach that takes the dis- +tance, azimuth, and elevation of the sound source, applies +an inverse a head-related transfer function (HRTF) to es- +timate the speech spoken by the speaker and then applies +another HRTF to estimate the audio at the target micro- +phone location. This baseline adjusts the loudness of the +left and right channels based on where the speaker is in the +target view. We supply GT coordinates for SoundSpaces- +NVAS and speakers’ head positions estimated with trian- +gulation on Replay-NVAS. 4. Visual Acoustic Matching +(VAM) [6], recently proposed for a related task of matching +acoustics of input audio with a target image. This task only +deals with single viewpoint and single-channel audio. We +adapt their model with minimal modification by feeding in +the image from the source viewpoint and concatenating the +position offset of the target microphone at the multimodal +fusion step. See Supp. for details. +6.1. Results on SoundSpaces-NVAS +Table 1 shows the results. For synthetic data, we con- +sider two evaluation setups: 1. single environment: train +and test on the same environment and 2. novel environ- +ment: train and test on multiple non-overlapping Gibson +environments (90/10/20 for train/val/test). +In the single environment setup, our model largely out- +performs all baselines as well as our audio-only ablation +on all metrics. TF Estimator performs poorly despite be- +ing indexed by the ground truth location values because es- + +SoundSpaces-NVAS +Replay-NVAS +Single Environment +Novel Environment +Single Environment +Mag +LRE +RTE +Mag +LRE +RTE +Mag +LRE +RTE +Input audio +0.225 +1.473 +0.032 +0.216 +1.408 +0.039 +0.159 +1.477 +0.046 +TF Estimator [56] +0.359 +2.596 +0.059 +0.440 +3.261 +0.092 +0.327 +2.861 +0.147 +DSP [8] +0.302 +3.644 +0.044 +0.300 +3.689 +0.047 +0.463 +1.300 +0.067 +VAM [6] +0.220 +1.198 +0.041 +0.235 +1.131 +0.051 +0.161 +0.924 +0.070 +ViGAS w/o visual +0.173 +0.973 +0.031 +0.181 +1.007 +0.036 +0.146 +0.877 +0.046 +ViGAS +0.159 +0.782 +0.029 +0.175 +0.971 +0.034 +0.142 +0.716 +0.048 +Table 1. Results on SoundSpaces-NVAS and Replay-NVAS. We report the magnitude spectrogram distance (Mag), left-right energy +ratio error (LRE), and RT60 error (RTE). Replay-NVAS does not have novel environment setup due to data being collected in a single +environment. For all metrics, lower is better. In addition to baselines, we also evaluate ViGAS w/o visual by removing the active speaker +localization and visual features. Note that reverberation time is mostly invariant of the receiver location in the same room and thus input +audio has low RTE. A good model should preserve this property while synthesizing the desired acoustics for the target viewpoint. +timating a transfer function directly from two audio clips +is non-trivial and noisy for low-energy parts of the signal. +DSP also performs badly despite having the ground truth +3D coordinates of the sound source. This is because head +related transfer functions are typically recorded in anechoic +chambers, which does not account for acoustics of differ- +ent environments, e.g., reverberation. Both traditional ap- +proaches perform worse than simply copying the input au- +dio, indicating that learning-based models are needed for +this challenging task. The recent model VAM [6] performs +much better compared to the traditional approaches but still +underperforms our model. There is a significant difference +between ViGAS w/o visual and the full model; this shows +that the visual knowledge about the speaker location and the +environment is important for this task. +Fig. 4 shows an example where given the same input +source viewpoint, our model synthesizes audio for three +different target viewpoints. The model reasons about how +the geometry and speaker locations changes based on the +source view and the target pose, and predicts the acoustic +difference accordingly. See Supp. video to listen to sounds. +For the novel environment setup, our model again out- +performs all baselines. Compared to ViGAS in the single +environment setup, both the magnitude spectrogram dis- +tance and the left-right energy ratio error increase. This is +expected because for novel (unseen) environments, single +images capture limited geometry and acoustic information. +The model fails sometime when there is a drastic viewpoint +change, e.g., target viewpoint 3 in Fig. 4. This setup re- +quires the model to reason or “imagine” the environment +based on single audio-visual observation, which poses great +challenge for NVAS as well as NVS, where typically syn- +thesis is performed in a fully observed environment. +Ablations. +Table 2 shows ablations on the model design. +To understand if the model uses visual information, we ab- +late the visual features VF and the active speaker feature +SS-NVAS +Replay-NVAS +ViGAS +Mag +LRE +Mag +LRE +full model +0.159 +0.782 +0.142 +0.716 +w/o visual features +0.171 +0.897 +0.146 +0.920 +w/o ASL +0.161 +0.814 +0.143 +0.757 +w/o alignment +0.176 +0.771 +0.144 +0.706 +w/o separation +0.165 +0.840 +0.182 +0.859 +Table 2. Ablations of the model on both datasets. +VL. Removing the active speaker feature leads to less dam- +age on the model performance, because without the explic- +itly localized active speaker, the model can still implicitly +reason about the active speaker location based on the im- +age and audio. If both are removed (“ViGAS w/o visual” in +Table 1), the performance suffers most. +To study the effectiveness of the temporal alignment and +ambient sound separation modules, we ablate them sepa- +rately. Removing the temporal alignment leads to higher +Mag error and slightly lower LRE. As for ambient sound +separation, the results show that optimizing for the high- +energy noise-like ambient sound degrades the performance. +6.2. Results on Replay-NVAS +Table 1 (right) shows the Replay-NVAS results. Com- +pared to SoundSpaces-NVAS, the magnitudes of all errors +are smaller because there are less drastic acoustic changes +between viewpoints (8 DLSR cameras form a circle around +the participants). Traditional approaches like TF Estima- +tor and DSP still perform poorly despite using the 3D co- +ordinates of the camera and the speaker (triangulated from +multiple cameras). VAM performs better due to end-to-end +learning; however, our model outperforms it. Compared to +ViGAS w/o visual, the full model has much lower left-right +energy ratio error and slightly higher reverberation time er- +ror, showing that the model takes into account the speaker +position and viewpoint change for synthesizing the audio. + +Source viewpoint +Target viewpoint 1 +Input +Prediction +Target viewpoint 2 +Target viewpoint 3 +Source viewpoint +Input +Target viewpoint +AV speech enhancement +Target +Prediction +Input/pred/target +Target +Target +Prediction +Target +Prediction +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +L +R +Figure 4. Qualitative examples. For all binaural audio, we show the left-channel and the right-channel waveforms side-by-side. Row +1: SoundSpaces-NVAS example where given the source viewpoint and input audio, the model synthesizes audio for three different target +viewpoints (target views are for reference only). In this case, the active speaker is the male speaker as indicated by the bounding box. +For target viewpoint 1, the view rotates about 90 degrees and the male speaker is on the left side and the predicted left channel is louder +than the right channel. Viewpoint 2 moves away from the speaker and thus yields lower amplitude compared to the first prediction. For +target viewpoint 3, it is completely located outside of the living room, in which case, the sound could only come from the door open on the +right (louder right channel) and the reverberation also greatly increases due to the vanishing direct sound. Row 2: Replay-NVAS example +where the speaker is located on the left in the source viewpoint which becomes the right and further from the camera in target viewpoint +2, the model also predicts lower amplitude and louder right channel. On the right side, we show an example of the audio-visual speech +enhancement for the active speaker. The model enhances the speech to largely match with the near-range audio (target). +Mag +RTE +Input +0.279 +0.376 +ViGAS (ours) +0.234 +0.122 +Table 3. Speech enhancement on Replay-NVAS. +Fig. 4 (row 2, left) shows a qualitative example. In the +source viewpoint, the active speaker is on the left, while in +the target viewpoint, he is further from the camera and on +the right. The model synthesizes an audio waveform that +captures the corresponding acoustic change, showing that +our model successfully learns from real videos. +Audio-visual speech enhancement. +In some real-world +applications, e.g., hearing aid devices, the goal is to obtain +the enhanced clean speech of the active speaker. This can +be seen as a special case of NVAS, where the target view- +point is the active speaker. Our model is capable of perform- +ing audio-visual speech enhancement without any modifica- +tion. We simply set the target audio to the near-range audio +recording for the active speaker. We show the results in Ta- +ble 3. Our model obtains cleaner audio compared to the +input audio (example in Fig. 4, row 2, right). +Human subject study. +To supplement the quantitative +metrics and evaluate how well our synthesized audio cap- +tures the acoustic change between viewpoints, we conduct +a human subject study. We show participants the image of +the target viewpoint VT as well as the audio AT as refer- +ence. We provide three audio samples: the input, the pre- +diction of ViGAS, and the prediction of DSP (the most nat- +Dataset +Input +DSP +ViGAS +SoundSpaces-NVAS +24% +2% +74% +Replay-NVAS +43% +6% +51% +Table 4. Human Study. Participants favor our approach over the +two most realistic sounding baselines, (1) copying the input signal, +and (2) a digital signal processing baseline. +urally sounding baseline) and ask them to select a clip that +sounds closest to the target audio. We select 20 examples +from SoundSpaces-NVAS and 20 examples from Replay- +NVAS and invite 10 participants to perform the study. +See Table 4 for the results. +On the synthetic dataset +SoundSpaces-NVAS, our approach is preferred over the +baselines by a large margin. This margin is lower on the +real-world Replay-NVAS dataset but is still significant. +7. Conclusion +We introduce the challenging novel-view acoustic syn- +thesis task and a related benchmark in form of both real and +synthetic datasets. We propose a neural rendering model +that learns to transform the sound from the source view- +point to the target viewpoint by reasoning about the ob- +served audio and visual stream. Our model surpasses all +baselines on both datasets. We believe this research un- +locks many potential applications and research in multi- +modal novel-view synthesis. In the future, we plan to incor- +porate active-speaker localization model into the approach +and let the model jointly learn to localize and synthesize. + ++光 +2References +[1] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisser- +man. The conversation: Deep audio-visual speech enhance- +ment. In INTERSPEECH, 2018. 3 +[2] Humam Alwassel, Dhruv Mahajan, Bruno Korbar, Lorenzo +Torresani, Bernard Ghanem, and Du Tran. Self-supervised +learning by cross-modal audio-video clustering. 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Vision-infused deep audio inpainting. In ICCV, 2019. +3 + +8. Supplementary Material +In this supplementary material, we provide additional +details about: +1. Supplementary video for qualitative assessment of our +model’s performance. +2. Replay-NVAS dataset details (referenced in Sec. 4.1 +of the main paper). +3. SoundSpaces-NVAS dataset details. +4. Implementation and training details (referenced in Sec. +5.6). +5. Baseline details (referenced in Sec. 6). +8.1. Supplementary Video +This video includes examples for the Replay-NVAS +dataset and the SoundSpaces-NVAS dataset as well our +model’s prediction on both datasets. Listen with a head- +phone for the spatial sound. +8.2. Replay-NVAS Dataset Details +Multi-view camera calibration. +We estimate camera +poses with COLMAP [49] Structure-from-Motion (SfM) +framework on each scene separately. Each scene is filmed +with 8 static DSLR cameras and 3 wearable GoPro cam- +eras (the latter are not used in our acoustic synthesis exper- +iments). We first run SfM on the segments of the GoPro +recordings where the wearers move significantly; followed +by registration of the static camera frames to the model and +a final round of bundle adjustment where we enforce con- +stant relative poses between static camera frames taken at +the same timestamp. This two-stage procedure greatly re- +duces the scale of the problem by making SfM focus first on +the most diverse part of the data. Upon feature extraction +stage, we cull the local features belonging to potentially dy- +namic object categories (such as persons or animals) as de- +tected by Detectron2 instance segmentation [57]. We then +exploit the stationarity of DSLRs by picking a medoid cam- +era pose among the frames filmed by each camera. Finally, +we rotate and scale the coordinate system so that Z axis is +pointing roughly upwards and the distances between cam- +eras match the approximate distances in centimeters. Fig. 5 +plots all camera coordinates and orientations projected to +XY plane. +Training data construction. +We align different DSLR +videos with the clapper sound, which gives us synchronized +multi-view audio-visual data. However, this data is not di- +rectly usable for training because some data are noisy (e.g., +people frequently talking over each other) or silent, which +leads to additional learning challenges for the model. Thus, +we design an automatic process for filtering out noisy clips. +More specifically, we first extract all one-second audio clips +X (cm) +Y (cm) +Figure 5. Camera coordinates estimated from COLMAP. +Figure 6. Environment mesh. +of all videos and obtain the corresponding near-range audio +clips and bounding boxes for each speaker. As described in +Sec. 5.2, we select the active speaker based on the maxi- +mum energy of near-range audio with ∆t = 0.2. For a one- +second video clip, we obtain 5 candidate bounding boxes. +We choose a threshold δ% and only keep clips where more +than δ% of the bounding boxes belong to the same person. +We set δ to 80. In this way, we keep clips where there is +one main speaker talking, and this speaker’s bounding box +is used as the localization feature VL. +8.3. SoundSpaces-NVAS Dataset Details +For the single environment experiment, we use an apart- +ment environment from the Gibson dataset [58]3. Fig. 6 +shows the mesh of the environment (the ceiling is removed). +For the novel environment experiment, we use the public +train/val/test splits. +For all images, we render with a resolution of 256 × 256 +and a field of view of 120 degrees. We render binaural audio +at a sample rate of 16000. +8.4. Implementation and Training Details +All audio clips during training are one second long with +a sample rate of 16000. The shape of AS and AT is thus +2 × 16000. The audio encoder is a conv1d layer that en- +codes audio from 2 channel (binaural) to latent features of +3http://gibsonenv.stanford.edu/models/?id=Oyens + +800 +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +300 +400 +500 +60064 channels, i.e., Ak +F is of shape 64 × 16000. For acoustic +synthesis, we have M = 30 gated multi-modal fusion lay- +ers, which are equally divided into 3 blocks. In each block, +the dilation of the dilated conv1d increases exponentially +with base 3. The kernel size for each dilated conv1d is also +3. Both the skip and residual layers are conv1d layers with +kernel size 1. The decoder network is a conv1d layer that +encodes the latent audio features from 64 channels back to +2 channels. +The image resolutions are downsampled to 216 × 384 +and 256 × 256 for Replay-NVAS (downsampled) and +SoundSpaces-NVAS respectively. After being processed by +a cond1d layer and flattened, the output visual feature VF +is of size 672 for Replay-NVAS and 512 for SoundSpaces- +NVAS. The fusion layer consists of two fully connected lay- +ers with the first output dimension being 512 and the second +being 256. +We +train +all +models +for +1000 +epochs +on +the +SoundSpaces-NVAS dataset and for 600 epochs +on +the Replay-NVAS dataset with a learning rate of 0.001. We +evaluate the checkpoint with the lowest validation loss on +the test set. +8.5. Baseline Details +For the Digital Signal Processing (DSP) baseline, we use +the head-related transfer function (HRTF) measured by a +KEMAR Dummy-Head Binaural Microphone. We apply a +Wiener filter [56] to estimate the inverse HRTF. We adjust +the gain of the HRTF by performing a binary search on the +validation dataset and selecting the best gain value for test- +ing. For the VAM [6] baseline, we take the original model +from the paper, and we make minimal modifications by con- +catenating the visual feature with the target viewpoint pose +PT . We train the model with the same hyper-parameters +described in the paper until convergence on both datasets. + diff --git a/bNFAT4oBgHgl3EQf5B4w/content/tmp_files/load_file.txt b/bNFAT4oBgHgl3EQf5B4w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..550f80c794bcabec953c832d40664f048888d32b --- /dev/null +++ b/bNFAT4oBgHgl3EQf5B4w/content/tmp_files/load_file.txt @@ -0,0 +1,1556 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf,len=1555 +page_content='Novel-View Acoustic Synthesis Changan Chen1,3 Alexander Richard2 Roman Shapovalov3 Vamsi Krishna Ithapu2 Natalia Neverova3 Kristen Grauman1,3 Andrea Vedaldi3 1University of Texas at Austin 2Reality Labs Research at Meta 3FAIR Abstract We introduce the novel-view acoustic synthesis (NVAS) task: given the sight and sound observed at a source view- point, can we synthesize the sound of that scene from an unseen target viewpoint?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' We propose a neural rendering approach: Visually-Guided Acoustic Synthesis (ViGAS) net- work that learns to synthesize the sound of an arbitrary point in space by analyzing the input audio-visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' To benchmark this task, we collect two first-of-their-kind large-scale multi-view audio-visual datasets, one synthetic and one real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' We show that our model successfully reasons about the spatial cues and synthesizes faithful audio on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' To our knowledge, this work represents the very first formulation, dataset, and approach to solve the novel- view acoustic synthesis task, which has exciting potential applications ranging from AR/VR to art and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' Un- locked by this work, we believe that the future of novel-view synthesis is in multi-modal learning from videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' Introduction Replaying a video recording from a new viewpoint1 has many applications in cinematography, video enhancement, and virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' For example, it can be used to edit a video, simulate a virtual camera, or, given a video of a per- sonal memory, even enable users to experience a treasured moment again—not just on a 2D screen, but in 3D in a vir- tual or augmented reality, thus ‘reliving’ the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' While the applications are exciting, there are still many unsolved technical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' Recent advances in 3D re- construction and novel-view synthesis (NVS) address the problem of synthesizing new images of a given scene [31, 33, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' However, thus far, the view synthesis problem is concerned with creating visuals alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' the output is silent or at best naively adopts the sounds of the original video (from the “wrong” viewpoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' Without sound, the emotional and cognitive significance of the replay is severely diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' In this work, we address this gap and introduce the new task of novel-view acoustic synthesis (NVAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQf5B4w/content/2301.08730v1.pdf'} +page_content=' The goal of 1We use “viewpoint” to mean a camera or microphone pose.' metadata={'source': 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(A.8) in appendix A. (Note, however, +that the formulae presented below are not limited to free-particle theories. They are robust against any +perturbations unless boundary conditions (8) are changed.) Below we shall construct time-evolution +kernels for a single walker on a circle, the half line, and a finite interval by gauging these discrete +symmetries. +Example 1. (A single walker on a circle) +Let us first consider a single walker on a periodic lattice +of 퐿 sites, {1, 2, ⋯ , 퐿 (mod 퐿)}. This lattice can be constructed from Z by making the identification +푥 ∼ 푥 + 푛퐿, where 푛 is an arbitrary integer. Hence the configuration space is the orbit space Z/퐿Z, +where 퐿Z = ⟨푡 ∣ ∅⟩ is the free group generated by a translation 푡. Its action on Z is defined by +푡푥 ≔ 푥 + 퐿. +(13) +Note that any element of 퐿Z can be written as the product 푡푛, whose action on Z is given by 푡푛푥 = 푥+푛퐿. +Now we need to find out one-dimensional unitary representations of 퐿Z. Since 퐿Z is the free group +generated by a single generator 푡, we have a one-parameter family of maps 퐷[휃] ∶ 퐿Z → 푈(1) labeled +by an angle parameter 휃: +퐷[휃](푡) = e푖휃, +(14) +6 + +where 휃 ∈ R/2휋R. It then follows from the formula (4) that the time-evolution kernel for a single +walker on Z/퐿Z takes the following form: +푈 [휃] +휏 (푥, 푦) = +∞ +∑ +푛=−∞ +퐷[휃](푡푛)̃푈휏(푥, 푡푛푦) += +∞ +∑ +푛=−∞ +e푖푛휃 ̃푈휏(푥, 푦 + 푛퐿). +(15) +Just as in the path integral on a circle (see, e.g., section 2.4 of ref. [24]), eq. (15) represents the summation +over winding numbers. Physically, eq. (15) describes the situation in which the walker acquires the +Aharonov-Bohm phase e푖휃 every time it winds around the circle, where 휃 plays the role of a magnetic +flux penetrating through the circle. This is the physical meaning of the weight factor (14) and the +summation over the orbit of initial point. +Now two remarks are in order. First, it follows from eq. (8) that 푈 [휃] +휏 ( ⋅ , ⋅ ) satisfies the identity +푈 [휃] +휏 (푥 + 퐿, 푦) = e푖휃 푈 [휃] +휏 (푥, 푦); that is, it satisfies the twisted boundary conditions 푈 [휃] +휏 (퐿 + 1, 푦) = +e푖휃 푈 [휃] +휏 (1, 푦) and 푈 [휃] +휏 (0, 푦) = e−푖휃 푈 [휃] +휏 (퐿, 푦). Namely, eq. (15) gives the universal formula of the time- +evolution kernel for a single walker on a circle subject to these twisted boundary conditions. +The second remark is that, under the reflection, eq. (15) satisfies 푈 [휃] +휏 (푧 − 푥, 푧 − 푦) = 푈 [−휃] +휏 +(푥, 푦). +Hence, at 휃 = 0 or 휋 (mod 2휋), eq. (15) becomes reflection invariant. We can use this invariance for +the construction of time-evolution kernels on a finite interval; see example 3. +Example 2. (A single walker on the half line) +Let us next consider a single walker on a semi-infinite +lattice {1, 2, ⋯}. This lattice can be constructed from the integer lattice Z by making the identification +푥 ∼ 1 − 푥. Hence the configuration space is the orbit space Z/Z2, where Z2 = ⟨푟 ∣ 푟2 = 푒⟩ is the cyclic +group of order 2. Here 푟 is the reflection whose action on Z is defined by +푟푥 ≔ 1 − 푥. +(16) +Note that 푟2푥 = 푥. Note also that the reflection (16) does not have a fixed point in the integer lattice. +(Its fixed point is 푥 = 1/2.) +Now, since 푟2 = 푒, any one-dimensional unitary representation 퐷 ∶ Z2 → 푈(1) must satisfy the +condition 퐷(푟)2 = 1, whose solution is 퐷(푟) = ±1. Hence there exist two distinct maps 퐷[휙] given by +퐷[휙](푟) = e푖휙, +(17) +where 휙 ∈ {0, 휋 (mod 2휋)}. Correspondingly, there exist the following two distinct time-evolution +kernels for a single walker on Z/Z2: +푈 [휙] +휏 +(푥, 푦) = +1 +∑ +푛=0 +퐷[휙](푟푛)̃푈휏(푥, 푟푛푦) += ̃푈휏(푥, 푦) + e푖휙 ̃푈휏(푥, 1 − 푦). +(18) +Again, just as in the path integral on the half line [21, 25, 26], eq. (18) represents the summation over +bouncing numbers off the boundary: the 푛 = 0 term is the contribution from the direct path, while the +푛 = 1 term is the contribution from the reflected path off the boundary. The physical meaning of the +weight factor (17) is now clear: it plays the role of the reflection amplitude off the boundary. In other +words, the walker acquires the phase shift 휙 when reflected from the boundary. +Notice that eq. (18) satisfies the identity 푈 [휙] +휏 +(1−푥, 푦) = e푖휙 푈 [휙] +휏 +(푥, 푦); that is, it satisfies the bound- +ary condition 푈 [휙] +휏 +(0, 푦) = e푖휙 푈 [휙] +휏 +(1, 푦). Hence, eq. (18) gives the universal form of the time-evolution +kernel for a single walker on the half line subject to this boundary condition. We emphasize that, as +noted at the end of section 2.1, if one wants a theory subject to the Dirichlet boundary condition at +푥 = 0, one should consider the reflection defined by 푟푥 ≔ −푥 and choose the representation 휙 = 휋. In +this case, one arrives at the formula 푈 [휙=휋] +휏 +(푥, 푦) = ̃푈휏(푥, 푦) − ̃푈휏(푥, −푦) which satisfies 푈 [휙=휋] +휏 +(0, 푦) = 0. +7 + +Example 3. (A single walker on a finite interval) +Let us next consider a single walker on a finite +interval of 퐿 sites, {1, 2, ⋯ , 퐿}. This lattice can be constructed from Z by making the identifications +푥 ∼ 푥 + 2푛퐿 and 푥 ∼ 2푛퐿 + 1 − 푥, where 푛 is an arbitrary integer. Hence, the configuration space is +the orbit space Z/퐷∞, where 퐷∞ = Z ⋊ Z2 = ⟨푡, 푟 ∣ 푟2 = 푒, 푟푡푟 = 푡−1⟩ is the infinite dihedral group +generated by a translation 푡 and a reflection 푟.4 The actions of these operators on Z are defined as +follows: +푡푥 ≔ 푥 + 2퐿 +and +푟푥 ≔ 1 − 푥. +(19) +Note that any element of 퐷∞ can be written as 푡푛푟푚, where 푛 = 0, ±1, ±2, ⋯ and 푚 = 0, 1. The action +of this operator on Z is given by 푡푛푟푚푥 = 푥 + 2푛퐿 for 푚 = 0 and 푡푛푟푚푥 = 2푛퐿 + 1 − 푥 for 푚 = 1, +respectively. Note also that, in contrast to the previous examples, 퐷∞ is a non-Abelian discrete group. +Now, since 푟2 = 푒 and 푟푡푟 = 푡−1, any one-dimensional unitary representation 퐷 ∶ 퐷∞ → 푈(1) +must satisfy the conditions 퐷(푟)2 = 1 and 퐷(푟)퐷(푡)퐷(푟) = 퐷(푡)−1, which leads to 퐷(푡)2 = 1. Thus we +have 퐷(푡) = ±1 and 퐷(푟) = ±1; that is, there exist 22 = 4 distinct maps 퐷[휃,휙] given by +퐷[휃,휙](푡) = e푖휃 +and +퐷[휃,휙](푟) = e푖휙, +(20) +where 휃, 휙 ∈ {0, 휋 (mod 2휋)}. Correspondingly, there exist the following four distinct time-evolution +kernels for a single walker on Z/퐷∞: +푈 [휃,휙] +휏 +(푥, 푦) = +∞ +∑ +푛=−∞ +1 +∑ +푚=0 +퐷[휃,휙](푡푛푟푚)̃푈휏(푥, 푡푛푟푚푦) += +∞ +∑ +푛=−∞ +[e푖푛휃 ̃푈휏(푥, 푦 + 2푛퐿) + e푖푛휃 e푖휙 ̃푈휏(푥, 2푛퐿 + 1 − 푦)] . +(21) +Once again, just as in the path integral on a finite interval [22, 27–29], eq. (21) represents the summation +over bouncing numbers off the two boundaries. Physically, e푖휙 and e푖(휃+휙) play the roles of the reflection +amplitudes off the boundaries 푥 = 1 and 푥 = 퐿, respectively. +Now, it follows from eq. (8) that eq. (21) satisfies the identities 푈 [휃,휙] +휏 +(푥 +2퐿, 푦) = e푖휃 푈 [휃,휙] +휏 +(푥, 푦) and +푈 [휃,휙] +휏 +(1 − 푥, 푦) = e푖휙 푈 [휃,휙] +휏 +(푥, 푦), which implies the boundary conditions 푈 [휃,휙] +휏 +(0, 푦) = e푖휙 푈 [휃,휙] +휏 +(1, 푦) +and 푈 [휃,휙] +휏 +(퐿 + 1, 푦) = e푖(휃+휙) 푈 [휃,휙] +휏 +(퐿, 푦). This means that eq. (21) gives the universal form of the time- +evolution kernel for a single walker on the finite interval subject to these boundary conditions. If one +wants a theory that satisfies the Dirichlet boundary conditions at 푥 = 0 and 푥 = 퐿 + 1, one should +redefine the translation and reflection as 푡푥 ≔ 푥 + 2(퐿 + 1) and 푟푥 ≔ −푥, respectively, and choose +the representation 휙 = 휋. In this case, one obtains 푈 [휃,휙=휋] +휏 +(푥, 푦) = ∑∞ +−∞ e푖푛휃[̃푈휏(푥, 푦 + 2푛(퐿 + 1)) − +̃푈휏(푥, 2푛(퐿 + 1) − 푦)] which satisfies 푈 [휃,휙=휋] +휏 +(0, 푦) = 0 and 푈 [휃,휙=휋] +휏 +(퐿 + 1, 푦) = 0. +We note in closing that eq. (21) can also be obtained from the time-evolution kernel on a circle +(15) by gauging the reflection invariance at 휃 = 0, 휋 (mod 2휋). In fact, eq. (21) can be written as +푈 [휃,휙] +휏 +(푥, 푦) = ∑1 +푚=0 퐷[휙](푟푚)푈 [휃] +휏 (푥, 푟푚푦) = ∑1 +푚=0 ∑∞ +푛=−∞ 퐷[휙](푟푚)퐷[휃](푡푛)̃푈휏(푥, 푡푛푟푚푦), where 퐷[휙] is the +one-dimensional unitary representation of Z2 given by eq. (17). An important lesson from this example +is that there could exist several ways to construct time-evolution kernels on orbit spaces. +3.2 +Identical walkers in one dimension +Now let us turn to the problem of multiple identical walkers on a lattice. The key to this problem +is the indistinguishability of identical particles, where physical observables must be invariant under +permutations of multiparticle coordinates. As is well known, this indistinguishability always makes the +multiparticle configuration space an orbit space [13, 17–20]. The basic idea behind this is to regard the +permutation invariance as a gauge symmetry (i.e., redundancy in description). From this perspective, +4The infinite dihedral group can also be written as the free product 퐷∞ ≅ Z2 ∗ Z2 = ⟨푟, 푟′ ∣ 푟2 = 푒, 푟′2 = 푒⟩, where 푟′(= 푡푟) +is another reflection defined by 푟′푥 ≔ 2퐿 + 1 − 푥. +8 + +the configuration space must be a collection of inequivalent gauge orbits because gauge-equivalent +configurations are physically equivalent. +To date, there exist two distinct formulations of this idea in identical-particle problems. The first +regards the configuration space of 푁 identical particles as the orbit space (푋 푁 − Δ푁 )/푆푁 , where 푋 푁 is +the 푁-fold Cartesian product of a single-particle configuration space 푋 and Δ푁 ⊂ 푋 푁 is the set of fixed +points under the action of the symmetric group 푆푁 [13, 17–19]. On the other hand, the second includes +the fixed points and regards the configuration space as the orbit space 푋 푁 /푆푁 [20]. The difference +between these two formulations is very subtle (especially in lattices) and beyond the scope of this +note. Fortunately, however, we can circumvent this issue and solve the 푁-identical-walker problems +as follows. +Suppose that 푋 itself is a nontrivial orbit space and takes the form 푋 = ̃푋/퐺, where 퐺 is a discrete +group whose action on ̃푋 has no fixed points. In this case, the configuration space can also be written +as (̃푋 푁 − ̃Δ푁)/(퐺 ≀ 푆푁 ) or ̃푋 푁 /(퐺 ≀ 푆푁 ).5 Here ≀ stands for the wreath product defined by the semidirect +product 퐺 ≀ 푆푁 ≔ 퐺푁 ⋊ 푆푁 and ̃Δ푁 ⊂ ̃푋 푁 is the set of fixed points of 푆푁 . Hence, irrespective of the +formulations, once given a time-evolution kernel on Λ = ̃푋 푁 − ̃Δ푁 or ̃푋 푁 , the problem just reduces to +the classification of one-dimensional unitary representations of the discrete group Γ = 퐺 ≀ 푆푁 . +In this section, we shall focus on the cases 푋 = Z, Z/퐿Z, Z/Z2, and Z/퐷∞ and construct time- +evolution kernels for 푁 identical walkers on the infinite line, a circle, the half line, and a finite interval. +In the following, ̃푈휏(푥, 푦) represents a time-evolution kernel on Z푁 − ̃Δ푁 or Z푁 that satisfies the trans- +lation invariance, reflection invariance, and permutation invariance. +Example 4. (푁 identical walkers on the infinite line) +Let us first consider 푁 identical walkers on +the integer lattice Z. In this case, the discrete group Γ = 푆푁 is just the symmetric group of order 푁!, +whose presentation is +푆푁 = ⟨휎1, ⋯ , 휎푁−1 || 휎2 +푖 = 푒, 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, 휎푖휎푗 = 휎푗휎푖 (|푖 − 푗| ≥ 2)⟩ . +(22) +Here 휎푖 = (푖, 푖+1) is the adjacent transposition that interchanges 푖 and 푖+1. Its action on 푥 = (푥1, ⋯ , 푥푁 ) ∈ +Z푁 is defined as follows: +휎푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖+1, 푥푖, 푥푖+2, ⋯ , 푥푁 ). +(23) +An arbitrary element 휎 ∈ 푆푁 can be written as a product of the generators 휎1, ⋯ , 휎푁−1. Its action on +푥 = (푥1, ⋯ , 푥푁 ) can be written as 휎푥 = (푥휎(1), ⋯ , 푥휎(푁)), where 휎(푖) stands for the permutation of 푖 under +휎. +Now, there exist two distinct one-dimensional unitary representations of 푆푁 : the trivial represen- +tation and the sign representation. Though this result is well known, let us reproduce it here just for +later convenience. Since 휎2 +푖 = 푒 and 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, any one-dimensional unitary representation +퐷 ∶ 푆푁 → 푈(1) must satisfy the conditions 퐷(휎푖)2 = 1 and 퐷(휎푖)퐷(휎푖+1)퐷(휎푖) = 퐷(휎푖+1)퐷(휎푖)퐷(휎푖+1), +whose solutions are 퐷(휎푖) = ±1 and 퐷(휎푖) = 퐷(휎푖+1). Hence we have 퐷(휎1) = ⋯ = 퐷(휎푁−1) = ±1; that is, +there exist two distinct maps 퐷[±] given by +퐷[±](휎) = (±1)#휎, +(24) +where #휎 stands for the number of adjacent transpositions in the permutation 휎. In the standard +terminology, 퐷[+] is the trivial representation and 퐷[−] is the sign representation.6 Correspondingly, +5Here is the proof. First, the wreath product 퐺 ≀ 푆푁 = 퐺푁 ⋊ 푆푁 can be written as the set {푔휎 ∶ 푔 ∈ 퐺푁, 휎 ∈ 푆푁} equipped +with the group composition law (푔휎)(푔′휎′) = (푔휎푔′휎−1)(휎휎′) for any 푔, 푔′ ∈ 퐺푁 and 휎, 휎′ ∈ 푆푁 . Here 푔 ↦ 휎푔휎−1 is the +automorphism of the 푁 -fold direct-product group 퐺푁 = 퐺 × ⋯ × 퐺 defined by 휎푔휎−1 ≔ 푔휎(1) ⋯ 푔휎(푁) for any 푔 = 푔1 ⋯ 푔푁 ∈ +퐺 × ⋯ × 퐺. It is now obvious that first making the identification 푥 ∼ 푔푥 by 푔 ∈ 퐺푁 in ̃푋 푁 and then making the identification +푥 ∼ 휎푥 by 휎 ∈ 푆푁 in ̃푋 푁 /퐺푁 is equivalent to making the identification 푥 ∼ 휎푔푥 by 휎푔 = (휎푔휎−1)휎 ∈ 퐺 ≀ 푆푁 in ̃푋 푁 . Hence +(̃푋 푁 /퐺푁)/푆푁 is equivalent to ̃푋 푁/(퐺 ≀ 푆푁). By subtracting the set of fixed points of 푆푁, we also see that (̃푋 푁 /퐺푁 − Δ푁)/푆푁 is +equivalent to (̃푋 푁 − ̃Δ푁)/(퐺 ≀ 푆푁). See also refs. [20, 30] for similar results in continuous spaces. +6The sign representation can also be written as 퐷[−](휎) = sgn(휎), where sgn(휎) stands for the signature of 휎. It is defined +by sgn(휎) = ±1 for even (odd) permutations. +9 + +there exist the following two distinct time-evolution kernels for 푁 identical walkers on Z: +푈 [±] +휏 (푥, 푦) = ∑ +휎∈푆푁 +퐷[±](휎)̃푈휏(푥, 휎푦) += ∑ +휎∈푆푁 +(±1)#휎 ̃푈휏(푥, 휎푦). +(25) +Notice that eq. (25) satisfies the identity 푈 [±] +휏 (휎푥, 푦) = (±1)#휎푈 [±] +휏 (푥, 푦). The weight factors (24) thus +describe particle-exchange phases under the permutation of identical particles. It is now obvious that +the two distinct representations 퐷[±] correspond to two distinct particle statistics: 푈 [+] +휏 +describes the +time-evolution kernel for 푁 identical bosons, while 푈 [−] +휏 +describes that for 푁 identical fermions. +Example 5. (푁 identical walkers on a circle) +Let us next consider 푁 identical particleson the periodic +lattice of 퐿 sites. In this case, the discrete group is the wreath product Γ = 퐿Z ≀ 푆푁 , whose presentation +is given by +퐿Z ≀ 푆푁 = ⟨ +푡1, ⋯ , 푡푁 , +휎1, ⋯ , 휎푁−1 +||||||| +푡푖푡푗 = 푡푗푡푖, +휎2 +푖 = 푒, +휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, +휎푖휎푗 = 휎푗휎푖 +(|푖 − 푗| ≥ 2), +휎푖푡푖휎푖 = 푡푖+1, +휎푖푡푗휎푖 = 푡푗 +(푗 ≠ 푖, 푖 + 1) +⟩ . +(26) +Here the actions of the generators 푡푖 and 휎푖 are defined by eq. (23) and +푡푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖 + 퐿, 푥푖+1, ⋯ , 푥푁 ). +(27) +Note that any element of 퐿Z≀푆푁 can be written as 푡푛1 +1 ⋯ 푡푛푁 +푁 휎, where 휎 is a permutation and 푛1, ⋯ , 푛푁 = +0, ±1, ⋯. Its action on 푥 = (푥1, ⋯ , 푥푁 ) is given by 푡푛1 +1 ⋯ 푡푛푁 +푁 휎푥 = (푥휎(1) + 푛1퐿, ⋯ , 푥휎(푁) + 푛푁 퐿). +Now we have to classify one-dimensional unitary representation 퐷 ∶ 퐿Z ≀ 푆푁 → 푈(1). First, the +relations 휎2 +푖 = 푒 and 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1 imply 퐷(휎1) = ⋯ = 퐷(휎푁 ) = ±1. Second, the relation 휎푖푡푖휎푖 = +푡푖+1 implies 퐷(휎푖)퐷(푡푖)퐷(휎푖) = 퐷(푡푖+1), which, together with 퐷(휎푖)2 = 1, leads to 퐷(푡1) = ⋯ = 퐷(푡푁 ) = e푖휃, +where 휃 ∈ R/2휋R. Thus we have two distinct one-parameter families of the maps 퐷[휃,±] given by +퐷[휃,±](푡푛1 +1 ⋯ 푡푛푁 +푁 휎) = e푖(푛1+⋯+푛푁 )휃(±1)#휎. +(28) +The time-evolution kernel for 푁 identical walkers on Z/퐿Z is therefore +푈 [휃,±] +휏 +(푥, 푦) = +∞ +∑ +푛1=−∞ +⋯ +∞ +∑ +푛푁 =−∞ +∑ +휎∈푆푁 +퐷[휃,±](푡푛1 +1 ⋯ 푡푛푁 +푁 휎)̃푈휏(푥, 푡푛1 +1 ⋯ 푡푛푁 +푁 휎푦) += +∞ +∑ +푛1=−∞ +⋯ +∞ +∑ +푛푁 =−∞ +e푖(푛1+⋯+푛푁 )휃 푈 [±] +휏 (푥, 푡푛1 +1 ⋯ 푡푛푁 +푁 푦), +(29) +where 푈 [±] +휏 +is given by eq. (25). Notice that the kernel (29) satisfies the identities 푈 [휃,±] +휏 +(휎푥, 푦) = +(±1)#휎푈 [휃,±] +휏 +(푥, 푦) and 푈 [휃,±] +휏 +(푡푖푥, 푦) = e푖휃 푈 [휃,±] +휏 +(푥, 푦) for any 푖 = 1, ⋯ , 푁. Physically, 푈 [휃,+] +휏 +(푈 [휃,−] +휏 +) de- +scribes the system of 푁 identical bosons (fermions) on a circle with a nonzero magnetic flux. +Example 6. (푁 identical walkers on the half line) +Let us next consider 푁 identical particles on the +semi-infinite lattice. In this case, the discrete group is Γ = Z2 ≀ 푆푁 , where +Z2 ≀ 푆푁 = ⟨ +푟1, ⋯ , 푟푁 , +휎1, ⋯ , 휎푁−1 +||||||| +푟푖푟푗 = 푟푗푟푖, +푟2 +푖 = 휎2 +푖 = 푒, +휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, +휎푖휎푗 = 휎푗휎푖 +(|푖 − 푗| ≥ 2), +휎푖푟푖휎푖 = 푟푖+1, +휎푖푟푗휎푖 = 푟푗 +(푗 ≠ 푖, 푖 + 1) +⟩ . +(30) +The actions of the generators are defined by eq. (23) and +푟푖푥 ≔ (푥1, ⋯ , 푥푖−1, 1 − 푥푖, 푥푖+1, ⋯ , 푥푁 ). +(31) +10 + +Note that any element of Z2≀푆푁 can be written as the product 푟푛1 +1 ⋯ 푟푛푁 +푁 휎, where 휎 ∈ 푆푁 and 푛1, ⋯ , 푛푁 = +0, 1. Its action on 푥 = (푥1, ⋯ , 푥푁 ) is given by 푟푛1 +1 ⋯ 푟푛푁 +푁 휎푥 = (⋯ , 푥휎(푖), ⋯) for 푛푖 = 0 and 푟푛1 +1 ⋯ 푟푛푁 +푁 휎푥 = +(⋯ , 1 − 푥휎(푖), ⋯) for 푛푖 = 1. +By repeating the same procedure as above, one can show that one-dimensional unitary represen- +tation 퐷 ∶ Z2 ≀ 푆푁 → 푈(1) must satisfy 퐷(푟1) = ⋯ = 퐷(푟푁 ) = ±1 and 퐷(휎1) = ⋯ = 퐷(휎푁−1) = ±1. Hence +there exist 22 = 4 distinct maps 퐷[휙,±] given by +퐷[휙,±](푟푛1 +1 ⋯ 푟푛푁 +푁 휎) = e푖(푛1+⋯+푛푁 )휙(±1)#휎, +(32) +where 휙 ∈ {0, 휋 (mod 2휋)}. The time-evolution kernel for 푁 identical walkers on Z/Z2 is therefore +푈 [휙,±] +휏 +(푥, 푦) = +1 +∑ +푛1=0 +⋯ +1 +∑ +푛푁 =0 +∑ +휎∈푆푁 +퐷[휙,±](푟푛1 +1 ⋯ 푟푛푁 +푁 휎)̃푈휏(푥, 푟푛1 +1 ⋯ 푟푛푁 +푁 휎푦) += +1 +∑ +푛1=0 +⋯ +1 +∑ +푛푁 =0 +e푖(푛1+⋯+푛푁 )휙 푈 [±] +휏 (푥, 푟푛1 +1 ⋯ 푟푛푁 +푁 푦). +(33) +Notice that eq. (33) satisfies 푈 [휙,±] +휏 +(휎푥, 푦) = (±1)#휎푈 [휙,±] +휏 +(푥, 푦) and 푈 [휙,±] +휏 +(푟푖푥, 푦) = e푖휙 푈 [휙,±] +휏 +(푥, 푦) for any +푖 = 1, ⋯ , 푁. Hence, 푈 [휙,±] +휏 +describes the system of 푁 identical bosons (fermions) that acquire the phase +shift 휙 when reflected off the boundary. +Example 7. (푁 identical walkers on a finite interval) +Let us finally consider 푁 identical particles on +a finite interval. In this case, the discrete group is Γ = 퐷∞ ≀ 푆푁 , where +퐷∞ ≀ 푆푁 = ⟨ +푡1, ⋯ , 푡푁 , +푟1, ⋯ , 푟푁 , +휎1, ⋯ , 휎푁−1 +||||||||| +푡푖푡푗 = 푡푗푡푖, +푟푖푟푗 = 푟푗푟푖, +푟2 +푖 = 휎2 +푖 = 푒, +푟푖푡푖푟푖 = 푡−1 +푖 , +푟푖푡푗푟푖 = 푡푗 +(푗 ≠ 푖), +휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, +휎푖휎푗 = 휎푗휎푖 +(|푖 − 푗| ≥ 2), +휎푖푟푖휎푖 = 푟푖+1, +휎푖푟푗휎푖 = 푟푗 +(푗 ≠ 푖, 푖 + 1) +⟩ . +(34) +The actions of the generators are given by eqs. (23), (31), and +푡푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖 + 2퐿, 푥푖+1, ⋯ , 푥푁 ). +(35) +We note that any element of 퐷∞ ≀ 푆푁 can be written as the product 푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎, where 휎 ∈ 푆푁 , +푛1, ⋯ , 푛푁 = 0, ±1, ±2, ⋯, and 푚1, ⋯ , 푚푁 = 0, 1. Its action is given by 푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎푥 = (⋯ , 푥휎(푖) + +2푛푖퐿, ⋯) for 푚푖 = 0 and 푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎푥 = (⋯ , 2푛푖퐿 + 1 − 푥휎(푖), ⋯) for 푚푖 = 1. +Now it is a straightforward exercise to show that there exist 23 = 8 distinct one-dimensional unitary +representations of the wreath product 퐷∞ ≀ 푆푁 . The result is the following maps: +퐷[휃,휙,±](푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎) = e푖(푛1+⋯+푛푁 )휃 e푖(푚1+⋯+푚푁 )휙(±1)#휎, +(36) +where 휃, 휙 ∈ {0, 휋 (mod 2휋)}. Correspondingly, we have the following eight distinct time-evolution +kernels for 푁 identical walkers on Z/퐷∞: +푈 [휃,휙,±] +휏 +(푥, 푦) = +∞ +∑ +푛1=−∞ +1 +∑ +푚1=0 +⋯ +∞ +∑ +푛푁 =−∞ +1 +∑ +푚푁 =0 +∑ +휎∈푆푁 +퐷[휃,휙,±](푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎)̃푈휏(푥, 푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 휎푦) += +∞ +∑ +푛1=−∞ +1 +∑ +푚1=0 +⋯ +∞ +∑ +푛푁 =−∞ +1 +∑ +푚푁 =0 +e푖(푛1+⋯+푛푁 )휃 e푖(푚1+⋯+푚푁)휙 푈 [±] +휏 (푥, 푡푛1 +1 푟푚1 +1 +⋯ 푡푛푁 +푁 푟푚푁 +푁 푦). +(37) +Physically, 푈 [휃,휙,±] +휏 +describes the system of 푁 identical bosons (fermions) that acquire the phase shifts +휙 and 휃 + 휙 when reflected off the boundaries 푥 = 1 and 푥 = 퐿, respectively. +11 + +4 +Asides +Now, there exist several other quantities that can be constructed in exactly the same way as for the +time-evolution kernel (4). Examples include the resolvent kernel (Green’s function) and the canonical +density matrix (density matrix in the canonical ensemble). Another example is a unitary representation +of an arbitrary group 퐺 on a (tensor-product) Hilbert space, which includes the time-evolution kernel in +discrete-time quantum walk. In this section, we shall briefly discuss the construction of these quantities +on the orbit space Λ/Γ. +4.1 +Resolvent kernel +Let us first start with the resolvent kernel—a matrix element of the resolvent operator in position +space. Let 퐻 be the Hamiltonian operator of the system. Then, the resolvent operator 퐺퐸 = (퐸퐼 − 퐻)−1 +for Im 퐸 > 0 and the time-evolution operator 푈휏 = e−푖퐻휏 for 휏 > 0 are transformed into one another +through the Laplace transform 푖(퐸퐼 −퐻)−1 = ∫ ∞ +0 푑휏 e−푖퐻휏 e푖퐸휏 and the inverse Laplace transform e−푖퐻휏 = +∫ ∞+푖휖 +−∞+푖휖 +푑퐸 +2휋 푖(퐸퐼 − 퐻)−1 e−푖퐸휏, respectively, where 휖 is an arbitrary positive real. Consequently, the matrix +elements 푈휏(푥, 푦) = ⟨푥|푈휏|푦⟩ and 퐺퐸(푥, 푦) = ⟨푥|퐺퐸|푦⟩ are mutually related through the following: +푖퐺퐸(푥, 푦) = ∫ +∞ +0 +푑휏 푈휏(푥, 푦) e푖퐸휏 +for +Im 퐸 > 0, +(38a) +푈휏(푥, 푦) = ∫ +∞+푖휖 +−∞+푖휖 +푑퐸 +2휋 푖퐺퐸(푥, 푦) e−푖퐸휏 +for +휏 > 0. +(38b) +Hence, by applying the Laplace transform to the formula (4), we find that the resolvent kernel on Λ/Γ +takes the following form: +퐺퐸(푥, 푦) = ∑ +훾∈Γ +퐷(훾) ̃퐺퐸(푥, 훾푦), +(39) +where 푖 ̃퐺퐸(푥, 푦) = ∫ ∞ +0 푑휏 ̃푈휏(푥, 푦) e푖퐸휏 (Im 퐸 > 0) is the resolvent kernel on Λ. +An immediate application of the above formula is the local density of states given by 휌퐸(푥) = +⟨푥|훿(퐸퐼 − 퐻)|푥⟩. In fact, by using the identity +lim +Im 퐸→0+(퐸퐼 − 퐻)−1 = P(퐸퐼 − 퐻)−1 − 푖휋훿(퐸퐼 − 퐻), +(40) +where Pstands for the Cauchy principal value, we find Im 퐺퐸(푥, 푥) = Im⟨푥|(퐸퐼 −퐻)−1|푥⟩ = −휋⟨푥|훿(퐸퐼 − +퐻)|푥⟩ = −휋휌퐸(푥) in the limit Im 퐸 → 0+. Thus, +휌퐸(푥) = − 1 +휋 Im ∑ +훾∈Γ +퐷(훾) ̃퐺퐸(푥, 훾푥) +as +Im 퐸 → 0+. +(41) +The density of states 휌퐸 = tr 훿(퐸퐼 − 퐻) then takes the form 휌퐸 = −(1/휋) Im ∑푥∈Λ/Γ ∑훾∈Γ 퐷(훾) ̃퐺퐸(푥, 훾푥). +4.2 +Canonical density matrix +Let us next consider the canonical density matrix on Λ/Γ. In thermal equilibriumat temperature 훽−1, the +canonical density matrix is given by 휌훽 = 푈−푖훽/푍(훽), where 푈−푖훽 = e−훽퐻 is the Gibbs operator and 푍(훽) = +tr 푈−푖훽 is the canonical partition function. Note that the Gibbs operator satisfies the composition law +푈−푖훽1푈−푖훽2 = 푈−푖(훽1+훽2), the hermiticity 푈 † +−푖훽 = 푈−푖훽, and the initial condition 푈0 = 퐼. Its matrix elements +(heat kernel) 푈−푖훽(푥, 푦) = ⟨푥| e−훽퐻 |푦⟩ must then satisfy these conditions as well. Namely, we must +have ∑푧∈Λ/Γ 푈−푖훽1(푥, 푧)푈−푖훽2(푧, 푦) = 푈−푖(훽1+훽2), 푈−푖훽(푥, 푦) = 푈−푖훽(푦, 푥), and 푈0(푥, 푦) = 훿푥,푦. Under these +conditions, one can again show that 푈−푖훽(푥, 푦) can be written as 푈−푖훽(푥, 푦) = ∑훾∈Γ 퐷(훾)̃푈−푖훽(푥, 훾푦). +Hence the matrix elements of the canonical density matrix is +휌훽(푥, 푦) = +1 +푍(훽) ∑ +훾∈Γ +퐷(훾)̃푈−푖훽(푥, 훾푦), +(42) +12 + +where 휌훽(푥, 푦) = ⟨푥|휌훽|푦⟩. Here 푍(훽) is the canonical partition function given by +푍(훽) = ∑ +푥∈Λ/Γ +∑ +훾∈Γ +퐷(훾)̃푈−푖훽(푥, 훾푥). +(43) +We note that the partition function (43) can also be written as 푍(훽) = ∑푥∈Λ/Γ ∑훾∈Γ 퐷(훾)⟨푥| e−훽 ̃퐻 |훾푥⟩ = +∑훾∈Γ 퐷(훾) tr(e−훽 ̃퐻 푊훾), where ̃퐻 is the Hamiltonian operator on Λ and 푊훾 is a unitary operator defined +by 푊훾|푥⟩ = |훾푥⟩. +4.3 +Unitary representations of arbitrary groups on a tensor-product Hilbert space +As mentioned in the beginning of section 2, our main formula (4) is also applicable to discrete-time +quantum walk, where the time 휏 takes discrete values and the one-particle Hilbert space is the tensor- +product of the position and coin Hilbert spaces. In this section, we shall see this from a more general +perspective: the construction of matrix elements of a unitary representation of an arbitrary group +퐺 on a tensor-product Hilbert space. The time-evolution kernel in discrete-time quantum walk just +corresponds to the special case 퐺 = Z (the additive group of integers). +To begin with, let {푈푔 ∈ 푈(H) ∶ 푔 ∈ 퐺} be a unitary representation of 퐺 on the tensor-product +Hilbert space H = Hposition⊗Hcoin, where 푈(H) stands for the set of unitary operators on H, Hposition = +푙2(Λ/Γ) is the set of square-summable sequences on the orbit space Λ/Γ, and Hcoin = C푑 is the 푑- +dimensional complex vector space that describes internal degrees of freedom of particles. Let {|푥⟩} +and {|푖⟩} be complete orthonormal systems of Hposition and Hcoin, respectively. The set {|푥⟩ ⊗ |푖⟩} then +provides a complete orthonormal system of the total Hilbert space Hsuch that the matrix elements of +푈푔 can be defined as 푈푔(푥, 푖; 푦, 푗) = (⟨푥| ⊗ ⟨푖|)푈푔(|푦⟩ ⊗ |푗⟩). +We now define 푈푔(푥, 푦) as the following 푑 × 푑 matrix: +푈푔(푥, 푦) ≔ +⎛ +⎜ +⎜ +⎝ +푈푔(푥, 1; 푦, 1) +⋯ +푈푔(푥, 1; 푦, 푑) +⋮ +⋱ +⋮ +푈푔(푥, 푑; 푦, 1) +⋯ +푈푔(푥, 푑; 푦, 푑) +⎞ +⎟ +⎟ +⎠ +. +(44) +Since the unitary representation must satisfy the group composition law 푈푔1푈푔2 = 푈푔1푔2, the unitarity +푈 † +푔 (= 푈 −1 +푔 ) = 푈푔−1, and the initial condition 푈푒 = 퐼, the matrix (44) must also satisfy the following +properties: +∑ +푧∈Λ/Γ +푈푔1(푥, 푧)푈푔2(푧, 푦) = 푈푔1푔2(푥, 푦), +(45a) +푡푈푔(푥, 푦) = 푈푔−1(푦, 푥), +(45b) +푈푒(푥, 푦) = 훿푥,푦ퟏ, +(45c) +where 푥, 푦 ∈ Λ/Γ. Here 푡 and ퟏ stand for the matrix transpose and the 푑×푑 identity matrix, respectively. +Now it is a straightforward exercise to show that the matrix (44) can be written as +푈푔(푥, 푦) = ∑ +훾∈Γ +퐷(훾)̃푈푔(푥, 훾푦), +(46) +where ̃푈푔(푥, 푦) is a 푑×푑 matrix subject to the conditions ∑푧∈Λ ̃푈푔1(푥, 푧)̃푈푔2(푧, 푦) = ̃푈푔1푔2(푥, 푦), 푡 ̃푈푔(푥, 푦) = +̃푈푔−1(푦, 푥), ̃푈푒(푥, 푦) = 훿푥,푦ퟏ, and ̃푈푔(훾푥, 훾푦) = ̃푈푔(푥, 푦) for any 푥, 푦 ∈ Λ and 훾 ∈ Γ. It is also straightfor- +ward to show that eq. (46) satisfies the following boundary condition: +푈푔(훾푥, 푦) = 퐷(훾)푈푔(푥, 푦), +∀훾 ∈ Γ. +(47) +It is now obvious that eq. (46) provides the time-evolution kernel of continuous-time quantum walk +with internal degrees of freedom when 퐺 = R (the additive group of real numbers) and of discrete- +time quantum walk when 퐺 = Z (the additive group of integers). It is also obvious that the examples +presented in section 3 apply to discrete-time quantum walk as well. +13 + +5 +Conclusion +Inspired by the covering-space method in path integral on multiply-connected spaces, we have devel- +oped a general theory of quantum walk on orbit spaces. In this note, we have proved the universal +formulae for time-evolution kernels, resolvent kernels, canonical density matrices, and unitary repre- +sentations of arbitrary groups in continuous- and discrete-time quantum walks on the orbit space Λ/Γ, +where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. All +of these quantities are given by summations over the orbit of initial point on Λ, where each orbit is +weighted by a phase factor given by a one-dimensional unitary representations of Λ. +There are several advantages of this orbit-space method. A main advantage is its universality: our +formulae are just based on geometric and group-theoretic structures of configuration spaces so that +they are robust against any perturbations or interparticle interactions as far as boundary conditions +(8) are remained unchanged. Another advantage is its computational simplicity: in our formalism, one +just needs to compute matrix elements on Λ, which is generally much easier than computations on +Λ/Γ. +Finally, let us comment on one possible future direction of this work. A promising direction would +be a generalization of our formulae to the problem of identical walkers on graphs. Recent studies +have shown that exotic statistics may show up in many-body problems of identical particles on graphs +[31–35]. Such exotic statistics are generalizations of braid-group statistics in two dimensions. Hence, +just as in topological quantum computation using anyons [36], they would have potential applications +in quantum computer science. Our formalism and its generalization may well serve a basic tool for +studying dynamics as well as thermodynamics of such systems. +Acknowledgments +The author would like to thank Naoto Namekata for discussion. +A +Sample computations +Continuous-time quantum walk is just equivalent to tight-binding models in condensed matter physics. +The advantage of this perspective is that it is straightforward to study many-particle problems by using +the second-quantization formalism. In this section, we study tight-binding models for free spinless +particles in one dimension and present sample computations that justify the formulae in section 3. +A.1 +Tight-binding model on the infinite line +Let us first consider spinless particles on the integer lattice Z only with a nearest-neighbor coupling. +In the second-quantization formalism, the Hamiltonian operator is given by +̃퐻 = −휔 +2 ∑ +푥∈Z (푎† +푥+1푎푥 + 푎† +푥 푎푥+1) , +(A.1) +where 휔(> 0) is a hopping parameter. 푎푥 and 푎† +푥 are annihilation and creation operators for spinless +bosons (fermions) and subject to the following (anti-)commutation relations: +[푎푥, 푎† +푦]∓ = 훿푥,푦 +and +[푎푥, 푎푦]∓ = 0, +(A.2) +where [퐴, 퐵]∓ = 퐴퐵 ∓ 퐵퐴. +Let |0⟩ be the Fock vacuum that satisfies 푎푥|0⟩ = 0 for all 푥. The time-evolution kernel in the +one-particle sector of the model is then given by +̃푈휏(푥, 푦) = ⟨푥| e−푖 ̃퐻휏 |푦⟩, +(A.3) +14 + +where |푥⟩ = 푎† +푥|0⟩ is the position-space basis in the one-particle sector. It satisfies the orthonormality +⟨푥|푦⟩ = ⟨0|푎푥푎† +푦|0⟩ = 훿푥,푦 for both bosons and fermions. In order to calculate the matrix element (A.3), +we first diagonalize the Hamiltonian operator, which can be achieved by the following Fourier integral: +푎푥 = ∫ +휋 +−휋 +푑푝 +2휋 ̃푎푝 e푖푝푥, +(A.4) +where ̃푎푝 and ̃푎† +푝 satisfy [ ̃푎푝, ̃푎† +푞]∓ = 2휋훿(푝 − 푞) and [ ̃푎푝, ̃푎푞]∓ = 0 for any 푝, 푞 ∈ (−휋, 휋). By substituting +eq. (A.4) into eq. (A.3), we obtain +̃퐻 = ∫ +휋 +−휋 +푑푝 +2휋 휀푝 ̃푎† +푝 ̃푎푝, +(A.5) +where 휀푝 = −휔 cos(푝) is the single-particle energy eigenvalue. It is now easy to see that the time- +evolution kernel (A.3) takes the following form: +̃푈휏(푥, 푦) = ⟨0|푎푥 e−푖 ̃퐻휏 푎† +푦|0⟩ += ∫ +휋 +−휋 +푑푝 +2휋 ∫ +휋 +−휋 +푑푞 +2휋 ⟨0| ̃푎푝 e−푖 ̃퐻휏 ̃푎† +푞|0⟩ e푖푝푥−푖푞푦 += ∫ +휋 +−휋 +푑푝 +2휋 ∫ +휋 +−휋 +푑푞 +2휋 e−푖휀푞푡⟨0| ̃푎푝 ̃푎† +푞|0⟩ e푖푝푥−푖푞푦 += ∫ +휋 +−휋 +푑푝 +2휋 e푖휔휏 cos(푝) e푖푝(푥−푦), +(A.6) +where in the third equality we have used e−푖퐻휏 ̃푎† +푞|0⟩ = e−푖휀푞푡 ̃푎† +푞|0⟩, which follows from e−푖퐻휏 ̃푎† +푞 e푖퐻휏 = +e−푖휀푞푡 ̃푎† +푞 and e−푖퐻휏 |0⟩ = |0⟩ (or, equivalently, [퐻, ̃푎† +푞] = 휀푞 ̃푎† +푞 and 퐻|0⟩ = 0). The fourth equality follows +from ⟨0| ̃푎푝 ̃푎† +푞|0⟩ = 2휋훿(푝 − 푞) for both bosons and fermions. To evaluate the last integral in eq. (A.6), +we note that e푖휔휏 cos(푝) is a generating function of the Bessel function of the first kind 퐽푛. In fact, +e푖휔휏 cos(푝) = +∞ +∑ +푛=−∞ +e푖푛(푝+ 휋 +2 ) 퐽푛(휔휏) += +∞ +∑ +푛=−∞ +e푖 휋 +2 |푛| 퐽|푛|(휔휏) e−푖푛푝, +(A.7) +where the second equality follows from 퐽−푛(푥) = e푖푛휋 퐽푛(푥). By substituting eq. (A.7) into eq. (A.6) and +then using the orthogonal relation ∫ 휋 +−휋 +푑푝 +2휋 e푖푝(푥−푦−푛) = 훿푛,푥−푦, we obtain +̃푈휏(푥, 푦) = e푖 휋 +2 |푥−푦| 퐽|푥−푦|(휔휏), +∀푥, 푦 ∈ Z. +(A.8) +This is the well-known transition amplitude for a single walker on the lattice Z (see, e.g., ref. [37]). Note +that eq. (A.8) satisfies the composition law (3a), the unitarity (3b), and the initial condition (3c), which +follow from the addition theorem 퐽푛1−푛2(푥1 + 푥2) e푖 휋 +2 (푛1−푛2) = ∑푛∈Z 퐽푛1−푛(푥1)퐽푛2−푛(푥2) e푖 휋 +2 (푛1−푛) e푖 휋 +2 (푛2−푛) +(푛1, 푛2 ∈ Z), the analytic continuation 퐽푛(e푖휋 푥) = e푖푛휋 퐽푛(푥), and 퐽푛(0) = 훿푛,0, respectively. Note also +that eq. (A.8) enjoys the translation invariance ̃푈휏(푥 + 푧, 푦 + 푧) = ̃푈휏(푥, 푦) and the reflection invariance +̃푈휏(푧 − 푥, 푧 − 푦) = ̃푈휏(푥, 푦) for any 푥, 푦, 푧 ∈ Z. As we shall see shortly, eq. (A.8) provides the building +block for the construction of time-evolution kernels for a free particle on a circle, the half line, and a +finite interval. +Several comments are in order. +• Resolvent kernel for a single walker. As discussed in section 4.1, the resolvent kernel (Green’s +function) is given by the Laplace transform of ̃푈휏(푥, 푦). Let 퐸 be a complex number with Im 퐸 > 0. +15 + +Then we have +푖 ̃퐺퐸(푥, 푦) = ∫ +∞ +0 +푑휏 ̃푈휏(푥, 푦) e푖퐸휏 += ∫ +휋 +−휋 +푑푝 +2휋 +푖 e푖푝(푥−푦) +퐸 + 휔 cos(푝) += 2푖 +휔 ∮|푧|=1 +푑푧 +2휋푖 +푧|푥−푦| +푧2 + 2퐸 +휔 푧 + 1, +(A.9) +where in the second equality we have substituted the last line of eq. (A.6) and performed the +integration with respect to 휏. In the last equality we have changed the integration variable from +푝 to 푧 = e푖푝, where the integration is over the closed loop |푧| = 1 in the counter-clockwise +direction. By using the residue theorem we find +푖 ̃퐺퐸(푥, 푦) = e푖푝|푥−푦| +휔 sin(푝), +(A.10) +where we have parameterized the energy as 퐸 = −휔 cos(푝) with Re 푝 ∈ (0, 휋) and Im 푝 ∈ (0, ∞). +Eq. (A.10) provides the building block for the construction of single-particle resolvent kernels on +a circle, the half line, and a finite interval. +• Heat kernel for a single walker. The matrix element of the Gibbs operator e−훽 ̃퐻 can be calculated +in exactly the same way as for ̃푈휏(푥, 푦). Under the substitution 휏 → −푖훽 in eq. (A.6) we find +̃푈−푖훽(푥, 푦) = ∫ +휋 +−휋 +푑푝 +2휋 e훽휔 cos(푝) e푖푝(푥−푦) += 퐼푥−푦(훽휔), +(A.11) +where 퐼푛(푥) = 퐼−푛(푥) stands for the modified Bessel function of the first kind. Here in the last line +we have used the fact that e훽휔 cos(푝) is the generating function of 퐼푛(훽휔). In fact, +e훽휔 cos(푝) = +∞ +∑ +푛=−∞ +퐼푛(훽휔) e−푖푛푝 . +(A.12) +By substituting this into the first line and using the orthogonal relation ∫ 휋 +−휋 +푑푝 +2휋 e푖푝(푥−푦−푛) = 훿푛,푥−푦, +we arrive at eq. (A.11). As discussed in section 4.2, eq. (A.11) provides the building block for the +construction of canonical density matrices for free particles on a circle and a finite interval. +• Time-evolution kernel for 푁 identical walkers. In the second-quantization formalism, it is easy +to generalize the above results to many-particle problems. First, the position-space basis in the +푁-particle sector is given by +|푥1, ⋯ , 푥푁 ⟩ ≔ 푎† +푥1 ⋯ 푎† +푥푁 |0⟩. +(A.13) +Notice that eq. (A.13) satisfies the orthonormality condition on the orbit space (Z푁 − Δ푁 )/푆푁 ≅ +{(푥1, ⋯ , 푥푁 ) ∈ Z푁 ∶ 푥1 > ⋯ > 푥푁 }. In fact, for 푥1 > ⋯ > 푥푁 and 푦1 > ⋯ > 푦푁, we have +⟨푥1, ⋯ , 푥푁 |푦1, ⋯ , 푦푁 ⟩ = ⟨0|푎푥푁 ⋯ 푎푥1푎† +푦1 ⋯ 푎† +푦푁 |0⟩ += ∑ +휎∈푆푁 +(±1)#휎훿푥휎(1),푦1 ⋯ 훿푥휎(푁),푦푁 += 훿푥1,푦1 ⋯ 훿푥푁 ,푦푁 , +(A.14) +16 + +where the last line follows from the fact that (푥휎(1), ⋯ , 푥휎(푁)) and (푦1, ⋯ , 푦푁) cannot be equal +except for 휎 = 푒. It is now easy to show that the time-evolution kernels for 푁 identical bosons +and fermions take the following forms:7 +⟨푥1, ⋯ , 푥푁 | e−푖 ̃퐻휏 |푦1, ⋯ , 푦푁 ⟩ = ⟨0|푎푥푁 ⋯ 푎푥1 e−푖 ̃퐻휏 푎† +푦1 ⋯ 푎† +푦푁 |0⟩ += [ +푁 +∏ +푗=1 ∫ +휋 +−휋 +푑푝푗 +2휋 ∫ +휋 +−휋 +푑푞푗 +2휋 ] e푖휔휏(cos(푞1)+⋯+cos(푞푁 )) +× ⟨0| ̃푎푝푁 ⋯ ̃푎푝1 ̃푎† +푞1 ⋯ ̃푎† +푞푁 |0⟩ e푖푝1푥1+⋯+푖푝푁 푥푁 −푖푞1푦1−⋯−푖푞푁푦푁 += ∑ +휎∈푆푁 +(±1)#휎 +푁 +∏ +푗=1 +e푖 휋 +2 |푥푗−푦휎(푗)| 퐽|푥푗−푦휎(푗)|(휔휏). +(A.15) +As shown in eqs. (29), (33), and (37), eq. (A.15) can be used to construct the time-evolution kernels +for free identical walkers on a circle, the half line, and a finite interval. +A.2 +Tight-binding model on a circle +Let us next consider the tight-binding model for free spinless particles on the periodic lattice {1, 2, ⋯ , 퐿 +(mod 퐿)} subject to the twisted boundary condition 푎푥+퐿 = e푖휃 푎푥. As we shall see shortly, the following +Hamiltonian operator yields the desired results: +퐻 = −휔 +2 +퐿 +∑ +푥=1 (푎† +푥+1푎푥 + 푎† +푥 푎푥+1) , +where +푎퐿+1 ≡ e푖휃 푎1. +(A.16) +In the following, we assume that 휃 ranges from 0 to 2휋. +In order to compute the time-evolution kernel, we first have to diagonalize the Hamiltonian oper- +ator (A.16), which can be done by using the mode expansion. Under the twisted boundary condition, +the annihilation operator can be expanded into the following: +푎푥 = 1 +√퐿 +퐿−1 +∑ +푝=0 +̃푎푝 e푖 2푝휋+휃 +퐿 +푥, +(A.17) +where ̃푎푝 and ̃푎† +푝 satisfy [ ̃푎푝, ̃푎† +푞]∓ = 훿푝,푞 and [ ̃푎푝, ̃푎푞]∓ = 0 for any 푝, 푞 ∈ {0, 1, ⋯ , 퐿 − 1}. By substituting +eq. (A.17) into eq. (A.16), we find that the Hamiltonian operator is diagonalized as follows: +퐻 = +퐿−1 +∑ +푝=0 +휀푝 ̃푎† +푝 ̃푎푝, +(A.18) +where 휀푝 = −휔 cos(2푝휋+휃 +퐿 +) is the single-particle energy eigenvalue on the periodic lattice. +Now it is easy to compute the time-evolution kernel in the one-particle sector. A straightforward +7It should be noted that ̃푈휏(푥1, ⋯ , 푥푁 , 푦1, ⋯ , 푦푁) = ∏푁 +푗=1 e푖 휋 +2 |푥푗−푦푗| 퐽|푥푗−푦푗|(휔휏) is equivalent to a single-particle time-evolution +kernel on Z푁 rather than Z푁 − Δ푁. As noted in the beginning of section 3.2, in this note we will not touch upon this type of +issues related to the fixed points of 푆푁 . +17 + +calculation gives +푈 [휃] +휏 (푥, 푦) = ⟨푥| e−푖퐻휏 |푦⟩ += ⟨0|푎푥 e−푖퐻휏 푎† +푦|0⟩ += 1 +퐿 +퐿−1 +∑ +푝=0 +퐿−1 +∑ +푞=0 +⟨0| ̃푎푝 e−푖퐻휏 ̃푎† +푞|0⟩ e푖 2푝휋+휃 +퐿 +푥−푖 2푞휋+휃 +퐿 +푦 += 1 +퐿 +퐿−1 +∑ +푝=0 +퐿−1 +∑ +푞=0 +e−푖휀푞휏⟨0| ̃푎푝 ̃푎† +푞|0⟩ e푖 2푝휋+휃 +퐿 +푥−푖 2푞휋+휃 +퐿 +푦 += 1 +퐿 +퐿−1 +∑ +푝=0 +e푖휔휏 cos( 2푝휋+휃 +퐿 +) e푖 2푝휋+휃 +퐿 +(푥−푦), +(A.19) +where we have used e−푖퐻휏 ̃푎† +푞|0⟩ = e−푖휀푞휏 ̃푎† +푞|0⟩ in the fourth line and ⟨0| ̃푎푝 ̃푎† +푞|0⟩ = 훿푝,푞 in the last line. +Notice that eq. (A.19) is the summation over the energy spectrum. In order to obtain the summation +over winding numbers, we therefore have to perform a resummation, which can be done by using +eq. (A.7). By substituting e푖휔푡 cos( 2푝휋+휃 +퐿 +) = ∑푚∈Z e푖 휋 +2 |푚| 퐽|푚|(휔푡) e−푖푚 2푝휋+휃 +퐿 +into eq. (A.19) and using the +orthogonal relation 1 +퐿 ∑퐿−1 +푝=0 e푖 2푝휋+휃 +퐿 +(푥−푦−푚) = e푖푛휃 훿푚,푥−푦−푛퐿 (푛 ∈ Z), we find that the time-evolution kernel +(A.19) can be put into the following alternative equivalent form:8 +푈 [휃] +휏 (푥, 푦) = +∞ +∑ +푛=−∞ +e푖푛휃 e푖 휋 +2 |푥−푦−푛퐿| 퐽|푥−푦−푛퐿|(휔휏), +(A.20) +which exactly coincides with eq. (15) with ̃푈휏( ⋅ , ⋅ ) given by eq. (A.8). This sample computation im- +plies that there is an equivalence (or duality) between the summation over energy spectrum and the +summation over particle’s trajectories, which is the heart of the trace formula in harmonic analysis and +representation theory (see, e.g., ref. [39]). In this respect, one could say that our formula is a version +of the trace formula in lattice geometry. +Although we omit the details, it is not difficult to show that the resolvent kernel, the canonical +density matrix, and the time-evolution kernel for 푁 identical particles all coincide with the universal +formulae. +We note in closing that the parameter 휃 can be removed from the twisted boundary condition +under the gauge transformation 푎푥 ↦ 푉휃푎푥푉 −1 +휃 += e푖 휃 +퐿 푥 푎푥, where 푉휃 is a unitary operator given by +푉휃 = exp(−푖 휃 +퐿 ∑퐿 +푥=1 푥푎† +푥 푎푥) (see, e.g., ref. [40]). In fact, a straightforward calculation gives +푉휃퐻푉 −1 +휃 += −휔 +2 +퐿 +∑ +푥=1 (e−푖휃/퐿 푎† +푥+1푎푥 + e+푖휃/퐿 푎† +푥 푎푥+1) , +where +푎퐿+1 ≡ 푎1. +(A.21) +The time-evolution kernel in the one-particle sector for this Hamiltonian coincides with eq. (A.20) up +to a phase factor e푖 휃 +퐿 (푥−푦) and hence is physically equivalent. +A.3 +Tight-binding model on the half line +Let us next consider the tight-binding model on the semi-infinite lattice {1, 2, ⋯} with the boundary +condition 푎0 = 푒푖휙푎1, where 휙 ∈ {0, 휋}. The Hamiltonian operator that ensures this boundary condition +is given by +퐻 = −휔 +2 +∞ +∑ +푥=1 (푎† +푥+1푎푥 + 푎† +푥 푎푥+1) − 휔 +2 e푖휙 푎† +1 푎1. +(A.22) +8The case 휃 = 0 was noted in ref. [38]. +18 + +By substituting the mode expansion +푎푥 = ∫ +휋 +0 +푑푝 +2휋 ̃푎푝 (e−푖푝푥 + e푖휙 e−푖푝(1−푥)) , +(A.23) +we get the following diagonalized Hamiltonian operator: +퐻 = ∫ +휋 +0 +푑푝 +2휋 휀푝 ̃푎† +푝 ̃푎푝, +(A.24) +where 휀푝 = −휔 cos(푝) is the single-particle energy eigenvalue. The time-evolution kernel for a single +walker is given by +푈 [휙] +휏 +(푥, 푦) = ⟨0|푎푥 e−푖퐻휏 푎† +푦|0⟩ += ∫ +휋 +0 +푑푝 +2휋 e−푖휀푝휏 (e−푖푝푥 + e푖휙 e−푖푝(1−푥)) (e푖푝푦 + e푖휙 e푖푝(1−푦)) += ∫ +휋 +−휋 +푑푝 +2휋 e푖휔휏 cos(푝) (e푖푝(푥−푦) + e푖휙 e푖푝(푥−1+푦)) += e푖 휋 +2 |푥−푦| 퐽|푥−푦|(휔휏) + e푖휙 e푖 휋 +2 |푥−1+푦| 퐽|푥−1+푦|(휔휏), +(A.25) +which exactly coincides with eq. (18). Other quantities can be calculated in a similar way and coincide +with the universal formulae. +We note that the model that satisfies the Dirichlet boundary condition 푎푥 = 0 at 푥 = 0 is described +by the Hamiltonian operator 퐻 = −(휔/2) ∑∞ +푥=1(푎† +푥+1푎푥 + 푎† +푥 푎푥+1). In this case, the time-evolution kernel +coincides with another formula discussed in example 2 in section 3.1. +A.4 +Tight-binding model on a finite interval +Let us finally quickly study the tight-binding model on the finite lattice {1, 2, ⋯ , 퐿} with the boundary +conditions 푎0 = e푖휙 푎1 and 푎퐿+1 = e푖(휃+휙) 푎퐿, where 휃, 휙 ∈ {0, 휋}. The Hamiltonian operator is given by +퐻 = −휔 +2 +퐿−1 +∑ +푥=1 (푎† +푥+1푎푥 + 푎† +푥 푎푥+1) − 휔 +2 e푖휙 푎† +1 푎1 − 휔 +2 e푖(휃+휙) 푎† +퐿푎퐿. +(A.26) +This operator can be diagonalized by using the following mode expansions: +푎푥 = +⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪ +⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ +1 +√퐿 ̃푎0 + +1 +√2퐿 +퐿−1 +∑ +푝=1 +̃푎푝 (e−푖 2푝휋 +2퐿 푥 + e−푖 2푝휋 +2퐿 (1−푥)) +for +휃 = 0 & 휙 = 0; +1 +√ +2퐿 +퐿−1 +∑ +푝=1 +̃푎푝 (e−푖 2푝휋 +2퐿 푥 − e−푖 2푝휋 +2퐿 (1−푥)) + 1 +√ +퐿 ̃푎퐿(−1)푥 +for +휃 = 0 & 휙 = 휋; +1 +√ +2퐿 +퐿−1 +∑ +푝=0 +̃푎푝 (e−푖 2푝휋+휃 +2퐿 푥 + e푖휙 e−푖 2푝휋+휃 +2퐿 +(1−푥)) +otherwise. +(A.27) +In fact, by substituting these into eq. (A.26) we find +퐻 = +⎧⎪⎪⎪⎪⎪ +⎨⎪⎪⎪⎪⎪⎩ +퐿 +∑ +푝=1 +휀푝 ̃푎† +푝 ̃푎푝 +for +휃 = 0 & 휙 = 휋; +퐿−1 +∑ +푝=0 +휀푝 ̃푎† +푝 ̃푎푝 +otherwise, +(A.28) +19 + +where 휀푝 = −휔 cos(2푝휋+휃 +2퐿 ) for any 휃, 휙 ∈ {0, 휋}. It is not difficult to show that the time-evolution kernel +for a single walker can be put into the following expression irrespective of the values of 휃 and 휙: +푈 [휃,휙] +휏 +(푥, 푦) = ⟨0|푎푥 e−푖퐻휏 푎† +푦|0⟩ += 1 +2퐿 +2퐿−1 +∑ +푝=0 +e푖휔휏 cos( 2푝휋+휃 +2퐿 ) (e푖 2푝휋+휃 +2퐿 +(푥−푦) + e푖휙 e푖 2푝휋+휃 +2퐿 (푥−1+푦)) . +(A.29) +Note that this is the summation over the energy spectrum. However, as was done in appendix A.2, +this summation can be rewritten into the following summation over the bouncing numbers off the +boundaries: +푈 [휃,휙] +휏 +(푥, 푦) = +∞ +∑ +푛=−∞ +[e푖푛휃 e푖 휋 +2 |푥−푦−2푛퐿| 퐽|푥−푦−2푛퐿|(휔휏) + e푖푛휃 e푖휙 e푖 휋 +2 |푥−2푛퐿−1+푦| 퐽|푥−2푛퐿−1+푦|(휔휏)] , +(A.30) +which exactly coincides with the universal formula (21). +If one wants to study the model that satisfies the Dirichlet boundary conditions 푎푥 = 0 at 푥 = 0 +and 푥 = 퐿 + 1, one should use 퐻 = −(휔/2) ∑퐿−1 +푥=1(푎† +푥+1푎푥 + 푎† +푥 푎푥+1). 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Lee, “Topological invariants for interacting systems: from twisted +boundary condition to center-of-mass momentum”, arXiv:2211.07494 [quant-ph]. +22 + diff --git a/bdE1T4oBgHgl3EQfdQRv/content/tmp_files/load_file.txt b/bdE1T4oBgHgl3EQfdQRv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fe08d3b589624996fbda75d231c86bbe2377b45 --- /dev/null +++ b/bdE1T4oBgHgl3EQfdQRv/content/tmp_files/load_file.txt @@ -0,0 +1,1173 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf,len=1172 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='03193v1 [quant-ph] 9 Jan 2023 Quantum Walk on Orbit Spaces Satoshi Ohya Institute of Quantum Science, Nihon University, Kanda-Surugadai 1-8-14, Chiyoda, Tokyo 101-8308, Japan ohya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='satoshi@nihon-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='jp (Dated: January 10, 2023) Abstract Inspired by the covering-space method in path integral on multiply-connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this note, we focus on the case in which walkers’ configuration space is the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We show that the time-evolution kernel on Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Focusing on one dimension, we present a number of examples of the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We also present universal formulae of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on Λ/Γ, all of which are constructed in exactly the same way as for the time- evolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Time-evolution kernel 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 The formula .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 5 3 Examples 6 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Identical walkers in one dimension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 8 4 Asides 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 Resolvent kernel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3 Tight-binding model on the half line .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4 Tight-binding model on a finite interval .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 19 References 20 1 1 Introduction Quantum walk—a quantum-mechanical analog of classical random walk on lattices or graphs—has been the subject of intense study over the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Just as in classical random walk, there exist two distinct formulations in quantum walk: continuous-time quantum walk and discrete-timequantum walk, the former is equivalent to tight-binding models in condensed matter physics, while the latter is a natural generalization of classical random walk and formulated without recourse to Hamiltonian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' These two formulations have their own merits and their applications now appear in many disciplines, including quantum search algorithm [1, 2], universal quantum computation [3–6], and topological phases of matter [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' see refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [8–10] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In both formulations, the central object is the probability amplitude for finding particles (walkers), which is given by a matrix element of time-evolution operator in position space—the time-evolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 This time-evolution kernel is normally calculated through spectral decomposition or numerical calculation, which becomes harder as the matrix size becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It would therefore be desirable if there exists a simpler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The purpose of this note is to present such a method by generalizing the Dowker’s covering-space method [11] in path integral (see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [12–16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As is well known, in quantum mechanics on continuous spaces, the time-evolution kernel can be represented by the Feynman path integral, which provides a number of powerful methods to analyze quantum systems nonperturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Among them is the covering-space method: it provides a universal method to construct the time-evolution kernel on multiply-connected spaces of the form \ue239 = ̃ \ue239/휋1(\ue239), where ̃ \ue239 is the universal covering space of \ue239 and 휋1(\ue239) is the fundamental group of \ue239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this method, the path integral on \ue239 is given by a linear combination of partial amplitudes, where each partial amplitude is given by the path integral on the universal covering space ̃ \ue239 and linear-combination coefficients are given by a one-dimensional unitary representations of the fundamental group 휋1(\ue239).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Inspired by this method, we here present a universal formula for the time-evolution kernel in both continuous- and discrete-time quantum walks where walkers’ configuration space can be regarded as the orbit space Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A typical example for such configura- tion spaces is that for a single walker on a periodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Another typical example is the configuration space for identical walkers on an arbitrary lattice, where the indistinguishability of identical particles makes their configuration space an orbit space [13, 17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We show that the time-evolution kernel on the orbit space Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the sum- mation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This universal formula offers a simpler method to construct the time-evolution kernel on Λ/Γ because computation becomes generally much easier on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In what follows, we first set up the problem and then present our main formula and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We then present a number of examples of the formula in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In section 4, we present several other quantities that can be constructed in exactly the same way as for the time-evolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Examples include the resolvent kernel, the canonical density matrix, and a unitary representation of arbitrary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Section 5 is devoted to conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Appendix A presents some sample computations in continuous-time quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Throughout this note we will use the units in which ℏ = 푎 = 1, where 푎 is a lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 2 Time-evolution kernel To begin with, let us fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let Λ be an arbitrary lattice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', a discrete space spanned by a set of linearly independent vectors in a Euclidean space) and let Γ be a discrete group whose action on Λ has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We note that Γ must be a discrete subgroup of the isometry of the Euclidean space, which consists of reflections, translations, and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let Λ/Γ be the orbit space (quotient space) given by the identification 푥 ∼ 훾푥 in Λ, where 훾푥 stands for the action of 훾 ∈ Γ on 푥 ∈ Λ that 1The term “kernel” is a remnant of continuum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In quantum mechanics on continuous spaces, a matrix element of time-evolution operator is given by an integral kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 2 satisfies the compatibility condition 훾1(훾2푥) = (훾1훾2)푥 for any 훾1, 훾2 ∈ Γ and 푥 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' For the moment, we shall consider continuous-time quantum walk on the lattice Λ/Γ, where the Hilbert space His the set of square-summable sequences on Λ/Γ, H = 푙2(Λ/Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Note, however, that the formula presented below is turned out to be applicable to discrete-time quantum walk as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=') The action of the time-evolution operator 푈휏 on a state 휓0 ∈ His defined by (푈휏휓0)(푥) ≔ ∑ 푦∈Λ/Γ 푈휏(푥, 푦)휓0(푦), ∀푥 ∈ Λ/Γ, (1) where 푈휏(푥, 푦) is the time-evolution kernel and the subscript 휏 ∈ R represents the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The probability for finding a particle at the time 휏 and at the position 푥 is then given by 푃휏(푥) = |(푈휏휓0)(푥)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (2) In particular, if the particle is initially localized at 푥 = 푥0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', 휓0(푥) = 훿푥,푥0), the probability is simply given by 푃휏(푥) = |푈휏(푥, 푥0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the following, we shall construct 푈휏(푥, 푦) in terms of the time-evolution kernel on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The key is the group property of the time-evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 The formula The time-evolution operator 푈휏 is a one-parameter family of unitary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It satisfies the com- position law 푈휏1푈휏2 = 푈휏1+휏2, the unitarity 푈 † 휏 (= 푈 −1 휏 ) = 푈−휏, and the initial condition 푈0 = 퐼, where 퐼 stands for the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Correspondingly, the time-evolution kernel 푈휏( ⋅ , ⋅ ) must satisfy the following properties: Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Composition law) ∑ 푧∈Λ/Γ 푈휏1(푥, 푧)푈휏2(푧, 푦) = 푈휏1+휏2(푥, 푦), ∀푥, 푦 ∈ Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3a) Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Unitarity) 푈휏(푥, 푦) = 푈−휏(푦, 푥), ∀푥, 푦 ∈ Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3b) Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Initial condition) 푈0(푥, 푦) = 훿푥,푦, ∀푥, 푦 ∈ Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3c) Here the overline ( ) stands for the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As we shall prove shortly, such a kernel can be constructed as follows: 푈휏(푥, 푦) = ∑ 훾∈Γ 퐷(훾)̃푈휏(푥, 훾푦), (4) where 퐷 ∶ Γ → 푈(1) (훾 ↦ 퐷(훾)) is a one-dimensional unitary representation of Γ that satisfies the group composition law 퐷(훾)퐷(훾 ′) = 퐷(훾훾 ′) and the unitarity 퐷(훾) = 퐷(훾)−1 = 퐷(훾 −1) for any 훾, 훾 ′ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here ̃푈휏( ⋅ , ⋅ ) is a time-evolution kernel on Λ that satisfies the following assumptions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Composition law) ∑ 푧∈Λ ̃푈휏1(푥, 푧)̃푈휏2(푧, 푦) = ̃푈휏1+휏2(푥, 푦), ∀푥, 푦 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (5a) Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Unitarity) ̃푈휏(푥, 푦) = ̃푈−휏(푦, 푥), ∀푥, 푦 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (5b) 3 Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Initial condition) ̃푈0(푥, 푦) = 훿푥,푦, ∀푥, 푦 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (5c) Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Γ-invariance) ̃푈휏(훾푥, 훾푦) = ̃푈휏(푥, 푦), ∀푥, 푦 ∈ Λ, ∀훾 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (5d) We note that the Γ-invariance (5d) is guaranteed if the Hamiltonian operator on Λ is invariant under the action of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Before giving the proof, let us first present a quick derivation of the formula (4) by following the Dowker’s method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' To this end, let ̃휓휏(푥) be an equivariant function on Λ that satisfies ̃휓휏(훾푥) = 퐷(훾) ̃휓휏(푥) for any 푥 ∈ Λ and 훾 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (The reason for using this will be apparent shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=') Then we have ̃휓휏(푥) = ∑ 푦∈Λ ̃푈휏(푥, 푦) ̃휓0(푦) = ∑ 푦∈Λ/Γ ∑ 훾∈Γ ̃푈휏(푥, 훾푦) ̃휓0(훾푦) = ∑ 푦∈Λ/Γ ∑ 훾∈Γ ̃푈휏(푥, 훾푦)퐷(훾) ̃ 휓0(푦) = ∑ 푦∈Λ/Γ (∑ 훾∈Γ 퐷(훾)̃푈휏(푥, 훾푦)) ̃휓0(푦), (6) where in the second equality we have used the following identity: ∑ 푥∈Λ 푓 (푥) = ∑ 푥∈Λ/Γ ∑ 훾∈Γ 푓 (훾푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (7) Here 푓 (푥) is an arbitrary test function on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This identity just says that first summing over the orbit Γ ⋅ 푥 ≔ {훾푥 ∶ 훾 ∈ Γ} of 푥 ∈ Λ/Γ and then summing over all 푥 ∈ Λ/Γ yields the summation over the whole space Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By comparing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (6) with the definition (1), we arrive at the formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now, since ̃푈휏( ⋅ , ⋅ ) is defined on the lattice Λ, the domain of 푈휏( ⋅ , ⋅ ) defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (4) can be naturally extended from Λ/Γ to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In particular, it satisfies the following equation: 푈휏(훾푥, 푦) = 퐷(훾)푈휏(푥, 푦), ∀푥, 푦 ∈ Λ, ∀훾 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (8) In fact, a straightforward calculation gives 푈휏(훾푥, 푦) = ∑ 훾 ′∈Γ 퐷(훾 ′)̃푈휏(훾푥, 훾 ′푦) = ∑ 훾 ′∈Γ 퐷(훾훾 −1훾 ′)̃푈휏(훾 −1훾푥, 훾 −1훾 ′푦) = 퐷(훾) ∑ 훾 ′∈Γ 퐷(훾 −1훾 ′)̃푈휏(푥, 훾 −1훾 ′푦) = 퐷(훾) ∑ 훾 ′′∈Γ 퐷(훾 ′′)̃푈휏(푥, 훾 ′′푦) = 퐷(훾)푈휏(푥, 푦), (9) where the second equality follows from the Γ-invariance (5d) and the third equality follows from the group composition law 퐷(훾훾 −1훾 ′) = 퐷(훾)퐷(훾 −1훾 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the fourth equality, we have changed the sum- mation variable from 훾 ′ to 훾 ′′ ≔ 훾 −1훾 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is now obvious from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (8) that (푈휏휓0)(푥) defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (1) also satisfies (푈휏휓0)(훾푥) = 퐷(훾)(푈휏휓0)(푥) for any 푥 ∈ Λ and 훾 ∈ Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' that is, (푈휏휓0)(푥) becomes an equiv- ariant function on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This is the reason why we used the equivariant function in the above derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As we shall see in section 3, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (8) provides boundary conditions on Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 4 Finally, let us comment on the case where the action of Γ has fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' First, the identity (7) does not hold in general if there is a fixed point: if there is a point 푥 ∈ Λ that satisfies 훾푥 = 푥 for some 훾(≠ 푒) ∈ Γ, where 푒 stands for the identity element of Γ, the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (7) leads to an overcounting of the fixed point 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Note, however, that if 푓 (푥) is subject to the Dirichlet boundary condition at the fixed point, such an overcounting does not occur so that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (7) holds true even in the presence of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3 Note that the Dirichlet boundary condition 푓 (푥) = 0 at 푥 = 훾푥 can be deduced from the equivariant property 퐷(훾)푓 (푥) = 푓 (훾푥) = 푓 (푥) if 퐷(훾) ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, if 퐷 ∶ Γ → 푈(1) is not the trivial representation, our formula (4) can be applied equally well to the case in which the action of Γ has fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' For the case of the trivial representation, however, the equivariant property does not lead to any definite boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' For simplicity, in this note we will mainly focus on the case where Γ has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Proof Now we show that 푈휏( ⋅ , ⋅ ) given by the formula (4) satisfies the required properties (3a)–(3c) if 퐷 is a one-dimensional unitary representation of Γ and if ̃푈휏( ⋅ , ⋅ ) satisfies the assumptions (5a)–(5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The proof is by direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Each property is proved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (See also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [21–23] for similar proofs in path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=') Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Composition law) Let us first prove the composition law (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (4) into the left-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3a), we get ∑ 푧∈Λ/Γ 푈휏1(푥, 푧)푈휏2(푧, 푦) = ∑ 푧∈Λ/Γ ∑ 훾1∈Γ ∑ 훾2∈Γ 퐷(훾1)퐷(훾2)̃푈휏1(푥, 훾1푧)̃푈휏2(푧, 훾2푦) = ∑ 푧∈Λ/Γ ∑ 훾1∈Γ ∑ 훾2∈Γ 퐷(훾1훾2)̃푈휏1(푥, 훾1푧)̃푈휏2(훾1푧, 훾1훾2푦) = ∑ 훾∈Γ 퐷(훾) ∑ 푧∈Λ/Γ ∑ 훾1∈Γ ̃푈휏1(푥, 훾1푧)̃푈휏2(훾1푧, 훾푦) = ∑ 훾∈Γ 퐷(훾) ∑ 푧∈Λ ̃푈휏1(푥, 푧)̃푈휏2(푧, 훾푦) = ∑ 훾∈Γ 퐷(훾)̃푈휏1+휏2(푥, 훾푦) = 푈휏1+휏2(푥, 푦), (10) where the second equality follows from the group composition law 퐷(훾1)퐷(훾2) = 퐷(훾1훾2) and the Γ- invariance (5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The third equality follows from the change of the summation variable from 훾2 to 훾 ≔ 훾1훾2, and the fourth equality follows from the formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Finally, the fifth equality follows from the assumption (5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Unitarity) Let us next prove the unitarity (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (4) into the left-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3b), we get 푈휏(푥, 푦) = ∑ 훾∈Γ 퐷(훾) ̃푈휏(푥, 훾푦) = ∑ 훾∈Γ 퐷(훾 −1)̃푈−휏(훾푦, 푥) = ∑ 훾∈Γ 퐷(훾 −1)̃푈−휏(푦, 훾 −1푥) = 푈−휏(푦, 푥), (11) 2In general, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (7) becomes ∑푥∈Λ 푓 (푥) = ∑푥∈(Λ−Δ)/Γ ∑훾∈Γ 푓 (푥) + ∑푥∈Δ 푓 (푥), where Δ stands for the set of fixed points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 3More generally, such an overcounting does not occur if ∑푥∈Δ 푓 (푥) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 5 where the second equality follows from the unitarity properties 퐷(훾) = 퐷(훾 −1) and (5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The third equality follows from the Γ-invariance (5d), and the last equality follows from the definition (4) (where the summation is over 훾 −1 instead of 훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Initial condition) Let us finally prove the initial condition (3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (4) into the left-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (3c), we get 푈0(푥, 푦) = ∑ 훾∈Γ 퐷(훾)̃푈0(푥, 훾푦) = ∑ 훾∈Γ 퐷(훾)훿푥,훾푦 = 퐷(푒)훿푥,푒푦 = 훿푥,푦, (12) where the second equality follows from the assumption (5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The third equality follows from the fact that 푥 and 훾푦 cannot be equal for any 푥, 푦 ∈ Λ/Γ except for the case 훾 = 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Finally, the last equality follows from 퐷(푒) = 1 for any one-dimensional unitary representations of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Putting all the above things together, we see that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (4) is the sufficient condition to be the time- evolution kernel on the orbit space Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 3 Examples There exist a number of examples in which walkers’ configuration space can be regarded as an orbit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Typical examples are a single walker on a torus, the half space, and a cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Another typical example is identical walkers on an arbitrary lattice, where their configuration space always becomes an orbit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this section, we shall focus on one spatial dimension for simplicity and present several examples that fit into the formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let us start with single-walker examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 A single walker in one dimension Let ̃푈휏(푥, 푦) be a time-evolution kernel on the integer lattice Λ = Z that satisfies the composition law (5a), the unitarity (5b), and the initial condition (5c) as well as the translation invariance ̃푈휏(푥+푧, 푦+푧) = ̃푈휏(푥, 푦) and the reflection invariance ̃푈휏(푧 − 푥, 푧 − 푦) = ̃푈휏(푥, 푦) for any 푥, 푦, 푧 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A typical example of such a kernel is that of a free particle given by ̃푈휏(푥, 푦) = e푖 휋 2 |푥−푦| 퐽|푥−푦|(휔휏), where 퐽푛 is the Bessel function of the first kind and 휔(> 0) is a hopping parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8) in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Note, however, that the formulae presented below are not limited to free-particle theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' They are robust against any perturbations unless boundary conditions (8) are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=') Below we shall construct time-evolution kernels for a single walker on a circle, the half line, and a finite interval by gauging these discrete symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A single walker on a circle) Let us first consider a single walker on a periodic lattice of 퐿 sites, {1, 2, ⋯ , 퐿 (mod 퐿)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This lattice can be constructed from Z by making the identification 푥 ∼ 푥 + 푛퐿, where 푛 is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence the configuration space is the orbit space Z/퐿Z, where 퐿Z = ⟨푡 ∣ ∅⟩ is the free group generated by a translation 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action on Z is defined by 푡푥 ≔ 푥 + 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (13) Note that any element of 퐿Z can be written as the product 푡푛, whose action on Z is given by 푡푛푥 = 푥+푛퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now we need to find out one-dimensional unitary representations of 퐿Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Since 퐿Z is the free group generated by a single generator 푡, we have a one-parameter family of maps 퐷[휃] ∶ 퐿Z → 푈(1) labeled by an angle parameter 휃: 퐷[휃](푡) = e푖휃, (14) 6 where 휃 ∈ R/2휋R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It then follows from the formula (4) that the time-evolution kernel for a single walker on Z/퐿Z takes the following form: 푈 [휃] 휏 (푥, 푦) = ∞ ∑ 푛=−∞ 퐷[휃](푡푛)̃푈휏(푥, 푡푛푦) = ∞ ∑ 푛=−∞ e푖푛휃 ̃푈휏(푥, 푦 + 푛퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) Just as in the path integral on a circle (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [24]), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) represents the summation over winding numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Physically, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) describes the situation in which the walker acquires the Aharonov-Bohm phase e푖휃 every time it winds around the circle, where 휃 plays the role of a magnetic flux penetrating through the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This is the physical meaning of the weight factor (14) and the summation over the orbit of initial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now two remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' First, it follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (8) that 푈 [휃] 휏 ( ⋅ , ⋅ ) satisfies the identity 푈 [휃] 휏 (푥 + 퐿, 푦) = e푖휃 푈 [휃] 휏 (푥, 푦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' that is, it satisfies the twisted boundary conditions 푈 [휃] 휏 (퐿 + 1, 푦) = e푖휃 푈 [휃] 휏 (1, 푦) and 푈 [휃] 휏 (0, 푦) = e−푖휃 푈 [휃] 휏 (퐿, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Namely, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) gives the universal formula of the time- evolution kernel for a single walker on a circle subject to these twisted boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The second remark is that, under the reflection, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) satisfies 푈 [휃] 휏 (푧 − 푥, 푧 − 푦) = 푈 [−휃] 휏 (푥, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, at 휃 = 0 or 휋 (mod 2휋), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) becomes reflection invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We can use this invariance for the construction of time-evolution kernels on a finite interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' see example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A single walker on the half line) Let us next consider a single walker on a semi-infinite lattice {1, 2, ⋯}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This lattice can be constructed from the integer lattice Z by making the identification 푥 ∼ 1 − 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence the configuration space is the orbit space Z/Z2, where Z2 = ⟨푟 ∣ 푟2 = 푒⟩ is the cyclic group of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here 푟 is the reflection whose action on Z is defined by 푟푥 ≔ 1 − 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (16) Note that 푟2푥 = 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Note also that the reflection (16) does not have a fixed point in the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (Its fixed point is 푥 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=') Now, since 푟2 = 푒, any one-dimensional unitary representation 퐷 ∶ Z2 → 푈(1) must satisfy the condition 퐷(푟)2 = 1, whose solution is 퐷(푟) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence there exist two distinct maps 퐷[휙] given by 퐷[휙](푟) = e푖휙, (17) where 휙 ∈ {0, 휋 (mod 2휋)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Correspondingly, there exist the following two distinct time-evolution kernels for a single walker on Z/Z2: 푈 [휙] 휏 (푥, 푦) = 1 ∑ 푛=0 퐷[휙](푟푛)̃푈휏(푥, 푟푛푦) = ̃푈휏(푥, 푦) + e푖휙 ̃푈휏(푥, 1 − 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (18) Again, just as in the path integral on the half line [21, 25, 26], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (18) represents the summation over bouncing numbers off the boundary: the 푛 = 0 term is the contribution from the direct path, while the 푛 = 1 term is the contribution from the reflected path off the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The physical meaning of the weight factor (17) is now clear: it plays the role of the reflection amplitude off the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In other words, the walker acquires the phase shift 휙 when reflected from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (18) satisfies the identity 푈 [휙] 휏 (1−푥, 푦) = e푖휙 푈 [휙] 휏 (푥, 푦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' that is, it satisfies the bound- ary condition 푈 [휙] 휏 (0, 푦) = e푖휙 푈 [휙] 휏 (1, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (18) gives the universal form of the time-evolution kernel for a single walker on the half line subject to this boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We emphasize that, as noted at the end of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1, if one wants a theory subject to the Dirichlet boundary condition at 푥 = 0, one should consider the reflection defined by 푟푥 ≔ −푥 and choose the representation 휙 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, one arrives at the formula 푈 [휙=휋] 휏 (푥, 푦) = ̃푈휏(푥, 푦) − ̃푈휏(푥, −푦) which satisfies 푈 [휙=휋] 휏 (0, 푦) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 7 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A single walker on a finite interval) Let us next consider a single walker on a finite interval of 퐿 sites, {1, 2, ⋯ , 퐿}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This lattice can be constructed from Z by making the identifications 푥 ∼ 푥 + 2푛퐿 and 푥 ∼ 2푛퐿 + 1 − 푥, where 푛 is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, the configuration space is the orbit space Z/퐷∞, where 퐷∞ = Z ⋊ Z2 = ⟨푡, 푟 ∣ 푟2 = 푒, 푟푡푟 = 푡−1⟩ is the infinite dihedral group generated by a translation 푡 and a reflection 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4 The actions of these operators on Z are defined as follows: 푡푥 ≔ 푥 + 2퐿 and 푟푥 ≔ 1 − 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (19) Note that any element of 퐷∞ can be written as 푡푛푟푚, where 푛 = 0, ±1, ±2, ⋯ and 푚 = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The action of this operator on Z is given by 푡푛푟푚푥 = 푥 + 2푛퐿 for 푚 = 0 and 푡푛푟푚푥 = 2푛퐿 + 1 − 푥 for 푚 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Note also that, in contrast to the previous examples, 퐷∞ is a non-Abelian discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now, since 푟2 = 푒 and 푟푡푟 = 푡−1, any one-dimensional unitary representation 퐷 ∶ 퐷∞ → 푈(1) must satisfy the conditions 퐷(푟)2 = 1 and 퐷(푟)퐷(푡)퐷(푟) = 퐷(푡)−1, which leads to 퐷(푡)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Thus we have 퐷(푡) = ±1 and 퐷(푟) = ±1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' that is, there exist 22 = 4 distinct maps 퐷[휃,휙] given by 퐷[휃,휙](푡) = e푖휃 and 퐷[휃,휙](푟) = e푖휙, (20) where 휃, 휙 ∈ {0, 휋 (mod 2휋)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Correspondingly, there exist the following four distinct time-evolution kernels for a single walker on Z/퐷∞: 푈 [휃,휙] 휏 (푥, 푦) = ∞ ∑ 푛=−∞ 1 ∑ 푚=0 퐷[휃,휙](푡푛푟푚)̃푈휏(푥, 푡푛푟푚푦) = ∞ ∑ 푛=−∞ [e푖푛휃 ̃푈휏(푥, 푦 + 2푛퐿) + e푖푛휃 e푖휙 ̃푈휏(푥, 2푛퐿 + 1 − 푦)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) Once again, just as in the path integral on a finite interval [22, 27–29], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) represents the summation over bouncing numbers off the two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Physically, e푖휙 and e푖(휃+휙) play the roles of the reflection amplitudes off the boundaries 푥 = 1 and 푥 = 퐿, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now, it follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (8) that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) satisfies the identities 푈 [휃,휙] 휏 (푥 +2퐿, 푦) = e푖휃 푈 [휃,휙] 휏 (푥, 푦) and 푈 [휃,휙] 휏 (1 − 푥, 푦) = e푖휙 푈 [휃,휙] 휏 (푥, 푦), which implies the boundary conditions 푈 [휃,휙] 휏 (0, 푦) = e푖휙 푈 [휃,휙] 휏 (1, 푦) and 푈 [휃,휙] 휏 (퐿 + 1, 푦) = e푖(휃+휙) 푈 [휃,휙] 휏 (퐿, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This means that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) gives the universal form of the time- evolution kernel for a single walker on the finite interval subject to these boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' If one wants a theory that satisfies the Dirichlet boundary conditions at 푥 = 0 and 푥 = 퐿 + 1, one should redefine the translation and reflection as 푡푥 ≔ 푥 + 2(퐿 + 1) and 푟푥 ≔ −푥, respectively, and choose the representation 휙 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, one obtains 푈 [휃,휙=휋] 휏 (푥, 푦) = ∑∞ −∞ e푖푛휃[̃푈휏(푥, 푦 + 2푛(퐿 + 1)) − ̃푈휏(푥, 2푛(퐿 + 1) − 푦)] which satisfies 푈 [휃,휙=휋] 휏 (0, 푦) = 0 and 푈 [휃,휙=휋] 휏 (퐿 + 1, 푦) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We note in closing that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) can also be obtained from the time-evolution kernel on a circle (15) by gauging the reflection invariance at 휃 = 0, 휋 (mod 2휋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (21) can be written as 푈 [휃,휙] 휏 (푥, 푦) = ∑1 푚=0 퐷[휙](푟푚)푈 [휃] 휏 (푥, 푟푚푦) = ∑1 푚=0 ∑∞ 푛=−∞ 퐷[휙](푟푚)퐷[휃](푡푛)̃푈휏(푥, 푡푛푟푚푦), where 퐷[휙] is the one-dimensional unitary representation of Z2 given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' An important lesson from this example is that there could exist several ways to construct time-evolution kernels on orbit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Identical walkers in one dimension Now let us turn to the problem of multiple identical walkers on a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The key to this problem is the indistinguishability of identical particles, where physical observables must be invariant under permutations of multiparticle coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As is well known, this indistinguishability always makes the multiparticle configuration space an orbit space [13, 17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The basic idea behind this is to regard the permutation invariance as a gauge symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', redundancy in description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' From this perspective, 4The infinite dihedral group can also be written as the free product 퐷∞ ≅ Z2 ∗ Z2 = ⟨푟, 푟′ ∣ 푟2 = 푒, 푟′2 = 푒⟩, where 푟′(= 푡푟) is another reflection defined by 푟′푥 ≔ 2퐿 + 1 − 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 8 the configuration space must be a collection of inequivalent gauge orbits because gauge-equivalent configurations are physically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' To date, there exist two distinct formulations of this idea in identical-particle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The first regards the configuration space of 푁 identical particles as the orbit space (푋 푁 − Δ푁 )/푆푁 , where 푋 푁 is the 푁-fold Cartesian product of a single-particle configuration space 푋 and Δ푁 ⊂ 푋 푁 is the set of fixed points under the action of the symmetric group 푆푁 [13, 17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' On the other hand, the second includes the fixed points and regards the configuration space as the orbit space 푋 푁 /푆푁 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The difference between these two formulations is very subtle (especially in lattices) and beyond the scope of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Fortunately, however, we can circumvent this issue and solve the 푁-identical-walker problems as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Suppose that 푋 itself is a nontrivial orbit space and takes the form 푋 = ̃푋/퐺, where 퐺 is a discrete group whose action on ̃푋 has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the configuration space can also be written as (̃푋 푁 − ̃Δ푁)/(퐺 ≀ 푆푁 ) or ̃푋 푁 /(퐺 ≀ 푆푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='5 Here ≀ stands for the wreath product defined by the semidirect product 퐺 ≀ 푆푁 ≔ 퐺푁 ⋊ 푆푁 and ̃Δ푁 ⊂ ̃푋 푁 is the set of fixed points of 푆푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, irrespective of the formulations, once given a time-evolution kernel on Λ = ̃푋 푁 − ̃Δ푁 or ̃푋 푁 , the problem just reduces to the classification of one-dimensional unitary representations of the discrete group Γ = 퐺 ≀ 푆푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this section, we shall focus on the cases 푋 = Z, Z/퐿Z, Z/Z2, and Z/퐷∞ and construct time- evolution kernels for 푁 identical walkers on the infinite line, a circle, the half line, and a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the following, ̃푈휏(푥, 푦) represents a time-evolution kernel on Z푁 − ̃Δ푁 or Z푁 that satisfies the trans- lation invariance, reflection invariance, and permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (푁 identical walkers on the infinite line) Let us first consider 푁 identical walkers on the integer lattice Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the discrete group Γ = 푆푁 is just the symmetric group of order 푁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', whose presentation is 푆푁 = ⟨휎1, ⋯ , 휎푁−1 || 휎2 푖 = 푒, 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, 휎푖휎푗 = 휎푗휎푖 (|푖 − 푗| ≥ 2)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (22) Here 휎푖 = (푖, 푖+1) is the adjacent transposition that interchanges 푖 and 푖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action on 푥 = (푥1, ⋯ , 푥푁 ) ∈ Z푁 is defined as follows: 휎푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖+1, 푥푖, 푥푖+2, ⋯ , 푥푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (23) An arbitrary element 휎 ∈ 푆푁 can be written as a product of the generators 휎1, ⋯ , 휎푁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action on 푥 = (푥1, ⋯ , 푥푁 ) can be written as 휎푥 = (푥휎(1), ⋯ , 푥휎(푁)), where 휎(푖) stands for the permutation of 푖 under 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now, there exist two distinct one-dimensional unitary representations of 푆푁 : the trivial represen- tation and the sign representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Though this result is well known, let us reproduce it here just for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Since 휎2 푖 = 푒 and 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, any one-dimensional unitary representation 퐷 ∶ 푆푁 → 푈(1) must satisfy the conditions 퐷(휎푖)2 = 1 and 퐷(휎푖)퐷(휎푖+1)퐷(휎푖) = 퐷(휎푖+1)퐷(휎푖)퐷(휎푖+1), whose solutions are 퐷(휎푖) = ±1 and 퐷(휎푖) = 퐷(휎푖+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence we have 퐷(휎1) = ⋯ = 퐷(휎푁−1) = ±1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' that is, there exist two distinct maps 퐷[±] given by 퐷[±](휎) = (±1)#휎, (24) where #휎 stands for the number of adjacent transpositions in the permutation 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the standard terminology, 퐷[+] is the trivial representation and 퐷[−] is the sign representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6 Correspondingly, 5Here is the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' First, the wreath product 퐺 ≀ 푆푁 = 퐺푁 ⋊ 푆푁 can be written as the set {푔휎 ∶ 푔 ∈ 퐺푁, 휎 ∈ 푆푁} equipped with the group composition law (푔휎)(푔′휎′) = (푔휎푔′휎−1)(휎휎′) for any 푔, 푔′ ∈ 퐺푁 and 휎, 휎′ ∈ 푆푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here 푔 ↦ 휎푔휎−1 is the automorphism of the 푁 -fold direct-product group 퐺푁 = 퐺 × ⋯ × 퐺 defined by 휎푔휎−1 ≔ 푔휎(1) ⋯ 푔휎(푁) for any 푔 = 푔1 ⋯ 푔푁 ∈ 퐺 × ⋯ × 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is now obvious that first making the identification 푥 ∼ 푔푥 by 푔 ∈ 퐺푁 in ̃푋 푁 and then making the identification 푥 ∼ 휎푥 by 휎 ∈ 푆푁 in ̃푋 푁 /퐺푁 is equivalent to making the identification 푥 ∼ 휎푔푥 by 휎푔 = (휎푔휎−1)휎 ∈ 퐺 ≀ 푆푁 in ̃푋 푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence (̃푋 푁 /퐺푁)/푆푁 is equivalent to ̃푋 푁/(퐺 ≀ 푆푁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By subtracting the set of fixed points of 푆푁, we also see that (̃푋 푁 /퐺푁 − Δ푁)/푆푁 is equivalent to (̃푋 푁 − ̃Δ푁)/(퐺 ≀ 푆푁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' See also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [20, 30] for similar results in continuous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 6The sign representation can also be written as 퐷[−](휎) = sgn(휎), where sgn(휎) stands for the signature of 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is defined by sgn(휎) = ±1 for even (odd) permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 9 there exist the following two distinct time-evolution kernels for 푁 identical walkers on Z: 푈 [±] 휏 (푥, 푦) = ∑ 휎∈푆푁 퐷[±](휎)̃푈휏(푥, 휎푦) = ∑ 휎∈푆푁 (±1)#휎 ̃푈휏(푥, 휎푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (25) Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (25) satisfies the identity 푈 [±] 휏 (휎푥, 푦) = (±1)#휎푈 [±] 휏 (푥, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The weight factors (24) thus describe particle-exchange phases under the permutation of identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is now obvious that the two distinct representations 퐷[±] correspond to two distinct particle statistics: 푈 [+] 휏 describes the time-evolution kernel for 푁 identical bosons, while 푈 [−] 휏 describes that for 푁 identical fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (푁 identical walkers on a circle) Let us next consider 푁 identical particleson the periodic lattice of 퐿 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the discrete group is the wreath product Γ = 퐿Z ≀ 푆푁 , whose presentation is given by 퐿Z ≀ 푆푁 = ⟨ 푡1, ⋯ , 푡푁 , 휎1, ⋯ , 휎푁−1 ||||||| 푡푖푡푗 = 푡푗푡푖, 휎2 푖 = 푒, 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, 휎푖휎푗 = 휎푗휎푖 (|푖 − 푗| ≥ 2), 휎푖푡푖휎푖 = 푡푖+1, 휎푖푡푗휎푖 = 푡푗 (푗 ≠ 푖, 푖 + 1) ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (26) Here the actions of the generators 푡푖 and 휎푖 are defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (23) and 푡푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖 + 퐿, 푥푖+1, ⋯ , 푥푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (27) Note that any element of 퐿Z≀푆푁 can be written as 푡푛1 1 ⋯ 푡푛푁 푁 휎, where 휎 is a permutation and 푛1, ⋯ , 푛푁 = 0, ±1, ⋯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action on 푥 = (푥1, ⋯ , 푥푁 ) is given by 푡푛1 1 ⋯ 푡푛푁 푁 휎푥 = (푥휎(1) + 푛1퐿, ⋯ , 푥휎(푁) + 푛푁 퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now we have to classify one-dimensional unitary representation 퐷 ∶ 퐿Z ≀ 푆푁 → 푈(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' First, the relations 휎2 푖 = 푒 and 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1 imply 퐷(휎1) = ⋯ = 퐷(휎푁 ) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Second, the relation 휎푖푡푖휎푖 = 푡푖+1 implies 퐷(휎푖)퐷(푡푖)퐷(휎푖) = 퐷(푡푖+1), which, together with 퐷(휎푖)2 = 1, leads to 퐷(푡1) = ⋯ = 퐷(푡푁 ) = e푖휃, where 휃 ∈ R/2휋R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Thus we have two distinct one-parameter families of the maps 퐷[휃,±] given by 퐷[휃,±](푡푛1 1 ⋯ 푡푛푁 푁 휎) = e푖(푛1+⋯+푛푁 )휃(±1)#휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (28) The time-evolution kernel for 푁 identical walkers on Z/퐿Z is therefore 푈 [휃,±] 휏 (푥, 푦) = ∞ ∑ 푛1=−∞ ⋯ ∞ ∑ 푛푁 =−∞ ∑ 휎∈푆푁 퐷[휃,±](푡푛1 1 ⋯ 푡푛푁 푁 휎)̃푈휏(푥, 푡푛1 1 ⋯ 푡푛푁 푁 휎푦) = ∞ ∑ 푛1=−∞ ⋯ ∞ ∑ 푛푁 =−∞ e푖(푛1+⋯+푛푁 )휃 푈 [±] 휏 (푥, 푡푛1 1 ⋯ 푡푛푁 푁 푦), (29) where 푈 [±] 휏 is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Notice that the kernel (29) satisfies the identities 푈 [휃,±] 휏 (휎푥, 푦) = (±1)#휎푈 [휃,±] 휏 (푥, 푦) and 푈 [휃,±] 휏 (푡푖푥, 푦) = e푖휃 푈 [휃,±] 휏 (푥, 푦) for any 푖 = 1, ⋯ , 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Physically, 푈 [휃,+] 휏 (푈 [휃,−] 휏 ) de- scribes the system of 푁 identical bosons (fermions) on a circle with a nonzero magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (푁 identical walkers on the half line) Let us next consider 푁 identical particles on the semi-infinite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the discrete group is Γ = Z2 ≀ 푆푁 , where Z2 ≀ 푆푁 = ⟨ 푟1, ⋯ , 푟푁 , 휎1, ⋯ , 휎푁−1 ||||||| 푟푖푟푗 = 푟푗푟푖, 푟2 푖 = 휎2 푖 = 푒, 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, 휎푖휎푗 = 휎푗휎푖 (|푖 − 푗| ≥ 2), 휎푖푟푖휎푖 = 푟푖+1, 휎푖푟푗휎푖 = 푟푗 (푗 ≠ 푖, 푖 + 1) ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (30) The actions of the generators are defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (23) and 푟푖푥 ≔ (푥1, ⋯ , 푥푖−1, 1 − 푥푖, 푥푖+1, ⋯ , 푥푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (31) 10 Note that any element of Z2≀푆푁 can be written as the product 푟푛1 1 ⋯ 푟푛푁 푁 휎, where 휎 ∈ 푆푁 and 푛1, ⋯ , 푛푁 = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action on 푥 = (푥1, ⋯ , 푥푁 ) is given by 푟푛1 1 ⋯ 푟푛푁 푁 휎푥 = (⋯ , 푥휎(푖), ⋯) for 푛푖 = 0 and 푟푛1 1 ⋯ 푟푛푁 푁 휎푥 = (⋯ , 1 − 푥휎(푖), ⋯) for 푛푖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By repeating the same procedure as above, one can show that one-dimensional unitary represen- tation 퐷 ∶ Z2 ≀ 푆푁 → 푈(1) must satisfy 퐷(푟1) = ⋯ = 퐷(푟푁 ) = ±1 and 퐷(휎1) = ⋯ = 퐷(휎푁−1) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence there exist 22 = 4 distinct maps 퐷[휙,±] given by 퐷[휙,±](푟푛1 1 ⋯ 푟푛푁 푁 휎) = e푖(푛1+⋯+푛푁 )휙(±1)#휎, (32) where 휙 ∈ {0, 휋 (mod 2휋)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The time-evolution kernel for 푁 identical walkers on Z/Z2 is therefore 푈 [휙,±] 휏 (푥, 푦) = 1 ∑ 푛1=0 ⋯ 1 ∑ 푛푁 =0 ∑ 휎∈푆푁 퐷[휙,±](푟푛1 1 ⋯ 푟푛푁 푁 휎)̃푈휏(푥, 푟푛1 1 ⋯ 푟푛푁 푁 휎푦) = 1 ∑ 푛1=0 ⋯ 1 ∑ 푛푁 =0 e푖(푛1+⋯+푛푁 )휙 푈 [±] 휏 (푥, 푟푛1 1 ⋯ 푟푛푁 푁 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (33) Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (33) satisfies 푈 [휙,±] 휏 (휎푥, 푦) = (±1)#휎푈 [휙,±] 휏 (푥, 푦) and 푈 [휙,±] 휏 (푟푖푥, 푦) = e푖휙 푈 [휙,±] 휏 (푥, 푦) for any 푖 = 1, ⋯ , 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, 푈 [휙,±] 휏 describes the system of 푁 identical bosons (fermions) that acquire the phase shift 휙 when reflected off the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (푁 identical walkers on a finite interval) Let us finally consider 푁 identical particles on a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the discrete group is Γ = 퐷∞ ≀ 푆푁 , where 퐷∞ ≀ 푆푁 = ⟨ 푡1, ⋯ , 푡푁 , 푟1, ⋯ , 푟푁 , 휎1, ⋯ , 휎푁−1 ||||||||| 푡푖푡푗 = 푡푗푡푖, 푟푖푟푗 = 푟푗푟푖, 푟2 푖 = 휎2 푖 = 푒, 푟푖푡푖푟푖 = 푡−1 푖 , 푟푖푡푗푟푖 = 푡푗 (푗 ≠ 푖), 휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1, 휎푖휎푗 = 휎푗휎푖 (|푖 − 푗| ≥ 2), 휎푖푟푖휎푖 = 푟푖+1, 휎푖푟푗휎푖 = 푟푗 (푗 ≠ 푖, 푖 + 1) ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (34) The actions of the generators are given by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (23), (31), and 푡푖푥 ≔ (푥1, ⋯ , 푥푖−1, 푥푖 + 2퐿, 푥푖+1, ⋯ , 푥푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (35) We note that any element of 퐷∞ ≀ 푆푁 can be written as the product 푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎, where 휎 ∈ 푆푁 , 푛1, ⋯ , 푛푁 = 0, ±1, ±2, ⋯, and 푚1, ⋯ , 푚푁 = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its action is given by 푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎푥 = (⋯ , 푥휎(푖) + 2푛푖퐿, ⋯) for 푚푖 = 0 and 푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎푥 = (⋯ , 2푛푖퐿 + 1 − 푥휎(푖), ⋯) for 푚푖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now it is a straightforward exercise to show that there exist 23 = 8 distinct one-dimensional unitary representations of the wreath product 퐷∞ ≀ 푆푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The result is the following maps: 퐷[휃,휙,±](푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎) = e푖(푛1+⋯+푛푁 )휃 e푖(푚1+⋯+푚푁 )휙(±1)#휎, (36) where 휃, 휙 ∈ {0, 휋 (mod 2휋)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Correspondingly, we have the following eight distinct time-evolution kernels for 푁 identical walkers on Z/퐷∞: 푈 [휃,휙,±] 휏 (푥, 푦) = ∞ ∑ 푛1=−∞ 1 ∑ 푚1=0 ⋯ ∞ ∑ 푛푁 =−∞ 1 ∑ 푚푁 =0 ∑ 휎∈푆푁 퐷[휃,휙,±](푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎)̃푈휏(푥, 푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 휎푦) = ∞ ∑ 푛1=−∞ 1 ∑ 푚1=0 ⋯ ∞ ∑ 푛푁 =−∞ 1 ∑ 푚푁 =0 e푖(푛1+⋯+푛푁 )휃 e푖(푚1+⋯+푚푁)휙 푈 [±] 휏 (푥, 푡푛1 1 푟푚1 1 ⋯ 푡푛푁 푁 푟푚푁 푁 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (37) Physically, 푈 [휃,휙,±] 휏 describes the system of 푁 identical bosons (fermions) that acquire the phase shifts 휙 and 휃 + 휙 when reflected off the boundaries 푥 = 1 and 푥 = 퐿, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 11 4 Asides Now, there exist several other quantities that can be constructed in exactly the same way as for the time-evolution kernel (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Examples include the resolvent kernel (Green’s function) and the canonical density matrix (density matrix in the canonical ensemble).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Another example is a unitary representation of an arbitrary group 퐺 on a (tensor-product) Hilbert space, which includes the time-evolution kernel in discrete-time quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this section, we shall briefly discuss the construction of these quantities on the orbit space Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 Resolvent kernel Let us first start with the resolvent kernel—a matrix element of the resolvent operator in position space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let 퐻 be the Hamiltonian operator of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Then, the resolvent operator 퐺퐸 = (퐸퐼 − 퐻)−1 for Im 퐸 > 0 and the time-evolution operator 푈휏 = e−푖퐻휏 for 휏 > 0 are transformed into one another through the Laplace transform 푖(퐸퐼 −퐻)−1 = ∫ ∞ 0 푑휏 e−푖퐻휏 e푖퐸휏 and the inverse Laplace transform e−푖퐻휏 = ∫ ∞+푖휖 −∞+푖휖 푑퐸 2휋 푖(퐸퐼 − 퐻)−1 e−푖퐸휏, respectively, where 휖 is an arbitrary positive real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Consequently, the matrix elements 푈휏(푥, 푦) = ⟨푥|푈휏|푦⟩ and 퐺퐸(푥, 푦) = ⟨푥|퐺퐸|푦⟩ are mutually related through the following: 푖퐺퐸(푥, 푦) = ∫ ∞ 0 푑휏 푈휏(푥, 푦) e푖퐸휏 for Im 퐸 > 0, (38a) 푈휏(푥, 푦) = ∫ ∞+푖휖 −∞+푖휖 푑퐸 2휋 푖퐺퐸(푥, 푦) e−푖퐸휏 for 휏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (38b) Hence, by applying the Laplace transform to the formula (4), we find that the resolvent kernel on Λ/Γ takes the following form: 퐺퐸(푥, 푦) = ∑ 훾∈Γ 퐷(훾) ̃퐺퐸(푥, 훾푦), (39) where 푖 ̃퐺퐸(푥, 푦) = ∫ ∞ 0 푑휏 ̃푈휏(푥, 푦) e푖퐸휏 (Im 퐸 > 0) is the resolvent kernel on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' An immediate application of the above formula is the local density of states given by 휌퐸(푥) = ⟨푥|훿(퐸퐼 − 퐻)|푥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, by using the identity lim Im 퐸→0+(퐸퐼 − 퐻)−1 = P(퐸퐼 − 퐻)−1 − 푖휋훿(퐸퐼 − 퐻), (40) where Pstands for the Cauchy principal value, we find Im 퐺퐸(푥, 푥) = Im⟨푥|(퐸퐼 −퐻)−1|푥⟩ = −휋⟨푥|훿(퐸퐼 − 퐻)|푥⟩ = −휋휌퐸(푥) in the limit Im 퐸 → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Thus, 휌퐸(푥) = − 1 휋 Im ∑ 훾∈Γ 퐷(훾) ̃퐺퐸(푥, 훾푥) as Im 퐸 → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (41) The density of states 휌퐸 = tr 훿(퐸퐼 − 퐻) then takes the form 휌퐸 = −(1/휋) Im ∑푥∈Λ/Γ ∑훾∈Γ 퐷(훾) ̃퐺퐸(푥, 훾푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Canonical density matrix Let us next consider the canonical density matrix on Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In thermal equilibriumat temperature 훽−1, the canonical density matrix is given by 휌훽 = 푈−푖훽/푍(훽), where 푈−푖훽 = e−훽퐻 is the Gibbs operator and 푍(훽) = tr 푈−푖훽 is the canonical partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Note that the Gibbs operator satisfies the composition law 푈−푖훽1푈−푖훽2 = 푈−푖(훽1+훽2), the hermiticity 푈 † −푖훽 = 푈−푖훽, and the initial condition 푈0 = 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Its matrix elements (heat kernel) 푈−푖훽(푥, 푦) = ⟨푥| e−훽퐻 |푦⟩ must then satisfy these conditions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Namely, we must have ∑푧∈Λ/Γ 푈−푖훽1(푥, 푧)푈−푖훽2(푧, 푦) = 푈−푖(훽1+훽2), 푈−푖훽(푥, 푦) = 푈−푖훽(푦, 푥), and 푈0(푥, 푦) = 훿푥,푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Under these conditions, one can again show that 푈−푖훽(푥, 푦) can be written as 푈−푖훽(푥, 푦) = ∑훾∈Γ 퐷(훾)̃푈−푖훽(푥, 훾푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence the matrix elements of the canonical density matrix is 휌훽(푥, 푦) = 1 푍(훽) ∑ 훾∈Γ 퐷(훾)̃푈−푖훽(푥, 훾푦), (42) 12 where 휌훽(푥, 푦) = ⟨푥|휌훽|푦⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here 푍(훽) is the canonical partition function given by 푍(훽) = ∑ 푥∈Λ/Γ ∑ 훾∈Γ 퐷(훾)̃푈−푖훽(푥, 훾푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (43) We note that the partition function (43) can also be written as 푍(훽) = ∑푥∈Λ/Γ ∑훾∈Γ 퐷(훾)⟨푥| e−훽 ̃퐻 |훾푥⟩ = ∑훾∈Γ 퐷(훾) tr(e−훽 ̃퐻 푊훾), where ̃퐻 is the Hamiltonian operator on Λ and 푊훾 is a unitary operator defined by 푊훾|푥⟩ = |훾푥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3 Unitary representations of arbitrary groups on a tensor-product Hilbert space As mentioned in the beginning of section 2, our main formula (4) is also applicable to discrete-time quantum walk, where the time 휏 takes discrete values and the one-particle Hilbert space is the tensor- product of the position and coin Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this section, we shall see this from a more general perspective: the construction of matrix elements of a unitary representation of an arbitrary group 퐺 on a tensor-product Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The time-evolution kernel in discrete-time quantum walk just corresponds to the special case 퐺 = Z (the additive group of integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' To begin with, let {푈푔 ∈ 푈(H) ∶ 푔 ∈ 퐺} be a unitary representation of 퐺 on the tensor-product Hilbert space H = Hposition⊗Hcoin, where 푈(H) stands for the set of unitary operators on H, Hposition = 푙2(Λ/Γ) is the set of square-summable sequences on the orbit space Λ/Γ, and Hcoin = C푑 is the 푑- dimensional complex vector space that describes internal degrees of freedom of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let {|푥⟩} and {|푖⟩} be complete orthonormal systems of Hposition and Hcoin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The set {|푥⟩ ⊗ |푖⟩} then provides a complete orthonormal system of the total Hilbert space Hsuch that the matrix elements of 푈푔 can be defined as 푈푔(푥, 푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푦, 푗) = (⟨푥| ⊗ ⟨푖|)푈푔(|푦⟩ ⊗ |푗⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We now define 푈푔(푥, 푦) as the following 푑 × 푑 matrix: 푈푔(푥, 푦) ≔ ⎛ ⎜ ⎜ ⎝ 푈푔(푥, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푦, 1) ⋯ 푈푔(푥, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푦, 푑) ⋮ ⋱ ⋮ 푈푔(푥, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푦, 1) ⋯ 푈푔(푥, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푦, 푑) ⎞ ⎟ ⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (44) Since the unitary representation must satisfy the group composition law 푈푔1푈푔2 = 푈푔1푔2, the unitarity 푈 † 푔 (= 푈 −1 푔 ) = 푈푔−1, and the initial condition 푈푒 = 퐼, the matrix (44) must also satisfy the following properties: ∑ 푧∈Λ/Γ 푈푔1(푥, 푧)푈푔2(푧, 푦) = 푈푔1푔2(푥, 푦), (45a) 푡푈푔(푥, 푦) = 푈푔−1(푦, 푥), (45b) 푈푒(푥, 푦) = 훿푥,푦ퟏ, (45c) where 푥, 푦 ∈ Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here 푡 and ퟏ stand for the matrix transpose and the 푑×푑 identity matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now it is a straightforward exercise to show that the matrix (44) can be written as 푈푔(푥, 푦) = ∑ 훾∈Γ 퐷(훾)̃푈푔(푥, 훾푦), (46) where ̃푈푔(푥, 푦) is a 푑×푑 matrix subject to the conditions ∑푧∈Λ ̃푈푔1(푥, 푧)̃푈푔2(푧, 푦) = ̃푈푔1푔2(푥, 푦), 푡 ̃푈푔(푥, 푦) = ̃푈푔−1(푦, 푥), ̃푈푒(푥, 푦) = 훿푥,푦ퟏ, and ̃푈푔(훾푥, 훾푦) = ̃푈푔(푥, 푦) for any 푥, 푦 ∈ Λ and 훾 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is also straightfor- ward to show that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (46) satisfies the following boundary condition: 푈푔(훾푥, 푦) = 퐷(훾)푈푔(푥, 푦), ∀훾 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (47) It is now obvious that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (46) provides the time-evolution kernel of continuous-time quantum walk with internal degrees of freedom when 퐺 = R (the additive group of real numbers) and of discrete- time quantum walk when 퐺 = Z (the additive group of integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is also obvious that the examples presented in section 3 apply to discrete-time quantum walk as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 13 5 Conclusion Inspired by the covering-space method in path integral on multiply-connected spaces, we have devel- oped a general theory of quantum walk on orbit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this note, we have proved the universal formulae for time-evolution kernels, resolvent kernels, canonical density matrices, and unitary repre- sentations of arbitrary groups in continuous- and discrete-time quantum walks on the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' All of these quantities are given by summations over the orbit of initial point on Λ, where each orbit is weighted by a phase factor given by a one-dimensional unitary representations of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' There are several advantages of this orbit-space method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A main advantage is its universality: our formulae are just based on geometric and group-theoretic structures of configuration spaces so that they are robust against any perturbations or interparticle interactions as far as boundary conditions (8) are remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Another advantage is its computational simplicity: in our formalism, one just needs to compute matrix elements on Λ, which is generally much easier than computations on Λ/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Finally, let us comment on one possible future direction of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A promising direction would be a generalization of our formulae to the problem of identical walkers on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Recent studies have shown that exotic statistics may show up in many-body problems of identical particles on graphs [31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Such exotic statistics are generalizations of braid-group statistics in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Hence, just as in topological quantum computation using anyons [36], they would have potential applications in quantum computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Our formalism and its generalization may well serve a basic tool for studying dynamics as well as thermodynamics of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Acknowledgments The author would like to thank Naoto Namekata for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A Sample computations Continuous-time quantum walk is just equivalent to tight-binding models in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The advantage of this perspective is that it is straightforward to study many-particle problems by using the second-quantization formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this section, we study tight-binding models for free spinless particles in one dimension and present sample computations that justify the formulae in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1 Tight-binding model on the infinite line Let us first consider spinless particles on the integer lattice Z only with a nearest-neighbor coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the second-quantization formalism, the Hamiltonian operator is given by ̃퐻 = −휔 2 ∑ 푥∈Z (푎† 푥+1푎푥 + 푎† 푥 푎푥+1) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1) where 휔(> 0) is a hopping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 푎푥 and 푎† 푥 are annihilation and creation operators for spinless bosons (fermions) and subject to the following (anti-)commutation relations: [푎푥, 푎† 푦]∓ = 훿푥,푦 and [푎푥, 푎푦]∓ = 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2) where [퐴, 퐵]∓ = 퐴퐵 ∓ 퐵퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let |0⟩ be the Fock vacuum that satisfies 푎푥|0⟩ = 0 for all 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The time-evolution kernel in the one-particle sector of the model is then given by ̃푈휏(푥, 푦) = ⟨푥| e−푖 ̃퐻휏 |푦⟩, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3) 14 where |푥⟩ = 푎† 푥|0⟩ is the position-space basis in the one-particle sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It satisfies the orthonormality ⟨푥|푦⟩ = ⟨0|푎푥푎† 푦|0⟩ = 훿푥,푦 for both bosons and fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In order to calculate the matrix element (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3), we first diagonalize the Hamiltonian operator, which can be achieved by the following Fourier integral: 푎푥 = ∫ 휋 −휋 푑푝 2휋 ̃푎푝 e푖푝푥, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4) where ̃푎푝 and ̃푎† 푝 satisfy [ ̃푎푝, ̃푎† 푞]∓ = 2휋훿(푝 − 푞) and [ ̃푎푝, ̃푎푞]∓ = 0 for any 푝, 푞 ∈ (−휋, 휋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3), we obtain ̃퐻 = ∫ 휋 −휋 푑푝 2휋 휀푝 ̃푎† 푝 ̃푎푝, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='5) where 휀푝 = −휔 cos(푝) is the single-particle energy eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is now easy to see that the time- evolution kernel (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3) takes the following form: ̃푈휏(푥, 푦) = ⟨0|푎푥 e−푖 ̃퐻휏 푎† 푦|0⟩ = ∫ 휋 −휋 푑푝 2휋 ∫ 휋 −휋 푑푞 2휋 ⟨0| ̃푎푝 e−푖 ̃퐻휏 ̃푎† 푞|0⟩ e푖푝푥−푖푞푦 = ∫ 휋 −휋 푑푝 2휋 ∫ 휋 −휋 푑푞 2휋 e−푖휀푞푡⟨0| ̃푎푝 ̃푎† 푞|0⟩ e푖푝푥−푖푞푦 = ∫ 휋 −휋 푑푝 2휋 e푖휔휏 cos(푝) e푖푝(푥−푦), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6) where in the third equality we have used e−푖퐻휏 ̃푎† 푞|0⟩ = e−푖휀푞푡 ̃푎† 푞|0⟩, which follows from e−푖퐻휏 ̃푎† 푞 e푖퐻휏 = e−푖휀푞푡 ̃푎† 푞 and e−푖퐻휏 |0⟩ = |0⟩ (or, equivalently, [퐻, ̃푎† 푞] = 휀푞 ̃푎† 푞 and 퐻|0⟩ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The fourth equality follows from ⟨0| ̃푎푝 ̃푎† 푞|0⟩ = 2휋훿(푝 − 푞) for both bosons and fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' To evaluate the last integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6), we note that e푖휔휏 cos(푝) is a generating function of the Bessel function of the first kind 퐽푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, e푖휔휏 cos(푝) = ∞ ∑ 푛=−∞ e푖푛(푝+ 휋 2 ) 퐽푛(휔휏) = ∞ ∑ 푛=−∞ e푖 휋 2 |푛| 퐽|푛|(휔휏) e−푖푛푝, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='7) where the second equality follows from 퐽−푛(푥) = e푖푛휋 퐽푛(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='7) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6) and then using the orthogonal relation ∫ 휋 −휋 푑푝 2휋 e푖푝(푥−푦−푛) = 훿푛,푥−푦, we obtain ̃푈휏(푥, 푦) = e푖 휋 2 |푥−푦| 퐽|푥−푦|(휔휏), ∀푥, 푦 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8) This is the well-known transition amplitude for a single walker on the lattice Z (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8) satisfies the composition law (3a), the unitarity (3b), and the initial condition (3c), which follow from the addition theorem 퐽푛1−푛2(푥1 + 푥2) e푖 휋 2 (푛1−푛2) = ∑푛∈Z 퐽푛1−푛(푥1)퐽푛2−푛(푥2) e푖 휋 2 (푛1−푛) e푖 휋 2 (푛2−푛) (푛1, 푛2 ∈ Z), the analytic continuation 퐽푛(e푖휋 푥) = e푖푛휋 퐽푛(푥), and 퐽푛(0) = 훿푛,0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Note also that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8) enjoys the translation invariance ̃푈휏(푥 + 푧, 푦 + 푧) = ̃푈휏(푥, 푦) and the reflection invariance ̃푈휏(푧 − 푥, 푧 − 푦) = ̃푈휏(푥, 푦) for any 푥, 푦, 푧 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As we shall see shortly, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8) provides the building block for the construction of time-evolution kernels for a free particle on a circle, the half line, and a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Several comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Resolvent kernel for a single walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1, the resolvent kernel (Green’s function) is given by the Laplace transform of ̃푈휏(푥, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Let 퐸 be a complex number with Im 퐸 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 15 Then we have 푖 ̃퐺퐸(푥, 푦) = ∫ ∞ 0 푑휏 ̃푈휏(푥, 푦) e푖퐸휏 = ∫ 휋 −휋 푑푝 2휋 푖 e푖푝(푥−푦) 퐸 + 휔 cos(푝) = 2푖 휔 ∮|푧|=1 푑푧 2휋푖 푧|푥−푦| 푧2 + 2퐸 휔 푧 + 1, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='9) where in the second equality we have substituted the last line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6) and performed the integration with respect to 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the last equality we have changed the integration variable from 푝 to 푧 = e푖푝, where the integration is over the closed loop |푧| = 1 in the counter-clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By using the residue theorem we find 푖 ̃퐺퐸(푥, 푦) = e푖푝|푥−푦| 휔 sin(푝), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='10) where we have parameterized the energy as 퐸 = −휔 cos(푝) with Re 푝 ∈ (0, 휋) and Im 푝 ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='10) provides the building block for the construction of single-particle resolvent kernels on a circle, the half line, and a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Heat kernel for a single walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The matrix element of the Gibbs operator e−훽 ̃퐻 can be calculated in exactly the same way as for ̃푈휏(푥, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Under the substitution 휏 → −푖훽 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='6) we find ̃푈−푖훽(푥, 푦) = ∫ 휋 −휋 푑푝 2휋 e훽휔 cos(푝) e푖푝(푥−푦) = 퐼푥−푦(훽휔), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='11) where 퐼푛(푥) = 퐼−푛(푥) stands for the modified Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Here in the last line we have used the fact that e훽휔 cos(푝) is the generating function of 퐼푛(훽휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, e훽휔 cos(푝) = ∞ ∑ 푛=−∞ 퐼푛(훽휔) e−푖푛푝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='12) By substituting this into the first line and using the orthogonal relation ∫ 휋 −휋 푑푝 2휋 e푖푝(푥−푦−푛) = 훿푛,푥−푦, we arrive at eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='11) provides the building block for the construction of canonical density matrices for free particles on a circle and a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Time-evolution kernel for 푁 identical walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In the second-quantization formalism, it is easy to generalize the above results to many-particle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' First, the position-space basis in the 푁-particle sector is given by |푥1, ⋯ , 푥푁 ⟩ ≔ 푎† 푥1 ⋯ 푎† 푥푁 |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='13) Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='13) satisfies the orthonormality condition on the orbit space (Z푁 − Δ푁 )/푆푁 ≅ {(푥1, ⋯ , 푥푁 ) ∈ Z푁 ∶ 푥1 > ⋯ > 푥푁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, for 푥1 > ⋯ > 푥푁 and 푦1 > ⋯ > 푦푁, we have ⟨푥1, ⋯ , 푥푁 |푦1, ⋯ , 푦푁 ⟩ = ⟨0|푎푥푁 ⋯ 푎푥1푎† 푦1 ⋯ 푎† 푦푁 |0⟩ = ∑ 휎∈푆푁 (±1)#휎훿푥휎(1),푦1 ⋯ 훿푥휎(푁),푦푁 = 훿푥1,푦1 ⋯ 훿푥푁 ,푦푁 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='14) 16 where the last line follows from the fact that (푥휎(1), ⋯ , 푥휎(푁)) and (푦1, ⋯ , 푦푁) cannot be equal except for 휎 = 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is now easy to show that the time-evolution kernels for 푁 identical bosons and fermions take the following forms:7 ⟨푥1, ⋯ , 푥푁 | e−푖 ̃퐻휏 |푦1, ⋯ , 푦푁 ⟩ = ⟨0|푎푥푁 ⋯ 푎푥1 e−푖 ̃퐻휏 푎† 푦1 ⋯ 푎† 푦푁 |0⟩ = [ 푁 ∏ 푗=1 ∫ 휋 −휋 푑푝푗 2휋 ∫ 휋 −휋 푑푞푗 2휋 ] e푖휔휏(cos(푞1)+⋯+cos(푞푁 )) × ⟨0| ̃푎푝푁 ⋯ ̃푎푝1 ̃푎† 푞1 ⋯ ̃푎† 푞푁 |0⟩ e푖푝1푥1+⋯+푖푝푁 푥푁 −푖푞1푦1−⋯−푖푞푁푦푁 = ∑ 휎∈푆푁 (±1)#휎 푁 ∏ 푗=1 e푖 휋 2 |푥푗−푦휎(푗)| 퐽|푥푗−푦휎(푗)|(휔휏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='15) As shown in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (29), (33), and (37), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='15) can be used to construct the time-evolution kernels for free identical walkers on a circle, the half line, and a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2 Tight-binding model on a circle Let us next consider the tight-binding model for free spinless particles on the periodic lattice {1, 2, ⋯ , 퐿 (mod 퐿)} subject to the twisted boundary condition 푎푥+퐿 = e푖휃 푎푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As we shall see shortly, the following Hamiltonian operator yields the desired results: 퐻 = −휔 2 퐿 ∑ 푥=1 (푎† 푥+1푎푥 + 푎† 푥 푎푥+1) , where 푎퐿+1 ≡ e푖휃 푎1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='16) In the following, we assume that 휃 ranges from 0 to 2휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In order to compute the time-evolution kernel, we first have to diagonalize the Hamiltonian oper- ator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='16), which can be done by using the mode expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Under the twisted boundary condition, the annihilation operator can be expanded into the following: 푎푥 = 1 √퐿 퐿−1 ∑ 푝=0 ̃푎푝 e푖 2푝휋+휃 퐿 푥, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='17) where ̃푎푝 and ̃푎† 푝 satisfy [ ̃푎푝, ̃푎† 푞]∓ = 훿푝,푞 and [ ̃푎푝, ̃푎푞]∓ = 0 for any 푝, 푞 ∈ {0, 1, ⋯ , 퐿 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='17) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='16), we find that the Hamiltonian operator is diagonalized as follows: 퐻 = 퐿−1 ∑ 푝=0 휀푝 ̃푎† 푝 ̃푎푝, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='18) where 휀푝 = −휔 cos(2푝휋+휃 퐿 ) is the single-particle energy eigenvalue on the periodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Now it is easy to compute the time-evolution kernel in the one-particle sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A straightforward 7It should be noted that ̃푈휏(푥1, ⋯ , 푥푁 , 푦1, ⋯ , 푦푁) = ∏푁 푗=1 e푖 휋 2 |푥푗−푦푗| 퐽|푥푗−푦푗|(휔휏) is equivalent to a single-particle time-evolution kernel on Z푁 rather than Z푁 − Δ푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' As noted in the beginning of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2, in this note we will not touch upon this type of issues related to the fixed points of 푆푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 17 calculation gives 푈 [휃] 휏 (푥, 푦) = ⟨푥| e−푖퐻휏 |푦⟩ = ⟨0|푎푥 e−푖퐻휏 푎† 푦|0⟩ = 1 퐿 퐿−1 ∑ 푝=0 퐿−1 ∑ 푞=0 ⟨0| ̃푎푝 e−푖퐻휏 ̃푎† 푞|0⟩ e푖 2푝휋+휃 퐿 푥−푖 2푞휋+휃 퐿 푦 = 1 퐿 퐿−1 ∑ 푝=0 퐿−1 ∑ 푞=0 e−푖휀푞휏⟨0| ̃푎푝 ̃푎† 푞|0⟩ e푖 2푝휋+휃 퐿 푥−푖 2푞휋+휃 퐿 푦 = 1 퐿 퐿−1 ∑ 푝=0 e푖휔휏 cos( 2푝휋+휃 퐿 ) e푖 2푝휋+휃 퐿 (푥−푦), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='19) where we have used e−푖퐻휏 ̃푎† 푞|0⟩ = e−푖휀푞휏 ̃푎† 푞|0⟩ in the fourth line and ⟨0| ̃푎푝 ̃푎† 푞|0⟩ = 훿푝,푞 in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Notice that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='19) is the summation over the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In order to obtain the summation over winding numbers, we therefore have to perform a resummation, which can be done by using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' By substituting e푖휔푡 cos( 2푝휋+휃 퐿 ) = ∑푚∈Z e푖 휋 2 |푚| 퐽|푚|(휔푡) e−푖푚 2푝휋+휃 퐿 into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='19) and using the orthogonal relation 1 퐿 ∑퐿−1 푝=0 e푖 2푝휋+휃 퐿 (푥−푦−푚) = e푖푛휃 훿푚,푥−푦−푛퐿 (푛 ∈ Z), we find that the time-evolution kernel (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='19) can be put into the following alternative equivalent form:8 푈 [휃] 휏 (푥, 푦) = ∞ ∑ 푛=−∞ e푖푛휃 e푖 휋 2 |푥−푦−푛퐿| 퐽|푥−푦−푛퐿|(휔휏), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='20) which exactly coincides with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (15) with ̃푈휏( ⋅ , ⋅ ) given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' This sample computation im- plies that there is an equivalence (or duality) between the summation over energy spectrum and the summation over particle’s trajectories, which is the heart of the trace formula in harmonic analysis and representation theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this respect, one could say that our formula is a version of the trace formula in lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Although we omit the details, it is not difficult to show that the resolvent kernel, the canonical density matrix, and the time-evolution kernel for 푁 identical particles all coincide with the universal formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We note in closing that the parameter 휃 can be removed from the twisted boundary condition under the gauge transformation 푎푥 ↦ 푉휃푎푥푉 −1 휃 = e푖 휃 퐿 푥 푎푥, where 푉휃 is a unitary operator given by 푉휃 = exp(−푖 휃 퐿 ∑퐿 푥=1 푥푎† 푥 푎푥) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=', ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In fact, a straightforward calculation gives 푉휃퐻푉 −1 휃 = −휔 2 퐿 ∑ 푥=1 (e−푖휃/퐿 푎† 푥+1푎푥 + e+푖휃/퐿 푎† 푥 푎푥+1) , where 푎퐿+1 ≡ 푎1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='21) The time-evolution kernel in the one-particle sector for this Hamiltonian coincides with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='20) up to a phase factor e푖 휃 퐿 (푥−푦) and hence is physically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='3 Tight-binding model on the half line Let us next consider the tight-binding model on the semi-infinite lattice {1, 2, ⋯} with the boundary condition 푎0 = 푒푖휙푎1, where 휙 ∈ {0, 휋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The Hamiltonian operator that ensures this boundary condition is given by 퐻 = −휔 2 ∞ ∑ 푥=1 (푎† 푥+1푎푥 + 푎† 푥 푎푥+1) − 휔 2 e푖휙 푎† 1 푎1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='22) 8The case 휃 = 0 was noted in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 18 By substituting the mode expansion 푎푥 = ∫ 휋 0 푑푝 2휋 ̃푎푝 (e−푖푝푥 + e푖휙 e−푖푝(1−푥)) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='23) we get the following diagonalized Hamiltonian operator: 퐻 = ∫ 휋 0 푑푝 2휋 휀푝 ̃푎† 푝 ̃푎푝, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='24) where 휀푝 = −휔 cos(푝) is the single-particle energy eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The time-evolution kernel for a single walker is given by 푈 [휙] 휏 (푥, 푦) = ⟨0|푎푥 e−푖퐻휏 푎† 푦|0⟩ = ∫ 휋 0 푑푝 2휋 e−푖휀푝휏 (e−푖푝푥 + e푖휙 e−푖푝(1−푥)) (e푖푝푦 + e푖휙 e푖푝(1−푦)) = ∫ 휋 −휋 푑푝 2휋 e푖휔휏 cos(푝) (e푖푝(푥−푦) + e푖휙 e푖푝(푥−1+푦)) = e푖 휋 2 |푥−푦| 퐽|푥−푦|(휔휏) + e푖휙 e푖 휋 2 |푥−1+푦| 퐽|푥−1+푦|(휔휏), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='25) which exactly coincides with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' Other quantities can be calculated in a similar way and coincide with the universal formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' We note that the model that satisfies the Dirichlet boundary condition 푎푥 = 0 at 푥 = 0 is described by the Hamiltonian operator 퐻 = −(휔/2) ∑∞ 푥=1(푎† 푥+1푎푥 + 푎† 푥 푎푥+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the time-evolution kernel coincides with another formula discussed in example 2 in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='4 Tight-binding model on a finite interval Let us finally quickly study the tight-binding model on the finite lattice {1, 2, ⋯ , 퐿} with the boundary conditions 푎0 = e푖휙 푎1 and 푎퐿+1 = e푖(휃+휙) 푎퐿, where 휃, 휙 ∈ {0, 휋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' The Hamiltonian operator is given by 퐻 = −휔 2 퐿−1 ∑ 푥=1 (푎† 푥+1푎푥 + 푎† 푥 푎푥+1) − 휔 2 e푖휙 푎† 1 푎1 − 휔 2 e푖(휃+휙) 푎† 퐿푎퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='26) This operator can be diagonalized by using the following mode expansions: 푎푥 = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 1 √퐿 ̃푎0 + 1 √2퐿 퐿−1 ∑ 푝=1 ̃푎푝 (e−푖 2푝휋 2퐿 푥 + e−푖 2푝휋 2퐿 (1−푥)) for 휃 = 0 & 휙 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 1 √ 2퐿 퐿−1 ∑ 푝=1 ̃푎푝 (e−푖 2푝휋 2퐿 푥 − e−푖 2푝휋 2퐿 (1−푥)) + 1 √ 퐿 ̃푎퐿(−1)푥 for 휃 = 0 & 휙 = 휋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 1 √ 2퐿 퐿−1 ∑ 푝=0 ̃푎푝 (e−푖 2푝휋+휃 2퐿 푥 + e푖휙 e−푖 2푝휋+휃 2퐿 (1−푥)) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='27) In fact, by substituting these into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='26) we find 퐻 = ⎧⎪⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎪⎩ 퐿 ∑ 푝=1 휀푝 ̃푎† 푝 ̃푎푝 for 휃 = 0 & 휙 = 휋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' 퐿−1 ∑ 푝=0 휀푝 ̃푎† 푝 ̃푎푝 otherwise, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='28) 19 where 휀푝 = −휔 cos(2푝휋+휃 2퐿 ) for any 휃, 휙 ∈ {0, 휋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' It is not difficult to show that the time-evolution kernel for a single walker can be put into the following expression irrespective of the values of 휃 and 휙: 푈 [휃,휙] 휏 (푥, 푦) = ⟨0|푎푥 e−푖퐻휏 푎† 푦|0⟩ = 1 2퐿 2퐿−1 ∑ 푝=0 e푖휔휏 cos( 2푝휋+휃 2퐿 ) (e푖 2푝휋+휃 2퐿 (푥−푦) + e푖휙 e푖 2푝휋+휃 2퐿 (푥−1+푦)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='29) Note that this is the summation over the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' However, as was done in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='2, this summation can be rewritten into the following summation over the bouncing numbers off the boundaries: 푈 [휃,휙] 휏 (푥, 푦) = ∞ ∑ 푛=−∞ [e푖푛휃 e푖 휋 2 |푥−푦−2푛퐿| 퐽|푥−푦−2푛퐿|(휔휏) + e푖푛휃 e푖휙 e푖 휋 2 |푥−2푛퐿−1+푦| 퐽|푥−2푛퐿−1+푦|(휔휏)] , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content='30) which exactly coincides with the universal formula (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' If one wants to study the model that satisfies the Dirichlet boundary conditions 푎푥 = 0 at 푥 = 0 and 푥 = 퐿 + 1, one should use 퐻 = −(휔/2) ∑퐿−1 푥=1(푎† 푥+1푎푥 + 푎† 푥 푎푥+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE1T4oBgHgl3EQfdQRv/content/2301.03193v1.pdf'} +page_content=' In this case, the 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LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +1 +Few-shot Semantic Segmentation with +Support-induced Graph Convolutional +Network +Jie Liu1 +j.liu5@uva.nl +Yanqi Bao2 +yanqibao1997@gmail.com +Wenzhe Yin1 +w.yin@uva.nl +Haochen Wang1 +h.wang3@uva.nl +Yang Gao2 +gaoy@nju.edu.cn +Jan-Jakob Sonke3 +j.sonke@nki.nl +Efstratios Gavves1 +egavves@uva.nl +1 University of Amsterdam +Amsterdam, Netherlands +2 Nanjing University +Nanjing, China +3 The Netherlands Cancer Institute +Amsterdam, Netherlands +Abstract +Few-shot semantic segmentation (FSS) aims to achieve novel objects segmentation +with only a few annotated samples and has made great progress recently. +Most of +the existing FSS models focus on the feature matching between support and query to +tackle FSS. However, the appearance variations between objects from the same cate- +gory could be extremely large, leading to unreliable feature matching and query mask +prediction. To this end, we propose a Support-induced Graph Convolutional Network +(SiGCN) to explicitly excavate latent context structure in query images. Specifically, +we propose a Support-induced Graph Reasoning (SiGR) module to capture salient query +object parts at different semantic levels with a Support-induced GCN. Furthermore, an +instance association (IA) module is designed to capture high-order instance context from +both support and query instances. By integrating the proposed two modules, SiGCN +can learn rich query context representation, and thus being more robust to appearance +variations. Extensive experiments on PASCAL-5i and COCO-20i demonstrate that our +SiGCN achieves state-of-the-art performance. +1 +Introduction +The development of deep convolutional neural networks has achieved great success in many +computer vision tasks, such as image classification [8, 26], semantic segmentation [5, 35, +© 2022. The copyright of this document resides with its authors. +It may be distributed unchanged freely in print or electronic forms. +arXiv:2301.03194v1 [cs.CV] 9 Jan 2023 + +2 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +43, 45], and object detection [6, 7, 9, 21]. Despite the effectiveness of deep networks, +their over-reliance on large-scale annotated dataset still remains a fundamental limitation +as data annotation requires large amount of human efforts, especially for dense prediction +tasks, e.g., semantic segmentation. To cope with these challenges, some weakly-supervised +and semi-supervised semantic segmentation techniques [3, 11, 24] try to introduce weak +annotations (e.g, points, scribbles, and bounding box) and unlabelled samples. However, +these techniques usually fail to work when novel classes that never appear in the training set +emerge. In such case, few-shot semantic segmentation (FSS) methods [1, 12, 13, 17, 19, +20, 25, 27, 29, 30, 31, 34, 37, 39, 40] struggle to segment novel objects with only a few +pixel-wise annotated samples. +… +Prototypes +Features +Support Image +Query Image +Support Mask +Freature +Matching +Prediction +Support Image +Query Image +Support Mask +Prediction +… +Prototypes +Activation Map +Graph +Updating +Guide +(a) Previous FSS Methods +(b) Our Support-induced GCN (SiGCN) +Figure 1: Comparison between (a) previous typical few-shot segmentation (FSS) methods +and (b) the proposed support-induced graph convolutional network (SiGCN). (a) Previous +FSS methods focus on designing effective feature matching mechanism between support pro- +totypes/features and query feature. However, such matching techniques usually fail to work +when the support and query objects exhibit large appearance variations. (b) Our SiGCN aims +to address the appearance variation problem by explicitly mining query context information. +Based on the activation map, SiGCN constructs a query graph to associate salient query ob- +ject parts. Then the support prototypes are introduced to guide the graph updating process +for effective query context excavation. +Meta-learning [23] is widely adopted in FSS to ensure better model generalization to +novel classes. Specifically, both training and testing sets are composed of many sub-tasks +(episodes), and each specific sub-task is divided into a support set and a query set, which +share the same object classes. The FSS models aim to segment images from the query set +given a few pixel-wise annotated images from the support set. Thus, the key challenge of the +FSS task is how to leverage the information from the support set and the query set effectively, +especially when the support and query objects exhibit large variations (e.g., appearance, +scale, and location). +Although existing FSS methods [17, 27, 40] have achieved extensive progress, we ob- +serve a significant drop in performance when the support and query images share large ap- +pearance variance. For instance, as shown in Figure 1, the birds in the support and query +images have completely different appearance, which makes it difficult to segment the query +image. Existing methods (Figure 1 (a)) usually generate support foreground features or +prototypes and then focus on designing various feature matching mechanisms between the +support foreground features/prototypes and the query feature to achieve the query mask pre- +diction. However, such matching with information coming only from the support set is +suboptimal when it comes to predicting an accurate mask for the query image. By con- +trast, different query object parts (e.g., the wings and the main body of the bird) contain rich + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +3 +contextual information, which is beneficial to the query object segmentation. +In this work, we propose a support-induced graph convolutional network (SiGCN) to +tackle the aforementioned challenge. Our main idea is to explicitly enrich the query context +representations with the guidance of support information in two ways. Firstly, we propose +a novel support-induced graph reasoning (SiGR) module to extract query context from dif- +ferent query object parts. Specifically, as shown in Figure 1(b), we utilize the activation +map, which is generated by the feature matching between the support and the query fea- +tures, to capture local object parts that are discriminative and easily localized in the query +image. These salient parts are associated together in the query graph by modeling the in- +teraction between them, thus learning rich context in the query image. The query graph +is then updated by the graph convolutional network, in which the state updating matrix is +generated by the support prototypes to effectively extract query context. Secondly, an in- +stance association (IA) module is introduced to aggregate query and support context from +instance level. Specifically, given activation maps produced from different semantic levels, +we get corresponding updated query graphs (i.e, query features) in the SiGR module. These +updated query features, which can be seen as different object instances, contain comple- +mentary context information. Therefore, we associate the query instances and the support +instance together to learn high-order instance context. By proposing the SiGR and IA mod- +ule, our SiGCN aggregates rich complementary context from both support and query images, +thus being more robust to appearance variations. +In summary, our contributions are summarized as follows: +• We propose a support-induced graph convolution network (SiGCN), which utilizes the +proposed IA and SiGR modules to capture complementary context from the query and +the support set, for addressing the appearance variation problem in the FSS task. +• We propose a novel graph updating mechanism, in which support prototypes guide the +node updating in the query graphs. This mechanism brings a significant performance +gain to the proposed SiGCN. +• Our method achieves state-of-the-art results on two FSS benchmarks, i.e., PASCAL- +5i and COCO-20i. Our SiGCN tackles the appearance variation problem from the +perspective of extracting query contextual information, which sheds light for future +research in this field. +2 +Related Work +2.1 +Semantic Segmentation +Semantic Segmentation is a fundamental computer vision task which aims to assign each +pixel with a class label. Fully convolutional network (FCN) [14] adopts fully convolutional +layers for dense prediction and achieves superior performance. Inspired by FCN, many net- +works have been invented to boost the performance of semantic segmentation. Unet [22] +presents an encoder-decoder structure to capture multi-scale context information. PSPNet +[45] utilizes the pyramid pooling to aggregate more object details. Dilated convolution [35] +and deformable convolution [5] are introduced to FCN to further improve the semantic seg- +mentation performance. Recently, K-Net [43] proposes to segment both instance and seman- +tic categories by a group of learnable kernels. However, these methods still require tons of + +4 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +samples with pixel-wise annotations in the training stage. In our work, we focus on few-shot +semantic segmentation. +2.2 +Few-shot Semantic Segmentation +Few-shot semantic segmentation is proposed to segment novel objects given only few anno- +tated samples. Most existing methods adopt episode-base meta-learning [31] strategy to en- +sure the efficient generalization of the model to the novel classes with few examples. Shaban +et al. [23] proposes the first few-shot semantic segmentation work, which generates param- +eters from the conditioning branch and then segments the query image in the segmentation +branch. Later on, prototype-based methods are proposed including CANet [39], PANet [29], +SG-One [44], PPNet [13], PMMs [33]. The intuition behind these methods lies in extracting +representative foreground or background prototypes for support samples and then interact- +ing these prototypes with query features using different strategies. Besides, matching-based +methods [17, 27, 31, 38, 40] also achieve great success in the few shot-segmentation task. +PFENet [27] introduces a prior mask by implementing pixel-to-pixel matching between high- +level support and query features, while PGNet [38] and SAGNN [31] propose to implement +element-wise matching between support and query by graph neural networks. Most recently, +HSNet [17] adopts multi-scale dense matching to construct hypercorrelations and uses 4D +convolutions to capture contextual information. CyCTR [40] designs a cycle consistency +transformer to aggregate pixel-wise support feature to query ones and achieves great suc- +cess. Different from previous work, we not only consider the matching between support and +query, but also highlight context excavation in the query feature, our model is thus more +robust to the object appearance variations. +2.3 +Graph Convolutional Network +Graph convolutional networks (GCNs) [10], first proposed to tackle the semi-supervised +classification, have seen massive application in computer vision tasks, e.g., action localiza- +tion [18, 36], object detection [16], and semantic segmentation [42]. Recently, GCNs are +introduced to few-shot learning and achieve superior performance. Zhang et al [41] propose +an adaptive aggregation GCN (AdarGCN) to model the information passing between noisy +samples and underlying training samples. Yan et al [32] deign a GCN based data selection +policy to compute a context-sensitive representation for each unlabeled data by graph mes- +sage passing. Furthermore, han et al [7] introduce a heterogeneous GCN with three different +types of edges to promote the pairwise matching for the few-shot object detection task. In- +spired by the powerful relation modelling capacity of GCNs, we design a support-induced +graph convolutional network (SiGCN) to enrich query contextual representation for address- +ing the object appearance variation problem in FSS. +3 +Method +3.1 +Problem Setting +To improve the model generalizability over previously unseen classes, most methods adopt +the episode-based meta-learning paradigm [23, 28]. Models are trained on the base classes +Cbase and tested on novel classes Cnovel. Cbase and Cnovel are disjoint, i.e., Cbase ∩Cnovel = /0. + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +5 +𝐼! +𝐼" +Backbone +(a) Feature Embedding & Matching +𝑀! +𝑀"! +Mask +Evaluator +Mask +Evaluator +Graph +Updating +Graph +Updating +𝑥" +𝑣!" +𝑣!# +𝑣$ +𝐴" +# +𝐴$ +# +Salient Parts +Indexing +Indexing +𝐺! = (𝑉!, 𝐸!) +𝐺" = (𝑉", 𝐸") +… +𝑥! +𝑥$! +(c) Instance Association +Decoder +(d) Mask Prediction +(b) Support-induced Graph Reasoing +𝑚# +𝑚$ +𝑚#$ +𝑚$# +Salient Parts +Figure 2: Overall architecture of our proposed network. Firstly, the feature embedding +and matching network are introduced to generate support feature xs, query feature xq, middle- +level activation map Ar +m, and high-level activation map Ar +h. Then, the support-induced graph +reasoning module captures salient query object parts and associates them by a graph convo- +lutoinal network. The support prototypes ˜xs is introduced to the graph updating for effective +query context excavation. In addition, the instance association module aims to mine instance- +level context by the massage passing between query instances (i.e., v0 +q and v1 +q) and a support +instance vs. Finally, the updated features are concatenated and fed into a decoder for the final +query mask prediction. +Both training and inference are based on the episode data (S,Q), which is composed of a +support set S, and a query set Q for a specific class c. For the 1-way K-shot setting, the +support set S = {(Ii +s,Mi +s)}K +i=1 contains K image-mask pairs, where Ii +s represents ith support +image and Mi +s indicates corresponding binary mask. Similarly, we define the query set as +Q = {(Iq,Mq)}, where Iq is query image and its binary mask Mq is only available in the +model training phase. In the meta-training stage, the FSS model takes input S and Iq from +a specific class c and generates a predicted mask ˆMq for the query image. Then, the model +can be trained with the supervision of a binary cross-entropy loss between Mq and ˆMq, i.e., +LBCE(Mq, ˆMq). Finally, multiple episodes (Sts +i ,Qts +i )Nts +i=1 are randomly sampled from the test- +ing set Dts for evaluation. +3.2 +Overview +Figure 2 illustrates the overall architecture of our proposed network, which consists of +three main components: feature embedding and matching module, support-induced graph +reasoning (SiGR) module and instance association (IA) module. +Given the support set +S = {(Ii +s,Mi +s)}K +i=1 and query image Iq, we first encode them into feature maps by a weight- +sharing backbone. Then, the feature embedding and matching network is introduced to +generate support/query features xs ∈ RC×H×W and xq ∈ RC×H×W, middle-level/high-level +activation map Ar +m ∈ RH×W and Ar +h ∈ RH×W, where C and HW are the channel and spatial +dimension, respectively. We obtain activation maps by region-to-region matching between +support and query features [15]. Afterwards, the support-induced graph reasoning module +is designed to enrich the query contextual representation by a novel support-induced graph +convolutional network. Finally, we aggregate complementary context from both support and +query by instance-level message passing in the instance association module. + +6 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +3.3 +Support-induced Graph Reasoning +The proposed SiGR module consists of two parallel branches. Taking the upper branch as +shown in Figure 2 as an example, we explain the support-induced graph reasoning process +in details. +Mask Evaluation. The coarse activation map for the query feature is usually not accurate +enough and contains noise and non-salient parts. Therefore, we design a mask evaluator to +select salient objects parts. Given the activation map Ar +m, we first process it with a min-max +normalization to normalize the value to between 0 and 1. Then, we select salient query object +parts by introducing a salience matrix Sr +m ∈ RH×W: +Sr +m(i, j) = +� +1, +Ar +m(i, j) ≥ t +0, +Ar +m(i, j) < t , +(1) +where Sr +m is a sparse matrix, in which the elements with value 1 represent target location, +otherwise noise or undiscriminating parts location. t is a threshold which decides whether a +pixel belongs to salient parts or not. +Graph Construction. Based on the defined salience matrix Sr +m, we select Nm = ∑HW +i, j Sr +m(i, j) +most salient pixels in the query feature xq as the set of salient parts Pm = {pn +m}Nm +n=1, which are +discriminative local parts of the object for segmentation. The captured salient object parts +are further associated together by a fully connected graph to learn rich query contextual +information. Specifically, we treat each pixel in the query feature as a node and build the +graph G0 = (V0,E0), where V0 and E0 are the node set the edge set, respectively. For the +connecting edges, We adopt fully-connected edges between Nm salient parts specified in Pm. +To achieve better generalization over novel objects, the edge weights are designed to be +the similarity score between two connected nodes. Thus, the adjacency matrix A0 ∈ RN×N, +N = HW, for the graph G0 is constructed as: +A0(i, j) = +� +vT +i v j +∥vi∥∥v j∥, +if edge < i, j >∈ E +0, +else +, +(2) +where vi ∈ RC and vj ∈ RC are two node vectors in the connection, E is the edge connection +set of all the salient parts feature vectors. +Graph Updating. To update graphs by aggregating the salient parts’ information, we pro- +pose a support-induced graph convolution network (SiGCN) to achieve effective information +aggregation with the guidance of support foreground prototypes. For the query graph V0, we +first construct a two-layer GCN to perform the node state updating. Formally, the l-th layer +graph convolution is formulated as: +Xl+1 = σl( ˆAXlW l), +(3) +where Xl ∈ RN×dl and Xl+1 ∈ RN×dl+1 are the input and output feature of all nodes at the +l-th layer, respectively, dl and dl+1 are the corresponding feature dimensions, d0 is equal to +the node feature dimension C. ˆA = ˜D− 1 +2 ˜A ˜D +1 +2 is the normalized adjacency matrix [10], where +˜A = A+I and ˜D is the diagonal degree matrix of ˜A. W l ∈ Rdl×dl+1 and σl(·) denote the state +updating matrix and activation function at the layer l, respectively. +Normally, the state updating matrix W is optimized with the model training proceeds. +Chen et. al [4] propose to adopt a 1D convolution as state updating function. Inspired by + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +7 +this, we implement the state updating matrix as a 1D convolution with fixed kernel generated +from the support foreground vectors. +Xl+1 = σl( ˆAFl +c(Xl|θ l)), +(4) +where Fl +c(·) is the 1D convolution with fixed kernel parameters θ l at the l-th layer. θ ∈ Rk×C, +where k is the number of support prototypes (i.e., kernel size of 1D convolution), is generated +by applying an average pooling to the support foreground feature. Given the high-level +activation map Ar +h, we can similarly select salient query object parts, construct the query +graph G1 = (V1,E1), and implement support-induced graph updating. Finally, we can obtain +two updated query graphs, which aggregate rich query contextual representation with the +guidance of support prototypes. +3.4 +Instance Association +Although rich context has been aggregated into query graphs, the complementary inter-graph +context still remains unexploited. Therefore, the updated query graphs are further trans- +formed into query instance features v0 +q ∈ RC×H×W and v1 +q ∈ RC×H×W, respectively. Mean- +while, we can also adopt the 1D average pooling and the reshape operation over the support +foreground prototypes ˜xs to get the support instance feature vs ∈ RC×s×s, where s is the spa- +tial size. Then, the the query instance features and support instance feature are associated +together by the instance association module to learn instance-level contextual information. +For the query instance feature v0 +q, the information gathered from the support instance feature +vs and the query instance feature v1 +q can be formulated as: +m0 = R(vs)R(vs)TR(v0 +q),m10 = R(v1 +q)R(v1 +q)TR(v0 +q), +(5) +where m0 ∈ RC×HW and m10 ∈ RC×HW are the instance-level contextual information from +support instance and query instance, respectively. R(·) is the vectorization operation. Then, +the query instance feature can be updated as ˜v0 +q by: +˜v0 +q = 1 +2(v0 +q +(αm0 +βm10)), +(6) +where α and β are two hyperparameters to weight the contribution of the contextual informa- +tion from the support and query, respectively. Similarly, the other query instance feature v1 +q +can be updated as ˜v1 +q. Compared with previous state, ˜v0 +q and ˜v1 +q aggregate more instance-level +context information from both support and query, thus being more robust to the appearance +variations. +3.5 +Mask Prediction and Model Training +Finally, we feed the updated query instance features as well as activation maps specified in +Section 3.2 to make pixel-wise prediction for query. +ˆMq = Fdecoder(C(˜v0 +q, ˜v1 +q,Ap +m,Ar +m,Ap +h,Ar +h)), +(7) +where C(·) is the channel-wise concatenation. Fdecoder(·) is the decoder network, which +consists of a Atrous Spatial Pyramid Pooling (ASPP) module [2] and three consecutive con- +volution layers with residual connection. For the activation maps Ap +m and Ap +h, the superscript + +8 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +Table 1: Comparison with the state-of-the-art methods on PASCAL-5i dataset under both +1-shot and 5-shot settings. The mIoU of each fold, and the averaged mIoU & FB-IoU of all +folds are reported. +Methods +Backbone +1-shot +5-shot +Fold-0 +Fold-1 +Fold-2 +Fold-3 +Mean +FB-IoU +Fold-0 +Fold-1 +Fold-2 +Fold-3 +Mean +FB-IoU +OSLSM (BMVC’17) [23] +VGG16 +33.6 +55.3 +40.9 +33.5 +40.8 +61.3 +35.9 +58.1 +42.7 +39.1 +43.9 +61.5 +co-FCN (ICLRW’18) [20] +VGG16 +36.7 +50.6 +44.9 +32.4 +41.1 +60.1 +37.5 +50.0 +44.1 +33.9 +41.4 +60.2 +PFENet (TPAMI’20) [27] +VGG16 +56.9 +68.2 +54.4 +52.4 +58.0 +72.0 +59.0 +69.1 +54.8 +52.9 +59.0 +72.3 +HSNet (ICCV’21) [17] +VGG16 +59.6 +65.7 +59.6 +54.0 +59.7 +73.4 +64.9 +69.0 +64.1 +58.6 +64.1 +76.6 +PFENet (TPAMI’20) [27] +ResNet50 +61.7 +69.5 +55.4 +56.3 +60.8 +73.3 +63.1 +70.7 +55.8 +57.9 +61.9 +73.9 +RePRI (CVPR’21) [1] +ResNet50 +59.8 +68.3 +62.1 +48.5 +59.7 +- +64.6 +71.4 +71.1 +59.3 +66.6 +- +SAGNN (CVPR’21) [31] +ResNet50 +64.7 +69.6 +57.0 +57.3 +62.1 +73.2 +64.9 +70.0 +57.0 +59.3 +62.8 +73.3 +MMNet (ICCV’21) [30] +ResNet50 +62.7 +70.2 +57.3 +57.0 +61.8 +- +62.2 +71.5 +57.5 +62.4 +63.4 +- +HSNet (ICCV’21) [17] +ResNet50 +64.3 +70.7 +60.3 +60.5 +64.0 +76.7 +70.3 +73.2 +67.4 +67.1 +69.5 +80.6 +CyCTR (NIPS’21) [40] +ResNet50 +67.8 +72.8 +58.0 +58.0 +64.2 +- +71.1 +73.2 +60.5 +57.5 +65.6 +- +Baseline +ResNet50 +62.5 +69.4 +58.9 +55.9 +61.7 +72.0 +63.2 +70.5 +60.1 +57.6 +62.9 +74.4 +SiGCN +ResNet50 +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +68.9 +72.6 +66.8 +65.8 +68.5 +78.3 +Table 2: Comparison with the state-of-the-art methods on COCO-20i dataset under both 1- +shot and 5-shot settings. The mIoU of each fold, and the averaged mIoU & FB-IoU of all +folds are reported. +Methods +Backbone +1-shot +5-shot +Fold-0 +Fold-1 +Fold-2 +Fold-3 +Mean +FB-IoU +Fold-0 +Fold-1 +Fold-2 +Fold-3 +Mean +FB-IoU +FWB(ICCV’19) [19] +VGG16 +18.4 +16.7 +19.6 +25.4 +20.0 +- +20.9 +19.2 +21.9 +28.4 +22.6 +- +PFENet(TPAMI’20) [27] +VGG16 +33.4 +36.0 +34.1 +32.8 +34.1 +60.0 +35.9 +40.7 +38.1 +36.1 +37.7 +61.6 +SAGNN(CVPR’21) [31] +VGG16 +35.0 +40.5 +37.6 +36.0 +37.3 +61.2 +37.2 +45.2 +40.4 +40.0 +40.7 +63.1 +RePRI(CVPR’21) [1] +ResNet50 +31.2 +38.1 +33.3 +33.0 +34.0 +- +38.5 +46.2 +40.0 +43.6 +42.1 +- +SAGNN(CVPR’21) [31] +ResNet101 +36.1 +41.0 +38.2 +33.5 +37.2 +60.9 +40.9 +48.3 +42.6 +38.9 +42.7 +63.4 +MMNet(ICCV’21) [30] +ResNet50 +34.9 +41.0 +37.2 +37.0 +37.5 +- +37.0 +40.3 +39.3 +36.0 +38.2 +- +HSNet(ICCV’21) [17] +ResNet50 +36.3 +43.1 +38.7 +38.7 +39.2 +68.2 +43.3 +51.3 +48.2 +45.0 +46.9 +70.7 +CyCTR(NIPS’21) [40] +ResNet50 +38.9 +43.0 +39.6 +39.8 +40.3 +- +41.1 +48.9 +45.2 +47.0 +45.6 +- +Baseline +ResNet50 +33.6 +39.2 +36.5 +34.2 +35.9 +60.7 +35.3 +43.1 +38.4 +38.7 +38.9 +62.0 +SiGCN +ResNet50 +38.7 +46.3 +43.1 +37.5 +41.4 +62.7 +44.9 +54.5 +46.5 +45.9 +48.0 +66.2 +indicates that the activation map comes from pixel-to-pixel matching (p) [27] between sup- +port and query features. To generate the final segmentation map, we upsample ˆMq by bilinear +interpolation. Finally, we adopt the binary cross entropy loss to train our model on all the +episodes in the training set Dtr. +4 +Experiments +4.1 +Implementation Details +Following previous methods [17, 27], we conduct experiments on PASCAL-5i and COCO- +20i. Classes in each dataset are divided into four base-novel class folds for cross validation. +Our model is based on PFENet [27] and we also adopt the same training and evaluation +setting as PFENet. We train our model with SGD optimizer on PASCAL-5i for 200 epochs +and COCO-20i for 50 epochs. The learning rate is initialized as 0.005 with batch size 8 and +decays according to the poly learning rate scheduler. During evaluation, we adopt the mIoU +as main metric and also report FBIoU. For the K-shot setting, we fuse foreground prototypes +from each support image-mask pair to generate θ in Eq. (4) and the support instance feature +vs in the instance association module. +4.2 +Comparison with the State-of-the-art methods +PASCAL-5i. +In Table 1, we show the performance comparison between the proposed +SiGCN and the leading methods on the PASCAL-5i dataset. With the ResNet50 as back- +bone, we can conclude that the proposed SiGCN achieves the state-of-the-art performance + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +9 +Support +Query +Prediction +Figure 3: Qualitative results on the PASCAL-5i (left four columns) and COCO-20i (right +four columns) under the 1-shot setting. Our SiGCN can make accurate segmentation predic- +tion, even though the support and query objects exhibit larger appearance variations. +under the 1-shot setting and competitive result under the 5-shot setting. Specifically, SiGCN +surpasses previous SOTA (i.e., CyCTR) by 1.1% in the 1-shot setting, while achieving com- +petitive results as previous SOTA HSNet (68.5 vs 69.5) in the 5-shot setting. Additionally, +compared with the baseline model, which is constructed by removing the SiGR and IA mod- +ules from SiGCN (see complementary materials for more details), our SiGCN achieves sig- +nificant improvement, i.e., 3.6% and 6.4% under the 1-shot and 5-shot settings, respectively. +Considering the comparison with previous SOTA methods using the FB-IoU, SiGCN also +obtain competitive results (77.5 and 78.3 under the 1-shot and 5-shot setting, respectively). +COCO-20i. Table 2 shows the comparison between previous SOTA methods and our SiGCN +on the COCO-20i dataset. Although the COCO-20i is more challenging (larger appearance +and scale variations than PASCAL-5i), SiGCN also achieves state-of-the art results (41.4 and +48.0 under 1-shot and 5-shot settings, respectively). It demonstrates the superiority of our +proposed SiGCN in addressing the appearance variation problem. Furthermore, SiGCN with +ResNet50 backbone achieves significant mIoU improvement (i.e., 5.5% and 9.1% under the +1-shot and 5-shot settings, respectively) over the baseline model. +Qualitative Results. In Figure 3, we report some typical qualitative results generated by our +SiGCN. As can be seen, the support and query object exhibits large appearance variations, +but our approach can also produce accurate mask prediction for the query image, which +thanks to the query context mining by our method. More experimental and quantitative re- +sults can be found in the supplementary material. +Table 3: Effect of key modules in SiGCN. +SiGR +IA +1-shot mIoU +50 +51 +52 +53 +mean +FB-IoU +62.5 +69.4 +58.9 +55.9 +61.7 +72.0 +✓ +62.0 +70.0 +58.5 +59.4 +62.7 +73.5 +✓ +64.6 +69.8 +64.1 +60.8 +64.8 +77.4 +✓ +✓ +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +Table 4: Effect of different GCN variants. +1-shot mIoU +GCN Variants +50 +51 +52 +53 +mean +FB-IoU +baseline +62.0 +70.0 +58.5 +59.4 +62.7 +73.5 +GCN [10] +63.0 +69.4 +59.7 +57.1 +62.3 +73.3 +SiGCN-1 +63.4 +69.6 +62.2 +60.0 +63.8 +75.4 +SiGCN-2 +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +4.3 +Ablation Study +Component analysis. There are two main components in our model, namely the support- +induced graph reasoning (SiGR) module and the instance association (IA) module . We +validate the effectiveness of each component and present the results in Table 3. It can be +summarized that the SiGR module plays the most important role in the performance im- +provement while the IA module is indispensable. With SiGR and IA module, our SiGCN + +A-SR-650074 +SANTUA +02-5198-34A-SR-650074 +02-5198-34BOMBEROS +BOMBEROS +998386BlinkainBlinkainGRACEGRACE +lsneh10 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +(a) Support instance size s +(b) Threshold t +Figure 4: Ablation studies of (a) support instance size s and (b) mask threshold t on +PASCAL-5i under the 1-shot setting. +achieves state-of-the-art performance. +Effect of different GCN variants. To investigate the effect of different GCN variants +in the target association module, we adopt three GCN variants, i.e., plain GCN, SiGCN with +one and two layers (denoted as SiGCN-1 and SiGCN-2, respectively). As shown in Table +4, our model with plain GCN achieves sightly better performance than the baseline model, +while our model with SiGCN makes substantial performance improvement as the number of +the SiGCN layer increasing from one to two. +Table 5: Effect of k prototypes. +1-shot mIoU +k prototypes +50 +51 +52 +53 +mean +FB-IoU +1 +64.8 +69.7 +63.5 +59.0 +64.3 +75.9 +3 +64.1 +70.2 +64.3 +59.5 +64.5 +76.2 +5 +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +7 +63.7 +69.8 +63.8 +60.2 +64.4 +75.5 +Table 6: Effect of different backbones. +1-shot mIoU +Backbone +50 +51 +53 +53 +mean +FB-IoU +VGG16 +58.0 +68.0 +62.9 +54.2 +60.8 +73.5 +ResNet-50 +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +ResNet-101 +65.4 +71.0 +64.6 +61.7 +65.7 +78.3 +Influence of k support prototypes. The number of support prototypes k in Eq. (4) is +a key parameter in SiGCN, which determines the quality of support foreground used for +guiding the query graph updating. We vary k in {1,3,5,7} to observe the performance +of our model. In Table 5, k = 5 achieves best performance, which indicates that SiGCN +with 5 support foreground prototypes provides better guidance to query graphs for capturing +contextual information. +Influence of support instance size s. The support instance feature size s in the instance +association module (section 3.4) determines the context information in the support instance. +The value of s is taken from {1,3,5,10,20} to observe the performance variations. As in +Figure 4(a), SiGCN with s = 10 exhibits the best performance, thus we set s = 10 in all of +our experiments. +Influence of mask threshold t. The mask threshold t in Eq.(1) is essential for selecting +discriminative parts, i.e., targets in the query graph. Thus we take its value from 0,0.1, 0.3, +0.5, 0.7, 0.9 to investigate the performance of our SiGCN. As can be seen in Figure 4(b), +SiGCN with t = 0.7 performs best. Additionally, SiGCN without mask threshold in Eq.(5) +(i.e., t = 0) also shows competitive results. +Effect of different backbones. To evaluate the performance of our model with different +backbones, we adopt three backbone networks (i.e., VGG16, ResNet50, and ResNet101) +to implement experiments. As shown in Table 6, our SiGCN with ResNet101 as backbone +yields superior performance. + +75 +70 +65 +IoU +60 +55 +foldo +fold2 +MIoU +fold 1 +fold3 +50 +0 +0.1 +0.3 +0.5 +0.7 +0.9 +t75 +70 +65 +IoU +60 +55 +foldo +fold2 +MIoU +fold1 +fold3 +50 +3 +5 +10 +20 +SJ. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +11 +5 +Conclusion +In this work, we propose a support-induced graph convolutional network (SiGCN) to tackle +the challenging appearance variation problem for the few-shot semantic segmentation (FSS) +task. Our main idea is to enrich query context representation with the discriminative parts in +the query objects and support objects. To achieve this, we propose a support-induced graph +reasoning (SiGR) module to associate salient query object parts with the graph convolution +network, in which support prototypes is introduced to guide the graph updating process. +Additionally, an instance association (IA) module is designed to capture high-order context +from the support instances and complementary query instances simultaneously. Extensive +experiments on two FSS benchmarks prove that our SiGCN achieves state-of-the-art perfor- +mance under both 1-shot and 5-shot settings. We believe that the idea of explicitly mining +query context will shed light for future research in this field. +Acknowledgement +This work was partially funded by Elekta Oncology Systems AB and +a RVO public-private partnership grant (PPS2102). +References +[1] Malik Boudiaf, Hoel Kervadec, Ziko Imtiaz Masud, Pablo Piantanida, Ismail Ben Ayed, +and Jose Dolz. 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Pyramid +scene parsing network. In Proceedings of the IEEE conference on computer vision and +pattern recognition, pages 2881–2890, 2017. + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +15 +6 +Supplementary Materials +In this supplementary material, we firstly present extensive implementation details about our +experiments, and then give two more additional ablation studies and analysis about different +level features and matching methods for activation maps. Finally, we provide more qualita- +tive visualization results for PASCAL-5i and COCO-20i benchmarks under the large object +appearance and scale variations scenarios. +Implementation Details In our experiments, we employ ResNet-50 (VGG-16 and ResNet- +101) pre-trained on ImageNet as our backbone network. For ResNet-50 and ResNet-101, the +dilation convolution is introduced to ensure that the feature receptive fields of layer2, layer3, +and layer4 preserve the same spatial resolution. The backbone weights are frozen except +for layer4, which is required to learn more robust activation maps. The proposed model is +validated on PASCAL-5i and COCO-20i benchmarks, which are widely used in the few-shot +semantic segmentation for cross validations. The detailed split of testing classes for each +cross validation (fold) is shown in Table 7 and Table 8, respectively. +The model is trained with a SGD optimizer for 200 and 50 epochs on the PASCAL-5i +and the COCO-20i benchmarks, respectively. The learning rates are initialized as 0.005 and +0.002 with a poly learning rate schedule in PASCAL-5i and COCO-20i, respectively. Our +entire network is trained with the same learning rate during each epoch, while layer4 of the +backbone network should be ensured a lower learning rate for fine-tuning, thus its parameters +starts back-propagation after training for multiple epochs. And the batch size is set as 8 on +PASCAL-5i and 32 on COCO-20i. Data augmentation strategies like random rotation and +flip are adopted in the training stage, and all images are cropped to 473×473 patches for two +benchmarks. Besides, no post-processing is used on PASCAL-5i, while for COCO-20i, we +adopt the multi-scale testing strategy for the model evaluation due to extremely large object +appearance and scale variations. By the way, the original Groudtruth of the query image +without any resize operations is adopted for the model evaluation. +In addition, the support instance size s in the IA module are set as 10, and mask threshold +t are set as 0.7. For the SiGR module, we set the number of support prototypes k = 5 in each +SiGCN layer. Finally, we implement our model with PyTorch 1.10.0 and conduct all the +experiments with Nvidia Tesla A100 GPUs and CUDA11.3. +Table 7: Testing classes split for each fold in PASCAL-5i dataset. +Fold +Testing (novel) classes +Fold-0 +Aeroplane, Bicycle, Bird, Boat, Bottle +Fold-1 +Bus, Car, Cat, Chair, Cow +Fold-2 +Diningtable, Dog, Horse, Motorbike, Person +Fold-3 +Potted plant, Sheep, Sofa, Train, Tvmonitor +Effect of activation maps from different feature levels. To evaluate the quality of ac- +tivation maps generated from different feature levels, we adopt three different-level features, +i.e., block3 (middle-level), block4 (high-level), and block3&4 (middle-level& high-level). +Support and query features from each level are matched by pixel-to-pixel matching to gen- +erate corresponding activation map. As shown in Table 9, our model with the activation map +generated from block3&4 yields superior performance, while our model with the activation +map generated from block3 or block4 also achieves competitive results. + +16 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +Table 8: Testing classes split for each fold in COCO-20i dataset. +Fold +Testing (novel) classes +Fold-0 +Person, Airplane, Boat, Parking meter, Dog, Elephant, Backpack,Suitcase, Sports Ball, +Skateboard, Wine glass, Spoon, Sandwich, Hot dog, Chair, Dining table, Mouse, Microwave, Scissorse +Fold-1 +Bicycle, Bus, Traffic light, Bench, Horse, Bear, Umbrella, Frisbee, Kite, Surfboard , +Cup, Bowl, Orange, Pizza, Couch, Toilet, Remote, Oven, Book, Teddy bear +Fold-2 +Car, Train, Fire hydrant, Bird, Sheep, Zebra, Handbag, Skis, Baseball bat, Tennis racket, +Fork, Banana, Broccoli, Donut, Potted plant, Tv, Keyboard, Sink, Toaster, Clock, Hair drier +Fold-3 +Motorcycle, Truck, Stop sign, Cat, Cow, Giraffe, Tie, Snowboard, Baseball glove, Bottle, +Knife, Apple, Carrot, Cake, Bed, Laptop, Cell phone, Refrigerator, Vase, Toothbrush +Effect of region matching. We append the activation maps generated by region-region +matching to evaluate the performance of our model with region matching. As shown in Table +9, we can conclude that the activation maps generated from region-to-region matching bring +overall performance improvement. Compared with pixel-to-pixel matching, region-to-region +matching can capture more contextual information, thus leading to improved performance. +Table 9: Ablation study on the different activation maps on the PASCAL-5i dataset. +1-shot mIoU +activation maps +50 +51 +52 +53 +mean +FB-IoU +block3 +64.7 +70.3 +61.1 +60.6 +64.2 +75.0 +block4 +63.3 +70.0 +64.0 +60.2 +64.5 +75.4 +block3&4 +64.5 +70.1 +64.3 +60.4 +64.8 +75.9 +block3+region +64.5 +69.4 +64.0 +60.6 +64.6 +75.4 +block4+region +62.9 +69.5 +65.1 +60.1 +64.4 +75.2 +block3&4+region +65.1 +70.1 +65.2 +60.8 +65.3 +77.5 +Additional Qualitative Results. We show more quantitative results of the proposed +SiGCN in this section to further demonstrate its few-shot semantic segmentation perfor- +mance. SiGCN aims to address the appearance variations problem in the few-shot semantic +segmentation task, thus we show some examples sampled from PASCAL-5i and COCO-20i +benchmarks with large appearance variations in Figure 5 and Figure 7, respectively. Further- +more, to demonstrate that our model can effectively address the scale variations problem, we +give some examples from PASCAL-5i and COCO-20i benchmarks in Figure 6 and Figure +8, respectively. According to the quantitative results, we can conclude that our proposed +SiGCN can effectively address the appearance and scale variations problem. + +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +17 +Support +Query +Prediction +Figure 5: Qualitative results of the proposed SiGCN on PASCAL-5i benchmark with large +object appearance variations. From top to bottom: Support Image & Ground truth, Query +Image & Ground truth, and Query Prediction. Zoom in for details. +Support +Query +Prediction +Figure 6: Qualitative results of the proposed SiGCN on PASCAL-5i benchmark with large +object scale variations. From top to bottom: Support Image & Ground truth, Query Image +& Ground truth, and Query Prediction. Zoom in for details. +Support +Query +Prediction +Figure 7: Qualitative results of the proposed SiGCN on COCO-20i benchmark with large +object appearance variations. From top to bottom: Support Image & Ground truth, Query +Image & Ground truth, and Query Prediction. Zoom in for details. + +RRLINGENHAUENRRLINGENHAUENTEPRERON +白口口WO +NRWO +NRTECHTTAN +LTAREUSEAMTI-VEUS +DTECHTITA +TECHTITAN +TTECHTITA +B1218RIDTECHTITAN +LITIATEUSEAMITI-VES +PECHTA +TECHTITAN +TECH TTAN +B1218RBELHUS +PeleCleeler20318 +J. LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN +Support +Query +Prediction +Figure 8: Qualitative results of the proposed SiGCN on COCO-20i benchmark with large +object scale variations. From top to bottom: Support Image & Ground truth, Query Image +& Ground truth, and Query Prediction. Zoom in for details. + +90189018OTOT \ No newline at end of file diff --git a/btE1T4oBgHgl3EQfdQTN/content/tmp_files/load_file.txt b/btE1T4oBgHgl3EQfdQTN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b627de2c83db34281fee88e078c2a2e3d249e518 --- /dev/null +++ b/btE1T4oBgHgl3EQfdQTN/content/tmp_files/load_file.txt @@ -0,0 +1,975 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf,len=974 +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 1 Few-shot Semantic Segmentation with Support-induced Graph Convolutional Network Jie Liu1 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='liu5@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='nl Yanqi Bao2 yanqibao1997@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='com Wenzhe Yin1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='yin@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='nl Haochen Wang1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='wang3@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='nl Yang Gao2 gaoy@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='cn Jan-Jakob Sonke3 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='sonke@nki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='nl Efstratios Gavves1 egavves@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='nl 1 University of Amsterdam Amsterdam, Netherlands 2 Nanjing University Nanjing, China 3 The Netherlands Cancer Institute Amsterdam, Netherlands Abstract Few-shot semantic segmentation (FSS) aims to achieve novel objects segmentation with only a few annotated samples and has made great progress recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Most of the existing FSS models focus on the feature matching between support and query to tackle FSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' However, the appearance variations between objects from the same cate- gory could be extremely large, leading to unreliable feature matching and query mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To this end, we propose a Support-induced Graph Convolutional Network (SiGCN) to explicitly excavate latent context structure in query images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, we propose a Support-induced Graph Reasoning (SiGR) module to capture salient query object parts at different semantic levels with a Support-induced GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Furthermore, an instance association (IA) module is designed to capture high-order instance context from both support and query instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' By integrating the proposed two modules, SiGCN can learn rich query context representation, and thus being more robust to appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Extensive experiments on PASCAL-5i and COCO-20i demonstrate that our SiGCN achieves state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 1 Introduction The development of deep convolutional neural networks has achieved great success in many computer vision tasks, such as image classification [8, 26], semantic segmentation [5, 35, © 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The copyright of this document resides with its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' It may be distributed unchanged freely in print or electronic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='03194v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='CV] 9 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 43, 45], and object detection [6, 7, 9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Despite the effectiveness of deep networks, their over-reliance on large-scale annotated dataset still remains a fundamental limitation as data annotation requires large amount of human efforts, especially for dense prediction tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To cope with these challenges, some weakly-supervised and semi-supervised semantic segmentation techniques [3, 11, 24] try to introduce weak annotations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='g, points, scribbles, and bounding box) and unlabelled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' However, these techniques usually fail to work when novel classes that never appear in the training set emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In such case, few-shot semantic segmentation (FSS) methods [1, 12, 13, 17, 19, 20, 25, 27, 29, 30, 31, 34, 37, 39, 40] struggle to segment novel objects with only a few pixel-wise annotated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' … Prototypes Features Support Image Query Image Support Mask Freature Matching Prediction Support Image Query Image Support Mask Prediction … Prototypes Activation Map Graph Updating Guide (a) Previous FSS Methods (b) Our Support-induced GCN (SiGCN) Figure 1: Comparison between (a) previous typical few-shot segmentation (FSS) methods and (b) the proposed support-induced graph convolutional network (SiGCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (a) Previous FSS methods focus on designing effective feature matching mechanism between support pro- totypes/features and query feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' However, such matching techniques usually fail to work when the support and query objects exhibit large appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (b) Our SiGCN aims to address the appearance variation problem by explicitly mining query context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Based on the activation map, SiGCN constructs a query graph to associate salient query ob- ject parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then the support prototypes are introduced to guide the graph updating process for effective query context excavation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Meta-learning [23] is widely adopted in FSS to ensure better model generalization to novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, both training and testing sets are composed of many sub-tasks (episodes), and each specific sub-task is divided into a support set and a query set, which share the same object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The FSS models aim to segment images from the query set given a few pixel-wise annotated images from the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Thus, the key challenge of the FSS task is how to leverage the information from the support set and the query set effectively, especially when the support and query objects exhibit large variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', appearance, scale, and location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Although existing FSS methods [17, 27, 40] have achieved extensive progress, we ob- serve a significant drop in performance when the support and query images share large ap- pearance variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For instance, as shown in Figure 1, the birds in the support and query images have completely different appearance, which makes it difficult to segment the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Existing methods (Figure 1 (a)) usually generate support foreground features or prototypes and then focus on designing various feature matching mechanisms between the support foreground features/prototypes and the query feature to achieve the query mask pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' However, such matching with information coming only from the support set is suboptimal when it comes to predicting an accurate mask for the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' By con- trast, different query object parts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', the wings and the main body of the bird) contain rich J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 3 contextual information, which is beneficial to the query object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In this work, we propose a support-induced graph convolutional network (SiGCN) to tackle the aforementioned challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our main idea is to explicitly enrich the query context representations with the guidance of support information in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Firstly, we propose a novel support-induced graph reasoning (SiGR) module to extract query context from dif- ferent query object parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, as shown in Figure 1(b), we utilize the activation map, which is generated by the feature matching between the support and the query fea- tures, to capture local object parts that are discriminative and easily localized in the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' These salient parts are associated together in the query graph by modeling the in- teraction between them, thus learning rich context in the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The query graph is then updated by the graph convolutional network, in which the state updating matrix is generated by the support prototypes to effectively extract query context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Secondly, an in- stance association (IA) module is introduced to aggregate query and support context from instance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, given activation maps produced from different semantic levels, we get corresponding updated query graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e, query features) in the SiGR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' These updated query features, which can be seen as different object instances, contain comple- mentary context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Therefore, we associate the query instances and the support instance together to learn high-order instance context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' By proposing the SiGR and IA mod- ule, our SiGCN aggregates rich complementary context from both support and query images, thus being more robust to appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In summary, our contributions are summarized as follows: We propose a support-induced graph convolution network (SiGCN), which utilizes the proposed IA and SiGR modules to capture complementary context from the query and the support set, for addressing the appearance variation problem in the FSS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We propose a novel graph updating mechanism, in which support prototypes guide the node updating in the query graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' This mechanism brings a significant performance gain to the proposed SiGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our method achieves state-of-the-art results on two FSS benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', PASCAL- 5i and COCO-20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our SiGCN tackles the appearance variation problem from the perspective of extracting query contextual information, which sheds light for future research in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 Semantic Segmentation Semantic Segmentation is a fundamental computer vision task which aims to assign each pixel with a class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Fully convolutional network (FCN) [14] adopts fully convolutional layers for dense prediction and achieves superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Inspired by FCN, many net- works have been invented to boost the performance of semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Unet [22] presents an encoder-decoder structure to capture multi-scale context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' PSPNet [45] utilizes the pyramid pooling to aggregate more object details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Dilated convolution [35] and deformable convolution [5] are introduced to FCN to further improve the semantic seg- mentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Recently, K-Net [43] proposes to segment both instance and seman- tic categories by a group of learnable kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' However, these methods still require tons of 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN samples with pixel-wise annotations in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In our work, we focus on few-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 Few-shot Semantic Segmentation Few-shot semantic segmentation is proposed to segment novel objects given only few anno- tated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Most existing methods adopt episode-base meta-learning [31] strategy to en- sure the efficient generalization of the model to the novel classes with few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Shaban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' [23] proposes the first few-shot semantic segmentation work, which generates param- eters from the conditioning branch and then segments the query image in the segmentation branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Later on, prototype-based methods are proposed including CANet [39], PANet [29], SG-One [44], PPNet [13], PMMs [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The intuition behind these methods lies in extracting representative foreground or background prototypes for support samples and then interact- ing these prototypes with query features using different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Besides, matching-based methods [17, 27, 31, 38, 40] also achieve great success in the few shot-segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' PFENet [27] introduces a prior mask by implementing pixel-to-pixel matching between high- level support and query features, while PGNet [38] and SAGNN [31] propose to implement element-wise matching between support and query by graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Most recently, HSNet [17] adopts multi-scale dense matching to construct hypercorrelations and uses 4D convolutions to capture contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' CyCTR [40] designs a cycle consistency transformer to aggregate pixel-wise support feature to query ones and achieves great suc- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Different from previous work, we not only consider the matching between support and query, but also highlight context excavation in the query feature, our model is thus more robust to the object appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 Graph Convolutional Network Graph convolutional networks (GCNs) [10], first proposed to tackle the semi-supervised classification, have seen massive application in computer vision tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', action localiza- tion [18, 36], object detection [16], and semantic segmentation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Recently, GCNs are introduced to few-shot learning and achieve superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Zhang et al [41] propose an adaptive aggregation GCN (AdarGCN) to model the information passing between noisy samples and underlying training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Yan et al [32] deign a GCN based data selection policy to compute a context-sensitive representation for each unlabeled data by graph mes- sage passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Furthermore, han et al [7] introduce a heterogeneous GCN with three different types of edges to promote the pairwise matching for the few-shot object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In- spired by the powerful relation modelling capacity of GCNs, we design a support-induced graph convolutional network (SiGCN) to enrich query contextual representation for address- ing the object appearance variation problem in FSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 3 Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 Problem Setting To improve the model generalizability over previously unseen classes, most methods adopt the episode-based meta-learning paradigm [23, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Models are trained on the base classes Cbase and tested on novel classes Cnovel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Cbase and Cnovel are disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', Cbase ∩Cnovel = /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 5 𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 𝐼" Backbone (a) Feature Embedding & Matching 𝑀!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 𝑀"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Mask Evaluator Mask Evaluator Graph Updating Graph Updating 𝑥" 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='" 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='# 𝑣$ 𝐴" # 𝐴$ # Salient Parts Indexing Indexing 𝐺!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' = (𝑉!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', 𝐸!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=') 𝐺" = (𝑉", 𝐸") … 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 𝑥$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (c) Instance Association Decoder (d) Mask Prediction (b) Support-induced Graph Reasoing 𝑚# 𝑚$ 𝑚#$ 𝑚$# Salient Parts Figure 2: Overall architecture of our proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Firstly, the feature embedding and matching network are introduced to generate support feature xs, query feature xq, middle- level activation map Ar m, and high-level activation map Ar h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, the support-induced graph reasoning module captures salient query object parts and associates them by a graph convo- lutoinal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The support prototypes ˜xs is introduced to the graph updating for effective query context excavation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In addition, the instance association module aims to mine instance- level context by the massage passing between query instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', v0 q and v1 q) and a support instance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, the updated features are concatenated and fed into a decoder for the final query mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Both training and inference are based on the episode data (S,Q), which is composed of a support set S, and a query set Q for a specific class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the 1-way K-shot setting, the support set S = {(Ii s,Mi s)}K i=1 contains K image-mask pairs, where Ii s represents ith support image and Mi s indicates corresponding binary mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Similarly, we define the query set as Q = {(Iq,Mq)}, where Iq is query image and its binary mask Mq is only available in the model training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In the meta-training stage, the FSS model takes input S and Iq from a specific class c and generates a predicted mask ˆMq for the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, the model can be trained with the supervision of a binary cross-entropy loss between Mq and ˆMq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', LBCE(Mq, ˆMq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, multiple episodes (Sts i ,Qts i )Nts i=1 are randomly sampled from the test- ing set Dts for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 Overview Figure 2 illustrates the overall architecture of our proposed network, which consists of three main components: feature embedding and matching module, support-induced graph reasoning (SiGR) module and instance association (IA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Given the support set S = {(Ii s,Mi s)}K i=1 and query image Iq, we first encode them into feature maps by a weight- sharing backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, the feature embedding and matching network is introduced to generate support/query features xs ∈ RC×H×W and xq ∈ RC×H×W, middle-level/high-level activation map Ar m ∈ RH×W and Ar h ∈ RH×W, where C and HW are the channel and spatial dimension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We obtain activation maps by region-to-region matching between support and query features [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Afterwards, the support-induced graph reasoning module is designed to enrich the query contextual representation by a novel support-induced graph convolutional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, we aggregate complementary context from both support and query by instance-level message passing in the instance association module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 Support-induced Graph Reasoning The proposed SiGR module consists of two parallel branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Taking the upper branch as shown in Figure 2 as an example, we explain the support-induced graph reasoning process in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Mask Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The coarse activation map for the query feature is usually not accurate enough and contains noise and non-salient parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Therefore, we design a mask evaluator to select salient objects parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Given the activation map Ar m, we first process it with a min-max normalization to normalize the value to between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, we select salient query object parts by introducing a salience matrix Sr m ∈ RH×W: Sr m(i, j) = � 1, Ar m(i, j) ≥ t 0, Ar m(i, j) < t , (1) where Sr m is a sparse matrix, in which the elements with value 1 represent target location, otherwise noise or undiscriminating parts location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' t is a threshold which decides whether a pixel belongs to salient parts or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Graph Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Based on the defined salience matrix Sr m, we select Nm = ∑HW i, j Sr m(i, j) most salient pixels in the query feature xq as the set of salient parts Pm = {pn m}Nm n=1, which are discriminative local parts of the object for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The captured salient object parts are further associated together by a fully connected graph to learn rich query contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, we treat each pixel in the query feature as a node and build the graph G0 = (V0,E0), where V0 and E0 are the node set the edge set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the connecting edges, We adopt fully-connected edges between Nm salient parts specified in Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To achieve better generalization over novel objects, the edge weights are designed to be the similarity score between two connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Thus, the adjacency matrix A0 ∈ RN×N, N = HW, for the graph G0 is constructed as: A0(i, j) = � vT i v j ∥vi∥∥v j∥, if edge < i, j >∈ E 0, else , (2) where vi ∈ RC and vj ∈ RC are two node vectors in the connection, E is the edge connection set of all the salient parts feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Graph Updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To update graphs by aggregating the salient parts’ information, we pro- pose a support-induced graph convolution network (SiGCN) to achieve effective information aggregation with the guidance of support foreground prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the query graph V0, we first construct a two-layer GCN to perform the node state updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Formally, the l-th layer graph convolution is formulated as: Xl+1 = σl( ˆAXlW l), (3) where Xl ∈ RN×dl and Xl+1 ∈ RN×dl+1 are the input and output feature of all nodes at the l-th layer, respectively, dl and dl+1 are the corresponding feature dimensions, d0 is equal to the node feature dimension C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' ˆA = ˜D− 1 2 ˜A ˜D 1 2 is the normalized adjacency matrix [10], where ˜A = A+I and ˜D is the diagonal degree matrix of ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' W l ∈ Rdl×dl+1 and σl(·) denote the state updating matrix and activation function at the layer l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Normally, the state updating matrix W is optimized with the model training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Chen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' al [4] propose to adopt a 1D convolution as state updating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Inspired by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 7 this, we implement the state updating matrix as a 1D convolution with fixed kernel generated from the support foreground vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Xl+1 = σl( ˆAFl c(Xl|θ l)), (4) where Fl c(·) is the 1D convolution with fixed kernel parameters θ l at the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' θ ∈ Rk×C, where k is the number of support prototypes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', kernel size of 1D convolution), is generated by applying an average pooling to the support foreground feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Given the high-level activation map Ar h, we can similarly select salient query object parts, construct the query graph G1 = (V1,E1), and implement support-induced graph updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, we can obtain two updated query graphs, which aggregate rich query contextual representation with the guidance of support prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 Instance Association Although rich context has been aggregated into query graphs, the complementary inter-graph context still remains unexploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Therefore, the updated query graphs are further trans- formed into query instance features v0 q ∈ RC×H×W and v1 q ∈ RC×H×W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Mean- while, we can also adopt the 1D average pooling and the reshape operation over the support foreground prototypes ˜xs to get the support instance feature vs ∈ RC×s×s, where s is the spa- tial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, the the query instance features and support instance feature are associated together by the instance association module to learn instance-level contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the query instance feature v0 q, the information gathered from the support instance feature vs and the query instance feature v1 q can be formulated as: m0 = R(vs)R(vs)TR(v0 q),m10 = R(v1 q)R(v1 q)TR(v0 q), (5) where m0 ∈ RC×HW and m10 ∈ RC×HW are the instance-level contextual information from support instance and query instance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' R(·) is the vectorization operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Then, the query instance feature can be updated as ˜v0 q by: ˜v0 q = 1 2(v0 q +(αm0 +βm10)), (6) where α and β are two hyperparameters to weight the contribution of the contextual informa- tion from the support and query, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Similarly, the other query instance feature v1 q can be updated as ˜v1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Compared with previous state, ˜v0 q and ˜v1 q aggregate more instance-level context information from both support and query, thus being more robust to the appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 Mask Prediction and Model Training Finally, we feed the updated query instance features as well as activation maps specified in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 to make pixel-wise prediction for query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' ˆMq = Fdecoder(C(˜v0 q, ˜v1 q,Ap m,Ar m,Ap h,Ar h)), (7) where C(·) is the channel-wise concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Fdecoder(·) is the decoder network, which consists of a Atrous Spatial Pyramid Pooling (ASPP) module [2] and three consecutive con- volution layers with residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the activation maps Ap m and Ap h, the superscript 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN Table 1: Comparison with the state-of-the-art methods on PASCAL-5i dataset under both 1-shot and 5-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The mIoU of each fold, and the averaged mIoU & FB-IoU of all folds are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Methods Backbone 1-shot 5-shot Fold-0 Fold-1 Fold-2 Fold-3 Mean FB-IoU Fold-0 Fold-1 Fold-2 Fold-3 Mean FB-IoU OSLSM (BMVC’17) [23] VGG16 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 co-FCN (ICLRW’18) [20] VGG16 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 Table 2: Comparison with the state-of-the-art methods on COCO-20i dataset under both 1- shot and 5-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The mIoU of each fold, and the averaged mIoU & FB-IoU of all folds are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Methods Backbone 1-shot 5-shot Fold-0 Fold-1 Fold-2 Fold-3 Mean FB-IoU Fold-0 Fold-1 Fold-2 Fold-3 Mean FB-IoU FWB(ICCV’19) [19] VGG16 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 PFENet(TPAMI’20) [27] VGG16 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 SAGNN(CVPR’21) [31] VGG16 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 RePRI(CVPR’21) [1] ResNet50 31.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 SAGNN(CVPR’21) [31] ResNet101 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 MMNet(ICCV’21) [30] ResNet50 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 HSNet(ICCV’21) [17] ResNet50 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 CyCTR(NIPS’21) [40] ResNet50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 Baseline ResNet50 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 SiGCN ResNet50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 indicates that the activation map comes from pixel-to-pixel matching (p) [27] between sup- port and query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To generate the final segmentation map, we upsample ˆMq by bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, we adopt the binary cross entropy loss to train our model on all the episodes in the training set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 Implementation Details Following previous methods [17, 27], we conduct experiments on PASCAL-5i and COCO- 20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Classes in each dataset are divided into four base-novel class folds for cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our model is based on PFENet [27] and we also adopt the same training and evaluation setting as PFENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We train our model with SGD optimizer on PASCAL-5i for 200 epochs and COCO-20i for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The learning rate is initialized as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='005 with batch size 8 and decays according to the poly learning rate scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' During evaluation, we adopt the mIoU as main metric and also report FBIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the K-shot setting, we fuse foreground prototypes from each support image-mask pair to generate θ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (4) and the support instance feature vs in the instance association module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 Comparison with the State-of-the-art methods PASCAL-5i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In Table 1, we show the performance comparison between the proposed SiGCN and the leading methods on the PASCAL-5i dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' With the ResNet50 as back- bone, we can conclude that the proposed SiGCN achieves the state-of-the-art performance J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 9 Support Query Prediction Figure 3: Qualitative results on the PASCAL-5i (left four columns) and COCO-20i (right four columns) under the 1-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our SiGCN can make accurate segmentation predic- tion, even though the support and query objects exhibit larger appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' under the 1-shot setting and competitive result under the 5-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Specifically, SiGCN surpasses previous SOTA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', CyCTR) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1% in the 1-shot setting, while achieving com- petitive results as previous SOTA HSNet (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 vs 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5) in the 5-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Additionally, compared with the baseline model, which is constructed by removing the SiGR and IA mod- ules from SiGCN (see complementary materials for more details), our SiGCN achieves sig- nificant improvement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4% under the 1-shot and 5-shot settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Considering the comparison with previous SOTA methods using the FB-IoU, SiGCN also obtain competitive results (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 under the 1-shot and 5-shot setting, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' COCO-20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Table 2 shows the comparison between previous SOTA methods and our SiGCN on the COCO-20i dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Although the COCO-20i is more challenging (larger appearance and scale variations than PASCAL-5i), SiGCN also achieves state-of-the art results (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 under 1-shot and 5-shot settings, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' It demonstrates the superiority of our proposed SiGCN in addressing the appearance variation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Furthermore, SiGCN with ResNet50 backbone achieves significant mIoU improvement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1% under the 1-shot and 5-shot settings, respectively) over the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In Figure 3, we report some typical qualitative results generated by our SiGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As can be seen, the support and query object exhibits large appearance variations, but our approach can also produce accurate mask prediction for the query image, which thanks to the query context mining by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' More experimental and quantitative re- sults can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Table 3: Effect of key modules in SiGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' SiGR IA 1-shot mIoU 50 51 52 53 mean FB-IoU 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 ✓ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 ✓ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 ✓ ✓ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 Table 4: Effect of different GCN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 1-shot mIoU GCN Variants 50 51 52 53 mean FB-IoU baseline 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 GCN [10] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 SiGCN-1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 SiGCN-2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 Ablation Study Component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' There are two main components in our model, namely the support- induced graph reasoning (SiGR) module and the instance association (IA) module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We validate the effectiveness of each component and present the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' It can be summarized that the SiGR module plays the most important role in the performance im- provement while the IA module is indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' With SiGR and IA module, our SiGCN A-SR-650074 SANTUA 02-5198-34A-SR-650074 02-5198-34BOMBEROS BOMBEROS 998386BlinkainBlinkainGRACEGRACE lsneh10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN (a) Support instance size s (b) Threshold t Figure 4: Ablation studies of (a) support instance size s and (b) mask threshold t on PASCAL-5i under the 1-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' achieves state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Effect of different GCN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To investigate the effect of different GCN variants in the target association module, we adopt three GCN variants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', plain GCN, SiGCN with one and two layers (denoted as SiGCN-1 and SiGCN-2, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As shown in Table 4, our model with plain GCN achieves sightly better performance than the baseline model, while our model with SiGCN makes substantial performance improvement as the number of the SiGCN layer increasing from one to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Table 5: Effect of k prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 1-shot mIoU k prototypes 50 51 52 53 mean FB-IoU 1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 Table 6: Effect of different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 1-shot mIoU Backbone 50 51 53 53 mean FB-IoU VGG16 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 ResNet-50 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 ResNet-101 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 Influence of k support prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The number of support prototypes k in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (4) is a key parameter in SiGCN, which determines the quality of support foreground used for guiding the query graph updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We vary k in {1,3,5,7} to observe the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In Table 5, k = 5 achieves best performance, which indicates that SiGCN with 5 support foreground prototypes provides better guidance to query graphs for capturing contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Influence of support instance size s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The support instance feature size s in the instance association module (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4) determines the context information in the support instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The value of s is taken from {1,3,5,10,20} to observe the performance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As in Figure 4(a), SiGCN with s = 10 exhibits the best performance, thus we set s = 10 in all of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Influence of mask threshold t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The mask threshold t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (1) is essential for selecting discriminative parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', targets in the query graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Thus we take its value from 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 to investigate the performance of our SiGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As can be seen in Figure 4(b), SiGCN with t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Additionally, SiGCN without mask threshold in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' (5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', t = 0) also shows competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Effect of different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To evaluate the performance of our model with different backbones, we adopt three backbone networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', VGG16, ResNet50, and ResNet101) to implement experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As shown in Table 6, our SiGCN with ResNet101 as backbone yields superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 75 70 65 IoU 60 55 foldo fold2 MIoU fold 1 fold3 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 t75 70 65 IoU 60 55 foldo fold2 MIoU fold1 fold3 50 3 5 10 20 SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 11 5 Conclusion In this work, we propose a support-induced graph convolutional network (SiGCN) to tackle the challenging appearance variation problem for the few-shot semantic segmentation (FSS) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our main idea is to enrich query context representation with the discriminative parts in the query objects and support objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To achieve this, we propose a support-induced graph reasoning (SiGR) module to associate salient query object parts with the graph convolution network, in which support prototypes is introduced to guide the graph updating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Additionally, an instance association (IA) module is designed to capture high-order context from the support instances and complementary query instances simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Extensive experiments on two FSS benchmarks prove that our SiGCN achieves state-of-the-art perfor- mance under both 1-shot and 5-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We believe that the idea of explicitly mining query context will shed light for future research in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Acknowledgement This work was partially funded by Elekta Oncology Systems AB and a RVO public-private partnership grant (PPS2102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' References [1] Malik Boudiaf, Hoel Kervadec, Ziko Imtiaz Masud, Pablo Piantanida, Ismail Ben Ayed, and Jose Dolz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Few-shot segmentation without meta-learning: A good transductive inference is all you need?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13979–13988, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' [2] Liang-Chieh Chen, George Papandreou, Florian Schroff, and Hartwig Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Re- thinking atrous convolution for semantic image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='05587, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' [3] Xiaokang Chen, Yuhui Yuan, Gang Zeng, and Jingdong Wang.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 15 6 Supplementary Materials In this supplementary material, we firstly present extensive implementation details about our experiments, and then give two more additional ablation studies and analysis about different level features and matching methods for activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, we provide more qualita- tive visualization results for PASCAL-5i and COCO-20i benchmarks under the large object appearance and scale variations scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Implementation Details In our experiments, we employ ResNet-50 (VGG-16 and ResNet- 101) pre-trained on ImageNet as our backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For ResNet-50 and ResNet-101, the dilation convolution is introduced to ensure that the feature receptive fields of layer2, layer3, and layer4 preserve the same spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The backbone weights are frozen except for layer4, which is required to learn more robust activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The proposed model is validated on PASCAL-5i and COCO-20i benchmarks, which are widely used in the few-shot semantic segmentation for cross validations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The detailed split of testing classes for each cross validation (fold) is shown in Table 7 and Table 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The model is trained with a SGD optimizer for 200 and 50 epochs on the PASCAL-5i and the COCO-20i benchmarks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' The learning rates are initialized as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='005 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='002 with a poly learning rate schedule in PASCAL-5i and COCO-20i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Our entire network is trained with the same learning rate during each epoch, while layer4 of the backbone network should be ensured a lower learning rate for fine-tuning, thus its parameters starts back-propagation after training for multiple epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' And the batch size is set as 8 on PASCAL-5i and 32 on COCO-20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Data augmentation strategies like random rotation and flip are adopted in the training stage, and all images are cropped to 473×473 patches for two benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Besides, no post-processing is used on PASCAL-5i, while for COCO-20i, we adopt the multi-scale testing strategy for the model evaluation due to extremely large object appearance and scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' By the way, the original Groudtruth of the query image without any resize operations is adopted for the model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' In addition, the support instance size s in the IA module are set as 10, and mask threshold t are set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' For the SiGR module, we set the number of support prototypes k = 5 in each SiGCN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Finally, we implement our model with PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 and conduct all the experiments with Nvidia Tesla A100 GPUs and CUDA11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Table 7: Testing classes split for each fold in PASCAL-5i dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Fold Testing (novel) classes Fold-0 Aeroplane, Bicycle, Bird, Boat, Bottle Fold-1 Bus, Car, Cat, Chair, Cow Fold-2 Diningtable, Dog, Horse, Motorbike, Person Fold-3 Potted plant, Sheep, Sofa, Train, Tvmonitor Effect of activation maps from different feature levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' To evaluate the quality of ac- tivation maps generated from different feature levels, we adopt three different-level features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=', block3 (middle-level), block4 (high-level), and block3&4 (middle-level& high-level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Support and query features from each level are matched by pixel-to-pixel matching to gen- erate corresponding activation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As shown in Table 9, our model with the activation map generated from block3&4 yields superior performance, while our model with the activation map generated from block3 or block4 also achieves competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN Table 8: Testing classes split for each fold in COCO-20i dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Fold Testing (novel) classes Fold-0 Person,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Airplane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Boat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Parking meter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Dog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Elephant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Backpack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='Suitcase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Sports Ball,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Skateboard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Wine glass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Spoon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Sandwich,' 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+page_content=' Bus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Traffic light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Bench,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Horse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Bear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Umbrella,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Frisbee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} 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+page_content=' Refrigerator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Vase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Toothbrush Effect of region matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We append the activation maps generated by region-region matching to evaluate the performance of our model with region matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' As shown in Table 9, we can conclude that the activation maps generated from region-to-region matching bring overall performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Compared with pixel-to-pixel matching, region-to-region matching can capture more contextual information, thus leading to improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Table 9: Ablation study on the different activation maps on the PASCAL-5i dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' 1-shot mIoU activation maps 50 51 52 53 mean FB-IoU block3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 block4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 block3&4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 block3+region 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 block4+region 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 block3&4+region 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content='5 Additional Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' We show more quantitative results of the proposed SiGCN in this section to further demonstrate its few-shot semantic segmentation perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' SiGCN aims to address the appearance variations problem in the few-shot semantic segmentation task, thus we show some examples sampled from PASCAL-5i and COCO-20i benchmarks with large appearance variations in Figure 5 and Figure 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Further- more, to demonstrate that our model can effectively address the scale variations problem, we give some examples from PASCAL-5i and COCO-20i benchmarks in Figure 6 and Figure 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' According to the quantitative results, we can conclude that our proposed SiGCN can effectively address the appearance and scale variations problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN 17 Support Query Prediction Figure 5: Qualitative results of the proposed SiGCN on PASCAL-5i benchmark with large object appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' From top to bottom: Support Image & Ground truth, Query Image & Ground truth, and Query Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Zoom in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Support Query Prediction Figure 6: Qualitative results of the proposed SiGCN on PASCAL-5i benchmark with large object scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' From top to bottom: Support Image & Ground truth, Query Image & Ground truth, and Query Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Zoom in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Support Query Prediction Figure 7: Qualitative results of the proposed SiGCN on COCO-20i benchmark with large object appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' From top to bottom: Support Image & Ground truth, Query Image & Ground truth, and Query Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' Zoom in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' RRLINGENHAUENRRLINGENHAUENTEPRERON 白口口WO NRWO NRTECHTTAN LTAREUSEAMTI-VEUS DTECHTITA TECHTITAN TTECHTITA B1218RIDTECHTITAN LITIATEUSEAMITI-VES PECHTA TECHTITAN TECH TTAN B1218RBELHUS PeleCleeler20318 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} +page_content=' LIU, ETAL: FEW-SHOT SEMANTIC SEGMENTATION WITH SUPPORT-INDUCED GCN Support Query Prediction Figure 8: Qualitative results of the proposed SiGCN on COCO-20i benchmark with large object scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfdQTN/content/2301.03194v1.pdf'} 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b/dNFJT4oBgHgl3EQfRyyj/content/tmp_files/2301.11497v1.pdf.txt @@ -0,0 +1,1041 @@ +DualCSG: Learning Dual CSG Trees for General and Compact CAD Modeling +Fenggen Yu1 +Qimin Chen1 +Maham Tanveer1 +Ali Mahdavi Amiri1 +Hao Zhang1 +1Simon Fraser University +Abstract +We present DualCSG, a novel neural network composed +of two dual and complementary branches for unsupervised +learning of constructive solid geometry (CSG) representa- +tions of 3D CAD shapes. Our network is trained to re- +construct a given 3D CAD shape through a compact as- +sembly of quadric surface primitives via fixed-order CSG +operations along two branches. The key difference between +our method and all previous neural CSG models is that +DualCSG has a dedicated branch, the residual branch, to +assemble the potentially complex, complement or residual +shape that is to be subtracted from an overall cover shape. +The cover shape is modeled by the other branch, the cover +branch. Both branches construct a union of primitive inter- +sections, where the only difference is that the residual branch +also learns primitive inverses while operating in the com- +plement space. With the shape complements, our network +is provably general. We demonstrate both quantitatively +and qualitatively that our network produces CSG reconstruc- +tions with superior quality, more natural trees, and better +quality-compactness tradeoff than all existing alternatives, +especially over complex and high-genus CAD shapes. +1. Introduction +CAD shapes have played a central role in the advance- +ment of geometric deep learning, with most neural models to +date trained on datasets such as ModelNet [43], ShapeNet [2], +and PartNet [28] for classification, reconstruction, and gener- +ation tasks. These shape collections all possess well-defined +category or class labels, and more often than not, the effec- +tiveness of the data-driven methods is tied to how well the +class-specific shape features can be learned. Recently, the +emergence of datasets of CAD parts and assemblies such as +ABC [18] and Fusion360 [41] has fueled the need for learn- +ing shape representations that are agnostic to class labels, +without any reliance on class priors. Case in point, the ABC +dataset does not provide any category labels, while another +challenge to the ensuing representation learning problem is +the rich topological varieties exhibited by the CAD shapes. +U +ー +U +CAPRI +U +U +DualCSG +GT +ー +Figure 1. Comparing CSG trees and shape reconstructions obtained +by our network, DualCSG, and by CAPRI-Net, the current state of +the art. A natural CSG construction necessitates a difference oper- +ation involving a complex (residual) shape, which DualCSG can +predict with compactness (three primitives) and quality. CAPRI- +Net can only build it using convexes, requiring unnecessarily many +primitives and results in poor reconstruction quality. +Constructive solid geometry (CSG) is a classical CAD +representation; it models a 3D shape as a recursive assembly +of solid primitives, e.g., cuboids, cylinders, etc., through +Boolean operations including union, intersection, and dif- +ference. Of particular note is the indispensable role the +difference operation plays when modeling holes and high- +genus shapes, as shown in Fig. 1. Recently, there have +been increased interests in 3D representation learning using +CSG [4,7,16,33,34,42,47], striving for generality, compact- +ness, and reconstruction quality of the learned models. +In terms of primitive counts, a direct indication of +compactness of the CSG trees, and reconstruction quality, +CAPRI-Net [47] represents the state of the art. However, it +is not a general neural model, e.g., it is unable to represent +1 +arXiv:2301.11497v1 [cs.CV] 27 Jan 2023 + +0Union +... +Feature +Vector +Guassian +Noise +Primitive +Prediction +Difference +Cover Branch +Residual Branch +Intersection +Union +Intersection +Recovered Shape +Given Shape S +Convex Primitives +Convex Primitives +Primitive Inverses +SR +SC +... +... +... +... +... +... +Cover Loss +Residual Loss +Figure 2. For a given 3D CAD shape S (shown at the right end), our network DualCSG is trained to optimize both its network parameters +and a feature code to reconstruct S by optimizing an occupancy loss. The network parameters define a CSG assembly over a set of quadric +primitives. The assembly is build using two branches: a cover branch (top), producing shape SC, and a residual branch (bottom), producing +shape SR. Primitives learned by the cover branch all define convex spaces, e.g., the half space for a plane, and the space enclosed by a +bi-infinite cylinder. In the residual branch, the primitive prediction results in both convexes and inverses (or complements) of convexes. +After applying intersections and a union to obtain the cover shape SC and residual shape SR, by optimizing their respective occupancy +losses, the recovered shape is obtained via a difference operation. +CAD shapes whose assembly necessitates nested difference +operations (i.e., needing to subtract a part that requires primi- +tive differences to build itself, e.g., see the CAD model in the +first column of Fig. 8 as an example). Since both operands +of the (single) difference operation in CAPRI-Net can only +model intersections and unions, their network cannot pro- +duce a natural and compact CSG assembly for relatively +complex CAD shapes with intricate concavities and topolog- +ical details, such as the CAD part shown in Fig. 1. +In this paper, we present DualCSG, a novel neural net- +work composed of two dual and complementary branches +for unsupervised learning of CSG tree representations of +3D CAD shapes. As shown in Fig. 2, our network follows +a fixed-order CSG assembly, like most previous unsuper- +vised CSG representation learning models [4, 7, 47]. The +key difference to all of them however, is that our network +has a dedicated branch, the residual branch, to assemble the, +potentially complex, complement or residual shape that is to +be subtracted from an overall cover shape. In turn, the cover +shape is modeled by the other branch, the cover branch. The +two branches both construct a union of primitive intersec- +tions, but the residual branch also learns primitive inverses +while operating over the complement space. Architecturally, +the two branches almost replicate each other, but they have +independent network parameters. +Given the challenge of unsupervised learning of CSG +tree assemblies amid significant structural diversity among +CAD shapes, our network is not designed to learn a unified +model over a shape collection. Rather, it overfits a given +3D CAD shape by optimizing a compact CSG assembly of +quadric surface primitives to approximate the shape. The +learning problem is still challenging since the number, se- +lection, and assembly of the primitives are unknown and +involve a complex search space. +In contrast to CAPRI-Net, our method is provably gen- +eral, meaning that any CSG tree can be converted into an +equivalent DualCSG representation. Our dual-branch net- +work is fully differentiable and can be trained end-to-end +with only the conventional occupancy loss for neural implicit +models [4, 5, 47]. With both operands of the final differ- +ence operation capable of learning general CAD assemblies, +our network excels at representing complex and high-genus +CAD shapes which challenge state-of-the-art methods, as +shown in Fig. 1. We demonstrate both quantitatively and +qualitatively that our network, when trained on ABC [18] or +ShapeNet [2], produces CSG reconstructions with superior +quality, more natural trees, and better quality-compactness +tradeoff than all existing alternatives, including BSP-Net [4], +CSG-Stump [7], and CAPRI-Net [47]. +2. Related Work +We discuss related works on learning structured represen- +tations, CAD shapes, CSG trees, and overfit models. Aside +from these, it is worth mentioning that there are heuristic- +based techniques attempting to solve a similar problem as +ours, e.g., using RANSAC. The most relevant work in this +category to DualCSG is InverseCSG [10], which requires ad- +ditional knowledge about each shape including the number of +surface segments and pre-defined primitive types. Moreover, +different parameter settings are used for different shapes. +Hence in this section, we focus on learning techniques that +use the same setup across shape categories to better contrast +2 + +S +S +(d) +(b) +(a) +(c) +Figure 3. To reconstruct a “negative S" shape, CSG-Stump [7] can +only approximate it by a union of small pieces (a) or by subtracting, +from a box, a union of basic primitives (b). Similarly, CAPRI- +Net [47] must employ a union of convexes to approximate the S +(c). None of these are natural or compact solutions. Only DualCSG +can reconstruct the most compact solution (d) as the S can be well +produced by its general residual branch using quadric primitives. +our work with the most comparable methods. +Structured models. +A shape can be represented as a set +of primitives or parts assembled together. Primitive fitting to +point clouds has been extensively studied in [20,22,24]. For +shape abstraction, cuboid [37] and super quadratics [32] have +been previously employed and 3D Gaussian local functions +are used [13] for template fitting. Having a single RGB im- +age, cuboids have been also used to estimate object parts and +their relations using a convolutional-recursive auto-encoder. +More complex sub-shapes have been learned for shape as- +sembly such as elementary 3D structures [9], implicit con- +vex [4,8] and neural star components [17], as well as parts +in the form of learnable parametric patches [34], moving +or deformable primitives [25,31,45,48], point clouds [23], +or a part-aware latent space [11]. However, none of these +techniques directly addresses reconstructing a CSG tree for +a given 3D shape which is our problem of interest. +Deep CAD models. +Synthesizing and editing CAD mod- +els are difficult tasks since they tend to have many sharp +features and various topologies. Learning-based shape pro- +grams have been designed to perform these tasks by pro- +viding easy-to-use tools and editing capabilities [1,12,36]. +In addition, due to the complex topology and geometry of +CAD models, certain representations are often used for CAD +models. Boundary Representations (B-Reps) are quite com- +mon for modeling CAD shapes and there are previous at- +tempts to reverse engineer such representations given an +input mesh or point cloud [44]. For example, BRepNet [19], +UV-Net [14], and SBGCN [15] offer network architectures +capable of working with B-Reps and their topological infor- +mation through message passing. +Learning CSG. +Learning CSG representations, e.g., prim- +itive assembly [40] and sketch analysis [21,29], has become +an emerging topic of geometric deep learning. While most +approaches are supervised, e.g., CSG-Net [33], SPFN [22], +ParseNet [34], DeepCAD [42], and Point2Cyl [38], there +have been several recent attempts at unsupervised CSG tree +reconstruction, especially under the class-agnostic setting, +resorting to neural implicit representations [4,5,26,30]. +UCSG-Net [16] is a relatively early method for recon- +structing CSG trees with arbitrary assembly orders. The +learning task is difficult due to the order flexibility, but can +be made feasible by limiting the primitives to boxes and +spheres only. More success in terms of reconstruction qual- +ity and compactness of the CSG trees has been obtained by +learning fixed-order assemblies, including DualCSG. +BSP-Net [4] learns plane primitives whose half-spaces +are assembled via intersections to obtain convexes, followed +by a union operation, to reconstruct concave shapes. CAPRI- +Net [47] extends BSP-Net by adding quadric surface primi- +tives and a difference operation after primitive unions. CSG- +Stump [7] also follows a fixed CSG assembly order while +including an inverse layer to model shape complements. +The complement operation helps attain generality of their +CSG reconstructions, in theory, but the inverse layer is non- +differentiable and the difference operations can only be ap- +plied to basic primitives, which can severely compromise +the compactness and quality of the reconstruction. +In Fig. 3, we use a 2D example to contrast how CSG- +Stump, CAPRI-Net, and DualCSG can reconstruct a shape +whose natural construction involves subtracting an S shape. +Overfit models. +Overfitting to the geometry of a shape is a +common approach. It has been used for applications such as +compression [6], reconstruction [39], and representing level +of details [35] as it significantly helps to recover intricate +details and features. The underlying geometry of a NeRF is +essentially an overfit (i.e., fixed) to the shape/scene although +the primary task of NeRF is novel view synthesis [27,46]. +Following a similar principle, to replicate fine geometric +details, DualCSG constructs a CSG tree for a given object +via optimizing a small neural network along with a randomly +initialized feature code (Fig. 2). We follow this optimiza- +tion/overfitting procedure as we did not find learning a prior +on CAD shapes very useful and generalizable due to the +structural and topological diversity in the CAD shapes. +3. Method +To reconstruct high-genus and complex target shapes with +various concavities, one effective approach is to produce a +volume containing the entire target shape and subtract a +residual volume from it. Here, one can observe an analogy +to CNC machining in mechanical engineering. In this set- +ting, the target shape can be constructed by producing the +cover and residual shapes via two separate cover and residual +branches and then subtracting the two to produce the final +result. By learning how to reconstruct cover and residual +shapes separately, details such as complex concavities are +better recovered by subtraction in the final stage. +3 + +In this work, called DualCSG, we reconstruct a given 3D +shape S by producing shape SC that contains or covers the +given shape S along with a residual shape SR that if it gets +subtracted from SC, S will be produced (i.e., S ≈ SC − +SR). SC and SR are obtained by a set of CSG operations. +Sampled from a Gaussian distribution, the input to DualCSG +is a feature code that is optimized along with the network’s +parameters to reconstruct a given 3D shape S and the output +is a set of occupancy values that are optimized to fit S. More +details about the network are explained in Section 3.1. +CAPRI-Net [47] also tries to perform a shape difference at +the end by predicting a set of primitives that undergo a fixed +order of intersection, union, and difference operations to ob- +tain the target shape. However, this sequence is not general +and cannot support all shapes (see supplementary material). +In addition, CAPRI-Net first uses an encoder that is pre- +trained on the entire dataset before it is optimized/overfit to +each input shape. However, we avoid using an encoder in +DualCSG as we realized that it does not help to converge +more accurately or faster. Since DualCSG is optimized for a +single shape, it benefits from a smaller network with fewer +parameters compared to CAPRI-Net (see Table 1). +There are approaches such as CSG-Stump [7] utilizing a +fixed CSG order of inverse, intersection, and union. This se- +quence is theoretically able to reproduce any CSG sequence +as the difference operation can be achieved by inverse and +intersection. However, this way, difference operations can +be only applied on basic primitives at the early level of the +CSG sequence. To reproduce a shape such as S in a box +illustrated in Fig. 3, CSG-Stump has to approximate it by +assembling a union of several small pieces of primitives as +(a) or by subtracting a union of basic primitives (e.g. circles, +boxes and triangles) as (b) instead of a natural difference of +a box and S shape. Therefore, CSG-Stump fails to produce +compact results and causes irregular artifacts/bumps on the +reconstructed shape. CAPRI-Net does apply the difference +as the last operation but uses a union of convex shapes to +approximate the S shape as (c). Thus the solution of CAPRI- +Net is not compact either. Only DualCSG can produce the +compact solution as (d) by introducing two separated CSG +branches, while the cover branch can produce the box and the +residual branch can produce the S shape since it is capable +of producing arbitrary and complex residual shapes. +Formally, in DualCSG, we utilize two branches: cover +branch that produces cover shape SC, which covers the tar- +get shape S with a combination of convex shapes (Fig. 2 +Top) and residual branch (Fig. 2 Bottom) that produces the +residual shape SR that is subtracted from SC to produce S. +To support generality, the set of primitives in the residual in- +cludes both convex primitives and complimentary primitives +(see Section 3.2). This allows incorporating difference oper- +ations at early stages of the residual branch using inverse and +intersection operations. Such alteration produces complex +residual shapes that can be used to reconstruct general, high- +genus, and complex concave shapes. See the supplementary +material for a formal proof of the generality of DualCSG. +3.1. DualCSG +SC, the output of the cover branch, is the union of convex +shapes. SR, the output of the residual branch, can be quite +complex since it is the union of possibly concave and con- +vex shapes. This property that was missing in CAPRI-Net +significantly empowers the network as many complicated +shapes can be constructed by finding the difference between +a simple convex shape and an arbitrary concave or convex +shape (see Figures 2 and 7). +In DualCSG, we only optimize a feature code and net- +work weights to fit the target shape (similar to Auto-decoder +in DeepSDF [30]). This is efficient since DualCSG is a light +network that quickly converges to each shape. Starting from +a feature code, we pass it to the primitive prediction network +and generate two matrices that hold the primitives’ param- +eters. Each matrix is used to determine the approximated +signed distance (ASD) of a set of query points sampled in +the space in which the shape is embedded. These two sets +of ASD values are separately passed to cover and residual +branches, and each branch has an intersection and a union +layer. The cover branch (Fig. 2 Top) and the residual branch +(Fig. 2 Bottom) generate point occupancy values indicating +whether a point is inside or outside SC and SR respectively. +The difference between SC and SR forms the final shape. +3.2. Primitive Representation +We use a more general primitive form for the residual +branch in comparison with the cover branch to generate +complex residual shapes. This is one of the key differences +between our method and CAPRI-Net [47]. The primitive +prediction network (an MLP) receives a code of size 256 and +outputs two matrices PC ∈ Rp×7 (fed to the cover branch) +and PR ∈ Rp×7 (fed to the residual branch), each contains +parameters of p primitives (see Fig. 2). Primitives in PC are +represented by a quadric equation same as CAPRI-Net [47]: +|a|x2 + |b|y2 + |c|z2 + dx + ey + fz + g = 0, +(1) +where the first three coefficients are constrained to be positive +to represent convex primitives. In DualCSG, while half of +the primitives in PR are the same as Equation (1), we require +the other half to be inverse convex primitives by constraining +the first three coefficients to be negative: +−|a|x2 − |b|y2 − |c|z2 + dx + ey + fz + g = 0. +(2) +We show two primitives produced by Equations (1) +and (2) along with 2D visualization of cross-section near the +surface of each primitive (Fig. 4). Considering the universal +4 + +� +� +� +� +� +� +� +� +��� +��� +Figure 4. Visualizations of convex primitive shown in (a) and +primitive inverse shown in (b). n denotes the normal direction +pointing outside the surface. +space as a cube, for convex primitives (Fig. 4 (a)), the ASDs +of query points enclosed by the curved surface are nega- +tive, meaning that these points are inside the primitive. For +primitive inverse (Fig. 4 (b)), on the other hand, the ASDs +of query points enclosed by the curved surface are positive, +meaning that the complement space is inside the primitive. +The negative constrains in Equation (2) possibly make the +residual shape complex with detailed concavities. +For reconstruction, n points near the shape’s surface are +sampled and their ASD to all primitives is calculated sim- +ilar to CAPRI-Net [47]. For each point qj = (xj, yj, zj), +its ASD is captured in matrix D ∈ Rn×p as: DC(j, :) = +Q(j, :)PT +C and DR(j, :) = Q(j, :)PT +R, where Q(j, :) = +(x2 +j, y2 +j , z2 +j , xj, yj, zj, 1) is the jth row of Q. +3.3. DualCSG Branches +We briefly discuss details about our DualCSG branches by +following similar notations used in BSP-Net [4] and CAPRI- +Net [47]. We input the ASD matrix DC into the cover branch +and DR into the residual branch, and output vector aC and +aR, indicating whether query points are inside/outside cover +shape SC and residual shape SR. Each branch contains an in- +tersection layer and a union layer adopted from BSP-Net [4]. +In addition, to encourage difference operation between cover +shape and residual shape to produce the final shape, we use +the same difference loss from CAPRI-Net [47] to encourage +SC from cover branch to cover the volume occupied by the +ground truth shape and SR from residual branch to subtract +a meaningful residual volume. +Our DualCSG branches aim to produce two vectors in- +dicating query points inside/outside cover shape SC and +residual shape SR from the predicted primitives. The CSG +operation order in each branch is the same as BSP-Net [4], +which contains an intersection layer and a union layer. Note +that the weights of the two branches are not shared. +During training, the inputs to intersection layers are two +ASD matrices DR ∈ Rn×p and DC ∈ Rn×p. Primitives +involved in forming intersected shapes are selected by two +learnable matrices TC ∈ Rp×c and TR ∈ Rp×c, where +c is the number of intersected solid shapes. We can ob- +tain Con ∈ Rn×c by the intersection layer and only when +Con(j, i) = 0, query point qj is inside the intersected solid +shape i (ConR is the same and only subscripts are R): +ConC = relu(DC)TC +� +0 +in, +> 0 +out. +(3) +Then all the shapes obtained by the intersection operation +are combined by two union layers to find the cover shape +SC as well as the residual shape SR. +The inside/outside indicators of the combined shape are +stored in the vector aR ∈ Rn×1 and aC ∈ Rn×1 indicating +whether a point is in/outside of the cover and residual shapes. +Similar to [47], aC and aR are computed in a multi-stage +fashion (a+ and a∗ for early and final stages). Specifically, +a+ +C is obtained by the following equation: +a+ +C(j) = +C ( +� +1≤i≤c +WC(i)C (1 − ConC(j, i))) +� +1 +≈ in, +< 1 +≈ out, +(4) +where WC ∈ Rc is a learnable weighting vector and C is +a clip operation to [0, 1], and a+ +R is defined similarly with +WR and ConR. In later stages, a∗ +C and a∗ +R are obtained by +finding min of each row of ConC and ConR: +a∗ +C(j) = min +1≤i≤c(ConC(j, i)) +� +0 +in, +> 0 +out. +(5) +3.4. Loss Functions and Training Strategy +We utilize the same reconstruction loss and multi-stage +training as CAPRI-Net’s to facilitate differentiable opera- +tions in the early stage and gradually achieve good results in +the end. The following loss function is used in our method: +L = Lrec + LT + LW, +(6) +where Lrec is the reconstruction loss applied to aC and aR, +it would force the subtracted result between cover shape and +residual shape to be close to the input shape. LT + LW are +the losses applied to the intersection and union layer weights. +Note that differently from CAPRI-Net [47], we have net- +work weights from DualCSG branches: T = [TC, TR] and +W = [WC, WR]. However, LT and LW and the difference +loss are the same as CAPRI-Net [47]. Please refer to more +details on the losses in the supplementary material. +4. Results +In our experiments, we use two public datasets ABC [18] +and ShapeNet [2]. On each dataset, we test the quality of +producing CSG operations from two inputs: mesh and point +cloud. We present qualitative and quantitative results of our +experiments to demonstrate the effectiveness of DualCSG. +5 + +Figure 5. Comparing results for 3D meshes in ABC. +4.1. Training Details +Since the methods we compare with require an additional +time-consuming optimization at test time to achieve satis- +factory results, (e.g., 30 min per shape for CSG-Stump), we +have randomly selected a moderately sized subset of shapes +as test set for evaluation: 500 shapes from ABC, and 50 from +each of the 13 categories of ShapeNet (650 shapes in total). +In addition, we ensured that 80% of the selected shapes from +ABC have genus larger than two with more than 10K vertices +to include complex structures. Experiments were performed +using an Nvidia GeForce RTX 2080 Ti GPU. +In our experiments, we set the number of maximum prim- +itives as p = 512 and the number of maximum intersections +as c = 32 for each branch to support complex shapes. Since +both these numbers are half of the setting in CAPRI-Net, the +size of our CSG layers is half of the CAPRI-Net’s. The size +of our latent code for all input types is 256 and a two-layer +MLPs is used to predict the parameters of the primitives +from the input feature code. We train DualCSG per shape +by optimizing the latent code, primitive prediction network, +intersection layer, and union layer. +4.2. Mesh to CSG Representation +Given a 3D mesh, the task is to learn an accurate and +compact CSG representation for this shape. To do so, we +first sample 24,576 points around the shape’s surface (i.e. +with a distance up to 1/64) and 4,096 random points in +3D space. All 28,672 points are then scaled into the range +[−0.5, 0.5], these points along with their occupancy values +are used to optimize the network. +We compare DualCSG with BSP-Net [4], CSG-Stump [7] +and CAPRI-Net [47], which output structured parametric +primitives. For a fair comparison, we optimize all of these +networks with the same number of iterations. BSP-Net, +CSG-Stump, and CAPRI-Net are pre-trained on the training +set provided in CAPRI-Net before optimization to achieve +better initialization. Note that CSG-Stump uses different net- +Figure 6. Comparing results for 3D meshes in ShapeNet. +work settings for shapes from ABC (with shape differences) +and ShapeNet (without shape difference); we therefore fol- +low the same settings in our comparisons. For each shape, +BSP-Net takes about 15 min, CSG-Stump about 30 min and +CAPRI-Net about 3 min to converge. The training process +in DualCSG will run 12,000 iterations for each stage, taking +about 5 minutes per shape. +Evaluation Metrics. +Quantitative metrics for shape recon- +struction are symmetric Chamfer Distance (CD), Normal +Consistency (NC), Edge Chamfer Distance [4] (ECD), and +Light Field Distance [3] (LFD). For ECD, we set the thresh- +old for normal cross products to 0.1 for extracting points +close to edges. CD and ECD are computed on 8K sample +points on the surface and multiplied by 1,000. For LFD, we +render each shape at ten different views and measure the +Light Field Distances. In addition, we compare the number +of primitives #P to evaluate the compactness of shapes since +all CSG-based modeling methods predict some primitives +that are combined with intersection operations. +Evaluation and Comparison. +We provide visual compar- +isons on representative examples from the ABC dataset in +Fig. 5 and the ShapeNet dataset in Fig. 6; more results can +be found in the supplementary material. +Our method consistently reconstructs more accurate +shapes with geometric details and concavities. BSP-Net +simply assembles convex shapes to fit the target shape and +obtain less compact results without a difference operation. +CSG-Stump tends to use considerably more difference op- +erations to reconstruct shapes. This also causes the shapes’ +surface to be carved by many redundant primitives (i.e., lack +of compactness). In addition, since CSG-Stump does not +support difference operations between complex shapes, it +fails to reconstruct small holes or intricate concavities. As +the intermediate shapes subtracted in CAPRI-Net’s fixed +6 + +BSP +STUMP +CAPRI +DualCSG +GTBSP +STUMP +CAPRI +DualCSG +GTTable 1. Comparing CSG rep learning from 3D meshes in ABC. +Methods +BSP +STUMP +CAPRI +Ours +CD ↓ +0.115 +0.383 +0.177 +0.071 +NC ↑ +0.921 +0.850 +0.903 +0.928 +ECD ↓ +4.047 +8.881 +3.990 +3.467 +LFD ↓ +774.57 +2201.43 +665.20 +490.48 +#P ↓ +359.38 +83.42 +66.26 +58.81 +Table 2. Comparing CSG rep learning from meshes in ShapeNet. +Methods +BSP +STUMP +CAPRI +Ours +CD ↓ +0.164 +2.214 +0.124 +0.119 +NC ↑ +0.882 +0.794 +0.890 +0.887 +ECD ↓ +3.899 +6.101 +2.035 +1.722 +LFD ↓ +2135.68 +4267.66 +1558.08 +1506.13 +#P ↓ +694.21 +228.58 +50.94 +63.78 +Table 3. Comparing methods under the same auto-decoder setting. +ABC +ShapeNet +Methods +CAPRI +Ours +CAPRI +Ours +CD ↓ +0.183 +0.071 +0.156 +0.119 +NC ↑ +0.907 +0.928 +0.882 +0.887 +ECD ↓ +3.918 +3.467 +2.359 +1.722 +LFD ↓ +635.01 +490.48 +1685.38 +1506.13 +#P ↓ +77.15 +58.81 +52.13 +63.78 +CSG sequence are only unions of convex shapes, it fails to +reproduce target shapes with complex concavities. +As shown in Table 1, DualCSG achieves the best recon- +struction quality and compactness in all metrics on ABC +dataset compared to other methods. As shown in Table 2, the +reconstruction accuracy of DualCSG on ShapeNet is mostly +better than CAPRI-Net (except for NC). DualCSG may use +slightly more but comparable primitives than CAPRI-Net on +shapes with complex cavities to produce better details and +concavities, such as the chair back in Fig. 6. +Ablation study. +DualCSG uses an auto-decoder network +architecture [30], requiring no pre-training. In contrast, +CAPRI-Net adopts the traditional auto-encoder network with +pre-training. To remove potential impacts of pre-training +from the evaluation, we compare DualCSG to CAPRI-Net +under the same auto-decoder setting, where the encoder of +CAPRI-Net is removed along with the pre-training. The re- +sults are shown in Table 3, where we can see that CAPRI-Net +still under-performs compared to DualCSG, as well as to its +version with pre-training as provided in Table 2. We believe +that the reason is that the limitation of only using convex +shapes by CAPRI-Net requires a better network parameter +initialization from the pre-training step, while DualCSG can +quickly overfit to the test shape from scratch owing to its +Table 4. Comparing CSG rep learning from 3D point cloud in ABC. +Methods +BSP +STUMP +CAPRI +Ours +CD ↓ +0.133 +0.695 +0.225 +0.085 +NC ↑ +0.919 +0.841 +0.894 +0.924 +ECD ↓ +3.899 +7.303 +3.308 +3.029 +LFD ↓ +773.29 +3149.82 +799.44 +558.08 +#P ↓ +360.87 +67.436 +68.57 +50.28 +more general representation capabilities. +CSG Trees Comparison. +Our network can learn to recon- +struct a 3D shape by producing a plausible CSG tree for a +given shape without any direct CSG supervision as shown in +Fig. 7. We simplify CSG trees from BSP-Net, CAPRI-Net, +and DualCSG by hiding their primitives to better illustrate +their learned CSG modeling process. +Fig. 7 clearly reveals the limitations of other methods. +BSP-Net combines many convex shapes (20 in this case) to +construct the final shape, which uses many primitives and +loses the compactness. CSG-Stump uses pre-defined limited +shapes as primitives, such as boxes and spheres, limiting +the reconstruction accuracy and requiring a large number of +primitives (53 for this shape, we only show a subset here). +Besides, CSG-Stump uses a primitive complement layer (il- +lustrated as operation C) to form the subtraction operation +for ABC shapes, which can result in many redundant differ- +ence operations. CAPRI-Net can only subtract the union of +convex shapes to approximate intricate concavities, which +limits its reconstruction accuracy and generality. +In contrast, shapes produced by our DualCSG are com- +posed of fewer intermediate shapes in comparison with other +methods, which makes our CSG tree more compact while +achieving better reconstruction quality. Note that the desired +task is not just to obtain a fewer number of parts with any +complexity, but rather to obtain fewer simple primitives or +convex shapes that are constructed via CSG operations. If the +former objective is favorable, other methods such as Neural +Parts [31] might be able to produce fewer parts. +4.3. Point Clouds to CSG +In our last experiment, we reconstruct CAD shapes from +point clouds, each containing 8,192 points. For each input +point, we sample 8 points along its normal with perturbations +sampled from Gaussian distribution (µ = 0, σ = 1/64). If +this point is at the opposite direction of normal vectors, the +occupancy value is 1, otherwise it is 0. This way, we sample +65, 536 points to fit the network to each shape. Similar to +mesh-to-CSG experiment, we only optimize the latent code, +primitive prediction network, and the selection matrix. +Quantitative comparisons in Table 4 and visual compar- +isons in Fig. 8 show that our network outperforms BSP-Net, +CSG-Stump, and CAPRI-Net in different reconstruction sim- +7 + +BSP +STUMP +CAPRI +DualCSG +GT +Figure 7. Comparing learned CSG Trees of 3D examples from ABC shown in the first row and ShapeNet in the second row. +����� +��� +����� +����� +������� +�� +����� +��� +����� +����� +������� +�� +Figure 8. Comparing CSG representation learning from 3D point cloud in ABC shown in column 1-6 and ShapeNet in column 7-12. +ilarity and compactness metrics on ABC dataset. Additional +ShapeNet results can be found in the supplementary material. +5. Discussion, limitation, and future work +We present DualCSG, a simple yet effective idea, for un- +supervised learning of general and compact CSG tree repre- +sentations of 3D CAD objects. Extensive experiments on the +ABC and ShapeNet datasets demonstrate that our network +outperforms state-of-the-art methods both in reconstruction +quality and compactness. We also have ample visual evi- +dence that the CSG trees obtained by our method tend to be +more natural than those produced by prior approaches. +Our network does not generalize over a shape collection; +it “overfits” to a single input shape and is in essence an opti- +mization to find a CSG assembly. While arguably limiting, +this is not entirely unjustified since the CAD shapes we seek +to handle, i.e., those from ABC, do not appear to possess +sufficient generalizability in their primitive assemblies. An- +other limitation is the lack of an explicit compactness loss +to enforce minimal CSG trees, like CAPRI-Net. In addi- +tion, incorporating interpretable CSG operations into the +network tends to cause gradient back-propagation issues and +limits the reconstruction accuracy of small details such as +decorative curves on chair legs. +Besides addressing the above limitations, we would like +to extend our method to structured CAD shape reconstruc- +tion from images and free-form sketches. Another interesting +direction for future work is to scale the primitive assembly +optimization from CAD parts to indoor scenes. +8 + +UUO容容References +[1] Dan Cascaval, Mira Shalah, Phillip Quinn, Rastislav Bodik, +Maneesh Agrawala, and Adriana Schulz. Differentiable 3d +cad programs for bidirectional editing. Computer Graphics +Forum, 41(2):309–323, 2022. 3 +[2] Angel X. Chang, Thomas Funkhouser, Leonidas Guibas, Pat +Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis +Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and +Fisher Yu. 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In ICCV, pages 900–909, 2017. 3 +10 + diff --git a/dNFJT4oBgHgl3EQfRyyj/content/tmp_files/load_file.txt b/dNFJT4oBgHgl3EQfRyyj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a864256b96aaca0671f831130c0c6f57e92fe8d4 --- /dev/null +++ b/dNFJT4oBgHgl3EQfRyyj/content/tmp_files/load_file.txt @@ -0,0 +1,582 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf,len=581 +page_content='DualCSG: Learning Dual CSG Trees for General and Compact CAD Modeling Fenggen Yu1 Qimin Chen1 Maham Tanveer1 Ali Mahdavi Amiri1 Hao Zhang1 1Simon Fraser University Abstract We present DualCSG, a novel neural network composed of two dual and complementary branches for unsupervised learning of constructive solid geometry (CSG) representa- tions of 3D CAD shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our network is trained to re- construct a given 3D CAD shape through a compact as- sembly of quadric surface primitives via fixed-order CSG operations along two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The key difference between our method and all previous neural CSG models is that DualCSG has a dedicated branch, the residual branch, to assemble the potentially complex, complement or residual shape that is to be subtracted from an overall cover shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The cover shape is modeled by the other branch, the cover branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Both branches construct a union of primitive inter- sections, where the only difference is that the residual branch also learns primitive inverses while operating in the com- plement space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' With the shape complements, our network is provably general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We demonstrate both quantitatively and qualitatively that our network produces CSG reconstruc- tions with superior quality, more natural trees, and better quality-compactness tradeoff than all existing alternatives, especially over complex and high-genus CAD shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Introduction CAD shapes have played a central role in the advance- ment of geometric deep learning, with most neural models to date trained on datasets such as ModelNet [43], ShapeNet [2], and PartNet [28] for classification, reconstruction, and gener- ation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' These shape collections all possess well-defined category or class labels, and more often than not, the effec- tiveness of the data-driven methods is tied to how well the class-specific shape features can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Recently, the emergence of datasets of CAD parts and assemblies such as ABC [18] and Fusion360 [41] has fueled the need for learn- ing shape representations that are agnostic to class labels, without any reliance on class priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Case in point, the ABC dataset does not provide any category labels, while another challenge to the ensuing representation learning problem is the rich topological varieties exhibited by the CAD shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' U ー U CAPRI U U DualCSG GT ー Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing CSG trees and shape reconstructions obtained by our network, DualCSG, and by CAPRI-Net, the current state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' A natural CSG construction necessitates a difference oper- ation involving a complex (residual) shape, which DualCSG can predict with compactness (three primitives) and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAPRI- Net can only build it using convexes, requiring unnecessarily many primitives and results in poor reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Constructive solid geometry (CSG) is a classical CAD representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' it models a 3D shape as a recursive assembly of solid primitives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', cuboids, cylinders, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', through Boolean operations including union, intersection, and dif- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Of particular note is the indispensable role the difference operation plays when modeling holes and high- genus shapes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Recently, there have been increased interests in 3D representation learning using CSG [4,7,16,33,34,42,47], striving for generality, compact- ness, and reconstruction quality of the learned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In terms of primitive counts, a direct indication of compactness of the CSG trees, and reconstruction quality, CAPRI-Net [47] represents the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, it is not a general neural model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', it is unable to represent 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='11497v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='CV] 27 Jan 2023 0Union .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Feature Vector Guassian Noise Primitive Prediction Difference Cover Branch Residual Branch Intersection Union Intersection Recovered Shape Given Shape S Convex Primitives Convex Primitives Primitive Inverses SR SC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Cover Loss Residual Loss Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For a given 3D CAD shape S (shown at the right end), our network DualCSG is trained to optimize both its network parameters and a feature code to reconstruct S by optimizing an occupancy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The network parameters define a CSG assembly over a set of quadric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The assembly is build using two branches: a cover branch (top), producing shape SC, and a residual branch (bottom), producing shape SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Primitives learned by the cover branch all define convex spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', the half space for a plane, and the space enclosed by a bi-infinite cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In the residual branch, the primitive prediction results in both convexes and inverses (or complements) of convexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' After applying intersections and a union to obtain the cover shape SC and residual shape SR, by optimizing their respective occupancy losses, the recovered shape is obtained via a difference operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAD shapes whose assembly necessitates nested difference operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', needing to subtract a part that requires primi- tive differences to build itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', see the CAD model in the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 8 as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Since both operands of the (single) difference operation in CAPRI-Net can only model intersections and unions, their network cannot pro- duce a natural and compact CSG assembly for relatively complex CAD shapes with intricate concavities and topolog- ical details, such as the CAD part shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In this paper, we present DualCSG, a novel neural net- work composed of two dual and complementary branches for unsupervised learning of CSG tree representations of 3D CAD shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2, our network follows a fixed-order CSG assembly, like most previous unsuper- vised CSG representation learning models [4, 7, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The key difference to all of them however, is that our network has a dedicated branch, the residual branch, to assemble the, potentially complex, complement or residual shape that is to be subtracted from an overall cover shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In turn, the cover shape is modeled by the other branch, the cover branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The two branches both construct a union of primitive intersec- tions, but the residual branch also learns primitive inverses while operating over the complement space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Architecturally, the two branches almost replicate each other, but they have independent network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Given the challenge of unsupervised learning of CSG tree assemblies amid significant structural diversity among CAD shapes, our network is not designed to learn a unified model over a shape collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Rather, it overfits a given 3D CAD shape by optimizing a compact CSG assembly of quadric surface primitives to approximate the shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The learning problem is still challenging since the number, se- lection, and assembly of the primitives are unknown and involve a complex search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In contrast to CAPRI-Net, our method is provably gen- eral, meaning that any CSG tree can be converted into an equivalent DualCSG representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our dual-branch net- work is fully differentiable and can be trained end-to-end with only the conventional occupancy loss for neural implicit models [4, 5, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' With both operands of the final differ- ence operation capable of learning general CAD assemblies, our network excels at representing complex and high-genus CAD shapes which challenge state-of-the-art methods, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We demonstrate both quantitatively and qualitatively that our network, when trained on ABC [18] or ShapeNet [2], produces CSG reconstructions with superior quality, more natural trees, and better quality-compactness tradeoff than all existing alternatives, including BSP-Net [4], CSG-Stump [7], and CAPRI-Net [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Related Work We discuss related works on learning structured represen- tations, CAD shapes, CSG trees, and overfit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Aside from these, it is worth mentioning that there are heuristic- based techniques attempting to solve a similar problem as ours, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', using RANSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The most relevant work in this category to DualCSG is InverseCSG [10], which requires ad- ditional knowledge about each shape including the number of surface segments and pre-defined primitive types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Moreover, different parameter settings are used for different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Hence in this section, we focus on learning techniques that use the same setup across shape categories to better contrast 2 S S (d) (b) (a) (c) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' To reconstruct a “negative S" shape, CSG-Stump [7] can only approximate it by a union of small pieces (a) or by subtracting, from a box, a union of basic primitives (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Similarly, CAPRI- Net [47] must employ a union of convexes to approximate the S (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' None of these are natural or compact solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Only DualCSG can reconstruct the most compact solution (d) as the S can be well produced by its general residual branch using quadric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' our work with the most comparable methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Structured models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' A shape can be represented as a set of primitives or parts assembled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Primitive fitting to point clouds has been extensively studied in [20,22,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For shape abstraction, cuboid [37] and super quadratics [32] have been previously employed and 3D Gaussian local functions are used [13] for template fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Having a single RGB im- age, cuboids have been also used to estimate object parts and their relations using a convolutional-recursive auto-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' More complex sub-shapes have been learned for shape as- sembly such as elementary 3D structures [9], implicit con- vex [4,8] and neural star components [17], as well as parts in the form of learnable parametric patches [34], moving or deformable primitives [25,31,45,48], point clouds [23], or a part-aware latent space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, none of these techniques directly addresses reconstructing a CSG tree for a given 3D shape which is our problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Deep CAD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Synthesizing and editing CAD mod- els are difficult tasks since they tend to have many sharp features and various topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Learning-based shape pro- grams have been designed to perform these tasks by pro- viding easy-to-use tools and editing capabilities [1,12,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, due to the complex topology and geometry of CAD models, certain representations are often used for CAD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Boundary Representations (B-Reps) are quite com- mon for modeling CAD shapes and there are previous at- tempts to reverse engineer such representations given an input mesh or point cloud [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For example, BRepNet [19], UV-Net [14], and SBGCN [15] offer network architectures capable of working with B-Reps and their topological infor- mation through message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Learning CSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Learning CSG representations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', prim- itive assembly [40] and sketch analysis [21,29], has become an emerging topic of geometric deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' While most approaches are supervised, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', CSG-Net [33], SPFN [22], ParseNet [34], DeepCAD [42], and Point2Cyl [38], there have been several recent attempts at unsupervised CSG tree reconstruction, especially under the class-agnostic setting, resorting to neural implicit representations [4,5,26,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' UCSG-Net [16] is a relatively early method for recon- structing CSG trees with arbitrary assembly orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The learning task is difficult due to the order flexibility, but can be made feasible by limiting the primitives to boxes and spheres only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' More success in terms of reconstruction qual- ity and compactness of the CSG trees has been obtained by learning fixed-order assemblies, including DualCSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' BSP-Net [4] learns plane primitives whose half-spaces are assembled via intersections to obtain convexes, followed by a union operation, to reconstruct concave shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAPRI- Net [47] extends BSP-Net by adding quadric surface primi- tives and a difference operation after primitive unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CSG- Stump [7] also follows a fixed CSG assembly order while including an inverse layer to model shape complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The complement operation helps attain generality of their CSG reconstructions, in theory, but the inverse layer is non- differentiable and the difference operations can only be ap- plied to basic primitives, which can severely compromise the compactness and quality of the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3, we use a 2D example to contrast how CSG- Stump, CAPRI-Net, and DualCSG can reconstruct a shape whose natural construction involves subtracting an S shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Overfit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Overfitting to the geometry of a shape is a common approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' It has been used for applications such as compression [6], reconstruction [39], and representing level of details [35] as it significantly helps to recover intricate details and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The underlying geometry of a NeRF is essentially an overfit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', fixed) to the shape/scene although the primary task of NeRF is novel view synthesis [27,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Following a similar principle, to replicate fine geometric details, DualCSG constructs a CSG tree for a given object via optimizing a small neural network along with a randomly initialized feature code (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We follow this optimiza- tion/overfitting procedure as we did not find learning a prior on CAD shapes very useful and generalizable due to the structural and topological diversity in the CAD shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Method To reconstruct high-genus and complex target shapes with various concavities, one effective approach is to produce a volume containing the entire target shape and subtract a residual volume from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Here, one can observe an analogy to CNC machining in mechanical engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In this set- ting, the target shape can be constructed by producing the cover and residual shapes via two separate cover and residual branches and then subtracting the two to produce the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' By learning how to reconstruct cover and residual shapes separately, details such as complex concavities are better recovered by subtraction in the final stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3 In this work, called DualCSG, we reconstruct a given 3D shape S by producing shape SC that contains or covers the given shape S along with a residual shape SR that if it gets subtracted from SC, S will be produced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', S ≈ SC − SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' SC and SR are obtained by a set of CSG operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Sampled from a Gaussian distribution, the input to DualCSG is a feature code that is optimized along with the network’s parameters to reconstruct a given 3D shape S and the output is a set of occupancy values that are optimized to fit S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' More details about the network are explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAPRI-Net [47] also tries to perform a shape difference at the end by predicting a set of primitives that undergo a fixed order of intersection, union, and difference operations to ob- tain the target shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, this sequence is not general and cannot support all shapes (see supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, CAPRI-Net first uses an encoder that is pre- trained on the entire dataset before it is optimized/overfit to each input shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, we avoid using an encoder in DualCSG as we realized that it does not help to converge more accurately or faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Since DualCSG is optimized for a single shape, it benefits from a smaller network with fewer parameters compared to CAPRI-Net (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' There are approaches such as CSG-Stump [7] utilizing a fixed CSG order of inverse, intersection, and union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This se- quence is theoretically able to reproduce any CSG sequence as the difference operation can be achieved by inverse and intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, this way, difference operations can be only applied on basic primitives at the early level of the CSG sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' To reproduce a shape such as S in a box illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3, CSG-Stump has to approximate it by assembling a union of several small pieces of primitives as (a) or by subtracting a union of basic primitives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' circles, boxes and triangles) as (b) instead of a natural difference of a box and S shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Therefore, CSG-Stump fails to produce compact results and causes irregular artifacts/bumps on the reconstructed shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAPRI-Net does apply the difference as the last operation but uses a union of convex shapes to approximate the S shape as (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Thus the solution of CAPRI- Net is not compact either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Only DualCSG can produce the compact solution as (d) by introducing two separated CSG branches, while the cover branch can produce the box and the residual branch can produce the S shape since it is capable of producing arbitrary and complex residual shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Formally, in DualCSG, we utilize two branches: cover branch that produces cover shape SC, which covers the tar- get shape S with a combination of convex shapes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2 Top) and residual branch (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2 Bottom) that produces the residual shape SR that is subtracted from SC to produce S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' To support generality, the set of primitives in the residual in- cludes both convex primitives and complimentary primitives (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This allows incorporating difference oper- ations at early stages of the residual branch using inverse and intersection operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Such alteration produces complex residual shapes that can be used to reconstruct general, high- genus, and complex concave shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' See the supplementary material for a formal proof of the generality of DualCSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' DualCSG SC, the output of the cover branch, is the union of convex shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' SR, the output of the residual branch, can be quite complex since it is the union of possibly concave and con- vex shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This property that was missing in CAPRI-Net significantly empowers the network as many complicated shapes can be constructed by finding the difference between a simple convex shape and an arbitrary concave or convex shape (see Figures 2 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In DualCSG, we only optimize a feature code and net- work weights to fit the target shape (similar to Auto-decoder in DeepSDF [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This is efficient since DualCSG is a light network that quickly converges to each shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Starting from a feature code, we pass it to the primitive prediction network and generate two matrices that hold the primitives’ param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Each matrix is used to determine the approximated signed distance (ASD) of a set of query points sampled in the space in which the shape is embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' These two sets of ASD values are separately passed to cover and residual branches, and each branch has an intersection and a union layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The cover branch (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2 Top) and the residual branch (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2 Bottom) generate point occupancy values indicating whether a point is inside or outside SC and SR respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The difference between SC and SR forms the final shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Primitive Representation We use a more general primitive form for the residual branch in comparison with the cover branch to generate complex residual shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This is one of the key differences between our method and CAPRI-Net [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The primitive prediction network (an MLP) receives a code of size 256 and outputs two matrices PC ∈ Rp×7 (fed to the cover branch) and PR ∈ Rp×7 (fed to the residual branch), each contains parameters of p primitives (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Primitives in PC are represented by a quadric equation same as CAPRI-Net [47]: |a|x2 + |b|y2 + |c|z2 + dx + ey + fz + g = 0, (1) where the first three coefficients are constrained to be positive to represent convex primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In DualCSG, while half of the primitives in PR are the same as Equation (1), we require the other half to be inverse convex primitives by constraining the first three coefficients to be negative: −|a|x2 − |b|y2 − |c|z2 + dx + ey + fz + g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' (2) We show two primitives produced by Equations (1) and (2) along with 2D visualization of cross-section near the surface of each primitive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Considering the universal 4 � � � � � � � � ��� ��� Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Visualizations of convex primitive shown in (a) and primitive inverse shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' n denotes the normal direction pointing outside the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' space as a cube, for convex primitives (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4 (a)), the ASDs of query points enclosed by the curved surface are nega- tive, meaning that these points are inside the primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For primitive inverse (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4 (b)), on the other hand, the ASDs of query points enclosed by the curved surface are positive, meaning that the complement space is inside the primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The negative constrains in Equation (2) possibly make the residual shape complex with detailed concavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For reconstruction, n points near the shape’s surface are sampled and their ASD to all primitives is calculated sim- ilar to CAPRI-Net [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For each point qj = (xj, yj, zj), its ASD is captured in matrix D ∈ Rn×p as: DC(j, :) = Q(j, :)PT C and DR(j, :) = Q(j, :)PT R, where Q(j, :) = (x2 j, y2 j , z2 j , xj, yj, zj, 1) is the jth row of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' DualCSG Branches We briefly discuss details about our DualCSG branches by following similar notations used in BSP-Net [4] and CAPRI- Net [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We input the ASD matrix DC into the cover branch and DR into the residual branch, and output vector aC and aR, indicating whether query points are inside/outside cover shape SC and residual shape SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Each branch contains an in- tersection layer and a union layer adopted from BSP-Net [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, to encourage difference operation between cover shape and residual shape to produce the final shape, we use the same difference loss from CAPRI-Net [47] to encourage SC from cover branch to cover the volume occupied by the ground truth shape and SR from residual branch to subtract a meaningful residual volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our DualCSG branches aim to produce two vectors in- dicating query points inside/outside cover shape SC and residual shape SR from the predicted primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The CSG operation order in each branch is the same as BSP-Net [4], which contains an intersection layer and a union layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Note that the weights of the two branches are not shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' During training, the inputs to intersection layers are two ASD matrices DR ∈ Rn×p and DC ∈ Rn×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Primitives involved in forming intersected shapes are selected by two learnable matrices TC ∈ Rp×c and TR ∈ Rp×c, where c is the number of intersected solid shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We can ob- tain Con ∈ Rn×c by the intersection layer and only when Con(j, i) = 0, query point qj is inside the intersected solid shape i (ConR is the same and only subscripts are R): ConC = relu(DC)TC � 0 in, > 0 out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' (3) Then all the shapes obtained by the intersection operation are combined by two union layers to find the cover shape SC as well as the residual shape SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The inside/outside indicators of the combined shape are stored in the vector aR ∈ Rn×1 and aC ∈ Rn×1 indicating whether a point is in/outside of the cover and residual shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Similar to [47], aC and aR are computed in a multi-stage fashion (a+ and a∗ for early and final stages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Specifically, a+ C is obtained by the following equation: a+ C(j) = C ( � 1≤i≤c WC(i)C (1 − ConC(j, i))) � 1 ≈ in, < 1 ≈ out, (4) where WC ∈ Rc is a learnable weighting vector and C is a clip operation to [0, 1], and a+ R is defined similarly with WR and ConR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In later stages, a∗ C and a∗ R are obtained by finding min of each row of ConC and ConR: a∗ C(j) = min 1≤i≤c(ConC(j, i)) � 0 in, > 0 out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Loss Functions and Training Strategy We utilize the same reconstruction loss and multi-stage training as CAPRI-Net’s to facilitate differentiable opera- tions in the early stage and gradually achieve good results in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The following loss function is used in our method: L = Lrec + LT + LW, (6) where Lrec is the reconstruction loss applied to aC and aR, it would force the subtracted result between cover shape and residual shape to be close to the input shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' LT + LW are the losses applied to the intersection and union layer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Note that differently from CAPRI-Net [47], we have net- work weights from DualCSG branches: T = [TC, TR] and W = [WC, WR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' However, LT and LW and the difference loss are the same as CAPRI-Net [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Please refer to more details on the losses in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Results In our experiments, we use two public datasets ABC [18] and ShapeNet [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' On each dataset, we test the quality of producing CSG operations from two inputs: mesh and point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We present qualitative and quantitative results of our experiments to demonstrate the effectiveness of DualCSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing results for 3D meshes in ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Training Details Since the methods we compare with require an additional time-consuming optimization at test time to achieve satis- factory results, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', 30 min per shape for CSG-Stump), we have randomly selected a moderately sized subset of shapes as test set for evaluation: 500 shapes from ABC, and 50 from each of the 13 categories of ShapeNet (650 shapes in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, we ensured that 80% of the selected shapes from ABC have genus larger than two with more than 10K vertices to include complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Experiments were performed using an Nvidia GeForce RTX 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In our experiments, we set the number of maximum prim- itives as p = 512 and the number of maximum intersections as c = 32 for each branch to support complex shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Since both these numbers are half of the setting in CAPRI-Net, the size of our CSG layers is half of the CAPRI-Net’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The size of our latent code for all input types is 256 and a two-layer MLPs is used to predict the parameters of the primitives from the input feature code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We train DualCSG per shape by optimizing the latent code, primitive prediction network, intersection layer, and union layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Mesh to CSG Representation Given a 3D mesh, the task is to learn an accurate and compact CSG representation for this shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' To do so, we first sample 24,576 points around the shape’s surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' with a distance up to 1/64) and 4,096 random points in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' All 28,672 points are then scaled into the range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='5], these points along with their occupancy values are used to optimize the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We compare DualCSG with BSP-Net [4], CSG-Stump [7] and CAPRI-Net [47], which output structured parametric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For a fair comparison, we optimize all of these networks with the same number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' BSP-Net, CSG-Stump, and CAPRI-Net are pre-trained on the training set provided in CAPRI-Net before optimization to achieve better initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Note that CSG-Stump uses different net- Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing results for 3D meshes in ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' work settings for shapes from ABC (with shape differences) and ShapeNet (without shape difference);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' we therefore fol- low the same settings in our comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For each shape, BSP-Net takes about 15 min, CSG-Stump about 30 min and CAPRI-Net about 3 min to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The training process in DualCSG will run 12,000 iterations for each stage, taking about 5 minutes per shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Quantitative metrics for shape recon- struction are symmetric Chamfer Distance (CD), Normal Consistency (NC), Edge Chamfer Distance [4] (ECD), and Light Field Distance [3] (LFD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For ECD, we set the thresh- old for normal cross products to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='1 for extracting points close to edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CD and ECD are computed on 8K sample points on the surface and multiplied by 1,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For LFD, we render each shape at ten different views and measure the Light Field Distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, we compare the number of primitives #P to evaluate the compactness of shapes since all CSG-based modeling methods predict some primitives that are combined with intersection operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Evaluation and Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We provide visual compar- isons on representative examples from the ABC dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 5 and the ShapeNet dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' more results can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our method consistently reconstructs more accurate shapes with geometric details and concavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' BSP-Net simply assembles convex shapes to fit the target shape and obtain less compact results without a difference operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CSG-Stump tends to use considerably more difference op- erations to reconstruct shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This also causes the shapes’ surface to be carved by many redundant primitives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', lack of compactness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addition, since CSG-Stump does not support difference operations between complex shapes, it fails to reconstruct small holes or intricate concavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' As the intermediate shapes subtracted in CAPRI-Net’s fixed 6 BSP STUMP CAPRI DualCSG GTBSP STUMP CAPRI DualCSG GTTable 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing CSG rep learning from 3D meshes in ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Methods BSP STUMP CAPRI Ours CD ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='071 NC ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='928 ECD ↓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='047 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='881 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='990 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='467 LFD ↓ 774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='57 2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='43 665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='20 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='48 #P ↓ 359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='38 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='42 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='26 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='81 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing CSG rep learning from meshes in ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Methods BSP STUMP CAPRI Ours CD ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='164 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='119 NC ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='887 ECD ↓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='899 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='035 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='722 LFD ↓ 2135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='68 4267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='66 1558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='08 1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='13 #P ↓ 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='21 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='58 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='94 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='78 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing methods under the same auto-decoder setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' ABC ShapeNet Methods CAPRI Ours CAPRI Ours CD ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='119 NC ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='887 ECD ↓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='918 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='467 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='359 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='722 LFD ↓ 635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='01 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='48 1685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='38 1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='13 #P ↓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='15 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='81 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='13 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='78 CSG sequence are only unions of convex shapes, it fails to reproduce target shapes with complex concavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' As shown in Table 1, DualCSG achieves the best recon- struction quality and compactness in all metrics on ABC dataset compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' As shown in Table 2, the reconstruction accuracy of DualCSG on ShapeNet is mostly better than CAPRI-Net (except for NC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' DualCSG may use slightly more but comparable primitives than CAPRI-Net on shapes with complex cavities to produce better details and concavities, such as the chair back in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' DualCSG uses an auto-decoder network architecture [30], requiring no pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In contrast, CAPRI-Net adopts the traditional auto-encoder network with pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' To remove potential impacts of pre-training from the evaluation, we compare DualCSG to CAPRI-Net under the same auto-decoder setting, where the encoder of CAPRI-Net is removed along with the pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' The re- sults are shown in Table 3, where we can see that CAPRI-Net still under-performs compared to DualCSG, as well as to its version with pre-training as provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We believe that the reason is that the limitation of only using convex shapes by CAPRI-Net requires a better network parameter initialization from the pre-training step, while DualCSG can quickly overfit to the test shape from scratch owing to its Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing CSG rep learning from 3D point cloud in ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Methods BSP STUMP CAPRI Ours CD ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='085 NC ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='924 ECD ↓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='899 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='303 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='308 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='029 LFD ↓ 773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='29 3149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='82 799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='44 558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='08 #P ↓ 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='87 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='436 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='57 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='28 more general representation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CSG Trees Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our network can learn to recon- struct a 3D shape by producing a plausible CSG tree for a given shape without any direct CSG supervision as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We simplify CSG trees from BSP-Net, CAPRI-Net, and DualCSG by hiding their primitives to better illustrate their learned CSG modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 7 clearly reveals the limitations of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' BSP-Net combines many convex shapes (20 in this case) to construct the final shape, which uses many primitives and loses the compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CSG-Stump uses pre-defined limited shapes as primitives, such as boxes and spheres, limiting the reconstruction accuracy and requiring a large number of primitives (53 for this shape, we only show a subset here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Besides, CSG-Stump uses a primitive complement layer (il- lustrated as operation C) to form the subtraction operation for ABC shapes, which can result in many redundant differ- ence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' CAPRI-Net can only subtract the union of convex shapes to approximate intricate concavities, which limits its reconstruction accuracy and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In contrast, shapes produced by our DualCSG are com- posed of fewer intermediate shapes in comparison with other methods, which makes our CSG tree more compact while achieving better reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Note that the desired task is not just to obtain a fewer number of parts with any complexity, but rather to obtain fewer simple primitives or convex shapes that are constructed via CSG operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' If the former objective is favorable, other methods such as Neural Parts [31] might be able to produce fewer parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Point Clouds to CSG In our last experiment, we reconstruct CAD shapes from point clouds, each containing 8,192 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' For each input point, we sample 8 points along its normal with perturbations sampled from Gaussian distribution (µ = 0, σ = 1/64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' If this point is at the opposite direction of normal vectors, the occupancy value is 1, otherwise it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' This way, we sample 65, 536 points to fit the network to each shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Similar to mesh-to-CSG experiment, we only optimize the latent code, primitive prediction network, and the selection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Quantitative comparisons in Table 4 and visual compar- isons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 8 show that our network outperforms BSP-Net, CSG-Stump, and CAPRI-Net in different reconstruction sim- 7 BSP STUMP CAPRI DualCSG GT Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing learned CSG Trees of 3D examples from ABC shown in the first row and ShapeNet in the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' ����� ��� ����� ����� ������� �� ����� ��� ����� ����� ������� �� Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Comparing CSG representation learning from 3D point cloud in ABC shown in column 1-6 and ShapeNet in column 7-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' ilarity and compactness metrics on ABC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Additional ShapeNet results can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Discussion, limitation, and future work We present DualCSG, a simple yet effective idea, for un- supervised learning of general and compact CSG tree repre- sentations of 3D CAD objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Extensive experiments on the ABC and ShapeNet datasets demonstrate that our network outperforms state-of-the-art methods both in reconstruction quality and compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' We also have ample visual evi- dence that the CSG trees obtained by our method tend to be more natural than those produced by prior approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Our network does not generalize over a shape collection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' it “overfits” to a single input shape and is in essence an opti- mization to find a CSG assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' While arguably limiting, this is not entirely unjustified since the CAD shapes we seek to handle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=', those from ABC, do not appear to possess sufficient generalizability in their primitive assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' An- other limitation is the lack of an explicit compactness loss to enforce minimal CSG trees, like CAPRI-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' In addi- tion, incorporating interpretable CSG operations into the network tends to cause gradient back-propagation issues and limits the reconstruction accuracy of small details such as decorative curves on chair legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Besides addressing the above limitations, we would like to extend our method to structured CAD shape reconstruc- tion from images and free-form sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Another interesting direction for future work is to scale the primitive assembly optimization from CAD parts to indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' 8 UUO容容References [1] Dan Cascaval, Mira Shalah, Phillip Quinn, Rastislav Bodik, Maneesh Agrawala, and Adriana Schulz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Differentiable 3d cad programs for bidirectional editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQfRyyj/content/2301.11497v1.pdf'} +page_content=' Computer Graphics Forum, 41(2):309–323, 2022.' 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Developing efficient and +cost-effective photocatalysts for water splitting is a growing need. For this purpose, semicon- +ductor photocatalysts have attracted much more attention due to their stability and low man- +ufacturing cost. +Here, we have systematically applied several state-of-the-art advanced first- +principles-based methodologies, viz., hybrid density functional theory, many-body perturbation +theory (G0W0) and density functional perturbation theory (DFPT), to understand the electronic +structure properties of ABX2Y perovskites. We have chosen the vast composition space of ABX2Y +type perovskites where A and B are cations and X and Y can be nitrogen, oxygen, or fluorine. +These perovskites exhibit direct band gaps ranging from 1.6 to 3.3 eV. Further, to evaluate the +feasibility of the visible light catalytic performance, we calculate the structural, electronic, and op- +tical properties of ABX2Y perovskites. In addition, from hydrogen evolution reaction (HER) and +oxygen evolution reaction (OER) mechanism, BaInO2F, InSnO2N, CsPbO2F and LaNbN2O are +found as probable photocatalysts. +1 +Introduction +The photocatalytic water splitting has become an intense research +area and an excellent way to capture and store energy from the +sun1–4. Over recent decades, the number of applications based +on photocatalysis increased acutely5–7. Although a wide range +of materials has been developed for photocatalytic performance +under visible light, most can only absorb light at a wavelength +of less than 500 nm, so only a small portion of the spectrum +can be utilized8–11. +Therefore, the design of a semiconductor +for the efficient generation of solar fuel requires a suitable band +gap (1.5 ≤ Eg ≤ 2.8 eV) to efficiently absorb visible light, high di- +electric constant, high charge carrier mobility and suitable band +positions in order to perform the hydrogen and oxygen evolution +half-reactions (HER/OER)12,13. +So far, most of the existing photocatalysts are oxides14. How- +ever, they have too large band gap to absorb visible light14,15. +This is mainly due to a very low valence band (VB) energy which +comes from the 2p orbitals of the oxygen atoms15. A require- +ment for visible light induced photocatalysts is that the optimum +band gap energy should be less than 3 eV16. In order to solve +aDepartment of Physics, Indian Institute of Technology Delhi, New Delhi, India. Fax: +91 11 2658 2037; Tel: 91 11 2659 1359; E-mail: saswata@physics.iitd.ac.in +this problem, non-oxides such as nitrides and sulfides have been +proposed as their VB position is usually higher in energy17,18. +Nitrogen is less electronegative than oxygen that leads to reduc- +tion in the band gap. Hence, the optical gap overlaps with the +solar spectrum which makes this class of materials interesting +for application as solar absorbers and visible light-driven photo- +catalysts19. Many useful oxynitrides have been reported, such +as: (i) CaTaO2N and LaTaON2 are non toxic solid solutions, (ii) +BaTaO2N has a high dielectric constant and acts as a photocata- +lyst in water decomposition, and (iii) EuNbO2N is ferromagnetic +and show colossal magneto resistance20–22. Also, it has been re- +ported that LaTaON2, LaTiO2N, SrTaO2N and BaTaO2N drive the +HER and OER half-reactions and can be used in Z-scheme con- +figurations capturing photons in the 600 to 750 nm range23–27. +Recently, Zhang et al. +reported that CoOx modified LaTiO2N +has a high quantum efficiency of 27% at 440 nm towards wa- +ter oxidation28. LaMgxTa1−xO1+3xN2−3x is found to be the first +oxynitride to utilize 600 nm photons in steady-state overall wa- +ter splitting25. Moreover, various oxyfluoride perovskites such as +BaFeO2F, SrFeO2F and PbFeO2F have been discovered. These per- +ovskites exhibit magnetic ordering until a temperature of around +645 K, 685 K, and >500 K, respectively29–31. Multiferroic behav- +ior was also shown by iron-based oxyfluoride perovskites32. Also, +Journal Name, [year], [vol.], +1–8 | 1 +arXiv:2301.04335v1 [cond-mat.mtrl-sci] 11 Jan 2023 + +ARTICLETYPEReceivedDate +AcceptedDate +D01:00.0000/xxxxxxxxxxSamir et al. proposed the potential existence of a nitrofluoride +(LaZrN2F) from the first-principles based DFT approach. They +showed that LaZrN2F composition exhibits semiconducting prop- +erties with iono-covalent behavior33. However, despite several +research endeavors in the field of oxynitride, oxyfluoride and ni- +trofluoride perovskites, no such in-depth theoretical work is avail- +able to address their electronic, optical, dielectric properties and +their application in photocatalytic water splitting. +In this work, we have used various advanced state-of-the-art +first-principles based methodologies under the framework of +density functional theory (DFT)34,35, many body perturbation +theory (MBPT)36,37 and density functional perturbation theory +(DFPT)38,39 to provide a comprehensive computational study of +oxynitride, oxyfluoride, and nitrofluoride perovskites. +Our ob- +jectives are: (i) to provide a list of compositions that could be +likely experimentally synthesized in the perovskite phase, (ii) to +study their electronic structure, (iii) to study their optical and ex- +citonic properties, and (iv) to find their application in photocat- +alytic water splitting. To achieve this goal, we report calculations +for the theoretical electrocatalytic HER and OER overpotentials +using DFT. +2 +Methodology +We have performed a systematic study to explore the struc- +tural, electronic and optical properties using DFT and beyond +approaches under the framework of MBPT. All calculations +are performed with Projector Augmented Wave (PAW) poten- +tials as implemented in Vienna ab initio simulation package +(VASP)40,41. +All the structures are optimized using general- +ized gradient approximation (GGA) as implemented in PBE42 +exchange-correlation (εxc) functional until the forces are smaller +than 0.001 eV/Å. The Γ-centered 2×2×2 k-mesh sampling is em- +ployed for optimization calculations (optimized structures are +shown in Fig. 1). +The electronic self-consistency loop conver- +gence is set to 0.01 meV, and the kinetic energy cutoff is set to +600 eV for plane wave basis set expansion. To explore the elec- +tronic properties, hybrid εxc functional (HSE06)43,44 is used. For +all the energy calculations 4×4×4 k-mesh has been used. The +phonon calculations are performed with 2×2×2 supercells using +the PHONOPY package45,46. Note that the surfaces (100 plane) +were constructed by cleaving the fully optimized bulk structure +with the lowest energy. Further, surface geometries are optimized +with a force convergence threshold of 0.001 eV/Å. Reciprocal +space is sampled by 4×4×1 k-mesh for the (100) surface. A 20 +Å vacuum is added to avoid artificial interaction between peri- +odic images, and a dipole correction is included. The two-body +Tkatchenko-Scheffler vdW scheme has been used to account for +van der Waals interactions47,48. Note that the spin-orbit coupling +(SOC) is not taken into consideration because it has negligible af- +fect on the electronic structure of ABX2Y perovskites (see section +I of supplemental information (SI)). +3 +Results and Discussion +3.1 +Crystal Structure +Here, we focus on the oxynitride, oxyfluoride, and nitrofluoride +perovskites i.e., ABX2Y (A = Ba, Ca, La, Sr, Ag, K, Cs, Pr, In ; B = +Nb, Ta, Zr, Ti, In, Pb, Mg, Sn ; and X, Y = O, N, F) perovskites. We +have considered in total 18 perovskite structures, out of which 9 +are ABO2N, 3 are ABN2O, 4 are ABO2F, 1 is ABF2O and 1 is ABF2N +type of perovskite. The oxynitride perovskite of type ABO2N e.g., +InSnO2N, crystallizes in a hexagonal cell and belongs to the polar +space group P63cm, as shown in Fig. 1(a). The optimized lat- +tice constants are a = b = 6.16 Å and c = 12.26 Å, which are +well in agreement with the previous findings18. The perovskite +type ABN2O e.g., LaNbN2O, crystallizes in an orthorhombic cell +with lattice constants a = 5.78 Å, b = 5.76 Å and c = 10.02 Å +(see Fig. 1(b)). AgTiO2F oxyfluoride perovskite has tetragonal +cell edges, a = b = 5.48 Å and c = 7.45 Å and belongs to the +space group I4/mcm as shown in Fig. 1(c). The system CsPbF2O +is an interesting perovskite because of the fact that the perovskite +CsPbI3 is the parent inorganic compound with an orthorhombic +crystal structure. This perovskite has lattice constants a = 13.03 +Å, b = 4.79 Å and c= 6.68 Å. Finally, LaMgF2N is the only ni- +trofluoride system we predict to have possibilities to be realized +experimentally with space group P121/m1. The lattice constants +of this perovskite structure are a = 6.33 Å, b = 3.69 Å and c = +8.25 Å. Similarly, the lattice parameters of other perovskite struc- +tures are mentioned in SI (see section II of SI). +Fig. +1 Optimized crystal structure of (a) InSnO2N, (b) LaNbN2O, +(c) AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N. Images produced with +VESTA 49. +3.2 +Structural Stability +The stability of perovskites is often discussed based on the Gold- +schmidt tolerance factor50 +t = +rA +ranion +√ +2(rB +ranion) +(1) +where rA, rB and ranion are the ionic radii of the A and B cations, +and of the anion, respectively. +A more recent study based on +2 | +1–8 +Journal Name, [year], [vol.], + +(a) InSnO2N +(b) LaNbN2O +(c) AgTiO2F +(d) CsPbF20 +(e) LaMgF2Nthe novel analytics approach has led to the proposition of a new +factor51 +τ = ranion +rB +nA +� +nA − +rA/rB +log(rA/rB) +� +(2) +where nA is the oxidation state of A cation. It has been reported +that for a large experimental dataset of perovskites, 0.825 < t < +1.08 gives a classification accuracy of 74%, while τ < 4.18 has an +accuracy of 92%. Now, to use these formulae for mixed anions, +we need to decide the value of ranion to be used. In line with +the suggestions provided, we decided to use the arithmetic aver- +age of the radii of two mixed anions, i.e., ranion = (2rX+rY)/3. +However, it has been noted that using ranion, the Goldschmidt tol- +erance factor fails to capture the stability trend of mixed anion +perovskites52,53. For this purpose, the geometric mean has also +been used to approximate the radius of the mixed anions54. Also, +more complicated factors like the octahedral factor and atomic +packing fraction have been proposed to understand the stability +of these perovskites54,55. But in our case, for simplicity, we de- +cided to consider traditional t and τ. +Table 1 Stability parameters of ABX2Y perovskites. +ABX2Y +t +τ +InSnO2N +0.84 +3.32 +LaNbN2O +0.84 +3.74 +LaTaN2O +0.84 +3.74 +PrTaN2O +0.87 +2.56 +AgTiO2F +0.90 +3.81 +BaInO2F +0.89 +4.20 +CsPbO2F +1.07 +3.49 +KTiO2F +0.98 +3.52 +LaMgF2N +0.81 +5.64 +From Table 1, we find that the value of t for all the selected per- +ovskites lies between 1.07 and 0.81, which is perfectly consistent +with the usual range reported for perovskites51. Now concerning +τ, we find that all selected perovskites have value τ < 4.18 except +LaMgF2N. For other ABO2N perovskites also, we have calculated +the structural stability parameters mentioned in section III of SI. +In addition, to further analyze the dynamic stability, we have +plotted the phonon band structures for the optimized perovskite +structures. The phonon band structures calculated along the high- +symmetry points of the Brillouin zone for InSnO2N, LaNbN2O, +AgTiO2F, CsPbF2O, and LaMgF2N are shown in Fig. 2. Notably, +the absence of negative frequencies confirms the dynamic stabil- +ity of these perovskites. Similarly, we have analyzed the phonon +band structures of other ABX2Y perovskites in section IV of SI. +Note that we have also checked the structural stability of different +ABX2Y perovskites at a higher temperature using ab initio molec- +ular dynamics (AIMD). We have obtained the radial distribution +function g(r) at T = 0 K and T = 300 K by a 6 ps long MD sim- +ulation run with NVT ensemble (Nose-Hoover thermostat)56. We +observe that the nature of the radial distribution function for the +nearest neighbors remains the same at room temperature (see +section V of SI). This confirms the stability of these perovskites at +Fig. +2 Phonon band structures of (a) InSnO2N, (b) LaNbN2O, (c) +AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N. +operational temperature of 300 K. +3.3 +Electronic Structure +We have plotted the electronic band structure and partial den- +sity of states (pDOS) for ABX2Y type of perovskites to better un- +derstand the role of A/B cations and X/Y anions near the va- +lence band maximum (VBM) and the conduction band minimum +(CBm). +Fig. 3(a) shows the band structure of InSnO2N with +a direct band gap of 1.60 eV at Γ using HSE06 εxc functional. +From Fig. 4(a), we can clearly observe that the valence states are +mainly composed of N and O with a small contribution coming +from Sn. While the conduction states have mostly N contribution. +Finally, we see very few states associated with In in the [-6, 6] eV +energy window, which is compatible with the interpretation that +the A atom is fully ionized in the perovskite structure. +The band structure of LaNbN2O is rather different from that of +Fig. +3 Electronic band structures of (a) InSnO2N, (b) LaNbN2O, (c) +AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N using HSE06 εxc functional. +InSnO2N. It has an indirect band gap of 2.04 eV with the bottom +of the CB at Γ and the top of the VB at M (see Fig. 3(b)). The VB is +mainly contributed by N and O states, while the CB comprises of +Nb states. Also, the La states are found only in the VB, indicating +that this atom is ionized in this structure (see Fig. 4(b)). For the +Journal Name, [year], [vol.], +1–8 | 3 + +6 +8 +11 +6 +(eV) +(eV) +10 +1 +2 +Energy ( +111 +1 +9 +0 +8 +2 +1 +-2 +7 +0 +r +ＩA +r +K +H +M +L +H +A +L +M +K +H +A +A +L +M +A +(a) InSnO2N +(b) LaNbN2O +(c) AgTiO2F +6 +8 +4 +6 +4 +- +1 +- +1 +0 +- +2 +-2 +- +1 +0 +- +-- +- +-4. +ZT +TY +Z +T +Y +S +RU +X +C +A +D +EOAO +(d) CsPbF2O +(e) LaMgF2N20 +20b +20 +15 +15 +10 +10 +5 +5 +0 +M K +A +H +A +r +A +K +H +A +TY +r +X +L +L +R +U +(a) InSnO2N +(b) LaNbN2O +(c) AgTiO2F +20 +20 +5 +10 +LO +5 +0 +Y +S +R +┌Y +r +Z +C +Y +r +B +EA +(d) CsPbF2O +(e) LaMgF2Nother cases of oxynitride perovskites, all the band structures and +PDOS plots are provided in section VI of SI. +From the AgTiO2F band structure (see Fig. 3(c)), we see that it +Fig. 4 Partial density of states (pDOS) of (a) InSnO2N, (b) LaNbN2O, (c) +AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N using HSE06 εxc functional. +has a direct band gap of 2.32 eV. Also, from Fig. 4(c), we can +clearly observe that the VB states are mostly composed of Ag and +O states, while Ti states mostly contribute to the CB. The HSE06 +band structure of CsPbF2O is depicted in Fig. 3(d). The band gap +is 1.87 eV which is indirect, with the top of the valence and bot- +tom of the conduction band (at Γ) composed of hybridized F, O, +and Pb states. In this case, the separation of the VB into two man- +ifolds is incomplete, leading to some overlap between the two sets +of bands. The bottom of the CB, on the other hand, is separated +by more than 1 eV from the rest of the CB (see Fig. 4(d)). Sim- +ilarly, for the other oxyfluoride perovskites, we have plotted the +band structure and PDOS (see section VI of SI). +In case of nitrofluoride perovskite LaMgF2N, we have obtained a +large band gap of 3.31 eV. The highly dispersive bottom of the +CB is mainly constructed from La states, while top of the VB has +mostly N states with smaller La contribution (see Fig. 3(e) and +4(e)). In LaMgF2N, we also see a clear splitting of the VB. How- +ever, due to a strong electronegativity difference of N and F, the +gap between the two manifolds is considerably larger than for the +oxyfluoride systems. +3.4 +Optoelectronics and Dielectrics +In order to get an in-depth insight into the suitability of a par- +ticular material in optoelectronic applications, a detailed study +of its optical properties like dielectric function, refractive index, +extinction coefficient and absorption coefficient is indispensable. +The absorption coefficient (α) of a material is computed from the +frequency dependent dielectric constant using the following for- +mulae57, where ε1 and ε2 are the real and imaginary terms of the +dielectric constant, respectively: +α = 4πκ(ω) +λ +(3) +where κ(ω) is the extinction coefficient, which is given by: +κ(ω) = +� +� +� +� +� +ε2 +2 +ε2 +1 −ε1 +2 +(4) +The absorption coefficient is one of the most essential properties +of a material in terms of its photovoltaic application as it depicts +key information regarding optimal solar energy conversion +efficiency. The typical α for direct semiconductors is of the order +105 58,59. +The theoretical results show that all the perovskites +exhibit high absorption coefficients (see Fig. 5 ). +Further, photovoltaic applications require high charge carrier +Fig. 5 Absorption coefficient of ABX2Y perovskites. +mobility to reduce nonradiative recombination losses and effi- +ciently transport the photogenerated carriers to the electrodes. +In order to estimate the carrier mobility, we have calculated the +effective mass of electrons and holes in ABX2Y perovskites from +their band structure obtained by using HSE06 εxc functional +(see Fig. 3 and section VI of SI). All the perovskites have light +electrons, with InSnO2N having me= 0.15m0. However, the holes +are heavier, as is typical for most metal oxides, with InSnO2N +having a hole effective mass of mh= 1.75m0, which is in well +agreement with the previous findings60. +Similarly, BaInO2F +has me= 0.41m0 and mh= 0.97m0. In this case also, the hole +effective mass is higher than the electron effective mass. Here, +the low effective mass of the electrons in ABX2Y perovskites +can likely make them promising semiconductors. Also, we have +calculated the electronic and ionic dielectric constant of these +perovskites and found that they have a large dielectric constant +value (see Table 2), which are in well agreement with the +previous findings7,60. To this point, the relevant properties for +solar energy conversion for some of the ABX2Y perovskites are +compiled and summarized in Table 2 and Table S3 of SI. +4 | +1–8 +Journal Name, [year], [vol.], + +15 +15 +15 +Total +Total +Total +La +Ag +10 +In +10 +10 +Nb +Sn +Ti + states/ev +0 + states/ev +0 +0 +e +N +N +5 +5 +F + states/ +0 +0 +No. +No. +-5 +.5 +-10F +-10 +-10 +-15 +-15 +-15 +4 +0 +4 +-4 +-2 +0 +-4 +-2 +0 +2 +-4 +-2 +2 +-6 +2 +4 +.6 +6 +6 +Energy (eV) +Energy (eV) +Energy (eV) +(a) InSnO2N +(b) LaNbN2O +c)A +TiO2F +15 +15 +Total +Total +La +10 +Cs +10 +Pb +Mg +No. of states/eV +No. of states/eV +0 +F +5 +F +5 +N +0 +-5 +.5 +-10 +-10 +-15 +6 +-4 +0 +2 +4 +6 +-4 +-2 +0 +2 +4 +6 +> +Energy (eV) +Energy (eV) +(d) CsPbF20 +(e) LaMgF2NAgTiO.F +BaInO.F +CsPbO.F +KTiO E +2 +CsPbF.O +3 +cm +LaMgF,N +LaNbN.O +LaTaN.O +2 +PrTaN.O +InSnO.N +0 +2 +3 +4 +Energy (eV)Table 2 DFT calculated properties of ABX2Y perovskites. Band gap (Eg), +electronic dielectric constant (ε∞), ionic dielectric constant (εr), effective +mass of electron (m∗ +e) and hole (m∗ +h) in terms of the rest mass of electron +(m0). +ABX2Y +Eg (eV) +ε∞ +εr +m∗e +m∗ +h +InSnO2N +1.60 +4.45 +10.12 +0.15 +1.75 +LaNbN2O +2.04 +7.27 +61.06 +0.33 +0.41 +LaTaN2O +2.11 +9.28 +70.58 +0.36 +0.51 +AgTiO2F +2.32 +2.74 +39.95 +0.82 +1.68 +BaInO2F +2.01 +4.14 +15.05 +0.41 +0.97 +CsPbF2O +1.87 +4.66 +14.97 +0.15 +0.17 +3.5 +Photocatalytic Water Splitting +In a photocatalytic process, the perovskite material absorbs light +as a result an electron at the VB is transferred to the CB, and a +photogenerated e-h pair is produced. Subsequently, for the ox- +idation or reduction reaction, the photogenerated electrons and +holes transfer to the corresponding reaction sites. The photocat- +alytic reaction can be described by two half-reactions for the hy- +drogen evolution and water oxidation which are summarized as +follows: +2H+ +2e− → H2 +(5) +2H2O+4h+ → O2 +4H+ +(6) +For the water splitting reaction, the CBm should be higher than +the reduction potential of H+/H2 (EH+/H2 = –4.44 eV at pH = +0.0). On the other hand, the VBM should be lower than the oxida- +tion potential of O2/H2O (EO2/H2O = –5.67 eV at pH = 0.0). From +the above discussion, we conclude that in order to evaluate the +feasibility of photocatalytic water splitting, it is crucial to calcu- +late precisely the edge positions of the materials. The calculation +of the band edges using DFT is quite complicated and compu- +tationally expensive because the surface chemistry and interfacial +effects are considered in the process. Galli et al. have summarized +recent progress and open theoretical challenges present in simu- +lations61. They use an appropriate method to treat the effect of +interface on the band edge position of the surface structure. How- +ever, it is still complex to predict the absolute energy positions of +CBm and VBM for the bulk structure. +Here, we have used a reliable empirical formula proposed by Xu +et al.62 to calculate the absolute band edges of ABX2Y bulk per- +ovskites using the following equation: +EVBM = χ − 1 +2Eg +(7) +ECBm = EVBM +Eg +(8) +where EVBM and ECBm represents the absolute potentials of VBM +and CBm. χ is the electronegativity of the perovskite which can be +determined by the absolute electronegativities of the constituent +atoms as: +χ(compound) = χa +1 χb +2 χc +3....χm +n +(9) +where χ1, χ2, χ3 and χn represent the electronegativities of the +constituent atoms; a, b, c and m are the molar fractions of the +atoms. The electronegativity of the constituent atoms is calcu- +lated using the Mulliken electronegativity63: +χ1 = (I +A)/2 +(10) +where I and A is the ionization energy and the electron affinity of +the atom, respectively. +Fig. 6 visually expresses the absolute energy positions of band +Fig. +6 Band edge alignment of ABX2Y perovskites w.r.t. water redox +potential levels (H+/H2, O2/H2O) +edges and the redox potentials of the water splitting reaction. +In case of AgTiO2F, the CBm is shifted downward by a large +amount, and hence, their reduction power is very low and could +not be utilized for hydrogen generation from water. For KTiO2F +and LaMgF2N cases, the band gap is very large thus not inducing +the visible-light response. In view of this, from the applicability +in photocatalytic water splitting, BaInO2F, CsPbO2F, LaNbN2O, +LaTaN2O and InSnO2N are the most desirable ones. For other +oxynitride perovskites also, we have plotted the band edge align- +ment (see section VIII of SI). Now to better classify these per- +ovskites for different applications, we have discussed two appli- +cations in detail. +3.5.1 +One-photon water splitting +The one-photon or overall water splitting process is schematized +in Fig. 7(a). In this process, a single photon creates an e-h pair. +Further, the electron and the hole reach two different regions of +the surface to avoid recombination, and evolve hydrogen and ox- +idize water, respectively. The criteria for a material to be used for +solar light capture that we have considered here are: (i) structural +stability, (ii) band gap in the visible light range, i.e., 1.5 ≤ Eg ≤ +3 eV, and (iii) band edges straddling with the oxygen and hydro- +gen evolution potentials i.e., VB > 1.23 eV and CB < 0 eV. From +Fig. 6, we have found various ABX2Y perovskites (e.g., BaInO2F, +CsPbO2F, LaNbN2O, InSnO2N, etc.) which are suitable for col- +lecting visible light and for the evolution of both hydrogen and +oxygen. +3.5.2 +Transparent shield +Photocorrosion is one of the major problems related to the use of +materials for oxygen production. In order to overcome this issue, +it is required to develop a highly stable and transparent thin film +to cover the oxygen evolution photocatalyst as a protective shield +Journal Name, [year], [vol.], +1–8 | 5 + +Vacuum E +ENHE +- Conduction band edge +Valence band edge +-2.5F +-2 +KTiO,F +-3.5 +-1 +BalnO.F +LaMgF,N +LaTaN,O +InSnO,N +-4.5 +0 +2.32 eV +2.44 eV +1.87 eV +2.04 eV +2.00eV +-5.5 +1 +3.21 eV +2.01 eV +1.60 eV +3.31 eV +2.11 eV +0/H.0 +LaNbN.O +CsPbF,O +CsPbO,F +PrTaN,O +-6.5 +2 +AgTiO,F +-7.5 +3Fig. 7 (a) Overall water splitting scheme, (b) Scheme of a tandem cell +with a transparent protecting shield on the oxygen evolution photocata- +lyst. +(see Fig. 7(b)). For this purpose, the material should have a large +band gap of, say, more than 3 eV. Fig. 7(b) illustrates a device +as a photoanode protecting transparent shield for a tandem cell. +Here, the electron with higher energy evolves hydrogen, while +the hole at lower energy moves through the transparent film and +evolves oxygen. The criteria for such material are: (i) a direct +band gap in the UV range in order to avoid absorbing a part of +the visible light spectrum, (ii) VB needs to be placed between the +edges of the photocatalyst and the potential of oxygen evolution +so that the hole reaches the reaction region with a small energy +loss i.e., 1.7 ≤ VBedge ≤ 2.8 eV. The same idea can be used to +develop a photocathode protecting shield64. The only change is +the position of the CB i.e., -0.7 ≤ CBedge ≤ 0 eV. From Fig. 6, +we find that KTiO2F and LaMgF2N can be used as a protective +transparent shield. +3.6 +HER and OER mechanism +The HER reaction mechanism is schematized in Fig. 8(a). In HER +mechanism, the reaction steps follow: (i) adsorption of H, (ii) +hydrogen reduction and (iii) desorption as H2 65. The Gibbs free +energy of hydrogen reaction (∆GH) is a reasonable descriptor for +HER reaction. In order to calculate the Gibbs free energy changes +(∆G) of the intermediates in HER, the following expression is +used: +∆GH = ∆EH +∆EZPE −T∆SH +(11) +where, +∆EH = Etotal −Esurface −1/2EH2 +(12) +In the above mentioned equations, ∆EH is the hydrogen binding +energy on the perovskite surface, Etotal is the total energy for the +adsorption state, Esurface is the energy of the pure surface, EH2 is +the energy of hydrogen molecule, ∆EZPE is the change in zero- +point energy and ∆SH is the difference in entropy. +At 298 K, +∆EZPE − T∆SH = 0.25 eV is well established in literature66. The +HER diagram along the reaction pathway H+ +e− → H∗ → 1/2H2 +is shown in Fig. 8(c). Under the conditions of pH = 0 and a stan- +dard hydrogen electrode (SHE) potential of 0 V, H+ +e− is equiv- +alent to 1/2H2. From the above discussion, we found BaInO2F, +InSnO2N, CsPbO2F and LaNbN2O as probable photocatalysts. +Similarly, the OER mechanism involves four proton-coupled elec- +tron transfer steps on metal ion centers with OH, O, and OOH re- +action intermediates, where ∗ indicates the active metal site (see +Fig. +8 (a) HER steps, (b) OER steps, (c) HER free energy diagram, +and (d) OER free energy diagram for ABX2Y perovskites at an electrode +potential ESHE=0 V and pH=0. +Fig. 8(b)): +∗ +H2O → ∗OH+H+ +e− +(13) +∗OH → ∗O+H+ +e− +(14) +∗O+H2O → ∗OOH+H+ +e− +(15) +∗OOH →∗ +O2 +H+ +e− +(16) +It has already been reported that for ABO3 perovskites, mostly +the B site is considered as the active site for OER reaction steps. +Hence, in the case of ABX2Y perovskites, we have considered the +B site as the active site67,68. Now, a computational analysis of +the OER is often simplified by only considering the ∗OH,∗O, and +∗OOH reaction intermediates appearing in the electrochemical +steps, as shown in equations 13- 1669–72. +Following this, we have estimated the catalytic activity of a spe- +cific reaction site by the largest electrochemical step in the OER +free energy profile +∆GOER = max[∆Gi] +(17) +where ∆Gi is the change in free energy of the electrochemical OER +step. Then, we derive the thermodynamic overpotential as +ηOER = ∆GOER/e−1.23V +(18) +where 1.23 V is the equilibrium potential73. Now, the reaction +steps (∆Gi) are calculated from differences in the adsorption en- +ergies of the various intermediates as given by: +∆Gi = Gtotal −Gsurface −nGH2O −mGH2 +(19) +where n and m are the stoichiometry coefficients that preserve the +number of atoms on both sides of the respective reaction. Here, +we have calculated the adsorption energies under standard con- +ditions (pH = 0, T= 298 K) and U = 0 V using the computational +hydrogen electrode, as the theoretical overpotential does not de- +pend on the pH or the potential74,75. In this case, the energy of +a proton and an electron equals half the energy of a hydrogen +molecule. Fig. 8(d) shows the free energy profile for each ele- +mentary step of OER and we have found that for each case, the +6 | +1–8 +Journal Name, [year], [vol.], + +Transparent thin layer +H2 +(a) +(b) +H2 +2H +e +2H+ +e +H+/H2 Potential +H+/H2 Potential +e +h+ +H2 evolution +photocatalyst +1/202+2H+ +O2/H2O Potential +1/202+2H +O2/H2O Potential +h+ +H20 +h+ +H20 +O2 evolution +photocatalyst(a) +(b) +2H+ + 2e' +H2 +Ax +OER +mechanism +HER +mechanism ++ +e +X +0.5 +(c) +(d) +02 +HOO +0 +0 +4 +H*+e +1/2H, + (eV) +-0.5 +(eV) +2 +HO +△G +BaInO,F +- -BalnO,F + InSnO,N +-1.5 +-InSnO.N +CsPbO,F +-2 +-CsPbO,F + LaNbN,O +-LaNbN,O +-2 +-4 +Reaction Coordinate +Reaction Coordinateoverall potential is determined by ∆G2 step (for details see section +IX of SI). From the above results, we obtain BaInO2F, InSnO2N, +CsPbO2F and LaNbN2O as probable photocatalysts for OER. +4 +Conclusion +In summary, from our extensive computational study of struc- +tural, electronic, optical and photocatalytic properties of quater- +nary oxynitride, oxyfluoride, and nitrofluoride perovskites, we +propose them as utmost prospective candidates for the efficient +absorption and conversion of solar energy into storable fuel. Ini- +tially, using different tolerance factor and phonopy calculations, +we have checked the structural and dynamical stability, respec- +tively. After that, from the band structure and pDOS calculations, +we have found that the varying electronegativity factor affects +the electronic properties. Further, the optoelectronic and dielec- +tric calculations reveal that these perovskites will enable several +promising applications in solar energy harvesting through effi- +cient solar cells and visible light optoelectronics. The results show +that BaInO2F, CsPbO2F, LaNbN2O, and InSnO2N have appropriate +band edges for the overall water splitting. However, KTiO2F and +LaMgF2N can be used as a protective transparent shield. Follow- +ing this, we have calculated the theoretical overpotentials for HER +and OER. Low calculated overpotentials for HER and OER suggest +that certain materials may merit further study as candidates for +good photocatalysts. The detailed theoretical investigation pre- +sented in this work will surely help future studies to improve the +stability, engineer the band gap of perovskite absorbers and good +photocatalysts for water splitting. +Conflicts of interest +There are no conflicts to declare. +Acknowledgements +MJ acknowledges CSIR, India, for the senior research fellow- +ship [grant no. +[09/086(1344)/2018-EMR-I]]. DG acknowl- +edges UGC, India, for the senior research fellowship [grant +no. [1268/(CSIR-UGC NET JUNE 2018)]]. +SM acknowledges +IIT Delhi for the financial support. +SB acknowledges finan- +cial support from SERB under a core research grant (grant no. +CRG/2019/000647) to set up his High Performance Computing +(HPC) facility “Veena” at IIT Delhi for computational resources. +Notes and references +Notes and references +1 J. H. Montoya, L. C. Seitz, P. Chakthranont, A. Vojvodic, T. F. +Jaramillo and J. K. Nørskov, Nat. Mater, 2016, 16, 70—81. +2 W. T. Hong, M. Risch, K. A. Stoerzinger, A. Grimaud, J. Sun- +tivich and Y. Shao-Horn, Energy Environ. Sci., 2015, 8, 1404– +1427. +3 J. Hwang, R. R. Rao, L. Giordano, Y. Katayama, Y. Yu and +Y. Shao-Horn, N. Y. sci. j., 2017, 358, 751—756. +4 A. Singh, M. Jain and S. Bhattacharya, Nanoscale Adv., 2021, +3, 2837–2845. +5 H.-C. Wang, J. Schmidt, S. Botti and M. A. L. Marques, J. +Mater. Chem. A, 2021, 9, 8501–8513. +6 Y. Wu, P. Lazic, G. Hautier, K. Persson and G. Ceder, Energy +Environ. Sci., 2013, 6, 157–168. +7 A. T. 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Rossmeisl, ChemCatChem, 2011, 3, 1159–1165. +8 | +1–8 +Journal Name, [year], [vol.], + +Theoretical evaluation of oxynitride, oxyfluoride +and nitrofluoride perovskites with promising +photon absorption properties for solar water +splitting† +Manjari Jain,∗ Deepika Gill, Sanchi Monga, and Saswata Bhattacharya∗ +Department of Physics, Indian Institute of Technology Delhi, New Delhi, India +E-mail: manjari.jain@physics.iitd.ac.in[MJ]; saswata@physics.iitd.ac.in[SB] +Phone: +91-11-2659 1359. Fax: +91-11-2658 2037 +S1 +arXiv:2301.04335v1 [cond-mat.mtrl-sci] 11 Jan 2023 + +Supplemental Information +I. Band structure plot with HSE06 and HSE06+SOC εxc functional +II. Optimized lattice parameters of ABX2Y perovskites +III. Stability parameters (t and τ) of ABO2N perovskites +IV. Phonon band structure +V. Radial distribution plot at 0 K and 300 K using MD simulation +VI. Electronic band structure and PDOS using HSE06 εxc functional +VII. DFT calculated properties of ABX2Y perovskites +VIII. Band edge alignment of ABO2N perovskites +IX. Theoretical overpotential value for OER mechanism +S2 + +I. Band structure plot with HSE06 and HSE06+SOC εxc +functional +Figure S1: Band structure of ABX2Y perovskites using HSE06 and HSE06+SOC εxc func- +tional. +S3 + +Energy (eV) +(eV) +Energy +OF +A +M +KH +A +L M +Al +A +M +H +AgTiO2F (w/o SOC) +AgTiO2F (with SOC) +BalnO2F (w/o SOC) +BalnO2F (with SOC) +Energy (eV) +(eV) +YS +KH +S +X +CsPbO2F (w/o SOC) +CsPbO2F (with SOC) +KTiO2F (w/o SOC) +KTiO2F (with SOC) +Energy (eV) +ZT +RU +ZT +RU +X +Z +CY +B D +A +E +A +CsPbF2O (w/o SOC) +CsPbF2O (with SOC) +LaMgF2N (wlo SOC) +LaMgF2N (with SOC) +10 +Energy (eV) +A +0 +M +M +K +M +H +A +H +A +A +InSnO2N (w/o SOC) +InSnO2N (with SOC) +LaNbN2O (w/o SOC) +LaNbN2O (with SOC)II. Optimized lattice parameters of ABX2Y perovskites +Table S1: Lattice parameters of ABX2Y perovskites. +ABX2Y +a (˚ +A) +b (˚ +A) +c (˚ +A) +BaNbO2N +5.91 +5.96 +10.24 +BaTaO2N +5.88 +5.88 +7.17 +CaNbO2N +5.61 +5.67 +9.95 +CaTaO2N +5.62 +5.65 +9.73 +LaTiO2N +5.61 +5.64 +9.68 +LaZrO2N +5.89 +5.89 +10.25 +SrNbO2N +5.79 +5.81 +10.02 +SrTaO2N +5.76 +5.76 +7.01 +LaTaN2O +5.74 +5.75 +9.93 +BaInO2F +6.10 +6.10 +7.49 +KTiO2F +6.48 +3.77 +8.23 +CsPbO2F +4.64 +19.76 +4.48 +S4 + +III. Stability parameters (t and τ) of ABO2N perovskites +Table S2: Stability parameters of ABO2N perovskites. +ABX2Y +t +τ +BaNbO2N +0.95 +3.54 +BaTaO2N +0.95 +3.54 +CaNbO2N +0.83 +5.34 +CaTaO2N +0.83 +5.34 +LaTiO2N +0.86 +3.13 +LaZrO2N +0.81 +5.70 +SrNbO2N +0.89 +4.07 +SrTaO2N +0.89 +4.07 +S5 + +IV. Phonon band structure +Figure S2: Phonon band structure of ABX2Y perovskites. +S6 + +20 +20 +Frequency (THz) +15 +10 +5 +0 +0 +L +M +r +K +H +A +L +M +K +H +A +A +(a) BalnO2F +(b) KTiO2F +50 +20 +Frequency (THz) +Frequency (THz) +15 +10 +5 +50 +0 +-100 +Z +T +Y +R +UX +R +S +T +U +X +Y +Z +(c) CsPbO2F +(d) PrTaO2NV. Radial distribution plot at 0 K and 300 K using MD +simulation +Figure S3: Radial distribution function for different ABX2Y perovskites at T = 0 K and T += 300 K. +S7 + +2.0 +2.0 +OK +300 K +1.5 +1.5 +0.5 +0.5 +0.% +0.0% +2 +1 +2 +r(A) +2 +r(A） +(a) InSnO2N +(b) LaNbN2O +2.0 +2.0 +1.5 +1.5 +1.0 +a +0.5 +0.5 +0.% + 0.0% +r(A) +2 +4 +r(A) +(c) LaTaN2O +(d) PrTaN20 +2.0 +2.0 +1.5 +1.5 +1.0 +bo +0.5 +0.5 +0.0% +%00 +r(A) +2 +2 +r(A)1 +3 +(f) BalnO2F +(e) AgTiO2F +2.0 +3.0 +2.5 +1.5 +2.0 +1.5 +1.0 +1.0 +0.5 +0.5 +0.0% +2 +2 +r(A) +2 +(g) CsPbO2F +(h) KTiO2F +3.0 +2.0 +2.5 +1.5 +2.0 +1.5 +1.0 +0.5 +0.5 +0.0 +0.05 +2 +2 +r(A) +r(A) +2 +(i) CsPbF2O +(i) LaMgF2NVI. Electronic band structure and PDOS using HSE06 +εxc functional +Figure S4: Band structure of (a) BaInO2F, (b) CsPbO2F, (c) KTiO2F, (d) LaTaN2O and +(e) PrTaN2O perovskites using HSE06 εxc functional. +S8 + +8 +4 +6 +6 +(eV) +(eV) +2 +4 +Energy +Energy +2 +2 +1 +0 +0 +0 +2 +2 +LM +H +r +Z +T +U +V +A +A +S +X +A +LM +KH +(b) CsPbO2F +(c) KTiO2F +(a) BalnO2F +1O +(eV) +8 +Energy +Energy +0 +CY +EA +Y +7 +B D +R +S +T +U +X +(d) LaTaN20 +(e) PrTaN20Figure S5: PDOS of (a) BaInO2F, (b) CsPbO2F, (c) KTiO2F, (d) LaTaN2O and (e) PrTaN2O +perovskites using HSE06 εxc functional. +S9 + +15 +15 +15 +Total +Total +Total +Cs +10 +10 +10 +K +Ba +Pb +Ti +In +No. of states/eV +OF + states/eV +0 +0 +F +F +JO +No. +5 +-10 +-10 +-10 +-15 +-15 +-15 +-6 +6 +6 +-4 +-2 +0 +2 +4 +6 +-4 +-2 +0 +2 +4 +6 +-4 +2 +2 +4 +6 +Energy (eV) +Energy (eV) +Energy (eV) +(a) BalnO2F +(b) CsPbO2F +(c) KTiO2F +15 +15 +Total +Total +La +Pr +10 +10 +Ta +Ta +No. of states/eV +No. of states/eV +0 +0 +N +N +0 +-10 +-10 +-15 +-15' +-4 +-2 +0 +-6 +4 +-6 +-4 +-2 +0 +2 +2 +4 +6 +9 +Energy (eV) +Energy (eV) +(d) LaTaN2O +(e) PrTaN20VII. DFT calculated properties of ABX2Y perovskites +Table S3: DFT calculated properties of ABX2Y perovskites. +Band gap (E g), +electronic dielectric constant (ε∞), ionic dielectric constant (εr), effective mass +of electron (m∗ +e) and hole (m∗ +h) in terms of the rest mass of electron (m0). +ABX2Y +E g (eV) +ε∞ +εr +m∗ +e +m∗ +h +BaNbO2N +1.70 +6.88 +34.59 +0.55 +0.39 +BaTaO2N +1.46 +7.07 +54.89 +0.79 +1.27 +CaNbO2N +1.94 +6.68 +27.25 +0.70 +1.21 +CaTaO2N +2.00 +6.42 +32.31 +0.49 +0.87 +LaTiO2N +1.64 +8.75 +62.56 +0.72 +1.00 +LaZrO2N +2.65 +6.47 +56.44 +0.63 +0.77 +SrNbO2N +1.66 +7.08 +40.86 +0.61 +2.24 +SrTaO2N +1.72 +6.77 +43.46 +0.99 +0.60 +S10 + +VIII. Band edge alignment of ABO2N perovskites +Figure S6: Band edge alignment of ABO2N perovskites w.r.t. water redox potential levels +(H+/H2, O2/H2O). +S11 + +Vacuum ENHE +-Conduction band edge +- Valence band edge +-2.5 +-2 +LaZrO.N +-3.5 +-1 +CaTaO,N +SrTaO.N +BaTaO,N +-4.5 +0 +1.70ev +1.94 eV +1.64 ev +1.66 eV +-5.5 +2.00 eV +1.46 eV +2.65 eV +1.72 eV +O./H.0 +BaNbO,N +LaTiO,N +SrNbO,N +CaNbO,N +-6.5 +2 +-7.5 +3IX. Theoretical overpotential value for OER mechanism +Table S4: Calculated value of overpotential of ABX2Y perovskites. +ABX2Y +ηOER (V) +InSnO2N +1.91 +LaNbN2O +0.92 +BaInO2F +0.71 +CsPbO2F +1.58 +S12 + diff --git a/h9E3T4oBgHgl3EQfIgmZ/content/tmp_files/load_file.txt b/h9E3T4oBgHgl3EQfIgmZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e30dbaea6e7b4b9ede0e5acc9e66e6bd556b0a9 --- /dev/null +++ b/h9E3T4oBgHgl3EQfIgmZ/content/tmp_files/load_file.txt @@ -0,0 +1,1360 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf,len=1359 +page_content='Journal Name Theoretical evaluation of oxynitride, oxyfluoride and nitrofluoride perovskites with promising photon ab- sorption properties for solar water splitting† Manjari Jaina, Deepika Gill, Sanchi Monga and Saswata Bhattacharya∗a Photocatalytic water splitting represents a very promising but at the same time very challenging contribution to a clean and renewable route to produce hydrogen fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Developing efficient and cost-effective photocatalysts for water splitting is a growing need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For this purpose, semicon- ductor photocatalysts have attracted much more attention due to their stability and low man- ufacturing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Here, we have systematically applied several state-of-the-art advanced first- principles-based methodologies, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', hybrid density functional theory, many-body perturbation theory (G0W0) and density functional perturbation theory (DFPT), to understand the electronic structure properties of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' We have chosen the vast composition space of ABX2Y type perovskites where A and B are cations and X and Y can be nitrogen, oxygen, or fluorine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' These perovskites exhibit direct band gaps ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='6 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Further, to evaluate the feasibility of the visible light catalytic performance, we calculate the structural, electronic, and op- tical properties of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In addition, from hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) mechanism, BaInO2F, InSnO2N, CsPbO2F and LaNbN2O are found as probable photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1 Introduction The photocatalytic water splitting has become an intense research area and an excellent way to capture and store energy from the sun1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Over recent decades, the number of applications based on photocatalysis increased acutely5–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Although a wide range of materials has been developed for photocatalytic performance under visible light, most can only absorb light at a wavelength of less than 500 nm, so only a small portion of the spectrum can be utilized8–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Therefore, the design of a semiconductor for the efficient generation of solar fuel requires a suitable band gap (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 ≤ Eg ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='8 eV) to efficiently absorb visible light, high di- electric constant, high charge carrier mobility and suitable band positions in order to perform the hydrogen and oxygen evolution half-reactions (HER/OER)12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' So far, most of the existing photocatalysts are oxides14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' How- ever, they have too large band gap to absorb visible light14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' This is mainly due to a very low valence band (VB) energy which comes from the 2p orbitals of the oxygen atoms15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' A require- ment for visible light induced photocatalysts is that the optimum band gap energy should be less than 3 eV16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In order to solve aDepartment of Physics, Indian Institute of Technology Delhi, New Delhi, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fax: 91 11 2658 2037;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Tel: 91 11 2659 1359;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' E-mail: saswata@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='in this problem, non-oxides such as nitrides and sulfides have been proposed as their VB position is usually higher in energy17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Nitrogen is less electronegative than oxygen that leads to reduc- tion in the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Hence, the optical gap overlaps with the solar spectrum which makes this class of materials interesting for application as solar absorbers and visible light-driven photo- catalysts19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Many useful oxynitrides have been reported, such as: (i) CaTaO2N and LaTaON2 are non toxic solid solutions, (ii) BaTaO2N has a high dielectric constant and acts as a photocata- lyst in water decomposition, and (iii) EuNbO2N is ferromagnetic and show colossal magneto resistance20–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, it has been re- ported that LaTaON2, LaTiO2N, SrTaO2N and BaTaO2N drive the HER and OER half-reactions and can be used in Z-scheme con- figurations capturing photons in the 600 to 750 nm range23–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Recently, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' reported that CoOx modified LaTiO2N has a high quantum efficiency of 27% at 440 nm towards wa- ter oxidation28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' LaMgxTa1−xO1+3xN2−3x is found to be the first oxynitride to utilize 600 nm photons in steady-state overall wa- ter splitting25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Moreover, various oxyfluoride perovskites such as BaFeO2F, SrFeO2F and PbFeO2F have been discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' These per- ovskites exhibit magnetic ordering until a temperature of around 645 K, 685 K, and >500 K, respectively29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Multiferroic behav- ior was also shown by iron-based oxyfluoride perovskites32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], 1–8 | 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='04335v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='mtrl-sci] 11 Jan 2023 ARTICLETYPEReceivedDate AcceptedDate D01:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0000/xxxxxxxxxxSamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' proposed the potential existence of a nitrofluoride (LaZrN2F) from the first-principles based DFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' They showed that LaZrN2F composition exhibits semiconducting prop- erties with iono-covalent behavior33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' However, despite several research endeavors in the field of oxynitride, oxyfluoride and ni- trofluoride perovskites, no such in-depth theoretical work is avail- able to address their electronic, optical, dielectric properties and their application in photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In this work, we have used various advanced state-of-the-art first-principles based methodologies under the framework of density functional theory (DFT)34,35, many body perturbation theory (MBPT)36,37 and density functional perturbation theory (DFPT)38,39 to provide a comprehensive computational study of oxynitride, oxyfluoride, and nitrofluoride perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Our ob- jectives are: (i) to provide a list of compositions that could be likely experimentally synthesized in the perovskite phase, (ii) to study their electronic structure, (iii) to study their optical and ex- citonic properties, and (iv) to find their application in photocat- alytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' To achieve this goal, we report calculations for the theoretical electrocatalytic HER and OER overpotentials using DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 2 Methodology We have performed a systematic study to explore the struc- tural, electronic and optical properties using DFT and beyond approaches under the framework of MBPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' All calculations are performed with Projector Augmented Wave (PAW) poten- tials as implemented in Vienna ab initio simulation package (VASP)40,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' All the structures are optimized using general- ized gradient approximation (GGA) as implemented in PBE42 exchange-correlation (εxc) functional until the forces are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='001 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The Γ-centered 2×2×2 k-mesh sampling is em- ployed for optimization calculations (optimized structures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The electronic self-consistency loop conver- gence is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='01 meV, and the kinetic energy cutoff is set to 600 eV for plane wave basis set expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' To explore the elec- tronic properties, hybrid εxc functional (HSE06)43,44 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For all the energy calculations 4×4×4 k-mesh has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The phonon calculations are performed with 2×2×2 supercells using the PHONOPY package45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Note that the surfaces (100 plane) were constructed by cleaving the fully optimized bulk structure with the lowest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Further, surface geometries are optimized with a force convergence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='001 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Reciprocal space is sampled by 4×4×1 k-mesh for the (100) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' A 20 Å vacuum is added to avoid artificial interaction between peri- odic images, and a dipole correction is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The two-body Tkatchenko-Scheffler vdW scheme has been used to account for van der Waals interactions47,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Note that the spin-orbit coupling (SOC) is not taken into consideration because it has negligible af- fect on the electronic structure of ABX2Y perovskites (see section I of supplemental information (SI)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='1 Crystal Structure Here, we focus on the oxynitride, oxyfluoride, and nitrofluoride perovskites i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', ABX2Y (A = Ba, Ca, La, Sr, Ag, K, Cs, Pr, In ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' B = Nb, Ta, Zr, Ti, In, Pb, Mg, Sn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' and X, Y = O, N, F) perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' We have considered in total 18 perovskite structures, out of which 9 are ABO2N, 3 are ABN2O, 4 are ABO2F, 1 is ABF2O and 1 is ABF2N type of perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The oxynitride perovskite of type ABO2N e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', InSnO2N, crystallizes in a hexagonal cell and belongs to the polar space group P63cm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The optimized lat- tice constants are a = b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='16 Å and c = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='26 Å, which are well in agreement with the previous findings18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The perovskite type ABN2O e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', LaNbN2O, crystallizes in an orthorhombic cell with lattice constants a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='78 Å, b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='76 Å and c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='02 Å (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' AgTiO2F oxyfluoride perovskite has tetragonal cell edges, a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='48 Å and c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='45 Å and belongs to the space group I4/mcm as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The system CsPbF2O is an interesting perovskite because of the fact that the perovskite CsPbI3 is the parent inorganic compound with an orthorhombic crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' This perovskite has lattice constants a = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='03 Å, b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='79 Å and c= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='68 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Finally, LaMgF2N is the only ni- trofluoride system we predict to have possibilities to be realized experimentally with space group P121/m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The lattice constants of this perovskite structure are a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='33 Å, b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='69 Å and c = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='25 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Similarly, the lattice parameters of other perovskite struc- tures are mentioned in SI (see section II of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 1 Optimized crystal structure of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Images produced with VESTA 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='2 Structural Stability The stability of perovskites is often discussed based on the Gold- schmidt tolerance factor50 t = rA +ranion √ 2(rB +ranion) (1) where rA, rB and ranion are the ionic radii of the A and B cations, and of the anion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' A more recent study based on 2 | 1–8 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], (a) InSnO2N (b) LaNbN2O (c) AgTiO2F (d) CsPbF20 (e) LaMgF2Nthe novel analytics approach has led to the proposition of a new factor51 τ = ranion rB nA � nA − rA/rB log(rA/rB) � (2) where nA is the oxidation state of A cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' It has been reported that for a large experimental dataset of perovskites, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='825 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='08 gives a classification accuracy of 74%, while τ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='18 has an accuracy of 92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Now, to use these formulae for mixed anions, we need to decide the value of ranion to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In line with the suggestions provided, we decided to use the arithmetic aver- age of the radii of two mixed anions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', ranion = (2rX+rY)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' However, it has been noted that using ranion, the Goldschmidt tol- erance factor fails to capture the stability trend of mixed anion perovskites52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For this purpose, the geometric mean has also been used to approximate the radius of the mixed anions54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, more complicated factors like the octahedral factor and atomic packing fraction have been proposed to understand the stability of these perovskites54,55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' But in our case, for simplicity, we de- cided to consider traditional t and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Table 1 Stability parameters of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ABX2Y t τ InSnO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='32 LaNbN2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='74 LaTaN2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='74 PrTaN2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='56 AgTiO2F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='81 BaInO2F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='20 CsPbO2F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='49 KTiO2F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='52 LaMgF2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='64 From Table 1, we find that the value of t for all the selected per- ovskites lies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='07 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='81, which is perfectly consistent with the usual range reported for perovskites51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Now concerning τ, we find that all selected perovskites have value τ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='18 except LaMgF2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For other ABO2N perovskites also, we have calculated the structural stability parameters mentioned in section III of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In addition, to further analyze the dynamic stability, we have plotted the phonon band structures for the optimized perovskite structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The phonon band structures calculated along the high- symmetry points of the Brillouin zone for InSnO2N, LaNbN2O, AgTiO2F, CsPbF2O, and LaMgF2N are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Notably, the absence of negative frequencies confirms the dynamic stabil- ity of these perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Similarly, we have analyzed the phonon band structures of other ABX2Y perovskites in section IV of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Note that we have also checked the structural stability of different ABX2Y perovskites at a higher temperature using ab initio molec- ular dynamics (AIMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' We have obtained the radial distribution function g(r) at T = 0 K and T = 300 K by a 6 ps long MD sim- ulation run with NVT ensemble (Nose-Hoover thermostat)56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' We observe that the nature of the radial distribution function for the nearest neighbors remains the same at room temperature (see section V of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' This confirms the stability of these perovskites at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 2 Phonon band structures of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' operational temperature of 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='3 Electronic Structure We have plotted the electronic band structure and partial den- sity of states (pDOS) for ABX2Y type of perovskites to better un- derstand the role of A/B cations and X/Y anions near the va- lence band maximum (VBM) and the conduction band minimum (CBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3(a) shows the band structure of InSnO2N with a direct band gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='60 eV at Γ using HSE06 εxc functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4(a), we can clearly observe that the valence states are mainly composed of N and O with a small contribution coming from Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' While the conduction states have mostly N contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Finally, we see very few states associated with In in the [-6, 6] eV energy window, which is compatible with the interpretation that the A atom is fully ionized in the perovskite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The band structure of LaNbN2O is rather different from that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3 Electronic band structures of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N using HSE06 εxc functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' InSnO2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' It has an indirect band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='04 eV with the bottom of the CB at Γ and the top of the VB at M (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The VB is mainly contributed by N and O states, while the CB comprises of Nb states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, the La states are found only in the VB, indicating that this atom is ionized in this structure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For the Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], 1–8 | 3 6 8 11 6 (eV) (eV) 10 1 2 Energy ( 111 1 9 0 8 2 1 2 7 0 r ＩA r K H M L H A L M K H A A L M A (a) InSnO2N (b) LaNbN2O (c) AgTiO2F 6 8 4 6 4 1 1 0 2 2 1 0 -- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ZT TY Z T Y S RU X C A D EOAO (d) CsPbF2O (e) LaMgF2N20 20b 20 15 15 10 10 5 5 0 M K A H A r A K H A TY r X L L R U (a) InSnO2N (b) LaNbN2O (c) AgTiO2F 20 20 5 10 LO 5 0 Y S R ┌Y r Z C Y r B EA (d) CsPbF2O (e) LaMgF2Nother cases of oxynitride perovskites, all the band structures and PDOS plots are provided in section VI of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From the AgTiO2F band structure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3(c)), we see that it Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4 Partial density of states (pDOS) of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N using HSE06 εxc functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' has a direct band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='32 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4(c), we can clearly observe that the VB states are mostly composed of Ag and O states, while Ti states mostly contribute to the CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The HSE06 band structure of CsPbF2O is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The band gap is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='87 eV which is indirect, with the top of the valence and bot- tom of the conduction band (at Γ) composed of hybridized F, O, and Pb states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In this case, the separation of the VB into two man- ifolds is incomplete, leading to some overlap between the two sets of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The bottom of the CB, on the other hand, is separated by more than 1 eV from the rest of the CB (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Sim- ilarly, for the other oxyfluoride perovskites, we have plotted the band structure and PDOS (see section VI of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In case of nitrofluoride perovskite LaMgF2N, we have obtained a large band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='31 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The highly dispersive bottom of the CB is mainly constructed from La states, while top of the VB has mostly N states with smaller La contribution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3(e) and 4(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In LaMgF2N, we also see a clear splitting of the VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' How- ever, due to a strong electronegativity difference of N and F, the gap between the two manifolds is considerably larger than for the oxyfluoride systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='4 Optoelectronics and Dielectrics In order to get an in-depth insight into the suitability of a par- ticular material in optoelectronic applications, a detailed study of its optical properties like dielectric function, refractive index, extinction coefficient and absorption coefficient is indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The absorption coefficient (α) of a material is computed from the frequency dependent dielectric constant using the following for- mulae57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' where ε1 and ε2 are the real and imaginary terms of the dielectric constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' respectively: α = 4πκ(ω) λ (3) where κ(ω) is the extinction coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' which is given by: κ(ω) = � � � � � ε2 2 +ε2 1 −ε1 2 (4) The absorption coefficient is one of the most essential properties of a material in terms of its photovoltaic application as it depicts key information regarding optimal solar energy conversion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The typical α for direct semiconductors is of the order 105 58,59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The theoretical results show that all the perovskites exhibit high absorption coefficients (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Further, photovoltaic applications require high charge carrier Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 5 Absorption coefficient of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' mobility to reduce nonradiative recombination losses and effi- ciently transport the photogenerated carriers to the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In order to estimate the carrier mobility, we have calculated the effective mass of electrons and holes in ABX2Y perovskites from their band structure obtained by using HSE06 εxc functional (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3 and section VI of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' All the perovskites have light electrons, with InSnO2N having me= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='15m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' However, the holes are heavier, as is typical for most metal oxides, with InSnO2N having a hole effective mass of mh= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='75m0, which is in well agreement with the previous findings60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Similarly, BaInO2F has me= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='41m0 and mh= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='97m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In this case also, the hole effective mass is higher than the electron effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Here, the low effective mass of the electrons in ABX2Y perovskites can likely make them promising semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Also, we have calculated the electronic and ionic dielectric constant of these perovskites and found that they have a large dielectric constant value (see Table 2), which are in well agreement with the previous findings7,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' To this point, the relevant properties for solar energy conversion for some of the ABX2Y perovskites are compiled and summarized in Table 2 and Table S3 of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4 | 1–8 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], 15 15 15 Total Total Total La Ag 10 In 10 10 Nb Sn Ti states/ev 0 states/ev 0 0 e N N 5 5 F states/ 0 0 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 10F 10 10 15 15 15 4 0 4 4 2 0 4 2 0 2 4 2 2 6 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='6 6 6 Energy (eV) Energy (eV) Energy (eV) (a) InSnO2N (b) LaNbN2O c)A TiO2F 15 15 Total Total La 10 Cs 10 Pb Mg No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' of states/eV No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' of states/eV 0 F 5 F 5 N 0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 10 10 15 6 4 0 2 4 6 4 2 0 2 4 6 > Energy (eV) Energy (eV) (d) CsPbF20 (e) LaMgF2NAgTiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='F BaInO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='F CsPbO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='F KTiO E 2 CsPbF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='O 3 cm LaMgF,N LaNbN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='O LaTaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='O 2 PrTaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='O InSnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='N 0 2 3 4 Energy (eV)Table 2 DFT calculated properties of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Band gap (Eg), electronic dielectric constant (ε∞), ionic dielectric constant (εr), effective mass of electron (m∗ e) and hole (m∗ h) in terms of the rest mass of electron (m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ABX2Y Eg (eV) ε∞ εr m∗e m∗ h InSnO2N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='45 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='75 LaNbN2O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='27 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='41 LaTaN2O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='28 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='51 AgTiO2F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='74 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='68 BaInO2F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='97 CsPbF2O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='87 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='66 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 Photocatalytic Water Splitting In a photocatalytic process, the perovskite material absorbs light as a result an electron at the VB is transferred to the CB, and a photogenerated e-h pair is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Subsequently, for the ox- idation or reduction reaction, the photogenerated electrons and holes transfer to the corresponding reaction sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The photocat- alytic reaction can be described by two half-reactions for the hy- drogen evolution and water oxidation which are summarized as follows: 2H+ +2e− → H2 (5) 2H2O+4h+ → O2 +4H+ (6) For the water splitting reaction, the CBm should be higher than the reduction potential of H+/H2 (EH+/H2 = –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='44 eV at pH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' On the other hand, the VBM should be lower than the oxida- tion potential of O2/H2O (EO2/H2O = –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='67 eV at pH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From the above discussion, we conclude that in order to evaluate the feasibility of photocatalytic water splitting, it is crucial to calcu- late precisely the edge positions of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The calculation of the band edges using DFT is quite complicated and compu- tationally expensive because the surface chemistry and interfacial effects are considered in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' have summarized recent progress and open theoretical challenges present in simu- lations61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' They use an appropriate method to treat the effect of interface on the band edge position of the surface structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' How- ever, it is still complex to predict the absolute energy positions of CBm and VBM for the bulk structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Here, we have used a reliable empirical formula proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='62 to calculate the absolute band edges of ABX2Y bulk per- ovskites using the following equation: EVBM = χ − 1 2Eg (7) ECBm = EVBM +Eg (8) where EVBM and ECBm represents the absolute potentials of VBM and CBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' χ is the electronegativity of the perovskite which can be determined by the absolute electronegativities of the constituent atoms as: χ(compound) = χa 1 χb 2 χc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='.χm n (9) where χ1, χ2, χ3 and χn represent the electronegativities of the constituent atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' a, b, c and m are the molar fractions of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The electronegativity of the constituent atoms is calcu- lated using the Mulliken electronegativity63: χ1 = (I +A)/2 (10) where I and A is the ionization energy and the electron affinity of the atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 6 visually expresses the absolute energy positions of band Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 6 Band edge alignment of ABX2Y perovskites w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' water redox potential levels (H+/H2, O2/H2O) edges and the redox potentials of the water splitting reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In case of AgTiO2F, the CBm is shifted downward by a large amount, and hence, their reduction power is very low and could not be utilized for hydrogen generation from water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For KTiO2F and LaMgF2N cases, the band gap is very large thus not inducing the visible-light response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In view of this, from the applicability in photocatalytic water splitting, BaInO2F, CsPbO2F, LaNbN2O, LaTaN2O and InSnO2N are the most desirable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For other oxynitride perovskites also, we have plotted the band edge align- ment (see section VIII of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Now to better classify these per- ovskites for different applications, we have discussed two appli- cations in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='1 One-photon water splitting The one-photon or overall water splitting process is schematized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In this process, a single photon creates an e-h pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Further, the electron and the hole reach two different regions of the surface to avoid recombination, and evolve hydrogen and ox- idize water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The criteria for a material to be used for solar light capture that we have considered here are: (i) structural stability, (ii) band gap in the visible light range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 ≤ Eg ≤ 3 eV, and (iii) band edges straddling with the oxygen and hydro- gen evolution potentials i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', VB > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='23 eV and CB < 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 6, we have found various ABX2Y perovskites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', BaInO2F, CsPbO2F, LaNbN2O, InSnO2N, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=') which are suitable for col- lecting visible light and for the evolution of both hydrogen and oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='2 Transparent shield Photocorrosion is one of the major problems related to the use of materials for oxygen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In order to overcome this issue, it is required to develop a highly stable and transparent thin film to cover the oxygen evolution photocatalyst as a protective shield Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], 1–8 | 5 Vacuum E ENHE Conduction band edge Valence band edge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5F 2 KTiO,F 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1 BalnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='F LaMgF,N LaTaN,O InSnO,N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='32 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='44 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='87 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='04 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='00eV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='21 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='01 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='60 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='31 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='11 eV 0/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 LaNbN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='O CsPbF,O CsPbO,F PrTaN,O 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 2 AgTiO,F 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 3Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 7 (a) Overall water splitting scheme, (b) Scheme of a tandem cell with a transparent protecting shield on the oxygen evolution photocata- lyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' For this purpose, the material should have a large band gap of, say, more than 3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 7(b) illustrates a device as a photoanode protecting transparent shield for a tandem cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Here, the electron with higher energy evolves hydrogen, while the hole at lower energy moves through the transparent film and evolves oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The criteria for such material are: (i) a direct band gap in the UV range in order to avoid absorbing a part of the visible light spectrum, (ii) VB needs to be placed between the edges of the photocatalyst and the potential of oxygen evolution so that the hole reaches the reaction region with a small energy loss i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='7 ≤ VBedge ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The same idea can be used to develop a photocathode protecting shield64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The only change is the position of the CB i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=', -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='7 ≤ CBedge ≤ 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 6, we find that KTiO2F and LaMgF2N can be used as a protective transparent shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='6 HER and OER mechanism The HER reaction mechanism is schematized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In HER mechanism, the reaction steps follow: (i) adsorption of H, (ii) hydrogen reduction and (iii) desorption as H2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The Gibbs free energy of hydrogen reaction (∆GH) is a reasonable descriptor for HER reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In order to calculate the Gibbs free energy changes (∆G) of the intermediates in HER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' the following expression is used: ∆GH = ∆EH +∆EZPE −T∆SH (11) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ∆EH = Etotal −Esurface −1/2EH2 (12) In the above mentioned equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ∆EH is the hydrogen binding energy on the perovskite surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Etotal is the total energy for the adsorption state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Esurface is the energy of the pure surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' EH2 is the energy of hydrogen molecule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ∆EZPE is the change in zero- point energy and ∆SH is the difference in entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' At 298 K, ∆EZPE − T∆SH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='25 eV is well established in literature66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The HER diagram along the reaction pathway H+ +e− → H∗ → 1/2H2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Under the conditions of pH = 0 and a stan- dard hydrogen electrode (SHE) potential of 0 V, H+ +e− is equiv- alent to 1/2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From the above discussion, we found BaInO2F, InSnO2N, CsPbO2F and LaNbN2O as probable photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Similarly, the OER mechanism involves four proton-coupled elec- tron transfer steps on metal ion centers with OH, O, and OOH re- action intermediates, where ∗ indicates the active metal site (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8 (a) HER steps, (b) OER steps, (c) HER free energy diagram, and (d) OER free energy diagram for ABX2Y perovskites at an electrode potential ESHE=0 V and pH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8(b)): ∗ +H2O → ∗OH+H+ +e− (13) ∗OH → ∗O+H+ +e− (14) ∗O+H2O → ∗OOH+H+ +e− (15) ∗OOH →∗ +O2 +H+ +e− (16) It has already been reported that for ABO3 perovskites, mostly the B site is considered as the active site for OER reaction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Hence, in the case of ABX2Y perovskites, we have considered the B site as the active site67,68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Now, a computational analysis of the OER is often simplified by only considering the ∗OH,∗O, and ∗OOH reaction intermediates appearing in the electrochemical steps, as shown in equations 13- 1669–72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Following this, we have estimated the catalytic activity of a spe- cific reaction site by the largest electrochemical step in the OER free energy profile ∆GOER = max[∆Gi] (17) where ∆Gi is the change in free energy of the electrochemical OER step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Then, we derive the thermodynamic overpotential as ηOER = ∆GOER/e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='23V (18) where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='23 V is the equilibrium potential73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Now, the reaction steps (∆Gi) are calculated from differences in the adsorption en- ergies of the various intermediates as given by: ∆Gi = Gtotal −Gsurface −nGH2O −mGH2 (19) where n and m are the stoichiometry coefficients that preserve the number of atoms on both sides of the respective reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Here, we have calculated the adsorption energies under standard con- ditions (pH = 0, T= 298 K) and U = 0 V using the computational hydrogen electrode, as the theoretical overpotential does not de- pend on the pH or the potential74,75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' In this case, the energy of a proton and an electron equals half the energy of a hydrogen molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8(d) shows the free energy profile for each ele- mentary step of OER and we have found that for each case, the 6 | 1–8 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=" ], Transparent thin layer H2 (a) (b) H2 2H e 2H+ e H+/H2 Potential H+/H2 Potential e h+ H2 evolution photocatalyst 1/202+2H+ O2/H2O Potential 1/202+2H O2/H2O Potential h+ H20 h+ H20 O2 evolution photocatalyst(a) (b) 2H+ + 2e' H2 Ax OER mechanism HER mechanism + e X 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 (c) (d) 02 HOO 0 0 4 H*+e 1/2H, (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 (eV) 2 HO △G BaInO,F -BalnO,F InSnO,N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 InSnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='N CsPbO,F 2 CsPbO,F LaNbN,O LaNbN,O 2 4 Reaction Coordinate Reaction Coordinateoverall potential is determined by ∆G2 step (for details see section IX of SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' From the above results, we obtain BaInO2F, InSnO2N, CsPbO2F and LaNbN2O as probable photocatalysts for OER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 4 Conclusion In summary, from our extensive computational study of struc- tural, electronic, optical and photocatalytic properties of quater- nary oxynitride, oxyfluoride, and nitrofluoride perovskites, we propose them as utmost prospective candidates for the efficient absorption and conversion of solar energy into storable fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Ini- tially, using different tolerance factor and phonopy calculations, we have checked the structural and dynamical stability, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' After that, from the band structure and pDOS calculations, we have found that the varying electronegativity factor affects the electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Further, the optoelectronic and dielec- tric calculations reveal that these perovskites will enable several promising applications in solar energy harvesting through effi- cient solar cells and visible light optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The results show that BaInO2F, CsPbO2F, LaNbN2O, and InSnO2N have appropriate band edges for the overall water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' However, KTiO2F and LaMgF2N can be used as a protective transparent shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Follow- ing this, we have calculated the theoretical overpotentials for HER and OER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Low calculated overpotentials for HER and OER suggest that certain materials may merit further study as candidates for good photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' The detailed theoretical investigation pre- sented in this work will surely help future studies to improve the stability, engineer the band gap of perovskite absorbers and good photocatalysts for water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Acknowledgements MJ acknowledges CSIR, India, for the senior research fellow- ship [grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' [09/086(1344)/2018-EMR-I]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' DG acknowl- edges UGC, India, for the senior research fellowship [grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' [1268/(CSIR-UGC NET JUNE 2018)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' SM acknowledges IIT Delhi for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' SB acknowledges finan- cial support from SERB under a core research grant (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' CRG/2019/000647) to set up his High Performance Computing (HPC) facility “Veena” at IIT Delhi for computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Notes and references Notes and references 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Montoya, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Seitz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Chakthranont, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Kitchin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Jaramillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Norskov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Rossmeisl, ChemCatChem, 2011, 3, 1159–1165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 8 | 1–8 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ], Theoretical evaluation of oxynitride, oxyfluoride and nitrofluoride perovskites with promising photon absorption properties for solar water splitting† Manjari Jain,∗ Deepika Gill, Sanchi Monga, and Saswata Bhattacharya∗ Department of Physics, Indian Institute of Technology Delhi, New Delhi, India E-mail: manjari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='jain@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='ac.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='25 SrNbO2N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='81 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='02 SrTaO2N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='76 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='01 LaTaN2O 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='93 BaInO2F 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='49 KTiO2F 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='77 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='23 CsPbO2F 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='64 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='48 S4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Stability parameters (t and τ) of ABO2N perovskites Table S2: Stability parameters of ABO2N perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ABX2Y t τ BaNbO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='54 BaTaO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='54 CaNbO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='34 CaTaO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='34 LaTiO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='13 LaZrO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='70 SrNbO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='07 SrTaO2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='07 S5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Phonon band structure Figure S2: Phonon band structure of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' S6 20 20 Frequency (THz) 15 10 5 0 0 L M r K H A L M K H A A (a) BalnO2F (b) KTiO2F 50 20 Frequency (THz) Frequency (THz) 15 10 5 50 0 100 Z T Y R UX R S T U X Y Z (c) CsPbO2F (d) PrTaO2NV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Radial distribution plot at 0 K and 300 K using MD simulation Figure S3: Radial distribution function for different ABX2Y perovskites at T = 0 K and T = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' S7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 OK 300 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0% 2 1 2 r(A) 2 r(A） (a) InSnO2N (b) LaNbN2O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0% r(A) 2 4 r(A) (c) LaTaN2O (d) PrTaN20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 bo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0% %00 r(A) 2 2 r(A)1 3 (f) BalnO2F (e) AgTiO2F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0% 2 2 r(A) 2 (g) CsPbO2F (h) KTiO2F 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='05 2 2 r(A) r(A) 2 (i) CsPbF2O (i) LaMgF2NVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Electronic band structure and PDOS using HSE06 εxc functional Figure S4: Band structure of (a) BaInO2F, (b) CsPbO2F, (c) KTiO2F, (d) LaTaN2O and (e) PrTaN2O perovskites using HSE06 εxc functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' S8 8 4 6 6 (eV) (eV) 2 4 Energy Energy 2 2 1 0 0 0 2 2 LM H r Z T U V A A S X A LM KH (b) CsPbO2F (c) KTiO2F (a) BalnO2F 1O (eV) 8 Energy Energy 0 CY EA Y 7 B D R S T U X (d) LaTaN20 (e) PrTaN20Figure S5: PDOS of (a) BaInO2F, (b) CsPbO2F, (c) KTiO2F, (d) LaTaN2O and (e) PrTaN2O perovskites using HSE06 εxc functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' S9 15 15 15 Total Total Total Cs 10 10 10 K Ba Pb Ti In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' of states/eV OF states/eV 0 0 F F JO No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' 5 10 10 10 15 15 15 6 6 6 4 2 0 2 4 6 4 2 0 2 4 6 4 2 2 4 6 Energy (eV) Energy (eV) Energy (eV) (a) BalnO2F (b) CsPbO2F (c) KTiO2F 15 15 Total Total La Pr 10 10 Ta Ta No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' of states/eV No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=" of states/eV 0 0 N N 0 10 10 15 15' 4 2 0 6 4 6 4 2 0 2 2 4 6 9 Energy (eV) Energy (eV) (d) LaTaN2O (e) PrTaN20VII." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' DFT calculated properties of ABX2Y perovskites Table S3: DFT calculated properties of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Band gap (E g), electronic dielectric constant (ε∞), ionic dielectric constant (εr), effective mass of electron (m∗ e) and hole (m∗ h) in terms of the rest mass of electron (m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ABX2Y E g (eV) ε∞ εr m∗ e m∗ h BaNbO2N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='70 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='5 3IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' Theoretical overpotential value for OER mechanism Table S4: Calculated value of overpotential of ABX2Y perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content=' ABX2Y ηOER (V) InSnO2N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='91 LaNbN2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='92 BaInO2F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E3T4oBgHgl3EQfIgmZ/content/2301.04335v1.pdf'} +page_content='71 CsPbO2F 1.' 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+INTERSECTIONS +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +Abstract. For a pair of finitely generated modules M and N over a codi- +mension c complete intersection ring R with ℓ(M ⊗R N) finite, we pay special +attention to the inequality dim M + dim N ≤ dim R + c. +In particular, we +develop an extension of Hochster’s theta invariant whose nonvanishing detects +equality. In addition, we consider a parallel theory where dimension and codi- +mension are replaced by depth and complexity, respectively. +introduction +Porting geometric ideas into homological algebra, Serre’s development of inter- +section multiplicity [20] provided a foundation for many important homological +conjectures. A basic result of Serre’s work is that the dimension inequality +dim M + dim N ≤ dim Q +(1) +holds for finitely generated modules M and N over a regular local ring Q, as long +as the length ℓ(M ⊗Q N) is finite. Hochster extended these ideas into the realm +of hypersurfaces, defining his theta invariant in [10] in part as a mechanism for +understanding when this dimension inequality holds for singular rings. One aim +of the present work is to continue this investigation for complete intersections of +arbitrary codimension. +Let R be a complete intersection ring of codimension c: that is R = Q/(f1, ..., fc), +where Q is a regular local ring and f1, ..., fc is a regular sequence. Let M, N be +a pair of finitely generated R-modules with ℓ(M ⊗R N) < ∞. +If the pair has +complexity r = cxR(M, N) (see 1.8 for the definition of complexity), we define in +Definition 2.2 an invariant θR +r (M, N) as the (r − 1)-st difference of the numerical +function underlying Hochster’s theta invariant [10]. This extends Hochster’s theta +invariant, and similar invariants have been considered previously by Dao [5], and +Moore, Piepmeyer, Spiroff, and Walker [15]; see Remark 2.3. +One of our first key results, proven in Theorem 2.8, is the existence of a complete +intersection R′ over which complexity is reduced and this invariant is preserved. +Namely, assuming r = cxR(M, N) ≥ 1, we show that there exists a principal lifting +Date: January 3, 2023. +2020 Mathematics Subject Classification. 13H15, 13D22, 13C40. +Key words and phrases. Complete intersection ring, depth inequality, dimension inequality, +intersection multiplicity, theta invariant. +P.A. Bergh would like to thank the organizers of the Representation Theory program hosted +by the Centre for Advanced Study at The Norwegian Academy of Science and Letters, where he +spent parts of fall 2022. P. Thompson is grateful for the support provided by the 2022 Summer +Research Grant at Niagara University. +1 + +2 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +R′ of R such that cxR′(M, N) = r − 1 and θR′ +r−1(M, N) = θR +r (M, N). This fact +plays a central role in the remainder of the paper. +As a natural extension of Serre’s dimension inequality (1) to complete intersec- +tions, one has that, for a pair of finitely generated R-modules M, N with ℓ(M ⊗R +N) < ∞, there is a dimension inequality (see Remark 2.16): +dim M + dim N ≤ dim R + c . +(2) +Motivated by Hochster’s result for hypersurfaces, we show in Theorem 2.15 that +if Q satisfies the conclusions of Serre’s intersection multiplicity conjectures, then +vanishing of θR +c (M, N) is equivalent to strict inequality in (2). +Although it is unknown whether one always has dim M + dim N ≤ dim R + r, +where r = cxR(M, N) (indeed, this is related to a long-standing open question +about modules of finite projective dimension; see the end of Section 2), we show in +Theorem 3.1 that +depth M + depth N ≤ dim R + r . +(3) +One naturally wonders whether vanishing of θR +r (M, N) might detect whether the +inequality (3) is strict: an answer is given in Theorem 4.1 and Remark 4.2. +Finally, although strictness in the inequality (3) is a weaker condition than van- +ishing of the invariant θR +r (M, N), for sufficiently nice pairs of modules much more +can be said. This is the investigation of Section 5. Under the assumption that a +pair of modules M, N is intersection liftable (this notion is defined in Definition 5.3 +and includes the assumption that ℓ(M ⊗R N) is finite), Theorem 5.4 shows that +dim M + dim N ≤ dim R + r holds, θR +r (M, N) ≥ 0 holds, and θR +r (M, N) vanishes +precisely when dim M + dim N < dim R + r. +The paper is outlined as follows. +The first section is devoted to developing +preliminary results and recalling definitions about complexity and Hilbert polyno- +mials. In Section 2 we define θR +r (M, N), and prove in Theorem 2.8 one of our main +tools for reducing complexity. This allows us to prove Theorem 2.15, motivated by +Hochster’s result. We also give new proofs of some basic properties of complexity, +in particular a new proof of a result of Avramov and Buchweitz regarding com- +plexity (see Theorem 2.10). Aiming to refine our extension of Hochster’s result, +we turn in Section 3 to replacing dimension by depth: the primary result here is +Theorem 3.1, and we obtain a number of applications, including an extension of +the Intersection Theorem (see Theorem 3.8). Section 4 considers versions of Serre’s +multiplicity conjectures in the situation of the previous two sections. A natural +observation that arises over the course of this investigation is that the main result +in Section 2 should hold at least for sufficiently nice pairs of modules: these are the +modules that are studied in Section 5. An appendix provides additional details for +some proofs. +This work was inspired by the paper of Hochster [10], and we found some ideas +from [5] useful. +1. Preliminaries +Let R = R[x1, . . . , xc] be a polynomial ring with indeterminates x1, . . . , xc, all of +the same degree d, and R a local ring with residue field k. Thus R is a Noetherian +graded ring. Throughout this paper, we use ℓ(−) to denote length. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +3 +We now want to define a dualizing functor which takes graded R-modules to +graded R-modules. The version of graded Matlis duality we give below has been +considered in more restrictive situations by Jørgensen [13] and van den Bergh [21]. +1.1. Let E denote the injective hull of k as an R-module. For any R-module N of +finite length we have +(1) ℓ(HomR(N, E)) = ℓ(N), in particular, HomR(N, E) has finite length; +(2) the natural map N → HomR(HomR(N, E), E) is an isomorphism. +(See [4, Proposition 3.2.12].) For a graded R-module G = � +n∈Z Gn we then define +the functor +D(G) = +� +n∈Z +HomR(Gn, E) +which returns a graded R-module. +The grading of D(G) is given by D(G)n = +HomR(G−n, E) for n ∈ Z. The action of xi on D(G) is given by xig(a) = g(xia), +for g ∈ HomR(Gn, E) = D(G)−n and a ∈ Gn−d. Thus xig ∈ HomR(Gn−d, E) = +D(G)−n+d. +For a homomorphism h : G → G′ of graded R-modules with h = +� +n∈Z hn and each hn : Gn → G′ +n R-linear, we have D(h) : D(G′) → D(G) +given by D(h) = � +n∈Z HomR(hn, E) and each HomR(hn, E) : HomR(G′ +n, E) → +HomR(Gn, E) is R-linear. +Now suppose that the graded R-module G is such that each Gn has finite length +as an R-module. +Then from the discussion above we see that D(G) is again a +graded R-module such that each D(G)n has finite length as an R-module. +Theorem 1.2. The functor D is exact and G ∼= D(D(G)) for all graded R-modules +G with Gn having finite length as an R-module for all n. +Proof. The exactness of the functor D follows directly from the exactness of the +functor HomR(−, E) from R-modules to R-modules. +For the second statement, the R-linear isomorphisms +ϕn : Gn → HomR(HomR(Gn, E), E) +in each degree, mentioned in 1.1, are given by ϕn(a)(g) = g(a) for a ∈ Gn and +g ∈ HomR(Gn, E). We claim moreover that these constitute an isomorphism of +R-modules. Indeed, for a ∈ Gn−d and g ∈ HomR(Gn, E) we have ϕn(xia)(g) = +g(xia) = xig(a) = ϕn−d(a)(xig) = xiϕn−d(a)(g). Thus ϕn(xia) = xiϕn−d(a), and +so ϕ = � +n∈Z ϕn is R-linear. +□ +Recall that a graded R-module G = � +n∈Z Gn is Artinian if it satisfies DCC on +graded submodules, and Noetherian if it satisfies ACC on graded submodules. We +have the following slight variation of results by Kirby [14], which give equivalent +conditions for Artinian and Noetherian graded R-modules G. +Theorem 1.3. Let G = � +n∈Z Gn be a graded R-module. Assume that the xi all +have negative degree d < 0. Then +(1) G is a Noetherian R-module if and only if there exist integers r and s such +that +(i) Gn = 0 for n > r; +(ii) Gn+d = x1Gn + · · · + xcGn for all n ≤ s; +(iii) Gn is a Noetherian R-module for all n. +(2) G is an Artinian R-module if and only if there exist integers r and s such +that + +4 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +(i) Gn = 0 for n < r; +(ii) (0 :Gn Rx1 + · · · + Rxc) = 0 for all n ≥ s; +(iii) Gn is an Artinian R-module for all n. +We provide a proof in the appendix. Note that Condition (1)(ii) of Theorem 1.3 is +equivalent to the R-linear maps ρG +n : Gc +n → Gn+d given by ρG +n (a1, . . . , ac) = x1a1 + +· · · + xcac being surjective for n ≤ s. Similarly, Condition (2)(ii) of Theorem 1.3 is +equivalent to the R-linear maps ιG +n : Gn → Gc +n+d given by ιG +n (a) = (x1a, . . . , xca) +being injective for n ≥ s. +One checks easily that D(ρG +n ) = ιD(G) +−n−d and D(ιG +n ) = ρD(G) +−n−d. +Theorem 1.4. Let G be a graded R-module such that each Gi has finite length +as an R-module. Then G is Noetherian (resp. Artinian) if and only if D(G) is +Artinian (resp. Noetherian). +Proof. Suppose that G is Noetherian. Then, according to Theorem 1.3, there exist +integers r and s such that +(i) Gn = 0 for n > r; +(ii) ρG +n is surjective for all n ≤ s; +(iii) Gn is a Noetherian R-module for all n. +Thus applying D we see that +(i) D(G)n = D(G−n) = 0 for n < −r; +(ii) ιD(G) +n += D(ρG +−n−d) is injective for all n ≥ −s + d; +(iii) D(G)n = D(G−n) is an Artinian R-module for all n. +Thus by Theorem 1.3 again, we see that D(G) is Artinian. The converse is clear. +□ +Notation 1.5. Throughout this paper, let R = Q/(f1, . . . , fc), where (Q, n, k) is +a local ring and f1, . . . , fc is a Q-regular sequence contained in the square of the +maximal ideal of Q. If we say that R is a complete intersection, we mean that Q is +a regular local ring. We also assume M and N are finitely generated R-modules. +We let TorR(M, N) denote � +n≥0 TorR +n (M, N). +Assumptions 1.6. We will assume throughout the paper that ℓ(M ⊗R N), the +length of M ⊗R N, is finite. In this case, each TorR +n (M, N) and TorQ +n (M, N) has +finite length, n ≥ 0. +From now on we assume that R = R[x1, . . . , xc] is the polynomial ring in c +variables, each of degree d = −2. Gulliksen shows in [9] that T = TorR(M, N) +can be given the structure of graded module over R such that T is an Artinian +R-module whenever TorQ(M, N) is an Artinian Q-module. We assume throughout +that this is the case. We note that since each TorQ +n (M, N) has finite length, the +condition that TorQ(M, N) is an Artinian Q-module is equivalent to the condition +that TorQ +n (M, N) = 0 for all n ≫ 0. This holds, for example, if R is a complete +intersection. +Since each xi ∈ R acts on T with degree −2, the graded R-module T naturally +decomposes as a direct sum of graded R-modules Te, the Tors with even index, and +To, the Tors with odd index. Obviously T is Artinian if and only if both Te and To +are Artinian. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +5 +1.7. Let HR +e (M, N) and HR +o (M, N) denote the even and odd Hilbert functions of +the pair M, N. That is, +HR +e (M, N)(n) = ℓ(TorR +2n(M, N)) +for n ≥ 0 +and +HR +o (M, N)(n) = ℓ(TorR +2n−1(M, N)) +for n ≥ 1 +Since Te and To are Artinian graded R-modules, D(Te) and D(To) are Noetherian +graded R-modules, and so their Hilbert functions are of polynomial type. Graded +Matlis duality 1.1 also gives the equalities +ℓ(TorR +n (M, N)) = ℓ(D(TorR +n (M, N))) +for n ≥ 0 +It follows that the numerical functions HR +e (M, N) and HR +o (M, N) are also of poly- +nomial type. +Thus there exist polynomials P R,M,N +e +(x), P R,M,N +o +(x) ∈ Q[x] such +that +P R,M,N +e +(n) = HR +e (M, N)(n) +and +P R,M,N +o +(n) = HR +o (M, N)(n) +for all n ≫ 0. We call P R,M,N +e +and P R,M,N +o +the even and odd Hilbert polynomials +of the pair M, N. +Definition 1.8. Assuming ℓ(M ⊗R N) < ∞ and TorQ +n (M, N) = 0 for all n ≫ 0, +we define the complexity, denoted cxR(M, N), of the pair M, N over R as +cxR(M, N) = max{deg P R,M,N +e +(x), deg P R,M,N +o +(x)} + 1 +where by convention we take the degree of the zero polynomial to be −1. In this +case we have cxR(M, N) = 0 if and only if P R,M,N +e +(x) = P R,M,N +o +(x) = 0. We +write cxR M to mean cxR(M, k). +Contrary to some extant literature, we write +cxR(M, N) instead of cxR(M, N) to reflect that it is Tor complexity, rather than +Ext complexity. +Thus for r = cxR(M, N) we may write +P R,M,N +e += axr−1 + g(x) +and +P R,M,N +o += bxr−1 + h(x) +where a and b are rational numbers, and deg g(x) < r − 1 and deg h(x) < r − 1. +Remark 1.9. Our definition of complexity of the pair M, N is not the ordinary +Ext complexity of the pair M, N defined in [1]. It is however the same as the length +complexity as defined in [5]. Both complexities agree over a complete intersection, +when defined, according to [5, Theorem 5.4]. +2. An extension of Hochster’s theta invariant +Let R = Q/(f1, ..., fc) be as in Notation 1.5, where Q is a local ring and f1, ..., fc +is a Q-regular sequence contained in the square of the maximal ideal of Q. +Consider an arbitrary numerical function F : Z → Z. +The (first backward) +difference operator ∇ applied to F is defined to be +∇F(n) = F(n) − F(n − 1) + +6 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +One extends this operator to higher order r ≥ 1 by setting ∇rF = ∇(∇r−1F), +where ∇1 = ∇, and ∇0F = F. A straightforward computation shows that +(2.0.1) +∇rF(n) = +r +� +i=0 +(−1)i +�r +i +� +F(n − i) +Note that ∇ takes a nonzero polynomial function to a polynomial function of degree +one less (we assume throughout that deg 0 = −1). +We extend Hochster’s theta invariant as follows. Consider the numerical function +θR(M, N) = HR +e (M, N) − HR +o (M, N), +that is, +(2.0.2) +θR(M, N)(n) = ℓ(TorR +2n(M, N)) − ℓ(TorR +2n−1(M, N)) +for n ≥ 1. +For r ≥ 0 consider the numerical function ∇rθR(M, N), whose explicit expression +is +(2.0.3) +∇rθR(M, N)(n) = +r +� +i=0 +(−1)i +�r +i +� � +ℓ(TorR +2(n−i)(M, N)) − ℓ(TorR +2(n−i)−1(M, N)) +� +The proof of the following is immediate from the definitions. +Proposition 2.1. Let M and N be finitely generated R-modules with ℓ(M ⊗R N) +finite and TorQ +n (M, N) = 0 for all n ≫ 0. Suppose that r = cxR(M, N). Then we +have: +(1) if s ≥ r, then ∇sθR(M, N)(n) = 0 for all n ≫ 0; +(2) ∇r−1θR(M, N)(n) = a−b for all n ≫ 0. Thus ∇r−1θR(M, N) is eventually +constant, and ∇r−1θR(M, N)(n) = 0 for all n ≫ 0 if and only if P R,M,N +e +(x) +and P R,M,N +o +(x) have the same degree and leading coefficient. +Definition 2.2. Let M and N be finitely generated R-modules with ℓ(M ⊗R N) +finite. Suppose that r = cxR(M, N) ≥ 1. Then we set +θR +s (M, N) = +� +∇s−1θR(M, N)(n) for n ≫ 0 +if s = r +0 +if s > r +Remark 2.3. In the case that R is a hypersurface, i.e., a complete intersection with +c = 1, the invariant θR +1 is precisely Hochster’s theta invariant from [10], which has +been studied in depth by others. For a complete intersection R more generally, and +a pair of R-modules M, N with r = cxR(M, N) and s ≥ r an integer, our invariant +θR +s (M, N) and Dao’s invariant ηR +s (M, N) from [5] vanish simultaneously. In fact, +they satisfy (2s · s!)ηs(M, N) = θR +s (M, N) for s ≥ r ≥ 1. Dao’s invariant is defined +in terms of limits. The invariant ηc(M, N) defined (only for the codimension c) in +[15] agrees with Dao’s invariant, yet, like our invariant, is defined in terms of the +(c − 1)-st difference of Hochster’s theta invariant. +Change of rings. +Definition 2.4. We say that R′ is a principal lifting of R if there exists a set of +generators g1, . . . , gc of (f1, . . . , fc) such that R′ = Q/(g1, . . . , gc−1). Note in this +case we have R = R′/(gc). + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +7 +The existence of short exact sequences of Tor as in the following lemma are +guaranteed by Theorem 2.8 below. +Lemma 2.5. Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite. +Let R′ be a principal lifting of R, and assume that for all large n we have short +exact sequences +0 → TorR′ +n (M, N) → TorR +n (M, N) → TorR +n−2(M, N) → 0 +Then for r ≥ 1 we have +∇r−1θR′(M, N)(n) = ∇rθR(M, N)(n) +for all n ≫ 0. +Proof. Abbreviate ℓ(TorR +i (M, N)) by ℓR +i . Then, using (2.0.3), for all n ≫ 0 we have +∇r−1θR′(M, N)(n) = +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR′ +2(n−i) − ℓR′ +2(n−i)−1 +� += +r−1 +� +i=0 +(−1)i�r−1 +i +� +ℓR′ +2(n−i) − +r−1 +� +i=0 +(−1)i�r−1 +i +� +ℓR′ +2(n−i)−1 += +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR +2(n−i) − ℓR +2(n−i)−2 +� +− +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR +2(n−i)−1 − ℓR +2(n−i)−3 +� +Now we exchange the second term in parentheses in the first sum with the first +term in parentheses in the second sum, getting += +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR +2(n−i) − ℓR +2(n−i)−1 +� +− +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR +2(n−i)−2 − ℓR +2(n−i)−3 +� += +r−1 +� +i=0 +(−1)i�r−1 +i +� � +ℓR +2(n−i) − ℓR +2(n−i)−1 +� +− +r +� +i=1 +(−1)i−1�r−1 +i−1 +� � +ℓR +2(n−i) − ℓR +2(n−i)−1 +� += +r +� +i=0 +(−1)i ��r−1 +i +� ++ +�r−1 +i−1 +�� � +ℓR +2(n−i) − ℓR +2(n−i)−1 +� += +r +� +i=0 +(−1)i�r +i +� � +ℓR +2(n−i) − ℓR +2(n−i)−1 +� +=∇rθR(M, N)(n) +□ +2.6. Let G be a graded R-module. It follows immediately from the exactness of D +that multiplication by a homogeneous element x ∈ R is eventually surjective on G, +meaning Gn +x−→ Gn+|x| → 0 is exact for all n ≫ 0, if and only if multiplication by +x on D(G) is eventually injective: 0 → D(G)n +x−→ D(G)n+|x| is exact for all n ≪ 0. +The following is a key result that allows for induction by passing to a principal +lifting; it is a slight generalization of [7, Theorem 3.1]. +Theorem 2.7. Let M and N be finitely generated R-modules with ℓ(M ⊗RN) finite +and TorQ +n (M, N) = 0 for all n ≫ 0. Consider the graded R-module T = � +n≥0 Tn, + +8 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +where Tn = TorR +n (M, N). +Then there exists a linear form x ∈ R−2 such that +Tn−2 = xTn for all n ≫ 0. +Proof. By 1.6 we have that T is an Artinian R-module. Therefore by Theorem 1.4, +D(T ) is a Noetherian R-module. Now a variant of a standard argument on the ex- +istence of eventually regular elements (see Theorem 6.5 in the appendix) applies to +show that there exists x ∈ R−2 such that x is eventually injective on D(T ). Thus x +is eventually surjective on T , by 2.6. (The proof of Theorem 6.5 is done for nonneg- +atively graded modules over a graded ring generated as an algebra over its degree +zero component by its degree 2 component. The obvious modification of Theorem +6.5 by replacing nonnegatively graded module by nonpositively graded module, and +generation by degree 2 component by generation by degree −2 component gives the +result we need.) +□ +Theorem 2.8. Let M and N be finitely generated R-modules with ℓ(M ⊗R N) +finite and TorQ +n (M, N) = 0 for all n ≫ 0. There exists a principle lifting R′ of +R such that the corresponding change of rings long exact sequence of Tor (6.1.1) +eventually breaks into short exact sequences +0 → TorR′ +n (M, N) → TorR +n (M, N) → TorR +n−2(M, N) → 0 +for all n ≫ 0. If moreover cxR(M, N) ≥ 1, then +cxR′(M, N) = cxR(M, N) − 1 +and for s ≥ cxR(M, N), +θR′ +s−1(M, N) = θR +s (M, N) +Proof. By Theorem 2.7 there exists a homogeneous element x ∈ R of degree −2 +which eventually surjective on TorR(M, N). By 6.1 and 6.2 in the appendix, there +exists a principal lifting R′ of R and a long exact sequence +· · · → TorR′ +n (M, N) → TorR +n (M, N) +x−→ TorR +n−2(M, N) → · · · +which eventually breaks up into short exact sequences, since multiplication by x is +eventually surjective. +For the moreover statement, the short exact sequences yield the equations +HR′ +e (M, N)(n) = HR +e (M, N)(n) − HR +e (M, N)(n − 1) = ∇HR +e (M, N)(n) +and +HR′ +o (M, N)(n) = HR +o (M, N)(n) − HR +o (M, N)(n − 1) = ∇HR +o (M, N)(n) +for all n ≫ 0. Thus deg P R′,M,N +e +(x) = deg P R,M,N +e +(x) − 1 and deg P R′,M,N +o +(x) = +deg P R,M,N +o +(x) − 1. The result on complexity follows. The final statement follows +from Lemma 2.5. +□ +Complexity. We state some properties of complexity. Analogous results for ordi- +nary Ext complexity are known. Our proofs for length Tor complexity are totally +different. +Lemma 2.9. Suppose that M1, . . . , Mm and N1, . . . , Nn are finitely generated R- +modules such that Mi ⊗R Nj has finite length and TorQ +n (Mi, Nj) = 0 for all i, j and +n ≫ 0, and cxR(Mi, Nj) > 0 for all i and j. Then there exists a principal lifting +R′ of R such that cxR′(Mi, Nj) = cxR(Mi, Nj) − 1 for all i and j. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +9 +Proof. One simply applies Theorem 2.7 to TorR(⊕m +i=1Mi, ⊕n +j=1Nj). Then an ele- +ment x ∈ R−2 eventually surjective on TorR(⊕m +i=1Mi, ⊕n +j=1Nj) implies x is eventu- +ally surjective on each summand TorR(Mi, Nj). The result follows from Theorem +2.8. +□ +The following inequality can be deduced from [1, Corollary 5.7] and [5, Theorem +5.4], but we include a proof here since it is short and fundamentally different, in +that it does not use support varieties. +Theorem 2.10. Assume that M and N are finitely generated R-modules with +ℓ(M ⊗R N) < ∞ and TorQ +n (M, N) = TorQ +n (M, k) = TorQ +n (N, k) = 0 for all n ≫ 0. +Then the inequality +cxR(M, N) ≥ cxR M + cxR N − c +holds. +Proof. We induct on r = cxR(M, N). +Suppose r = 0. +Since cxR(M, N) = +cxR(ΩiM, N) and cxR M = cxR ΩiM for any R-syzygy ΩiM of M (note we also +have TorQ +n (ΩiM, N ⊕ k) = 0 for all n ≫ 0), we can assume that TorR +n (M, N) = 0 +for all n > 0. In this case, if F and G are minimal free resolutions of M and N, +respectively, then F ⊗R G is a minimal free resolution of M ⊗R N. It follows that +P R,M⊗N,k +e +(n) = +� +i≥0 +P R,M,k +e +(i)P R,N,k +e +(n − i) + +� +i≥1 +P R,M,k +o +(i)P R,N,k +o +(n − i) +and +P R,M⊗N,k +o +(n) = +� +i≥0 +P R,M,k +e +(i)P R,N,k +o +(n − i) + +� +i≥1 +P R,M,k +o +(i)P R,N,k +e +(n − i) +Now using the well-known fact that if F(n) is a numerical function of polynomial +type of degree d and G(n) is a numerical function of polynomial type of degree e, +then (F ∗ G)(n) = � +i≥0 F(i)G(n − i) is a numerical function of polynomial type in +n of degree d + e + 1, we see that cxR(M ⊗R N) = cxR M + cxR N. Thus Corollary +2.11 gives the desired inequality in this case. +For the inductive step, assume now that r > 0. Then, necessarily, cxR M > 0 +and cxR N > 0. We now employ Lemma 2.9 to find a principal lifting R′ of R such +that cxR′(M, N) = cxR(M, N) − 1, cxR′ M = cxR M − 1, and cxR′ N = cxR N − 1. +Thus by induction, +cxR(M, N) = cxR′(M, N) + 1 +≥ cxR′ M + cxR′ N − (c − 1) + 1 += cxR M + cxR N − c +as claimed. +□ +Corollary 2.11. Let M and N be finitely generated R-modules with ℓ(M ⊗R N) +finite and TorQ +n (M, N ⊕ k) = 0 for all n ≫ 0. Then the following hold. +(1) cxR(M, N) ≤ cxR M ≤ c. +(2) If 0 → M1 → M2 → M3 → 0 and 0 → N1 → N2 → N3 → 0 are short exact +sequences of finitely generated R-modules with ℓ(Mi ⊗R N), ℓ(M ⊗R Nj) + +10 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +finite and TorQ +n (M ⊕ Mi, N ⊕ Nj) = 0 for all i, j and n ≫ 0, then for +{h, i, j} = {1, 2, 3} we have +cxR(Mh, N) ≤ max{cxR(Mi, N), cxR(Mj, N)} +and +cxR(M, Nh) ≤ max{cxR(M, Ni), cxR(M, Nj)} +(3) If ℓ(M) is finite and TorQ +n (M, M ⊕k) = 0 for all n ≫ 0, then cxR(M, M) = +cxR M. +(4) If R is a complete intersection and M is a Cohen-Macaulay module of grade +g, then cxR M = cxR Extg +R(M, R). +Proof. (1). The second inequality holds from Theorem 2.10 simply by letting N = +0. For the first inequality, we first note that if cxR M = 0, then cxR(M, N) = 0, +since the former condition is equivalent to M having finite projective dimension. +Now induct on r = cxR(M, N). When r = 0 the inequality certainly holds. Now +assume that r > 0. Then cxR M > 0, and by Lemma 2.9 one may find a principal +lifting R′ of R such that cxR′(M, N) = cxR(M, N) − 1 and cxR′ M = cxR M − 1. +Thus by induction +cxR(M, N) = cxR′(M, N) + 1 ≤ cxR′ M + 1 = cxR M +(2). For the first statement we induct on +r = max{cxR(M1, N), cxR(M2, N), cxR(M3, N)} +If r = 0, then the inequality holds. Now assume r > 0. If cxR(Mh, N) = 0, then the +induced long exact sequence of Tor shows that cxR(Mi, N) = cxR(Mj, N), and the +inequality holds for all possible values of h, i, j. Now assume all three complexities +are positive. Then by Lemma 2.9, one may find a principal lifting R′ of R such +that all three complexities are reduced by one. By induction, the inequality holds +over R′, thus also over R. The second inequality is proved similarly. +(3). We induct on r = cxR M. When r = 0 the equality certainly holds. Now +assume r > 0. We claim that cxR(M, M) > 0. Otherwise, according to [3, Theorem +3.3], there exists a residual algebraic closure Q♯ of Q and a nonzerodivisor f in the +square of the maximal ideal of Q♯ such that TorQ♯/(f) +n +(M ⊗Q Q♯, M ⊗Q Q♯) = 0 +for all n ≫ 0 but TorQ♯/(f) +n +(M ⊗Q Q♯, k ⊗Q Q♯) ̸= 0 for all n ≫ 0, and this +contradicts Theorem 2.10. Thus we know cxR(M, M) > 0. Therefore by Lemma +2.9 we can find a principal lifting R′ of R shuch that cxR′(M, M) = cxR(M, M)− 1 +and cxR′ M = cxR M − 1. The result follows by induction. +(4). If r = 0, then M is perfect of grade g. Thus, since R is Gorenstein, we have +that Extg +R(M, R) is also perfect of grade g. In particular, cxR Extg +R(M, R) = 0. If +r > 0, then cxR Extg +R(M, R) > 0. Otherwise M ∼= Extg +R(Extg +R(M, R), R) would be +perfect of grade g. Thus by Lemma 2.9 we may find a principal lifting R′ of R such +that cxR′ M = cxR M − 1 and cxR′ Extg +R(M, R) = cxR Extg +R(M, R) − 1. The result +follows by induction. +□ +Recall that Serre’s Euler characteristic χR(M, N) for intersection multiplicities +in the case where cxR(M, N) = 0 is given by +χR(M, N) = +� +i≥0 +(−1)iℓ(TorR +i (M, N)) + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +11 +Lemma 2.12. Let M and N be finitely generated R-modules with ℓ(M ⊗RN) finite +and TorQ +n (M, N) = 0 for all n ≫ 0. Suppose that cxR(M, N) ≤ 1. Then for any +principal lifting R′ of R for which cxR′(M, N) = 0we have +χR′(M, N) = θR +1 (M, N) +In particular, if cxR(M, N) = 0, then χR′(M, N) = 0. If cxR(M, N) = 0, then +cxR′(M, N) = 0. +Proof. For short, write T R +n for TorR +n (M, N). We have R = R′/(f) and the change +of rings long exact sequence of Tor (6.1.1) +· · · → T R +p+1 → T R +p−1 → T R′ +p +→ · · · +→ T R +3 →T R +1 → T R′ +2 +→ T R +2 → T R +0 → T R′ +1 +→ T R +1 → 0 +Since cxR′(M, N) = 0, we have T R′ +n += 0 for all n ≫ 0. Let q = sup{i | T R′ +i +̸= 0}. +It follows from the long exact sequence that T R +q+2n = T R +q and T R +q+1+2n = T R +q+1 for +all n ≥ 0. If we truncate the long exact sequence at T R′ +q+1 = 0, then the alternating +sum of terms equates to 0: +(−1)qT R +q+1 + +q +� +i=1 +(−1)i−1(ℓT R +i−1 − ℓT R′ +i ++ ℓT R +i ) = 0 +Since the sum is mostly telescoping, it is reduced to +(2.12.1) +� q +� +i=0 +(−1)iℓT R′ +i +� +− (−1)q(ℓT R +q − ℓT R +q+1) = 0 +Note that if cxR(M, N) = 0, then sup{i | T R +i +̸= 0} = q − 1. If q is even then +(−1)q(ℓT R +q − ℓT R +q+1) = ℓT R +q+2 − ℓT R +q+1 = θR +1 (M, N). If q is odd then (−1)q(ℓT R +q − +ℓT R +q+1) = ℓT R +q+1 − ℓT R +q = θ1(M, N). Thus (2.12.1) says +χR′(M, N) − θR +1 (M, N) = 0 +as desired. +The last statement is obvious from the long exact sequence of Tor above. +□ +Lemma 2.12 validates defining θR +0 (M, N) = χR(M, N) when cxR(M, N) = 0. +Thus we make this so. +Proposition 2.13. The invariant θR +r (−, −) is biadditive on short exact sequences +of finitely generated R-modules, when defined, whose pairs have complexity at most +r. +Proof. We induct on r. +When r = 0, we have θR +0 (−, −) = χR(−, −), which is +already known to be biadditive. +When r = 1 Tor modules are periodic. +Thus +Hochster’s proof in [10, p. 98] applies. Now assume that r > 1. Let N be a finitely +generated R-module and 0 → M1 → M2 → M3 → 0 be a short exact sequence of +finitely generated R-modules such that the complexities of the pairs (Mi, N) are at +most r, for i = 1, 2, 3. By Lemma 2.9, Theorem 2.8 and the last statement of Lemma +2.12, we may choose a principal lifting R′ of R such that θR′ +r−1(Mi, N) = θR +r (Mi, N) + +12 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +for i = 1, 2, 3, and all complexities over R′ are at most r − 1. Then by induction +we have +θR +r (M2, N) − θR +r (M1, N) − θR +r (M3, N) += θR′ +r−1(M2, N) − θR′ +r−1(M1, N) − θR′ +r−1(M3, N) = 0 +A similar argument applies in the other argument. +□ +An extension of Hochster’s result. +2.14. Recall Serre’s intersection multiplicity conjectures for finitely generated mod- +ules M and N over a regular local ring Q, satisfying ℓ(M ⊗Q N) < ∞: +(1) Dimension inequality: dim M + dim N ≤ dim Q. +(2) Non-negativity: χQ(M, N) ≥ 0. +(3) Vanishing: If dim M + dim N < dim Q, then χQ(M, N) = 0. +(4) Positivity: If dim M + dim N = dim Q, then χQ(M, N) > 0. +Serre proved (1)-(4) hold in the case where the completion �R is a formal power series +over a field or DVR, and showed that (1) holds in general [20]. Gabber (see [17]) +settled (2), and (3) was shown by Roberts [18] and Gillet-Soul´e [8] independently. +The positivity conjecture (4) is still open in general. +The main result of this section is the following extension of Hochster’s Theorem +[10, Theorem 1.4]. By saying that R is an admissible complete intersection, we mean +that Q satisfies the conclusions of Serre’s intersection multiplicity conjectures 2.14. +Theorem 2.15. Let R be an admissible complete intersection of codimension c, +and let M and N be finitely generated R-modules with ℓ(M ⊗R N) < ∞. +The +following are then equivalent: +(1) θR +c (M, N) = 0; +(2) dim M + dim N < dim R + c. +Proof. We induct on c. +For c = 1, the statement is just Hochster’s result on +admissible hypersurfaces. We include the proof for completeness. We have R = +Q/(f), and we can apply Lemma 2.12 to conclude that χQ(M, N) and θR +1 (M, N) +vanish simultaneously. By Serre’s theorem, The vanishing of χQ(M, N) is equivalent +to dim M + dim N < dim Q = dim R + 1, and we are done in this case. +Now assume that c > 1. Suppose that cxR(M, N) ≥ 1. Then by Theorem 2.8 +there exists a principal lifting R′ of R such that cxR′(M, N) = cxR(M, N)−1, and so +θR′ +c−1(M, N) = θR +c (M, N), regardless of whether cxR(M, N) = c or cxR(M, N) < c. +Now if cxR(M, N) = 0, then cxR′(M, N) = 0 for any principal lifting R′ of R. +Thus by definition, θR′ +c−1(M, N) = θR +c (M, N) = 0. Thus in any case, the vanishing +of θR +c (M, N) is equivalent to the vanishing of θR′ +c−1(M, N), and by induction this is +equivalent to dim M + dim N < dim R′ + c − 1 = dim R + c. +□ +Remark 2.16. Since dim Q = dim R + c, a natural higher codimension analogue +of Serre’s dimension inequality (1) above holds: for finitely generated R-modules +M and N satisfying ℓ(M ⊗R N) < ∞, one has +(2.16.1) +dim M + dim N ≤ dim R + c . +This theorem therefore provides higher codimension analogues of Serre’s conjectures +(3) and (4). Note, however, that the naive analogue of (2) fails: θR +r (M, N) can be +negative, see Section 4. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +13 +We end this section addressing a famous open question on intersections (see [16].) +2.17. Suppose that Q is a local ring and M and N are finitely generated Q-modules +with ℓ(M ⊗Q N) < ∞ and pdQ M < ∞. Then does the following inequality hold? +(2.17.1) +dim M + dim N ≤ dim Q +As is done in [5, Question 8.5], one can ask whether an asymptotic version of +(3.7.1) holds: +2.18. Suppose that M and N are finitely generated R-modules with ℓ(M ⊗R N) < +∞ and TorQ +n (M, N) = 0 for all n ≫ 0. Then does the following inequality hold? +(2.18.1) +dim M + dim N ≤ dim R + cxR M +Proposition 2.19. Let M and N be finitely generated R-modules with ℓ(M ⊗R +N) < ∞ and TorQ +n (M, N) = 0 for all n ≫ 0. Then (2.17.1) holds if and only if +(2.18.1) holds. +Proof. We induct on cxR M. +When cxR M = 0 both questions 2.17 and 2.18 +are the same question. +Now assume that cxR M > 0. +Then by Theorem 2.8 +there exists a principal lifting R′ of R such that cxR′ M = cxR M − 1. Therefore +induction gives that dim M +dim N ≤ dim Q holds if and only if dim M +dim N ≤ +dim R′ + cxR′ M holds, and the latter inequality is equivalent to dim M + dim N ≤ +dim R + cxR M. +□ +3. Depth instead of Dimension +If we use depth instead of dimension, we achieve a version of Theorem 2.15 where +we can replace codimension by complexity and do not need to assume the vanishing +of a theta invariant. In this section, R is assumed to be a complete intersection. +Recall from [11] the generalized Auslander-Buchsbaum formula: +sup{i | TorR +i (M, N) ̸= 0} += sup{depth Rp − depth Mp − depth Np | p ∈ supp M ∩ supp N} +which is shown to hold for finitely generated modules M and N over a complete +intersection R, provided the left hand side is finite. If ℓ(M ⊗R N) < ∞, then the +maximal ideal of R is the only prime in supp M ∩supp N, and so if cxR(M, N) = 0, +then the formula reduces to +(3.0.1) +sup{i | TorR +i (M, N) ̸= 0} = depth R − depth M − depth N +and one concludes that if ℓ(M ⊗R N) < ∞ and cxR(M, N) = 0 then +(3.0.2) +depth M + depth N ≤ depth R +and if R is a complete intersection, then it is Cohen-Macaulay, and depth R can be +replaced by dim R. Of course if M and N are also Cohen-Macaulay modules, then +depth can replaced by dim in all the results below. +Theorem 3.1. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Then +depth M + depth N ≤ dim R + cxR(M, N) . + +14 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +Proof. We induct on r = cxR(M, N). When r = 0, the statement is just (3.0.2). +Now assuming r > 0, by Theorem 2.8 there exists a principal lifting R′ of R such +that cxR′(M, N) = r − 1. Thus by induction we have +depth M + depth N ≤ depth R′ + r − 1 = depth R + r +as required. +□ +Consider the invariant qR(M, N) = sup{i | TorR +i (M, N) ̸= 0}. +3.2. By [11, Lemma 2.9(2)], when qR(M, N) is finite we have +qQ(M, N) = qR(M, N) + c +Using the invariant qQ(M, N), we have a more precise version of Theorem 3.1. +Theorem 3.3. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Then +qQ(M, N) − c = dim R − depth M − depth N +Proof. We have from (3.0.1) and 3.2 +qQ(M, N) = dim Q − depth M − depth N +(3.3.1) += (dim R + c) − depth M − depth N +(3.3.2) +which gives the result. +□ +Thus we get the following. +Corollary 3.4. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Then +depth M + depth N ≤ dim R +if and only if qQ(M, N) ≥ c. +From the definition of qQ(M, N) and 3.2, there are obvious bounds on this +invariant: +3.5. For a complete intersection R and finitely generated R-modules M and N +satisfying ℓ(M ⊗R N) < ∞, we have +c − cxR(M, N) ≤ qQ(M, N) ≤ dim Q . +There are examples showing that these bounds are sharp for pairs of modules M +and N satisfying ℓ(M ⊗R N) < ∞. Indeed, taking both M = k and N = k gives +qQ(M, N) = dim Q. For the other extreme we employ a well-used example from +[12, Example 4.1]. +Example 3.6. Let k be a field, Q = k[x1, . . . , xn, y1, . . . , yn] and +R = Q/(x1y1, . . . , xnyn) +Then for the R-modules M = R/(x1, . . . , xn) and N = R/(y1, . . . , yn) we have +ℓ(M ⊗R N) < ∞, cxR(M, N) = n = c and qQ(M, N) = 0. +The following is an extension of [20, Corollary V.B.6]. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +15 +Corollary 3.7. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Then +qQ(M, N) ≥ dim Q − dim M − dim N +and equality holds if and only if M and N are Cohen-Macaulay. +Proof. From 3.0.1 (with R = Q) we have +qQ(M, N) = dim Q − depth M − depth N += dim Q − dim M − dim N + (dim M − depth M) + (dim N − depth N) +Since the quantities in parentheses are nonnegative, the inequality follows; the +quantities are zero if and only if equality holds. +□ +We consider the Intersection Theorem of Peskine-Szpiro and Roberts: Suppose +that A is a local ring, and M and N are finitely generated A-modules such that +pdA M < ∞ and ℓ(M ⊗AN) < ∞. Then dim N ≤ pdA M, or, using the Auslander- +Buchsbaum Formula, this can be rewritten as +(3.7.1) +dim N + depth M ≤ depth A +We may extend this as follows. +Theorem 3.8. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Then +(3.8.1) +dim N + depth M ≤ dim R + cxR M . +Proof. We induct on s = cxR M. The case s = 0 is equivalent to pdR M < ∞. +Thus in this case (3.7.1) gives the desired inequality. +Now assume s > 0. Then by Theorem 2.8 there exists a principal lifting R′ of R +such that cxR′ M = s − 1. Thus by induction we have +dim N + depth M ≤ dim R′ + cxR′ M = (dim R + 1) + (s − 1) = dim R + s +as desired. +□ +4. Depth versions of Serre’s conjectures in higher codimension +In this section, assume that R = Q/(f1, ..., fc) is a complete intersection. Let M +and N be finitely generated R-modules with ℓ(M ⊗R N) < ∞ and r = cxR(M, N). +In Section 2, we suggest that θR +c should play the role of χQ in a higher codimension +analogue of Serre’s intersection multiplicity conjectures. On the other hand, one +observation of Section 3 is that shifting focus from dimension to depth, and from +codimension to complexity, provides another higher codimension analogue of Serre’s +dimension inequality (see Theorem 3.1): +(4.0.1) +depth M + depth N ≤ dim R + r +Thus we are naturally led to ask: Does equality in (4.0.1) imply positivity of +θR +r (M, N)? We confirm this here, and remark that the converse does not hold. +We also modify an example of Dutta, Hochster, and McLaughlin to show that +θR +r (M, N) be can be negative. +We start with a higher codimension “depth” analogue of the positivity conjec- +ture: + +16 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +Theorem 4.1. Assume that R is a complete intersection, and that M and N are +finitely generated R-modules with ℓ(M ⊗R N) < ∞. Set r = cxR(M, N). Consider +the following conditions +(1) θR +r (M, N) ≤ 0; +(2) depth M + depth N < dim R + r. +Then (1) implies (2). +Proof. We will show that if depth M + depth N = dim R + r, then θR +r (M, N) > 0. +First consider the case where r = 0. In this case, one has that the assumption +implied by (3.0.1) that TorR +i (M, N) = 0 for i > 0. Thus +θR +0 (M, N) = χR(M, N) = ℓ(M ⊗R N) > 0. +Now consider the case when r > 0. By Theorem 2.8, there exists a principal lifting +R′ of R such that +cxR′(M, N) = cxR(M, N) − 1 +and +θR′ +r−1(M, N) = θR +r (M, N). +Note also that dim R′ = dim R + 1. +Thus dim R + r = dim R′ + cxR′(M, N). +Inductively we may thus reduce to the case where cxR(M, N) = 0, and we are done +by the first part. +□ +Remark 4.2. The implication (2) ⇒ (1) in Theorem 4.1 does not always hold: over +a regular local ring Q, take any pair of finitely generated Q-modules M and N, at +least one of which is not Cohen-Macaulay, having χQ(M, N) > 0. (For example, +one could take Q = k[[x, y]], M = Q/(x2, xy), and N = Q/(y), and note that +χQ(M, N) = 1.) In this situation, one would have depth M + depth N < dim Q, +yet θQ +0 (M, N) > 0. Thus the corresponding analogue of the vanishing conjecture +for θR +r relative to the inequality (4.0.1) fails. +It is still natural to wonder whether an analogue of the vanishing conjecture for +θR +r relative to the inequality (4.0.1) holds for Cohen-Macaulay modules, but this +is also not the case. When r = 0, the example from Dutta-Hochster-McLaughlin +[6] provides a hypersurface R and R-modules M and N, where depth M = 0, +depth N = 2, dim R = 3, and r = 0, but θR +0 (M, N) = χR(M, N) < 0. One can +modify that same example to obtain a similar counterexample for θR +1 : +Example 4.3. Let R′ be the local hypersurface from [6]: +R′ = k[X1, X2, X3, X4](X1,X2,X3,X4)/(X1X4 − X2X3) +and consider the local complete intersection R = R′/(x2 +2) having codimension 2 +and dimension 2. Let M be the R′-module of finite projective dimension and finite +length from [6]; it is also an R-module as it is killed by x2 +2. Let N = R/(x1, x2) +and note depth N = 2 = dim N. +It is clear that r = cxR(M, N) is at most 1. To see that it cannot be 0, note +that R′ is a principal lifting of R such that cxR′(M, N) = 0. Now by Lemma 2.12 +and the fact that χR′(M, N) = −1, we see that cxR(M, N) ̸= 0. Again by Lemma +2.12 we have θR +1 (M, N) = χR′(M, N) = −1. Thus the implication +dim M + dim N < dim R + r +=⇒ +θR +1 (M, N) = 0 +fails for a codimension 2 complete intersection R. + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +17 +Remark 4.4. This example also shows that the implication (2) ⇒ (1) in Theorem +2.15 does not hold if you just replace the codimension c by the complexity r in both +conditions of that theorem, even for pairs of Cohen-Macaulay modules. +The following still seems reasonable to ask, however: +Question 4.5. Assume that ℓ(M ⊗R N) < ∞. If dim M + dim N < dim R + r, +does one always have θR +r (M, N) ≤ 0? +5. Lifting pairs of modules +In this section we introduce a condition on a pair of R-modules under which a +version of Theorem 2.15 involving the complexity of the pair of modules holds; cf. +Remark 4.4. Let R = Q/(f1, ..., fc) be as in Notation 1.5. +Definition 5.1. Let M, N be a pair of R-modules with cxR(M, N) = 0 and such +that ℓ(M ⊗R N) < ∞. An intersection lifting of M, N to Q is a pair of Q-modules +M ′, N ′ satisfying: +(1) dim M ′ + dim N ′ = dim M + dim N + c, and +(2) χQ(M ′, N ′) = χR(M, N). +Definition 5.2. A ring R′ is an intermediate complete intersection of R of codi- +mension c′ if there exists a set of generators g1, ..., gc of (f1, ..., fc) such that +R′ = Q/(g1, ..., gc′). In this case, we have R = R′/(gc′+1, ..., gc). +Definition 5.3. Let M, N be a pair of R-modules with cxR(M, N) = r. The pair +M, N is intersection liftable to Q if for any intermediate complete intersection R′ +of codimension c′ with cxR′(M, N) = 0, there exists an intersection lifting to Q of +the pair of R′-modules M, N. +One may compare the next result to Serre’s intersection multiplicity conjectures +recalled in 2.14. +Theorem 5.4. Assume that R = Q/(f1, ..., fc) is a complete intersection. +Let +M, N be a pair of R-modules with cxR(M, N) = r. If M, N is intersection liftable +to Q, then the following hold: +(1) dim M + dim N ≤ dim R + r, +(2) θR +r (M, N) ≥ 0, +(3) If dim M + dim N < dim R + r, then θR +r (M, N) = 0, +(4) If dim M + dim N = dim R + r and Q satisfies Serre’s positivity conjecture, +then θR +r (M, N) > 0. +Proof. Repeatedly applying Theorem 2.8, we obtain an intermediate complete in- +tersection R′ of codimension c − r such that cxR′(M, N) = 0. +By intersection +liftability, one has that there exist Q-modules M ′ and N ′ such that +dim M ′ + dim N ′ = dim M + dim N + c − r +and +χQ(M ′, N ′) = χR′(M, N) . +The proofs of (1)-(4) all reduce to the corresponding results in 2.14. First notice +that dim M + dim N ≤ dim R + r is equivalent to dim M ′ + dim N ′ ≤ dim Q, so (1) +holds. Next, as in the proof of Theorem 2.15, one has θR +r (M, N) = χR′(M, N), thus +θR +r (M, N) = χQ(M ′, N ′). Non-negativity of χQ implies the same for θR +r , so that (2) +holds. Since dim M +dim N < dim R+r is equivalent to dim M ′+dim N ′ < dim Q, +(3) and (4) follow similarly. +□ + +18 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +A first example of intersection liftability is when one of the modules is cyclic and +defined by a regular sequence. +Example 5.5. Let M, N be a pair of R-modules with cxR(M, N) = 0, and suppose +that M = R/(g1, ..., gn), where g1, ..., gn is an R-regular sequence. As Q is local +Noetherian, this lifts to a regular sequence in Q, which (by abuse of notation) we +also call g1, ..., gn. Set M ′ = Q/(g1, ..., gn) and N ′ = N. +We claim that M ′, N ′ is an intersection lifting of M, N to Q. As f1, ..., fc is M ′- +regular, and M = M ′/(f1, ..., fc)M ′, evidently dim M ′ = dim M + c, so condition +(1) of Definition 5.1 holds. The Koszul complex KQ(g1, ..., gn) is a free resolution of +M ′ over Q. In fact, KQ(g1, ..., gn)⊗QR ∼= KR(g1, ..., gn) since g1, ..., gn is R-regular. +Thus for i ≥ 0, +TorQ +i (M ′, N ′) = Hi(KQ(g1, ..., gn) ⊗Q N ′) +∼= Hi(KR(g1, ..., gn) ⊗R N) +∼= TorR +i (M, N) . +The second condition of Definition 5.1, that χQ(M ′, N ′) = χR(M, N), therefore +follows. +In the following discussion we give a common way in which a pair of R-modules +M, N has an intersection lifting. The approach generalizes the previous example. +Definition 5.6. We say that the R-module M lifts to Q if there exists a Q-module +M ′ such that f1, . . . , fc is a regular sequence on M ′ and M ′ ⊗Q R ∼= M. In this +case we call M ′ a lifting of M to Q. +Note that if M ′ is a lifting of M to Q, then +depth M ′ = depth M + c +We recall an important fact regarding change of rings for Tor, [19, 11.51]). +5.7. Suppose that A is a commutative ring, J an ideal of A, and set B = A/J. If +X is an A-module such that TorA +i (X, B) = 0 for all i ≥ 1, then for any B-module +Y we have +TorA +i (X, Y ) ∼= TorB +i (X ⊗A B, Y ) +for all i ≥ 0 . +Theorem 5.8. Let M and N be finitely generated R-modules. Suppose that for +some c′, 0 ≤ c′ ≤ c, M lifts to R1 = Q/(f1, . . . , fc′) with lifting M ′ and N lifts to +R2 = Q/(fc′+1, . . . , fc) with lifting N ′. Then +(1) dim M ′ + dim N ′ = dim M + dim N + c, and +(2) TorQ +i (M ′, N ′) = TorR +i (M, N) for all i ≥ 0. +In particular, cxR(M, N) = 0 if cxQ(M ′, N ′) = 0. +Proof. Since M ′ is a lifting of M to R1, we have depth M ′ = depth M + c − c′, and +since N ′ is a lifting of N to R2, we have depth N ′ = depth N +c′. Thus (1) follows. +For (2), we note that TorR1 +i (M ′, R) is the Koszul homology of the Koszul com- +plex KR1(f c′+1, . . . , fc) ⊗R1 M ′. +Since f c′+1, . . . , fc is regular on M ′, we have +TorR1 +i (M ′, R) = 0 for all i ≥ 1. Thus 5.7 says that +(5.8.1) +TorR1 +i (M ′, N) ∼= TorR +i (M, N) for all i ≥ 0 + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +19 +Since f 1, . . . , f c′ in R2 is regular on N ′, it is also true that f1, . . . , fc′ in Q is regular +on N ′. We have that TorQ +i (R1, N ′) is the Koszul homology of the Koszul complex +KQ(f1, . . . , fc′) ⊗Q N ′. Thus TorQ +i (R1, N ′) = 0 for all i ≥ 1. Now 5.7 gives +(5.8.2) +TorQ +i (M ′, N ′) ∼= TorR1 +i (M ′, N) for all i ≥ 0 +Thus (5.8.1) and (5.8.2) give (2). +□ +Corollary 5.9. Suppose that for some c′, 0 ≤ c′ ≤ c, M lifts to R1 = Q/(f1, . . . , fc′) +with lifting M ′ and N lifts to R2 = Q/(fc′+1, . . . , fc) with lifting N ′. Then the pair +M ′, N ′ is an intersection lifting of M, N. +Here is an example illustrating Theorem 5.8. We use the same ring as in Example +4.3. +Example 5.10. Consider the ring Q = k[X1, X2, X3, X4](X1,X2,X3,X4), and set +R = Q/(X1X4 − X2X3, X2 +2), M = R/(x1, x3, x4), and N = R/(x2). Then clearly +M⊗RN has finite length. One checks easily that M lifts to R1 = Q/(X1X4−X2X3), +with lifting M ′ = Q/(X1, X3, X4), and N lifts to R2 = Q/(X2 +2), with lifting N ′ = +Q/(X2). Thus Theorem 5.8 applies. +6. Appendix +Below are important results that were deferred from the main body of the paper. +6.1. Principal liftings. Let (R′, n′, k) be a local ring, f a nonzerodivisor in n′2, +and denote by R the quotient R′/(f). The following construction from [3, Theorem +3.2] yields an R′-free resolution of an R-module M from an R-free resolution. +Let F = (Fn, ∂n) be an R-free resolution of an R-module M. +We lift F to +a sequence �F = ( �Fn, �∂n) of free R′-modules and maps. +For each n ≥ 0 let +�tn+2 : �Fn+2 → �Fn be the R′-homomorphism with �∂n+1 ◦ �∂n+2 = f · �tn+2. Then +the sequence F ′ given by +· · · +� �F3 ⊕ �F2 +� �∂3 −f +�t3 −�∂2 +� +� �F2 ⊕ �F1 +� �∂2 −f +�t2 −�∂1 +� +� �F1 ⊕ �F0 +[ �∂1 −f ] +� �F0 +is an R′-free resolution of M. +As is proved in [7], the family of maps t = {tn = �tn ⊗ R : Fn → Fn−2} defines +a degree −2 chain endomorphism on F. For degree considerations, we write t : +Σ−1F → ΣF, where ΣsF denotes the shifted complex with (ΣsF)i = Fi−s. Let +C be the cone of t. Then we have the standard short exact sequence of complexes +0 → ΣF → C → F → 0. It is easy to see that C is isomorphic to F ′ ⊗R′ R as +complexes. Thus, for an R-module N, the change of rings long exact sequence of +homology of +0 → ΣF ⊗R N → C ⊗R N → F ⊗R N → 0 +is +... +(6.1.1) +TorR +2 (M, N) → TorR′ +3 (M, N) → TorR +3 (M, N) +x−→ +TorR +1 (M, N) → TorR′ +2 (M, N) → TorR +2 (M, N) +x−→ +TorR +0 (M, N) → TorR′ +1 (M, N) → TorR +1 (M, N) → 0 + +20 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +where the maps x are the TorR +i (t, N). +6.2. Eisenbud operators and the polynomial ring R = R[x1, . . . , xc]. +In +this subsection, Notation 1.5 is in force. We discuss the Eisenbud operators as +per [7]. +Let F = (Fn, ∂n) be a free resolution of the R-module M. Similar to +what we did in 6.1, we lift this resolution to Q as a sequence of homomorphisms +of Q-free modules �F = ( �Fn, �∂n). Since �F is a complex modulo (f1, ..., fc), we can +write �∂2 = � fi�ti, where the �ti are endomorphisms of �F of degree −2. Then as +is discussed in [7], the endomorphisms ti = �ti ⊗ R of F are of degree −2, are well +defined up to homotopy, and commute with each other, up to homotopy. Finally, +the action of xi on TorR(M, N) is given by xi = TorR(ti, N). +Now let x be a linear form of R guaranteed to exist by Theorem 6.5. Write +x = a1x1 + · · · + acxc for ai ∈ R. Then, as is shown in the proof of Theorem 6.5, at +least one of the ai must be a unit in R. Without loss of generality, assume ac is a +unit in R. Let �ai denote a preimage in Q of ai, 1 ≤ i ≤ c. Note that �ac is also a unit +in Q. Set gi = fi − (�ai/�ac)fc for 1 ≤ i ≤ c − 1 and gc = (1/�ac)fc. Then g1, . . . , gc +is another minimal generating set for (f1, ..., fc). Letting R′ = Q/(g1, . . . , gc−1) we +see that t from 6.1 is nothing more than a1t1 + · · · + actc. Thus the x from 6.1 is +eventually surjective. +6.3. Proof of Theorem 1.3. +Proof. Kirby gives a proof in the case where all the xi have degree 1. The obvi- +ous modifications are made when assuming all the xi have degree −1; this is the +statement of Theorem 1.3 assuming d = −1. For the general case where all the +xi have the same negative degree d < 0, the graded module G naturally decom- +poses as a direct sum of d graded submodules G = G(0) ⊕ · · · ⊕ G(d−1), where +G(j) = � +i∈Z Gdi+j. After reindexing, we can assume each G(j) is a module over +a polynomial ring in variables of degree −1, thus obtaining integers r(j) and s(j) +satisfying the conditions of the theorem for d = −1. By taking the appropriate +extreme values of these integers, and indexing back, we obtain the integers r and s +satisfying the conditions of Theorem 1.3 in the general case. +□ +6.4. Existence of eventually regular elements. +Let G be a graded R-module. We say that x ∈ R is eventually regular on G if +(0 :G x)n = 0 for all n ≫ 0. +The following is a variation of [22, Page 285]; therein eventually regular elements +are referred to as superficial. Another synonym of eventually regular occurring in +the literature is filter regular, see [2]. +Theorem 6.5. Assume that A = A0[A2] is a nonnegatively graded ring, generated +over A0 by its degree 2 elements. Assume (A0, m, k) is local with infinite residue +field k. Then A is Noetherian. Let E be a nonnegatively graded A-module, finitely +generated over A. Then there exists x ∈ A2 which is eventually regular on E. +Proof. Let +0 = Q1 ∩ · · · ∩ Qs +be a primary decomposition of 0 in E, and pi = +� +AnnA(E/Qi) be the prime ideal +associated to Qi. +First assume that A2 ⊂ pi for all 1 ≤ i ≤ s. It follows that there exists a ≥ 1 +such that +(A+)a ⊆ AnnA(E/Q1) ∩ · · · ∩ AnnA(E/Qs) + +DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS +21 +or, in other words, +(A+)aE ⊆ Q1 ∩ · · · ∩ Qs = 0 +This can only happen if En = 0 for all n ≫ 0, and so any x ∈ A2 will do the job. +Therefore we may assume that for some 1 ≤ h ≤ s, A2 ̸⊂ pi for 1 ≤ i ≤ h and +A2 ⊂ pi for h + 1 ≤ i ≤ s. Let Si = pi ∩ A2 for 1 ≤ i ≤ h. Then the Si + mA2/mA2 +are proper subspaces of A2/mA2 for each 1 ≤ i ≤ h. Thus there exists x ∈ A2 such +that x+ mA2 is not in Si + mA2/mA2 for all 1 ≤ i ≤ h. In other words, x ∈ A2 − pi +for all 1 ≤ i ≤ h. +As before, there exists an integer b > 0 such that En ⊆ Qh+1 ∩ · · · ∩ Qs for +all n ≥ b. +Let u ∈ (0 :E x)n for n ≥ b. +Then u ∈ Qh+1 ∩ · · · ∩ Qs. +Since +xu ∈ Q1 ∩· · ·∩Qh, and x is not in pi for 1 ≤ i ≤ h, it follows that u ∈ Q1 ∩· · ·∩Qh. +Thus u ∈ Q1 ∩ · · · ∩ Qs = 0. +□ +References +[1] L. L. Avramov and R.-O. Buchweitz. Support varieties and cohomology over complete inter- +sections. Invent. Math., 142(2):285–318, 2000. +[2] P. A. Bergh, S. B. Iyengar, H. Krause, and S. Oppermann. Dimensions of triangulated cate- +gories via Koszul objects. Math. Z., 265(4):849–864, 2010. +[3] P. A. Bergh and D. A. Jorgensen. A generalized Dade’s Lemma for local rings. Algebr. Rep- +resent. Theory 21 (2018), no. 6, 1369—1380, 2015. +[4] W. Bruns and J. Herzog. Cohen-Macaulay Rings, Revised Edition. Cambridge studies in +advanced mathematics, 39, Cambridge University Press, 1998. +[5] H. Dao. Asymptotic behavior of Tor over complete intersections and applications. Preprint, +arXiv:0710.5818, 2007. +[6] S. P. Dutta, M. Hochster, and J. E. McLaughlin. Modules of finite projective dimension with +negative intersection multiplicities. Invent. Math., 79(2):253–291, 1985. +[7] D. Eisenbud. Homological algebra on a complete intersection, with an application to group +representations. Trans. Amer. Math. Soc., 260(1):35–64, July 1980. +[8] H. Gillet and C. Soul´e. Intersection theory using Adams operations. Invent. Math., 90(2):243– +277, 1987. +[9] T. H. Gulliksen. A change of ring theorem with applications to Poincar´e series and intersection +multiplicity. Math. Scand., 34:167–183, 1974. +[10] M. Hochster. The dimension of an intersection in an ambient hypersurface. In Algebraic +geometry (Chicago, Ill., 1980), volume 862 of Lecture Notes in Math., pages 93–106. Springer, +Berlin-New York, 1981. +[11] D. A. Jorgensen. A generalization of the Auslander-Buchsbaum formula. J. Pure Appl. Al- +gebra, 144(2):145–155, 1999. +[12] D. A. Jorgensen. Complexity and Tor on a complete intersection. J. Algebra, 211:578–598, +1999. +[13] P. Jørgensen. Local cohomology for non-commutative graded algebras. Comm. Algebra, +25(2):575–591, 1997. +[14] D. Kirby. Artinian modules and Hilbert polynomials. Quar. J. Math., 24 (1973) 47-57. +[15] W. F. Moore, G. Piepmeyer, S. Spiroff, and M. E. Walker. The vanishing of a higher codi- +mension analogue of Hochster’s theta invariant. Math. Z., 273(3-4):907–920, 2013. +[16] C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. Applications `a +la d´emonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst. Hautes +´Etudes Sci. Publ. Math. No. 42 (1973), 47–119. +[17] P. Roberts. Recent developments on Serre’s multiplicity conjectures: Gabber’s proof of the +nonnegativity conjecture. Enseign. Math. (2), 44(3-4):305–324, 1998. +[18] P. Roberts. The vanishing of intersection multiplicities of perfect complexes. Bull. Amer. +Math. Soc. (N.S.), 13(2):127–130, 1985. +[19] J. Rotman. An introduction to homological algebra. Academic Press, New York, 1979. +[20] J.-P. Serre. Alg`ebre locale. Multiplicit´es. (French) Cours au Coll`ege de France, 1957–1958, +r´edig´e par Pierre Gabriel. Seconde ´edition, 1965. Lecture Notes in Mathematics, 11 Springer- +Verlag, Berlin-New York 1965 vii+188 pp. + +22 +PETTER ANDREAS BERGH, DAVID A. JORGENSEN, AND PEDER THOMPSON +[21] M. van den Bergh. Existence theorems for dualizing complexes over non-commutative graded +and filtered rings. J. Algebra, 195(2):662–679, 1997. +[22] O. Zariski and P. Samuel. Commutative Algebra II. Graduate Texts in Mathematics, Vol. 29. +Springer-Verlag, New York, 1960. +(P. A. Bergh) Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway +Email address: petter.bergh@ntnu.no +URL: https://www.ntnu.edu/employees/petter.bergh +(D. A. Jorgensen) Department of Mathematics, University of Texas at Arlington, 411 +S. Nedderman Drive, Pickard Hall 429, Arlington, TX 76019, U.S.A. +Email address: djorgens@uta.edu +URL: http://www.uta.edu/faculty/djorgens/ +(P. Thompson) Department of Mathematics, Niagara University, NY 14109, U.S.A. +Current address: Department of Mathematics and Physics, M¨alardalen University, V¨aster˚as, +Sweden +Email address: peder.thompson@mdu.se +URL: https://www.mdu.se/en/malardalen-university/staff?id=ptn04 + diff --git a/i9AzT4oBgHgl3EQfbPzh/content/tmp_files/load_file.txt b/i9AzT4oBgHgl3EQfbPzh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31a21638a6dfe2ab989f53047dcea8e75ff0d026 --- /dev/null +++ b/i9AzT4oBgHgl3EQfbPzh/content/tmp_files/load_file.txt @@ -0,0 +1,1178 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf,len=1177 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='01384v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='AC] 3 Jan 2023 DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For a pair of finitely generated modules M and N over a codi- mension c complete intersection ring R with ℓ(M ⊗R N) finite, we pay special attention to the inequality dim M + dim N ≤ dim R + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In particular, we develop an extension of Hochster’s theta invariant whose nonvanishing detects equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In addition, we consider a parallel theory where dimension and codi- mension are replaced by depth and complexity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' introduction Porting geometric ideas into homological algebra, Serre’s development of inter- section multiplicity [20] provided a foundation for many important homological conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A basic result of Serre’s work is that the dimension inequality dim M + dim N ≤ dim Q (1) holds for finitely generated modules M and N over a regular local ring Q, as long as the length ℓ(M ⊗Q N) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Hochster extended these ideas into the realm of hypersurfaces, defining his theta invariant in [10] in part as a mechanism for understanding when this dimension inequality holds for singular rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One aim of the present work is to continue this investigation for complete intersections of arbitrary codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let R be a complete intersection ring of codimension c: that is R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc), where Q is a regular local ring and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc is a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M, N be a pair of finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If the pair has complexity r = cxR(M, N) (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 for the definition of complexity), we define in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2 an invariant θR r (M, N) as the (r − 1)-st difference of the numerical function underlying Hochster’s theta invariant [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This extends Hochster’s theta invariant, and similar invariants have been considered previously by Dao [5], and Moore, Piepmeyer, Spiroff, and Walker [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One of our first key results, proven in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8, is the existence of a complete intersection R′ over which complexity is reduced and this invariant is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Namely, assuming r = cxR(M, N) ≥ 1, we show that there exists a principal lifting Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 13H15, 13D22, 13C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Complete intersection ring, depth inequality, dimension inequality, intersection multiplicity, theta invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Bergh would like to thank the organizers of the Representation Theory program hosted by the Centre for Advanced Study at The Norwegian Academy of Science and Letters, where he spent parts of fall 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thompson is grateful for the support provided by the 2022 Summer Research Grant at Niagara University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 1 2 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON R′ of R such that cxR′(M, N) = r − 1 and θR′ r−1(M, N) = θR r (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This fact plays a central role in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' As a natural extension of Serre’s dimension inequality (1) to complete intersec- tions, one has that, for a pair of finitely generated R-modules M, N with ℓ(M ⊗R N) < ∞, there is a dimension inequality (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='16): dim M + dim N ≤ dim R + c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) Motivated by Hochster’s result for hypersurfaces, we show in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15 that if Q satisfies the conclusions of Serre’s intersection multiplicity conjectures, then vanishing of θR c (M, N) is equivalent to strict inequality in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Although it is unknown whether one always has dim M + dim N ≤ dim R + r, where r = cxR(M, N) (indeed, this is related to a long-standing open question about modules of finite projective dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' see the end of Section 2), we show in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 that depth M + depth N ≤ dim R + r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (3) One naturally wonders whether vanishing of θR r (M, N) might detect whether the inequality (3) is strict: an answer is given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Finally, although strictness in the inequality (3) is a weaker condition than van- ishing of the invariant θR r (M, N), for sufficiently nice pairs of modules much more can be said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This is the investigation of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Under the assumption that a pair of modules M, N is intersection liftable (this notion is defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 and includes the assumption that ℓ(M ⊗R N) is finite), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4 shows that dim M + dim N ≤ dim R + r holds, θR r (M, N) ≥ 0 holds, and θR r (M, N) vanishes precisely when dim M + dim N < dim R + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The first section is devoted to developing preliminary results and recalling definitions about complexity and Hilbert polyno- mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In Section 2 we define θR r (M, N), and prove in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 one of our main tools for reducing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This allows us to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15, motivated by Hochster’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We also give new proofs of some basic properties of complexity, in particular a new proof of a result of Avramov and Buchweitz regarding com- plexity (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Aiming to refine our extension of Hochster’s result, we turn in Section 3 to replacing dimension by depth: the primary result here is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1, and we obtain a number of applications, including an extension of the Intersection Theorem (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Section 4 considers versions of Serre’s multiplicity conjectures in the situation of the previous two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A natural observation that arises over the course of this investigation is that the main result in Section 2 should hold at least for sufficiently nice pairs of modules: these are the modules that are studied in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' An appendix provides additional details for some proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This work was inspired by the paper of Hochster [10], and we found some ideas from [5] useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Preliminaries Let R = R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xc] be a polynomial ring with indeterminates x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xc, all of the same degree d, and R a local ring with residue field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus R is a Noetherian graded ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Throughout this paper, we use ℓ(−) to denote length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 3 We now want to define a dualizing functor which takes graded R-modules to graded R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The version of graded Matlis duality we give below has been considered in more restrictive situations by Jørgensen [13] and van den Bergh [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let E denote the injective hull of k as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For any R-module N of finite length we have (1) ℓ(HomR(N, E)) = ℓ(N), in particular, HomR(N, E) has finite length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) the natural map N → HomR(HomR(N, E), E) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (See [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=') For a graded R-module G = � n∈Z Gn we then define the functor D(G) = � n∈Z HomR(Gn, E) which returns a graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The grading of D(G) is given by D(G)n = HomR(G−n, E) for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The action of xi on D(G) is given by xig(a) = g(xia), for g ∈ HomR(Gn, E) = D(G)−n and a ∈ Gn−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus xig ∈ HomR(Gn−d, E) = D(G)−n+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For a homomorphism h : G → G′ of graded R-modules with h = � n∈Z hn and each hn : Gn → G′ n R-linear, we have D(h) : D(G′) → D(G) given by D(h) = � n∈Z HomR(hn, E) and each HomR(hn, E) : HomR(G′ n, E) → HomR(Gn, E) is R-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now suppose that the graded R-module G is such that each Gn has finite length as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then from the discussion above we see that D(G) is again a graded R-module such that each D(G)n has finite length as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The functor D is exact and G ∼= D(D(G)) for all graded R-modules G with Gn having finite length as an R-module for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The exactness of the functor D follows directly from the exactness of the functor HomR(−, E) from R-modules to R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the second statement, the R-linear isomorphisms ϕn : Gn → HomR(HomR(Gn, E), E) in each degree, mentioned in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1, are given by ϕn(a)(g) = g(a) for a ∈ Gn and g ∈ HomR(Gn, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We claim moreover that these constitute an isomorphism of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Indeed, for a ∈ Gn−d and g ∈ HomR(Gn, E) we have ϕn(xia)(g) = g(xia) = xig(a) = ϕn−d(a)(xig) = xiϕn−d(a)(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus ϕn(xia) = xiϕn−d(a), and so ϕ = � n∈Z ϕn is R-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Recall that a graded R-module G = � n∈Z Gn is Artinian if it satisfies DCC on graded submodules, and Noetherian if it satisfies ACC on graded submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We have the following slight variation of results by Kirby [14], which give equivalent conditions for Artinian and Noetherian graded R-modules G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let G = � n∈Z Gn be a graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that the xi all have negative degree d < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then (1) G is a Noetherian R-module if and only if there exist integers r and s such that (i) Gn = 0 for n > r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (ii) Gn+d = x1Gn + · · · + xcGn for all n ≤ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (iii) Gn is a Noetherian R-module for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) G is an Artinian R-module if and only if there exist integers r and s such that 4 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON (i) Gn = 0 for n < r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (ii) (0 :Gn Rx1 + · · · + Rxc) = 0 for all n ≥ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (iii) Gn is an Artinian R-module for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We provide a proof in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note that Condition (1)(ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 is equivalent to the R-linear maps ρG n : Gc n → Gn+d given by ρG n (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , ac) = x1a1 + · · + xcac being surjective for n ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Similarly, Condition (2)(ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 is equivalent to the R-linear maps ιG n : Gn → Gc n+d given by ιG n (a) = (x1a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xca) being injective for n ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One checks easily that D(ρG n ) = ιD(G) −n−d and D(ιG n ) = ρD(G) −n−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let G be a graded R-module such that each Gi has finite length as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then G is Noetherian (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Artinian) if and only if D(G) is Artinian (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Noetherian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that G is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then, according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3, there exist integers r and s such that (i) Gn = 0 for n > r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (ii) ρG n is surjective for all n ≤ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (iii) Gn is a Noetherian R-module for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus applying D we see that (i) D(G)n = D(G−n) = 0 for n < −r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (ii) ιD(G) n = D(ρG −n−d) is injective for all n ≥ −s + d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (iii) D(G)n = D(G−n) is an Artinian R-module for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 again, we see that D(G) is Artinian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The converse is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Throughout this paper, let R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc), where (Q, n, k) is a local ring and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc is a Q-regular sequence contained in the square of the maximal ideal of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If we say that R is a complete intersection, we mean that Q is a regular local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We also assume M and N are finitely generated R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We let TorR(M, N) denote � n≥0 TorR n (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assumptions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We will assume throughout the paper that ℓ(M ⊗R N), the length of M ⊗R N, is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case, each TorR n (M, N) and TorQ n (M, N) has finite length, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' From now on we assume that R = R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xc] is the polynomial ring in c variables, each of degree d = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Gulliksen shows in [9] that T = TorR(M, N) can be given the structure of graded module over R such that T is an Artinian R-module whenever TorQ(M, N) is an Artinian Q-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We assume throughout that this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We note that since each TorQ n (M, N) has finite length, the condition that TorQ(M, N) is an Artinian Q-module is equivalent to the condition that TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This holds, for example, if R is a complete intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since each xi ∈ R acts on T with degree −2, the graded R-module T naturally decomposes as a direct sum of graded R-modules Te, the Tors with even index, and To, the Tors with odd index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Obviously T is Artinian if and only if both Te and To are Artinian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let HR e (M, N) and HR o (M, N) denote the even and odd Hilbert functions of the pair M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' That is, HR e (M, N)(n) = ℓ(TorR 2n(M, N)) for n ≥ 0 and HR o (M, N)(n) = ℓ(TorR 2n−1(M, N)) for n ≥ 1 Since Te and To are Artinian graded R-modules, D(Te) and D(To) are Noetherian graded R-modules, and so their Hilbert functions are of polynomial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Graded Matlis duality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 also gives the equalities ℓ(TorR n (M, N)) = ℓ(D(TorR n (M, N))) for n ≥ 0 It follows that the numerical functions HR e (M, N) and HR o (M, N) are also of poly- nomial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus there exist polynomials P R,M,N e (x), P R,M,N o (x) ∈ Q[x] such that P R,M,N e (n) = HR e (M, N)(n) and P R,M,N o (n) = HR o (M, N)(n) for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We call P R,M,N e and P R,M,N o the even and odd Hilbert polynomials of the pair M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assuming ℓ(M ⊗R N) < ∞ and TorQ n (M, N) = 0 for all n ≫ 0, we define the complexity, denoted cxR(M, N), of the pair M, N over R as cxR(M, N) = max{deg P R,M,N e (x), deg P R,M,N o (x)} + 1 where by convention we take the degree of the zero polynomial to be −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case we have cxR(M, N) = 0 if and only if P R,M,N e (x) = P R,M,N o (x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We write cxR M to mean cxR(M, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Contrary to some extant literature, we write cxR(M, N) instead of cxR(M, N) to reflect that it is Tor complexity, rather than Ext complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus for r = cxR(M, N) we may write P R,M,N e = axr−1 + g(x) and P R,M,N o = bxr−1 + h(x) where a and b are rational numbers, and deg g(x) < r − 1 and deg h(x) < r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Our definition of complexity of the pair M, N is not the ordinary Ext complexity of the pair M, N defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It is however the same as the length complexity as defined in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Both complexities agree over a complete intersection, when defined, according to [5, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' An extension of Hochster’s theta invariant Let R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc) be as in Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5, where Q is a local ring and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc is a Q-regular sequence contained in the square of the maximal ideal of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Consider an arbitrary numerical function F : Z → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The (first backward) difference operator ∇ applied to F is defined to be ∇F(n) = F(n) − F(n − 1) 6 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON One extends this operator to higher order r ≥ 1 by setting ∇rF = ∇(∇r−1F), where ∇1 = ∇, and ∇0F = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A straightforward computation shows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) ∇rF(n) = r � i=0 (−1)i �r i � F(n − i) Note that ∇ takes a nonzero polynomial function to a polynomial function of degree one less (we assume throughout that deg 0 = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We extend Hochster’s theta invariant as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Consider the numerical function θR(M, N) = HR e (M, N) − HR o (M, N), that is, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2) θR(M, N)(n) = ℓ(TorR 2n(M, N)) − ℓ(TorR 2n−1(M, N)) for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For r ≥ 0 consider the numerical function ∇rθR(M, N), whose explicit expression is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3) ∇rθR(M, N)(n) = r � i=0 (−1)i �r i � � ℓ(TorR 2(n−i)(M, N)) − ℓ(TorR 2(n−i)−1(M, N)) � The proof of the following is immediate from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then we have: (1) if s ≥ r, then ∇sθR(M, N)(n) = 0 for all n ≫ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) ∇r−1θR(M, N)(n) = a−b for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus ∇r−1θR(M, N) is eventually constant, and ∇r−1θR(M, N)(n) = 0 for all n ≫ 0 if and only if P R,M,N e (x) and P R,M,N o (x) have the same degree and leading coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that r = cxR(M, N) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then we set θR s (M, N) = � ∇s−1θR(M, N)(n) for n ≫ 0 if s = r 0 if s > r Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In the case that R is a hypersurface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', a complete intersection with c = 1, the invariant θR 1 is precisely Hochster’s theta invariant from [10], which has been studied in depth by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For a complete intersection R more generally, and a pair of R-modules M, N with r = cxR(M, N) and s ≥ r an integer, our invariant θR s (M, N) and Dao’s invariant ηR s (M, N) from [5] vanish simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In fact, they satisfy (2s · s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' )ηs(M, N) = θR s (M, N) for s ≥ r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Dao’s invariant is defined in terms of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The invariant ηc(M, N) defined (only for the codimension c) in [15] agrees with Dao’s invariant, yet, like our invariant, is defined in terms of the (c − 1)-st difference of Hochster’s theta invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Change of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We say that R′ is a principal lifting of R if there exists a set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , gc of (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc) such that R′ = Q/(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , gc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note in this case we have R = R′/(gc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 7 The existence of short exact sequences of Tor as in the following lemma are guaranteed by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let R′ be a principal lifting of R, and assume that for all large n we have short exact sequences 0 → TorR′ n (M, N) → TorR n (M, N) → TorR n−2(M, N) → 0 Then for r ≥ 1 we have ∇r−1θR′(M, N)(n) = ∇rθR(M, N)(n) for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Abbreviate ℓ(TorR i (M, N)) by ℓR i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' for all n ≫ 0 we have ∇r−1θR′(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' N)(n) = r−1 � i=0 (−1)i�r−1 i � � ℓR′ 2(n−i) − ℓR′ 2(n−i)−1 � = r−1 � i=0 (−1)i�r−1 i � ℓR′ 2(n−i) − r−1 � i=0 (−1)i�r−1 i � ℓR′ 2(n−i)−1 = r−1 � i=0 (−1)i�r−1 i � � ℓR 2(n−i) − ℓR 2(n−i)−2 � − r−1 � i=0 (−1)i�r−1 i � � ℓR 2(n−i)−1 − ℓR 2(n−i)−3 � Now we exchange the second term in parentheses in the first sum with the first term in parentheses in the second sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' getting = r−1 � i=0 (−1)i�r−1 i � � ℓR 2(n−i) − ℓR 2(n−i)−1 � − r−1 � i=0 (−1)i�r−1 i � � ℓR 2(n−i)−2 − ℓR 2(n−i)−3 � = r−1 � i=0 (−1)i�r−1 i � � ℓR 2(n−i) − ℓR 2(n−i)−1 � − r � i=1 (−1)i−1�r−1 i−1 � � ℓR 2(n−i) − ℓR 2(n−i)−1 � = r � i=0 (−1)i ��r−1 i � + �r−1 i−1 �� � ℓR 2(n−i) − ℓR 2(n−i)−1 � = r � i=0 (−1)i�r i � � ℓR 2(n−i) − ℓR 2(n−i)−1 � =∇rθR(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' N)(n) □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let G be a graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It follows immediately from the exactness of D that multiplication by a homogeneous element x ∈ R is eventually surjective on G, meaning Gn x−→ Gn+|x| → 0 is exact for all n ≫ 0, if and only if multiplication by x on D(G) is eventually injective: 0 → D(G)n x−→ D(G)n+|x| is exact for all n ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following is a key result that allows for induction by passing to a principal lifting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' it is a slight generalization of [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗RN) finite and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Consider the graded R-module T = � n≥0 Tn, 8 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON where Tn = TorR n (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then there exists a linear form x ∈ R−2 such that Tn−2 = xTn for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6 we have that T is an Artinian R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Therefore by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4, D(T ) is a Noetherian R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now a variant of a standard argument on the ex- istence of eventually regular elements (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5 in the appendix) applies to show that there exists x ∈ R−2 such that x is eventually injective on D(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus x is eventually surjective on T , by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5 is done for nonneg- atively graded modules over a graded ring generated as an algebra over its degree zero component by its degree 2 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The obvious modification of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5 by replacing nonnegatively graded module by nonpositively graded module, and generation by degree 2 component by generation by degree −2 component gives the result we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=') □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' There exists a principle lifting R′ of R such that the corresponding change of rings long exact sequence of Tor (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) eventually breaks into short exact sequences 0 → TorR′ n (M, N) → TorR n (M, N) → TorR n−2(M, N) → 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If moreover cxR(M, N) ≥ 1, then cxR′(M, N) = cxR(M, N) − 1 and for s ≥ cxR(M, N), θR′ s−1(M, N) = θR s (M, N) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7 there exists a homogeneous element x ∈ R of degree −2 which eventually surjective on TorR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2 in the appendix, there exists a principal lifting R′ of R and a long exact sequence · · → TorR′ n (M, N) → TorR n (M, N) x−→ TorR n−2(M, N) → · · · which eventually breaks up into short exact sequences, since multiplication by x is eventually surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the moreover statement, the short exact sequences yield the equations HR′ e (M, N)(n) = HR e (M, N)(n) − HR e (M, N)(n − 1) = ∇HR e (M, N)(n) and HR′ o (M, N)(n) = HR o (M, N)(n) − HR o (M, N)(n − 1) = ∇HR o (M, N)(n) for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus deg P R′,M,N e (x) = deg P R,M,N e (x) − 1 and deg P R′,M,N o (x) = deg P R,M,N o (x) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The result on complexity follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The final statement follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We state some properties of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Analogous results for ordi- nary Ext complexity are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Our proofs for length Tor complexity are totally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , Mm and N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , Nn are finitely generated R- modules such that Mi ⊗R Nj has finite length and TorQ n (Mi, Nj) = 0 for all i, j and n ≫ 0, and cxR(Mi, Nj) > 0 for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then there exists a principal lifting R′ of R such that cxR′(Mi, Nj) = cxR(Mi, Nj) − 1 for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One simply applies Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7 to TorR(⊕m i=1Mi, ⊕n j=1Nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then an ele- ment x ∈ R−2 eventually surjective on TorR(⊕m i=1Mi, ⊕n j=1Nj) implies x is eventu- ally surjective on each summand TorR(Mi, Nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The result follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ The following inequality can be deduced from [1, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7] and [5, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4], but we include a proof here since it is short and fundamentally different, in that it does not use support varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞ and TorQ n (M, N) = TorQ n (M, k) = TorQ n (N, k) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then the inequality cxR(M, N) ≥ cxR M + cxR N − c holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since cxR(M, N) = cxR(ΩiM, N) and cxR M = cxR ΩiM for any R-syzygy ΩiM of M (note we also have TorQ n (ΩiM, N ⊕ k) = 0 for all n ≫ 0), we can assume that TorR n (M, N) = 0 for all n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case, if F and G are minimal free resolutions of M and N, respectively, then F ⊗R G is a minimal free resolution of M ⊗R N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It follows that P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M⊗N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k e (n) = � i≥0 P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k e (i)P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k e (n − i) + � i≥1 P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k o (i)P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k o (n − i) and P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M⊗N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k o (n) = � i≥0 P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k e (i)P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k o (n − i) + � i≥1 P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k o (i)P R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='k e (n − i) Now using the well-known fact that if F(n) is a numerical function of polynomial type of degree d and G(n) is a numerical function of polynomial type of degree e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' then (F ∗ G)(n) = � i≥0 F(i)G(n − i) is a numerical function of polynomial type in n of degree d + e + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' we see that cxR(M ⊗R N) = cxR M + cxR N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='11 gives the desired inequality in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the inductive step, assume now that r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then, necessarily, cxR M > 0 and cxR N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We now employ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9 to find a principal lifting R′ of R such that cxR′(M, N) = cxR(M, N) − 1, cxR′ M = cxR M − 1, and cxR′ N = cxR N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by induction, cxR(M, N) = cxR′(M, N) + 1 ≥ cxR′ M + cxR′ N − (c − 1) + 1 = cxR M + cxR N − c as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) finite and TorQ n (M, N ⊕ k) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (1) cxR(M, N) ≤ cxR M ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) If 0 → M1 → M2 → M3 → 0 and 0 → N1 → N2 → N3 → 0 are short exact sequences of finitely generated R-modules with ℓ(Mi ⊗R N), ℓ(M ⊗R Nj) 10 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON finite and TorQ n (M ⊕ Mi, N ⊕ Nj) = 0 for all i, j and n ≫ 0, then for {h, i, j} = {1, 2, 3} we have cxR(Mh, N) ≤ max{cxR(Mi, N), cxR(Mj, N)} and cxR(M, Nh) ≤ max{cxR(M, Ni), cxR(M, Nj)} (3) If ℓ(M) is finite and TorQ n (M, M ⊕k) = 0 for all n ≫ 0, then cxR(M, M) = cxR M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (4) If R is a complete intersection and M is a Cohen-Macaulay module of grade g, then cxR M = cxR Extg R(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The second inequality holds from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='10 simply by letting N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the first inequality, we first note that if cxR M = 0, then cxR(M, N) = 0, since the former condition is equivalent to M having finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now induct on r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 0 the inequality certainly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume that r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then cxR M > 0, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9 one may find a principal lifting R′ of R such that cxR′(M, N) = cxR(M, N) − 1 and cxR′ M = cxR M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by induction cxR(M, N) = cxR′(M, N) + 1 ≤ cxR′ M + 1 = cxR M (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the first statement we induct on r = max{cxR(M1, N), cxR(M2, N), cxR(M3, N)} If r = 0, then the inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If cxR(Mh, N) = 0, then the induced long exact sequence of Tor shows that cxR(Mi, N) = cxR(Mj, N), and the inequality holds for all possible values of h, i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume all three complexities are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9, one may find a principal lifting R′ of R such that all three complexities are reduced by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By induction, the inequality holds over R′, thus also over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The second inequality is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on r = cxR M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 0 the equality certainly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We claim that cxR(M, M) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Otherwise, according to [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3], there exists a residual algebraic closure Q♯ of Q and a nonzerodivisor f in the square of the maximal ideal of Q♯ such that TorQ♯/(f) n (M ⊗Q Q♯, M ⊗Q Q♯) = 0 for all n ≫ 0 but TorQ♯/(f) n (M ⊗Q Q♯, k ⊗Q Q♯) ̸= 0 for all n ≫ 0, and this contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus we know cxR(M, M) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Therefore by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9 we can find a principal lifting R′ of R shuch that cxR′(M, M) = cxR(M, M)− 1 and cxR′ M = cxR M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If r = 0, then M is perfect of grade g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus, since R is Gorenstein, we have that Extg R(M, R) is also perfect of grade g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In particular, cxR Extg R(M, R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If r > 0, then cxR Extg R(M, R) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Otherwise M ∼= Extg R(Extg R(M, R), R) would be perfect of grade g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9 we may find a principal lifting R′ of R such that cxR′ M = cxR M − 1 and cxR′ Extg R(M, R) = cxR Extg R(M, R) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Recall that Serre’s Euler characteristic χR(M, N) for intersection multiplicities in the case where cxR(M, N) = 0 is given by χR(M, N) = � i≥0 (−1)iℓ(TorR i (M, N)) DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗RN) finite and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that cxR(M, N) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then for any principal lifting R′ of R for which cxR′(M, N) = 0we have χR′(M, N) = θR 1 (M, N) In particular, if cxR(M, N) = 0, then χR′(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If cxR(M, N) = 0, then cxR′(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For short, write T R n for TorR n (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We have R = R′/(f) and the change of rings long exact sequence of Tor (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) · · → T R p+1 → T R p−1 → T R′ p → · · · → T R 3 →T R 1 → T R′ 2 → T R 2 → T R 0 → T R′ 1 → T R 1 → 0 Since cxR′(M, N) = 0, we have T R′ n = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let q = sup{i | T R′ i ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It follows from the long exact sequence that T R q+2n = T R q and T R q+1+2n = T R q+1 for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If we truncate the long exact sequence at T R′ q+1 = 0, then the alternating sum of terms equates to 0: (−1)qT R q+1 + q � i=1 (−1)i−1(ℓT R i−1 − ℓT R′ i + ℓT R i ) = 0 Since the sum is mostly telescoping, it is reduced to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) � q � i=0 (−1)iℓT R′ i � − (−1)q(ℓT R q − ℓT R q+1) = 0 Note that if cxR(M, N) = 0, then sup{i | T R i ̸= 0} = q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If q is even then (−1)q(ℓT R q − ℓT R q+1) = ℓT R q+2 − ℓT R q+1 = θR 1 (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If q is odd then (−1)q(ℓT R q − ℓT R q+1) = ℓT R q+1 − ℓT R q = θ1(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) says χR′(M, N) − θR 1 (M, N) = 0 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The last statement is obvious from the long exact sequence of Tor above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12 validates defining θR 0 (M, N) = χR(M, N) when cxR(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus we make this so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The invariant θR r (−, −) is biadditive on short exact sequences of finitely generated R-modules, when defined, whose pairs have complexity at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 0, we have θR 0 (−, −) = χR(−, −), which is already known to be biadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 1 Tor modules are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus Hochster’s proof in [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 98] applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume that r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let N be a finitely generated R-module and 0 → M1 → M2 → M3 → 0 be a short exact sequence of finitely generated R-modules such that the complexities of the pairs (Mi, N) are at most r, for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 and the last statement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12, we may choose a principal lifting R′ of R such that θR′ r−1(Mi, N) = θR r (Mi, N) 12 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON for i = 1, 2, 3, and all complexities over R′ are at most r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then by induction we have θR r (M2, N) − θR r (M1, N) − θR r (M3, N) = θR′ r−1(M2, N) − θR′ r−1(M1, N) − θR′ r−1(M3, N) = 0 A similar argument applies in the other argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ An extension of Hochster’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Recall Serre’s intersection multiplicity conjectures for finitely generated mod- ules M and N over a regular local ring Q, satisfying ℓ(M ⊗Q N) < ∞: (1) Dimension inequality: dim M + dim N ≤ dim Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) Non-negativity: χQ(M, N) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (3) Vanishing: If dim M + dim N < dim Q, then χQ(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (4) Positivity: If dim M + dim N = dim Q, then χQ(M, N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Serre proved (1)-(4) hold in the case where the completion �R is a formal power series over a field or DVR, and showed that (1) holds in general [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Gabber (see [17]) settled (2), and (3) was shown by Roberts [18] and Gillet-Soul´e [8] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The positivity conjecture (4) is still open in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The main result of this section is the following extension of Hochster’s Theorem [10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By saying that R is an admissible complete intersection, we mean that Q satisfies the conclusions of Serre’s intersection multiplicity conjectures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let R be an admissible complete intersection of codimension c, and let M and N be finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following are then equivalent: (1) θR c (M, N) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) dim M + dim N < dim R + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For c = 1, the statement is just Hochster’s result on admissible hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We include the proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We have R = Q/(f), and we can apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12 to conclude that χQ(M, N) and θR 1 (M, N) vanish simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By Serre’s theorem, The vanishing of χQ(M, N) is equivalent to dim M + dim N < dim Q = dim R + 1, and we are done in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume that c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that cxR(M, N) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 there exists a principal lifting R′ of R such that cxR′(M, N) = cxR(M, N)−1, and so θR′ c−1(M, N) = θR c (M, N), regardless of whether cxR(M, N) = c or cxR(M, N) < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now if cxR(M, N) = 0, then cxR′(M, N) = 0 for any principal lifting R′ of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by definition, θR′ c−1(M, N) = θR c (M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus in any case, the vanishing of θR c (M, N) is equivalent to the vanishing of θR′ c−1(M, N), and by induction this is equivalent to dim M + dim N < dim R′ + c − 1 = dim R + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since dim Q = dim R + c, a natural higher codimension analogue of Serre’s dimension inequality (1) above holds: for finitely generated R-modules M and N satisfying ℓ(M ⊗R N) < ∞, one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) dim M + dim N ≤ dim R + c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This theorem therefore provides higher codimension analogues of Serre’s conjectures (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note, however, that the naive analogue of (2) fails: θR r (M, N) can be negative, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 13 We end this section addressing a famous open question on intersections (see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that Q is a local ring and M and N are finitely generated Q-modules with ℓ(M ⊗Q N) < ∞ and pdQ M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then does the following inequality hold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) dim M + dim N ≤ dim Q As is done in [5, Question 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5], one can ask whether an asymptotic version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) holds: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞ and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then does the following inequality hold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) dim M + dim N ≤ dim R + cxR M Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) < ∞ and TorQ n (M, N) = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) holds if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on cxR M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When cxR M = 0 both questions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='17 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='18 are the same question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume that cxR M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 there exists a principal lifting R′ of R such that cxR′ M = cxR M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Therefore induction gives that dim M +dim N ≤ dim Q holds if and only if dim M +dim N ≤ dim R′ + cxR′ M holds, and the latter inequality is equivalent to dim M + dim N ≤ dim R + cxR M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Depth instead of Dimension If we use depth instead of dimension, we achieve a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15 where we can replace codimension by complexity and do not need to assume the vanishing of a theta invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this section, R is assumed to be a complete intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Recall from [11] the generalized Auslander-Buchsbaum formula: sup{i | TorR i (M, N) ̸= 0} = sup{depth Rp − depth Mp − depth Np | p ∈ supp M ∩ supp N} which is shown to hold for finitely generated modules M and N over a complete intersection R, provided the left hand side is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If ℓ(M ⊗R N) < ∞, then the maximal ideal of R is the only prime in supp M ∩supp N, and so if cxR(M, N) = 0, then the formula reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) sup{i | TorR i (M, N) ̸= 0} = depth R − depth M − depth N and one concludes that if ℓ(M ⊗R N) < ∞ and cxR(M, N) = 0 then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2) depth M + depth N ≤ depth R and if R is a complete intersection, then it is Cohen-Macaulay, and depth R can be replaced by dim R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Of course if M and N are also Cohen-Macaulay modules, then depth can replaced by dim in all the results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then depth M + depth N ≤ dim R + cxR(M, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 14 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 0, the statement is just (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assuming r > 0, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 there exists a principal lifting R′ of R such that cxR′(M, N) = r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by induction we have depth M + depth N ≤ depth R′ + r − 1 = depth R + r as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Consider the invariant qR(M, N) = sup{i | TorR i (M, N) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9(2)], when qR(M, N) is finite we have qQ(M, N) = qR(M, N) + c Using the invariant qQ(M, N), we have a more precise version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then qQ(M, N) − c = dim R − depth M − depth N Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2 qQ(M, N) = dim Q − depth M − depth N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) = (dim R + c) − depth M − depth N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2) which gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Thus we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then depth M + depth N ≤ dim R if and only if qQ(M, N) ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' From the definition of qQ(M, N) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2, there are obvious bounds on this invariant: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For a complete intersection R and finitely generated R-modules M and N satisfying ℓ(M ⊗R N) < ∞, we have c − cxR(M, N) ≤ qQ(M, N) ≤ dim Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' There are examples showing that these bounds are sharp for pairs of modules M and N satisfying ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Indeed, taking both M = k and N = k gives qQ(M, N) = dim Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the other extreme we employ a well-used example from [12, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let k be a field, Q = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , yn] and R = Q/(x1y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xnyn) Then for the R-modules M = R/(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xn) and N = R/(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , yn) we have ℓ(M ⊗R N) < ∞, cxR(M, N) = n = c and qQ(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following is an extension of [20, Corollary V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 15 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then qQ(M, N) ≥ dim Q − dim M − dim N and equality holds if and only if M and N are Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' From 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 (with R = Q) we have qQ(M, N) = dim Q − depth M − depth N = dim Q − dim M − dim N + (dim M − depth M) + (dim N − depth N) Since the quantities in parentheses are nonnegative, the inequality follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' the quantities are zero if and only if equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ We consider the Intersection Theorem of Peskine-Szpiro and Roberts: Suppose that A is a local ring, and M and N are finitely generated A-modules such that pdA M < ∞ and ℓ(M ⊗AN) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then dim N ≤ pdA M, or, using the Auslander- Buchsbaum Formula, this can be rewritten as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) dim N + depth M ≤ depth A We may extend this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) dim N + depth M ≤ dim R + cxR M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We induct on s = cxR M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The case s = 0 is equivalent to pdR M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus in this case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) gives the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now assume s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 there exists a principal lifting R′ of R such that cxR′ M = s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus by induction we have dim N + depth M ≤ dim R′ + cxR′ M = (dim R + 1) + (s − 1) = dim R + s as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Depth versions of Serre’s conjectures in higher codimension In this section, assume that R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc) is a complete intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules with ℓ(M ⊗R N) < ∞ and r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In Section 2, we suggest that θR c should play the role of χQ in a higher codimension analogue of Serre’s intersection multiplicity conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' On the other hand, one observation of Section 3 is that shifting focus from dimension to depth, and from codimension to complexity, provides another higher codimension analogue of Serre’s dimension inequality (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) depth M + depth N ≤ dim R + r Thus we are naturally led to ask: Does equality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) imply positivity of θR r (M, N)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We confirm this here, and remark that the converse does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We also modify an example of Dutta, Hochster, and McLaughlin to show that θR r (M, N) be can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We start with a higher codimension “depth” analogue of the positivity conjec- ture: 16 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R is a complete intersection, and that M and N are finitely generated R-modules with ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Set r = cxR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Consider the following conditions (1) θR r (M, N) ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (2) depth M + depth N < dim R + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We will show that if depth M + depth N = dim R + r, then θR r (M, N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' First consider the case where r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case, one has that the assumption implied by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) that TorR i (M, N) = 0 for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus θR 0 (M, N) = χR(M, N) = ℓ(M ⊗R N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now consider the case when r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8, there exists a principal lifting R′ of R such that cxR′(M, N) = cxR(M, N) − 1 and θR′ r−1(M, N) = θR r (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note also that dim R′ = dim R + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus dim R + r = dim R′ + cxR′(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Inductively we may thus reduce to the case where cxR(M, N) = 0, and we are done by the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The implication (2) ⇒ (1) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 does not always hold: over a regular local ring Q, take any pair of finitely generated Q-modules M and N, at least one of which is not Cohen-Macaulay, having χQ(M, N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (For example, one could take Q = k[[x, y]], M = Q/(x2, xy), and N = Q/(y), and note that χQ(M, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=') In this situation, one would have depth M + depth N < dim Q, yet θQ 0 (M, N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus the corresponding analogue of the vanishing conjecture for θR r relative to the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It is still natural to wonder whether an analogue of the vanishing conjecture for θR r relative to the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) holds for Cohen-Macaulay modules, but this is also not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' When r = 0, the example from Dutta-Hochster-McLaughlin [6] provides a hypersurface R and R-modules M and N, where depth M = 0, depth N = 2, dim R = 3, and r = 0, but θR 0 (M, N) = χR(M, N) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One can modify that same example to obtain a similar counterexample for θR 1 : Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let R′ be the local hypersurface from [6]: R′ = k[X1, X2, X3, X4](X1,X2,X3,X4)/(X1X4 − X2X3) and consider the local complete intersection R = R′/(x2 2) having codimension 2 and dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M be the R′-module of finite projective dimension and finite length from [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' it is also an R-module as it is killed by x2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let N = R/(x1, x2) and note depth N = 2 = dim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It is clear that r = cxR(M, N) is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' To see that it cannot be 0, note that R′ is a principal lifting of R such that cxR′(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12 and the fact that χR′(M, N) = −1, we see that cxR(M, N) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Again by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='12 we have θR 1 (M, N) = χR′(M, N) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus the implication dim M + dim N < dim R + r =⇒ θR 1 (M, N) = 0 fails for a codimension 2 complete intersection R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 17 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' This example also shows that the implication (2) ⇒ (1) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15 does not hold if you just replace the codimension c by the complexity r in both conditions of that theorem, even for pairs of Cohen-Macaulay modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following still seems reasonable to ask, however: Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If dim M + dim N < dim R + r, does one always have θR r (M, N) ≤ 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Lifting pairs of modules In this section we introduce a condition on a pair of R-modules under which a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15 involving the complexity of the pair of modules holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc) be as in Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M, N be a pair of R-modules with cxR(M, N) = 0 and such that ℓ(M ⊗R N) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' An intersection lifting of M, N to Q is a pair of Q-modules M ′, N ′ satisfying: (1) dim M ′ + dim N ′ = dim M + dim N + c, and (2) χQ(M ′, N ′) = χR(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A ring R′ is an intermediate complete intersection of R of codi- mension c′ if there exists a set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gc of (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc) such that R′ = Q/(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gc′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case, we have R = R′/(gc′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M, N be a pair of R-modules with cxR(M, N) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The pair M, N is intersection liftable to Q if for any intermediate complete intersection R′ of codimension c′ with cxR′(M, N) = 0, there exists an intersection lifting to Q of the pair of R′-modules M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One may compare the next result to Serre’s intersection multiplicity conjectures recalled in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that R = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc) is a complete intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M, N be a pair of R-modules with cxR(M, N) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If M, N is intersection liftable to Q, then the following hold: (1) dim M + dim N ≤ dim R + r, (2) θR r (M, N) ≥ 0, (3) If dim M + dim N < dim R + r, then θR r (M, N) = 0, (4) If dim M + dim N = dim R + r and Q satisfies Serre’s positivity conjecture, then θR r (M, N) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Repeatedly applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8, we obtain an intermediate complete in- tersection R′ of codimension c − r such that cxR′(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By intersection liftability, one has that there exist Q-modules M ′ and N ′ such that dim M ′ + dim N ′ = dim M + dim N + c − r and χQ(M ′, N ′) = χR′(M, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The proofs of (1)-(4) all reduce to the corresponding results in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' First notice that dim M + dim N ≤ dim R + r is equivalent to dim M ′ + dim N ′ ≤ dim Q, so (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Next, as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='15, one has θR r (M, N) = χR′(M, N), thus θR r (M, N) = χQ(M ′, N ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Non-negativity of χQ implies the same for θR r , so that (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since dim M +dim N < dim R+r is equivalent to dim M ′+dim N ′ < dim Q, (3) and (4) follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ 18 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON A first example of intersection liftability is when one of the modules is cyclic and defined by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M, N be a pair of R-modules with cxR(M, N) = 0, and suppose that M = R/(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn), where g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn is an R-regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' As Q is local Noetherian, this lifts to a regular sequence in Q, which (by abuse of notation) we also call g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Set M ′ = Q/(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn) and N ′ = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We claim that M ′, N ′ is an intersection lifting of M, N to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' As f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc is M ′- regular, and M = M ′/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc)M ′, evidently dim M ′ = dim M + c, so condition (1) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The Koszul complex KQ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn) is a free resolution of M ′ over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In fact, KQ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn)⊗QR ∼= KR(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn) since g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn is R-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus for i ≥ 0, TorQ i (M ′, N ′) = Hi(KQ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn) ⊗Q N ′) ∼= Hi(KR(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', gn) ⊗R N) ∼= TorR i (M, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The second condition of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1, that χQ(M ′, N ′) = χR(M, N), therefore follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In the following discussion we give a common way in which a pair of R-modules M, N has an intersection lifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The approach generalizes the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We say that the R-module M lifts to Q if there exists a Q-module M ′ such that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc is a regular sequence on M ′ and M ′ ⊗Q R ∼= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this case we call M ′ a lifting of M to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note that if M ′ is a lifting of M to Q, then depth M ′ = depth M + c We recall an important fact regarding change of rings for Tor, [19, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that A is a commutative ring, J an ideal of A, and set B = A/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' If X is an A-module such that TorA i (X, B) = 0 for all i ≥ 1, then for any B-module Y we have TorA i (X, Y ) ∼= TorB i (X ⊗A B, Y ) for all i ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let M and N be finitely generated R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that for some c′, 0 ≤ c′ ≤ c, M lifts to R1 = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc′) with lifting M ′ and N lifts to R2 = Q/(fc′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc) with lifting N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then (1) dim M ′ + dim N ′ = dim M + dim N + c, and (2) TorQ i (M ′, N ′) = TorR i (M, N) for all i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In particular, cxR(M, N) = 0 if cxQ(M ′, N ′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since M ′ is a lifting of M to R1, we have depth M ′ = depth M + c − c′, and since N ′ is a lifting of N to R2, we have depth N ′ = depth N +c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For (2), we note that TorR1 i (M ′, R) is the Koszul homology of the Koszul com- plex KR1(f c′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc) ⊗R1 M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since f c′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc is regular on M ′, we have TorR1 i (M ′, R) = 0 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7 says that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) TorR1 i (M ′, N) ∼= TorR i (M, N) for all i ≥ 0 DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 19 Since f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , f c′ in R2 is regular on N ′, it is also true that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc′ in Q is regular on N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We have that TorQ i (R1, N ′) is the Koszul homology of the Koszul complex KQ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc′) ⊗Q N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus TorQ i (R1, N ′) = 0 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='7 gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2) TorQ i (M ′, N ′) ∼= TorR1 i (M ′, N) for all i ≥ 0 Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2) give (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Suppose that for some c′, 0 ≤ c′ ≤ c, M lifts to R1 = Q/(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc′) with lifting M ′ and N lifts to R2 = Q/(fc′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , fc) with lifting N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then the pair M ′, N ′ is an intersection lifting of M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Here is an example illustrating Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We use the same ring as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Consider the ring Q = k[X1, X2, X3, X4](X1,X2,X3,X4), and set R = Q/(X1X4 − X2X3, X2 2), M = R/(x1, x3, x4), and N = R/(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then clearly M⊗RN has finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' One checks easily that M lifts to R1 = Q/(X1X4−X2X3), with lifting M ′ = Q/(X1, X3, X4), and N lifts to R2 = Q/(X2 2), with lifting N ′ = Q/(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='8 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Appendix Below are important results that were deferred from the main body of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Principal liftings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let (R′, n′, k) be a local ring, f a nonzerodivisor in n′2, and denote by R the quotient R′/(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following construction from [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2] yields an R′-free resolution of an R-module M from an R-free resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let F = (Fn, ∂n) be an R-free resolution of an R-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We lift F to a sequence �F = ( �Fn, �∂n) of free R′-modules and maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For each n ≥ 0 let �tn+2 : �Fn+2 → �Fn be the R′-homomorphism with �∂n+1 ◦ �∂n+2 = f · �tn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then the sequence F ′ given by · · � �F3 ⊕ �F2 � �∂3 −f �t3 −�∂2 � � �F2 ⊕ �F1 � �∂2 −f �t2 −�∂1 � � �F1 ⊕ �F0 [ �∂1 −f ] � �F0 is an R′-free resolution of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' As is proved in [7], the family of maps t = {tn = �tn ⊗ R : Fn → Fn−2} defines a degree −2 chain endomorphism on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For degree considerations, we write t : Σ−1F → ΣF, where ΣsF denotes the shifted complex with (ΣsF)i = Fi−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let C be the cone of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then we have the standard short exact sequence of complexes 0 → ΣF → C → F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It is easy to see that C is isomorphic to F ′ ⊗R′ R as complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus, for an R-module N, the change of rings long exact sequence of homology of 0 → ΣF ⊗R N → C ⊗R N → F ⊗R N → 0 is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1) TorR 2 (M, N) → TorR′ 3 (M, N) → TorR 3 (M, N) x−→ TorR 1 (M, N) → TorR′ 2 (M, N) → TorR 2 (M, N) x−→ TorR 0 (M, N) → TorR′ 1 (M, N) → TorR 1 (M, N) → 0 20 PETTER ANDREAS BERGH, DAVID A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' JORGENSEN, AND PEDER THOMPSON where the maps x are the TorR i (t, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Eisenbud operators and the polynomial ring R = R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , xc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In this subsection, Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5 is in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We discuss the Eisenbud operators as per [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let F = (Fn, ∂n) be a free resolution of the R-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Similar to what we did in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1, we lift this resolution to Q as a sequence of homomorphisms of Q-free modules �F = ( �Fn, �∂n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since �F is a complex modulo (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc), we can write �∂2 = � fi�ti, where the �ti are endomorphisms of �F of degree −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then as is discussed in [7], the endomorphisms ti = �ti ⊗ R of F are of degree −2, are well defined up to homotopy, and commute with each other, up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Finally, the action of xi on TorR(M, N) is given by xi = TorR(ti, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Now let x be a linear form of R guaranteed to exist by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Write x = a1x1 + · · · + acxc for ai ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then, as is shown in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5, at least one of the ai must be a unit in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Without loss of generality, assume ac is a unit in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let �ai denote a preimage in Q of ai, 1 ≤ i ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Note that �ac is also a unit in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Set gi = fi − (�ai/�ac)fc for 1 ≤ i ≤ c − 1 and gc = (1/�ac)fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , gc is another minimal generating set for (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=', fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Letting R′ = Q/(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' , gc−1) we see that t from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 is nothing more than a1t1 + · · · + actc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus the x from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='1 is eventually surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Kirby gives a proof in the case where all the xi have degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The obvi- ous modifications are made when assuming all the xi have degree −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' this is the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 assuming d = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' For the general case where all the xi have the same negative degree d < 0, the graded module G naturally decom- poses as a direct sum of d graded submodules G = G(0) ⊕ · · · ⊕ G(d−1), where G(j) = � i∈Z Gdi+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' After reindexing, we can assume each G(j) is a module over a polynomial ring in variables of degree −1, thus obtaining integers r(j) and s(j) satisfying the conditions of the theorem for d = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' By taking the appropriate extreme values of these integers, and indexing back, we obtain the integers r and s satisfying the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='3 in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Existence of eventually regular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let G be a graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' We say that x ∈ R is eventually regular on G if (0 :G x)n = 0 for all n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' The following is a variation of [22, Page 285];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' therein eventually regular elements are referred to as superficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Another synonym of eventually regular occurring in the literature is filter regular, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume that A = A0[A2] is a nonnegatively graded ring, generated over A0 by its degree 2 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Assume (A0, m, k) is local with infinite residue field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then A is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let E be a nonnegatively graded A-module, finitely generated over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then there exists x ∈ A2 which is eventually regular on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let 0 = Q1 ∩ · · · ∩ Qs be a primary decomposition of 0 in E, and pi = � AnnA(E/Qi) be the prime ideal associated to Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' First assume that A2 ⊂ pi for all 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' It follows that there exists a ≥ 1 such that (A+)a ⊆ AnnA(E/Q1) ∩ · · · ∩ AnnA(E/Qs) DIMENSION AND DEPTH INEQUALITIES OVER COMPLETE INTERSECTIONS 21 or, in other words, (A+)aE ⊆ Q1 ∩ · · · ∩ Qs = 0 This can only happen if En = 0 for all n ≫ 0, and so any x ∈ A2 will do the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Therefore we may assume that for some 1 ≤ h ≤ s, A2 ̸⊂ pi for 1 ≤ i ≤ h and A2 ⊂ pi for h + 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let Si = pi ∩ A2 for 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then the Si + mA2/mA2 are proper subspaces of A2/mA2 for each 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus there exists x ∈ A2 such that x+ mA2 is not in Si + mA2/mA2 for all 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' In other words, x ∈ A2 − pi for all 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' As before, there exists an integer b > 0 such that En ⊆ Qh+1 ∩ · · · ∩ Qs for all n ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Let u ∈ (0 :E x)n for n ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Then u ∈ Qh+1 ∩ · · · ∩ Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Since xu ∈ Q1 ∩· · ·∩Qh, and x is not in pi for 1 ≤ i ≤ h, it follows that u ∈ Q1 ∩· · ·∩Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thus u ∈ Q1 ∩ · · · ∩ Qs = 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Springer-Verlag, New York, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Bergh) Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway Email address: petter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='bergh@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='no URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='edu/employees/petter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='bergh (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Jorgensen) Department of Mathematics, University of Texas at Arlington, 411 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Nedderman Drive, Pickard Hall 429, Arlington, TX 76019, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Email address: djorgens@uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='edu URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='edu/faculty/djorgens/ (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Thompson) Department of Mathematics, Niagara University, NY 14109, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content=' Current address: Department of Mathematics and Physics, M¨alardalen University, V¨aster˚as, Sweden Email address: peder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='thompson@mdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='se URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='mdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='se/en/malardalen-university/staff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} +page_content='id=ptn04' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfbPzh/content/2301.01384v1.pdf'} diff --git a/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/2301.04566v1.pdf.txt b/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/2301.04566v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6d34dc52b62863a33a1085ff4cfc382cb2542f4 --- /dev/null +++ b/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/2301.04566v1.pdf.txt @@ -0,0 +1,935 @@ +Casimir effect in 2+1 Hoˇrava gravity +Claudio B´orquez1,a and Byron Droguett2,b +1Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta, Chile. +2Departamento de Ciencias B´asicas, Facultad de Ciencias, Universidad Santo +Tom´as, Sede Arica, Chile. +acl.borquezg@gmail.com, +bbyrondroguett@santotomas.cl +Abstract +We study the Casimir effect of a membrane embedded in 2 + 1 dimen- +sions flat cone generated by a massive particle located at the origin of the +coordinate system. The flat cone is an exact solution of the nonprojectable +Hoˇrava theory, similar to general relativity. We consider a scalar field satis- +fying Dirichlet boundary conditions, and regularize the spectrum using the +ζ–function technique. In addition, we include the effects of temperature in our +analysis. Our results show that the Casimir force depends on three factors: +the anisotropic scaling z, the mass of the point particle, and the temperature. +arXiv:2301.04566v1 [hep-th] 11 Jan 2023 + +1 +Introduction +The Casimir effect is a physical manifestation of the quantum fluctuations of empty +space. Casimir predicted the force per unit area between two uncharged, parallel, +and conducting plates is attractive in 3 + 1 dimensions [1]. Other configurations +also have been studied, such as two spheres, which exhibit the opposite sign for +the Casimir force [2]. +Theoretical results for the Casimir effect generally agree +with laboratory measurements [3, 4]. The effect depends on the geometry of the +boundaries and the structure of the manifold considered [5]. The temperature also +plays a crucial role in the results. It is often helpful to approach the problem within +the framework of quantum field theory, which it allows us to associate operators that +satisfy boundary conditions and incorporate temperature through an effective action +[6]. Then, the effective path integral can be expressed in terms of a ζ–function, +which allows us to calculate the spectral sum over all eigenvalues associated to +some operator. In this research, we focus on Dirichlet boundary conditions and +finite temperature in a 2 + 1 dimensional manifold. +Our aim is to study the Casimir effect in theories where the Lorentz symmetry is +broken, such as Hoˇrava-Lifshitz-like theories. There have been several studies on the +violation of Lorentz symmetry in this context [7, 8]. Several cases with anisotropic +behavior have been considered in the literature, such as the extensions of Klein- +Gordon and fermionic fields [9, 10, 11, 12]. Other studies on Lorentz violation have +included terms in the Lagrangian with a preferred direction [13, 14]. Additionally, +finite temperature problems in quantum field theory have been analyzed [15, 16]. +The framework in which we calculate the Casimir effect is in the context of the +Hoˇrava gravity theory [17, 18]. It is a proposal for quantizing the ultraviolet sector of +general relativity that involves Lorentz symmetry breaking through an anisotropic +scaling between space and time. The theory has a preferred foliation with absolute +physical meaning, and the diffeomorphisms that preserve this foliation (FDiff) en- +sure the existence of a vector invariant that depends on the lapse function (in the +nonprojectable case) [19]. In a previous study in 2 + 1 dimensions, one of us found +an exact solution that geometrically represents a cone with a deficit or excess angle, +with a massive point particle localized at the origin [20]. This solution is similar to +one found in general relativity [21], except for a dimensionless coupling constant. +It is a solution that helped to define the condition of asymptotic flatness exactly in +the same way as 2+1 general relativity [22]. Motivated by this solution, we place a +finite membrane on the cone which satisfies Dirichlet boundary conditions. In this +investigation, we consider the case of a scalar field, which generates the quantum +fluctuations of the vacuum. We examine how the mass of the particle located at the +origin affects the Casimir energy and force. Finally, we make an analytic extension +to explore the effects of the finite temperature on Casimir force. +This paper is organized as follows. In section 2, we introduce Hoˇrava gravity +and explain as was calculated the exact solution in 2 + 1 dimensions. In section 3, +we calculate the Casimir force of a finite membrane at zero temperature. In section + +4, we include finite temperature in the calculation of the Casimir force. Finally, in +section 5, we present our conclusions. +2 +Hoˇrava gravity +The Hoˇrava theory [17] is a proposal to complete the ultraviolet regime of general +relativity using quantum field theories techniques, making it unitary and power- +counting renormalizable. This theory has a symmetry under anisotropic scaling of +the coordinates, which is given by +[t] = −z , +[xi] = −1 . +(2.1) +The price to pay for this theory is the Lorentz symmetry breaking in the ultraviolet, +which arises as an accidental symmetry at large scales. The foliation of spacetime +has absolute physical meaning. The Arnowitt-Deser-Misner variables N, Ni, and +gij are used to describe the gravitational dynamics on the foliation. As a result, +the general diffeomorphisms of general relativity are broken, and it is possible to +introduce higher spatial derivatives into the Lagrangian while it keeps the second +order time derivative under control. The symmetry group characteristic of the the- +ory is given by the foliation-preserving diffeomorphisms (FDiff). The infinitesimal +transformations are +δt = f(t) , +δxi = ζi(t, ⃗x) . +(2.2) +These induce the transformations on the fields +δN += +ζk∂kN + f ˙N + ˙fN , +(2.3) +δNi += +ζk∂kNi + Nk∂iζk + ˙ζjgij + f ˙Ni + ˙fNi , +(2.4) +δgij += +ζk∂kgij + 2gk(i∂j)ζk + f ˙gij . +(2.5) +The transformation of the lapse function (2.3) guarantees the existence of two ver- +sions: a projectable version (the lapse function depends only on time) and a non- +projectable version (the lapse function depends on time and space). In this paper, +we work with the nonprojectable version since it is closer to general relativity. +A consequence is the theory propagates an instantaneous scalar mode in all +dimensions, yielding a nontopological theory in 2+1 dimensions. This scalar mode is +responsible for the gravitational interaction in this dimension, making it an excellent +laboratory for studying the perturbative quantization and black hole solutions in +the UV complete theory. +In a previous work [20] we coupled a relativistic particle to the Hoˇrava action in +2 + 1 dimension considering the infrared terms in the potential +V += +−βR − αakak , +(2.6) +where the vector ak = ∂k ln(N) is invariant under FDiff [19]. + +The combined system Hoˇrava gravity-point particle in 2 + 1 dimensions is given +by the action +S = 1 +2κ +� +dt d2x√gN +� +KijKij − λK2 + βR + αakak� +− M +� +dt +√ +L , +(2.7) +where +Kij = +1 +2N +� +˙gij − 2∇(iNj) +� +, +(2.8) +L = (N 2 − NkN k) +� +˙q0�2 − 2Nk ˙q0 ˙qk − gkl ˙qk ˙ql . +(2.9) +The coefficient M is the mass of the particle, κ and λ are coupling constants. +The tensor (2.8) is the extrinsic curvature. L is the squared line element of the +particle evaluated on the background of the ADM variables, and these variables +are evaluated at the position of the particle in L. The mechanic of the particle is +characterized by the embedding fields q0 = q0(t) and qi = qi(t), which define the +position of the particle in the foliation. +If we consider the particle at rest in the origin coordinate system, and all the +fields are considered static, then it is possible to obtain an exact solution similar +to topological general relativity, except for a dimensionless constant β. This exact +solution has the form +ds2 += +r− κM +πβ (dr2 + r2dθ2) . +(2.10) +This solution is not trivial due to the complexity of the equations of motion, and +because the nonprojectable version has second-class constraints, unlike general rel- +ativity. The dimensionless constant β is fixed to 1 by general symmetry in general +relativity. From a geometric point of view, it is convenient to make the following +coordinate change +ρ = 1 +γ rγ , +θ ′ = γθ , +γ ≡ 1 − κM +2πβ , +(2.11) +this leads to a flat cone solution with a deficit or excess angle +ds2 = dρ2 + ρ2dθ ′2 . +(2.12) +For the case where γ > 0, the solution represents a flat cone with a deficit or +excess angle, with the particle localized at the origin. The domain of this solution +is ρ ∈ [0, ∞) and θ ′ ∈ [0, 2πγ]. When M = 0, the geometry is globally plane. In +the case where γ < 0, there is no physical solution because distances are not well +defined. The physical solution γ > 0 motivated us to define the asymptotic flatness +conditions of the Hoˇrava theory in 2 + 1 dimensions, similar to general relativity +[22]. The solution is completely global and the Newtonian force is identically zero; +the cone is affected globally. In addition, the perturbative wave equation for scalar +mode does not determine the sign of the coupling constant associated to the Ricci +scalar [20]. Therefore, it is important to research other physical quantities in order +to determine this value. Motivated by this exact solution in 2 + 1 dimensions, we +calculate the Casimir energy and force of a membrane localized on the flat cone +which satisfies Dirichlet boundary conditions. + +3 +Casimir effect in Hoˇrava-Lifshitz theory +The modified Klein-Gordon Lagrangian for a scalar field invariant under the anisotropic +scaling is given by +Sφ += +1 +2 +� +dt ddx√g +� +∂tφ∂tφ − l2(z−1)∂i1∂i2 · · · ∂izφ∂i1∂i2 · · · ∂izφ +� +, +(3.1) +and the equation of motion for the scalar field is +(∂2 +t + (−1)zl2(z−1)∆z)φ = 0 , +(3.2) +where ∆ = gij∇i∇j and the parameter l has dimension of the inverse of mass. +The problem we aim to solve is the embedding of a membrane on a flat cone +with a deficit or excess angle in 2+1 dimensions. We solve the eigenvalues problem +for a scalar field with arbitrary z value satisfying Dirichlet boundary conditions +Pφ = (−1)zl2(z−1)∆zφ = ωzφ , +φ(θ1) = φ(θ2) = 0 , +φ(R1) = φ(R2) = 0 . +(3.3) +First, we consider the case z = 1 in order to obtain its eigenvalues, which will help +us to find the eigenvalues form for an arbitrary z value. Using the exact solution +(2.10), the partial differential equation is given by +− ∆φ = −rµ +�∂2φ +∂r2 + 1 +r +∂φ +∂r + 1 +r2 +∂2φ +∂θ2 +� += ω1φ , +(3.4) +where µ = − κM +πβ . +By using separable variables φ = R(r)Θ(θ), we obtain two +equations given by +Θ′′(θ) + k2Θ(θ) += +0 , +(3.5) +r2R′′(r) + rR′(r) + +� +ω1r2−µ − k2� +R += +0 , +(3.6) +To solve Eq. (3.6), we make the following change of variables: +x = σrγ , +σ = +√ω1 +γ +, +(3.7) +where γ = 1 − µ/2. This results in the standard Bessel differential equation, whose +solution can be expressed in terms of Bessel functions of the first and second kind +R(r) = C1Jp +�√ω1 +γ rγ +� ++ C2Yp +�√ω1 +γ rγ +� +, +(3.8) +with p = k/γ. For simplicity, we consider asymptotic behavior of the Bessel func- +tions +Jp +�√ω1 +γ rγ +� +∼ +� +2γ +π√ω1rγ cos +�√ω1rγ +γ +− pπ/2 − π/4 +� +, +(3.9) +Yp +�√ω1 +γ rγ +� +∼ +� +2γ +π√ω1rγ sin +�√ω1rγ +γ +− pπ/2 − π/4 +� +. +(3.10) + +Therefore, the Dirichlet boundary conditions determine the form of the eigenvalues +ω1 = +� +nπγ +Rγ +2 − Rγ +1 +�2 +, +n ∈ N . +(3.11) +With this result, we can generalize the eigenvalues solution of the operator P in +Eq. (3.3) for an arbitrary z value. Thus, the eigenvalues are given by +ωz = l2(z−1) +� +nπγ +Rγ +2 − Rγ +1 +�2z +. +(3.12) +The vacuum expectation value of the energy must be regularized using the ζ– +function associated to the spatial operator P +ζP(s) = l−2(z−1)s � +n∈N +� +nπγ +Rγ +2 − Rγ +1 +�−2sz +. +(3.13) +Therefore the Casimir energy is given by +EC = 1 +2ζP (−1/2) = 1 +2lz−1 +� +πγ +Rγ +2 − Rγ +1 +�z +ζR(−z) , +(3.14) +where ζR is the Riemann ζ–function. If we derive with respect to the separation of +the membrane dγ = 1 +γ(Rγ +2 − Rγ +1), we get the Casimir force +FC += +1 +2zπzlz−1 +� +γ +Rγ +2 − Rγ +1 +�z+1 +ζR(−z) . +(3.15) +The Riemann ζ–function is a global factor of the energy and force. Here, two cases +for integer values of z can occur: first, if z is an even number, the energy and force +are equal to zero. Second, if z is an odd number, the Casimir force can be either +attractive or repulsive. When the separation dγ tends to infinity the Casimir force +tends to zero for all γ. +In the Fig.1 (a), we consider the globally flat case γ = 1. The force as a function +of distance decays as expected. The unusual aspect is the change in the orientation +of the force each time we fixed different z values, such as it is presented in [10]. +This is due directly to the Riemann ζ–function. For the case where z takes on even +values, the force is zero. In the Fig.1 (b), the force tends to zero for different values +of the parameter γ. Here, we note that the magnitude of the force is modified, that +is, the force will be stronger (for a cone with deficit angle) or weaker (for a cone +with excess angle) depending on the γ factor. + +(a) +(b) +Figure 1: (a): Casimir force versus separation dγ, considering γ = 1. (b): Casimir +force versus separation dγ, considering z = 1. +4 +Casimir force at finite temperature +In order to study the effect of temperature, it is convenient to use the path integral +approach considering imaginary time, which is associated to finite temperature in +the system. The path integral for the scalar field is given by +Z = +� +Dφ exp (S(φ)) . +(4.1) +The effective action associated to operator O is +Γ = − ln(Z) = 1 +2 ln det[(−∂2 +τ + P)/ϵ] = 1 +2 ln det[O/ϵ] , +(4.2) +where ϵ is an arbitrary parameter with mass dimension, introduced to render the +ζ–function dimensionless. Eventually, the ζ–function will be independent of this +parameter ϵ hence we say ϵ = 1 for simplicity. Therefore, the Casimir energy is +given by +EC += +∂ +∂ξΓ = −1 +2 +∂ +∂ξ +� d +dsζO(s) +����� +s=0 +, +(4.3) +where ξ = 1/T is the inverse of the temperature. +The eigenvalues problem of the operator O is expressed by +(−∂2 +τ + (−1)zl2(z−1)∆z)φ = ωφ , +(4.4) +where τ ∈ C and it has periodic border. We propose the following solution to the +scalar field +φm,n(τ, xi) = 1 +ξ e +2πim +ξ +τϕn(xi) , +(4.5) + +Fc +0.15 上 +0.10 +上 +Z=3 +0.05 +dy +4 +8 +10 +-0.05 +-0.10 +-0.15 +Z=1 +Z=5 +-0.20Fc +8 +10 +y=1.5 +y=1 +-0.05 +y=0.5 +-0.10 +-0.15 +-0.20where the eigenvalues associated to the time derivative are ωm = +2πm +ξ , and the +spatial eigenvalues come from the asymptotic behavior of the functions (3.9) and +(3.10). Then, the ζ–function associated with the operator O is give by +ζO(s) += +∞ +� +m=−∞ +∞ +� +n=1 +��2πm +ξ +�2 ++ l2(z−1) +� +nπγ +Rγ +2 − Rγ +1 +�2z�−s +. +(4.6) +We use the integral representation ζ–function to rewrite the spectral function as +ζO(s) += +1 +Γ (s) +� ∞ +0 +dt ts−1 +∞ +� +m=−∞ +∞ +� +n=1 +exp +� +−t +��2πm +ξ +�2 ++ l2(z−1) +� +nπγ +Rγ +2 − Rγ +1 +�2z�� +. +(4.7) +A suitable representation is obtained by using the Poisson resummation [6] +ζO(s) += +ξ +√ +4π +Γ (s − 1/2) +Γ (s) +� +l2(z−1) +� +πγ +Rγ +2 − Rγ +1 +�2z�1/2−s +ζR (z(2s − 1)) ++ +ξ +√πΓ(s) +∞ +� +n,m=1 +� ∞ +0 +dt ts−3/2 exp +� +−ξ2m2 +4t +− tl2(z−1) +� +nπγ +Rγ +2 − Rγ +1 +�2z� +. +(4.8) +We can reduce the ζ–function by introducing the modified Bessel function through +the following change of variables +y = l2(z−1) +� +nπγ +Rγ +2 − Rγ +1 +�2z +t , +z = mξlz−1 +� +nπγ +Rγ +2 − Rγ +1 +�z +, +(4.9) +then, after performing the change, we have +ζO(s) += +ξ +√ +4π +Γ (s − 1/2) +Γ (s) +� +l2(z−1) +� +πγ +Rγ +2 − Rγ +1 +�2z� 1 +2 −s +ζR (z(2s − 1)) ++ +2ξ +√πΓ(s) +∞ +� +n,m=1 +��2lz−1 +mξ +� +nπγ +Rγ +2 − Rγ +1 +�z� 1 +2 −s +K 1 +2 −s +� +ξmlz−1 +� +nπγ +Rγ +2 − Rγ +1 +�z�� +. +(4.10) +Now, by expanding the derivative of the spectral function around s = 0, we have +d +dsζO(s) +���� +s=0 += −ξlz−1 +� +πγ +Rγ +2 − Rγ +1 +�z +ζR(−z) + 2 +∞ +� +m,n=1 +1 +m exp +� +−ξmlz−1 +� +nπγ +Rγ +2 − Rγ +1 +�z� +. +(4.11) + +The sum over m can be explicitly performed using a geometric serie, then the +Casimir energy is +EC = lz−1 +� +πγ +Rγ +2 − Rγ +1 +�z +� +�1 +2ζR(−z) + +∞ +� +n=1 +nz +exp +� +ξlz−1 +� +nπγ +Rγ +2−Rγ +1 +�z� +− 1 +� +� . +(4.12) +In the case of γ = 1, we recover the result for globally flat space. By taking the +derivative with respect to the separation dγ, we obtain the Casimir force +FC += +zπzlz−1 +� +γ +Rγ +2 − Rγ +1 +�z+1 +� +� +� +1 +2ζR(−z) + +∞ +� +n=1 +nz +exp +� +lz−1ξ +� +nπγ +Rγ +2−Rγ +1 +�z� +− 1 +− ξlz−1πz +� +γ +Rγ +2 − Rγ +1 +�z +∞ +� +n=1 +n2z exp +� +ξlz−1 � +nπγ +Rγ +2−Rγ +1 +�z� +� +exp +� +ξlz−1 +� +nπγ +Rγ +2−Rγ +1 +�z� +− 1 +�2 +� +� +� +� +� +. +(4.13) +In Fig.2 (a) the behavior of the force as a function of γ changes in magnitude +for different separation distances by considering a fixed temperature. When the +separation distance increases, the force decreases. The exponential factor γ deter- +mines how the force decays. In Fig.2 (b) we show how the force varies with respect +to γ, for different temperature values. At high temperatures, the magnitude of the +force is greater, and the decay is faster than in the case of nontemperature. It is +important to note that the temperature is independent of the factor γ. +(a) +(b) +Figure 2: (a): Casimir force versus γ, T = 100 and different separation dγ. (b): +Casimir force versus γ, for z = 1. The solid, dashed and dotted curves are T = +100, 120, 140, respectively. +In the Fig.3 (a), unlike what occurs in Fig. 1 (a), we can see that for a certain +finite temperature, and γ = 1, the contribution of the sums in n in Eq. (4.13) are +completely relevant. The orientation of the force for z = 1, 2, 3 is the same, that is, + +Fc +上 +0.5 +y +1.0 +.0 +2.0 +-1000 +2000 +-3000 +-4000 +5000Fc +0.2 +0.6 +1.2 +1.4 +0.4 +0.8 +1.0 +-1000 +2000 +3000 +-4000 +-5000 +6000there is no change in the sign of the force. In addition, for even values of z, there +is a contribution to the Casimir force. In Fig. 3 (b), we show how the force varies +as a function of distance for three different values of γ by considering a constant +temperature and anisotropic factor z = 1, such as in Fig. 1 (b). The force decays +to a finite value for different γ. This behavior was expected for temperatures other +than zero. The magnitude of the force is stronger with a deficit angle. +(a) +(b) +Figure 3: (a): Casimir force versus separation dγ, T = 100 and γ = 1. (b): Casimir +force versus separation dγ, z = 1, and T = 100 temperature. +5 +Conclusions +We study the case of a membrane embedded in a flat cone with a deficit or excess +angle generated by a massive particle located at the origin of the coordinate system. +This cone is an exact solution of the Hoˇrava theory in 2 + 1 dimensions. +The +vacuum fluctuations are described by a scalar field, which is expressed using the +Hoˇrava-Lifshitz theory, and satisfies Dirichlet boundary conditions. We have derived +the Casimir energy and force of this particular manifold at both zero and finite +temperature. +At zero temperature, the force is multiplied by a global Riemann ζ–function that +depends on the anisotropic scaling factor z. If z is even, the energy and force are +zero. For the special case of z = 3, the energy is positive, leading to a repulsive force. +When temperature is taken into account, additional terms appear in the Casimir +effect. If z is even, the Casimir energy and force are nonzero, and specifically, for +z = 1, 2, 3, the force is attractive due to thermal influences. The factor γ plays a +significant role in the Casimir effect. When the cone has a deficit angle (γ < 1), +the magnitude of the force is greater than in the case of a flat space or excess angle +(γ ≥ 1). +The results are consistent with those found in the literature about the Casimir +effect in Lorentz-violating theories: the Casimir effect depends on the anisotropic +scaling z. Furthermore, the topology has a strong influence on the decay of the + +Fc +dy +2 +3 +4 +5 +6 +7 +8 +Z=1 +- +-1000 +- +2000 +- +- +1 +3000 +- +Z=2 +................. +Z=3 +- +- +-4000 +- +- +- +! +5000Fc +L +dy +2 +3 +4 +6 +8 +-1000 +y=1.5 +2000 +y=1 +y=0.5 +-3000 +-4000 +5000energy and force. Therefore, the Casimir effect in Hoˇrava-Lifshitz gravity theories +will not only depend on boundary conditions, but also on the spatial configuration +determined by the γ parameter, that is, by the presence of a massive point particle. +Acknowledgements +C.B. is partially supported by Grant No. CONICYT PFCHA/DOCTORADO BE- +CAS CHILE /2019 – 21190960. C.B. is a graduate student in the “Doctorado en +F´ısica Menci´on F´ısica-Matem´atica” Ph.D. program at the Universidad de Antofa- +gasta. +References +[1] H. B. G. Casimir, On the Attraction Between Two Perfectly Conducting Plates, +Indag. Math. 10, 261-263 (1948). +[2] T. H. 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Jackiw and G. ’t Hooft, Three-Dimensional Einstein Gravity: +Dynamics of Flat Space, Annals Phys. 152, 220 (1984). +[22] A. Ashtekar and M. Varadarajan, A Striking property of the gravitational +Hamiltonian, Phys. Rev. D 50, 4944 (1994) [gr-qc/9406040]. + diff --git a/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/load_file.txt b/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3f733084e5231269ef02846222ede3f947f38d2 --- /dev/null +++ b/jNE3T4oBgHgl3EQfgwr4/content/tmp_files/load_file.txt @@ -0,0 +1,456 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf,len=455 +page_content='Casimir effect in 2+1 Hoˇrava gravity Claudio B´orquez1,a and Byron Droguett2,b 1Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 2Departamento de Ciencias B´asicas, Facultad de Ciencias, Universidad Santo Tom´as, Sede Arica, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' acl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='borquezg@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='com, bbyrondroguett@santotomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='cl Abstract We study the Casimir effect of a membrane embedded in 2 + 1 dimen- sions flat cone generated by a massive particle located at the origin of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The flat cone is an exact solution of the nonprojectable Hoˇrava theory, similar to general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' We consider a scalar field satis- fying Dirichlet boundary conditions, and regularize the spectrum using the ζ–function technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In addition, we include the effects of temperature in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Our results show that the Casimir force depends on three factors: the anisotropic scaling z, the mass of the point particle, and the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='04566v1 [hep-th] 11 Jan 2023 1 Introduction The Casimir effect is a physical manifestation of the quantum fluctuations of empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Casimir predicted the force per unit area between two uncharged, parallel, and conducting plates is attractive in 3 + 1 dimensions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Other configurations also have been studied, such as two spheres, which exhibit the opposite sign for the Casimir force [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Theoretical results for the Casimir effect generally agree with laboratory measurements [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The effect depends on the geometry of the boundaries and the structure of the manifold considered [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The temperature also plays a crucial role in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' It is often helpful to approach the problem within the framework of quantum field theory, which it allows us to associate operators that satisfy boundary conditions and incorporate temperature through an effective action [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Then, the effective path integral can be expressed in terms of a ζ–function, which allows us to calculate the spectral sum over all eigenvalues associated to some operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In this research, we focus on Dirichlet boundary conditions and finite temperature in a 2 + 1 dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Our aim is to study the Casimir effect in theories where the Lorentz symmetry is broken, such as Hoˇrava-Lifshitz-like theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' There have been several studies on the violation of Lorentz symmetry in this context [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Several cases with anisotropic behavior have been considered in the literature, such as the extensions of Klein- Gordon and fermionic fields [9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Other studies on Lorentz violation have included terms in the Lagrangian with a preferred direction [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Additionally, finite temperature problems in quantum field theory have been analyzed [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The framework in which we calculate the Casimir effect is in the context of the Hoˇrava gravity theory [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' It is a proposal for quantizing the ultraviolet sector of general relativity that involves Lorentz symmetry breaking through an anisotropic scaling between space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The theory has a preferred foliation with absolute physical meaning, and the diffeomorphisms that preserve this foliation (FDiff) en- sure the existence of a vector invariant that depends on the lapse function (in the nonprojectable case) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In a previous study in 2 + 1 dimensions, one of us found an exact solution that geometrically represents a cone with a deficit or excess angle, with a massive point particle localized at the origin [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This solution is similar to one found in general relativity [21], except for a dimensionless coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' It is a solution that helped to define the condition of asymptotic flatness exactly in the same way as 2+1 general relativity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Motivated by this solution, we place a finite membrane on the cone which satisfies Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In this investigation, we consider the case of a scalar field, which generates the quantum fluctuations of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' We examine how the mass of the particle located at the origin affects the Casimir energy and force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Finally, we make an analytic extension to explore the effects of the finite temperature on Casimir force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In section 2, we introduce Hoˇrava gravity and explain as was calculated the exact solution in 2 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In section 3, we calculate the Casimir force of a finite membrane at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In section 4, we include finite temperature in the calculation of the Casimir force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Finally, in section 5, we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 2 Hoˇrava gravity The Hoˇrava theory [17] is a proposal to complete the ultraviolet regime of general relativity using quantum field theories techniques, making it unitary and power- counting renormalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This theory has a symmetry under anisotropic scaling of the coordinates, which is given by [t] = −z , [xi] = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='1) The price to pay for this theory is the Lorentz symmetry breaking in the ultraviolet, which arises as an accidental symmetry at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The foliation of spacetime has absolute physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The Arnowitt-Deser-Misner variables N, Ni, and gij are used to describe the gravitational dynamics on the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' As a result, the general diffeomorphisms of general relativity are broken, and it is possible to introduce higher spatial derivatives into the Lagrangian while it keeps the second order time derivative under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The symmetry group characteristic of the the- ory is given by the foliation-preserving diffeomorphisms (FDiff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The infinitesimal transformations are δt = f(t) , δxi = ζi(t, ⃗x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2) These induce the transformations on the fields δN = ζk∂kN + f ˙N + ˙fN , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3) δNi = ζk∂kNi + Nk∂iζk + ˙ζjgij + f ˙Ni + ˙fNi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='4) δgij = ζk∂kgij + 2gk(i∂j)ζk + f ˙gij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5) The transformation of the lapse function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3) guarantees the existence of two ver- sions: a projectable version (the lapse function depends only on time) and a non- projectable version (the lapse function depends on time and space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In this paper, we work with the nonprojectable version since it is closer to general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' A consequence is the theory propagates an instantaneous scalar mode in all dimensions, yielding a nontopological theory in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This scalar mode is responsible for the gravitational interaction in this dimension, making it an excellent laboratory for studying the perturbative quantization and black hole solutions in the UV complete theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In a previous work [20] we coupled a relativistic particle to the Hoˇrava action in 2 + 1 dimension considering the infrared terms in the potential V = −βR − αakak , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='6) where the vector ak = ∂k ln(N) is invariant under FDiff [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The combined system Hoˇrava gravity-point particle in 2 + 1 dimensions is given by the action S = 1 2κ � dt d2x√gN � KijKij − λK2 + βR + αakak� − M � dt √ L , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='7) where Kij = 1 2N � ˙gij − 2∇(iNj) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='8) L = (N 2 − NkN k) � ˙q0�2 − 2Nk ˙q0 ˙qk − gkl ˙qk ˙ql .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='9) The coefficient M is the mass of the particle, κ and λ are coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='8) is the extrinsic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' L is the squared line element of the particle evaluated on the background of the ADM variables, and these variables are evaluated at the position of the particle in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The mechanic of the particle is characterized by the embedding fields q0 = q0(t) and qi = qi(t), which define the position of the particle in the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' If we consider the particle at rest in the origin coordinate system, and all the fields are considered static, then it is possible to obtain an exact solution similar to topological general relativity, except for a dimensionless constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This exact solution has the form ds2 = r− κM πβ (dr2 + r2dθ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10) This solution is not trivial due to the complexity of the equations of motion, and because the nonprojectable version has second-class constraints, unlike general rel- ativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The dimensionless constant β is fixed to 1 by general symmetry in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' From a geometric point of view, it is convenient to make the following coordinate change ρ = 1 γ rγ , θ ′ = γθ , γ ≡ 1 − κM 2πβ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='11) this leads to a flat cone solution with a deficit or excess angle ds2 = dρ2 + ρ2dθ ′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='12) For the case where γ > 0, the solution represents a flat cone with a deficit or excess angle, with the particle localized at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The domain of this solution is ρ ∈ [0, ∞) and θ ′ ∈ [0, 2πγ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' When M = 0, the geometry is globally plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In the case where γ < 0, there is no physical solution because distances are not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The physical solution γ > 0 motivated us to define the asymptotic flatness conditions of the Hoˇrava theory in 2 + 1 dimensions, similar to general relativity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The solution is completely global and the Newtonian force is identically zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' the cone is affected globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In addition, the perturbative wave equation for scalar mode does not determine the sign of the coupling constant associated to the Ricci scalar [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Therefore, it is important to research other physical quantities in order to determine this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Motivated by this exact solution in 2 + 1 dimensions, we calculate the Casimir energy and force of a membrane localized on the flat cone which satisfies Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 3 Casimir effect in Hoˇrava-Lifshitz theory The modified Klein-Gordon Lagrangian for a scalar field invariant under the anisotropic scaling is given by Sφ = 1 2 � dt ddx√g � ∂tφ∂tφ − l2(z−1)∂i1∂i2 · · · ∂izφ∂i1∂i2 · · · ∂izφ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='1) and the equation of motion for the scalar field is (∂2 t + (−1)zl2(z−1)∆z)φ = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2) where ∆ = gij∇i∇j and the parameter l has dimension of the inverse of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The problem we aim to solve is the embedding of a membrane on a flat cone with a deficit or excess angle in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' We solve the eigenvalues problem for a scalar field with arbitrary z value satisfying Dirichlet boundary conditions Pφ = (−1)zl2(z−1)∆zφ = ωzφ , φ(θ1) = φ(θ2) = 0 , φ(R1) = φ(R2) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3) First, we consider the case z = 1 in order to obtain its eigenvalues, which will help us to find the eigenvalues form for an arbitrary z value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Using the exact solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10), the partial differential equation is given by − ∆φ = −rµ �∂2φ ∂r2 + 1 r ∂φ ∂r + 1 r2 ∂2φ ∂θ2 � = ω1φ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='4) where µ = − κM πβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' By using separable variables φ = R(r)Θ(θ), we obtain two equations given by Θ′′(θ) + k2Θ(θ) = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5) r2R′′(r) + rR′(r) + � ω1r2−µ − k2� R = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='6) To solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='6), we make the following change of variables: x = σrγ , σ = √ω1 γ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='7) where γ = 1 − µ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This results in the standard Bessel differential equation, whose solution can be expressed in terms of Bessel functions of the first and second kind R(r) = C1Jp �√ω1 γ rγ � + C2Yp �√ω1 γ rγ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='8) with p = k/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' For simplicity, we consider asymptotic behavior of the Bessel func- tions Jp �√ω1 γ rγ � ∼ � 2γ π√ω1rγ cos �√ω1rγ γ − pπ/2 − π/4 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='9) Yp �√ω1 γ rγ � ∼ � 2γ π√ω1rγ sin �√ω1rγ γ − pπ/2 − π/4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10) Therefore, the Dirichlet boundary conditions determine the form of the eigenvalues ω1 = � nπγ Rγ 2 − Rγ 1 �2 , n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='11) With this result, we can generalize the eigenvalues solution of the operator P in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3) for an arbitrary z value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Thus, the eigenvalues are given by ωz = l2(z−1) � nπγ Rγ 2 − Rγ 1 �2z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='12) The vacuum expectation value of the energy must be regularized using the ζ– function associated to the spatial operator P ζP(s) = l−2(z−1)s � n∈N � nπγ Rγ 2 − Rγ 1 �−2sz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='13) Therefore the Casimir energy is given by EC = 1 2ζP (−1/2) = 1 2lz−1 � πγ Rγ 2 − Rγ 1 �z ζR(−z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='14) where ζR is the Riemann ζ–function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' If we derive with respect to the separation of the membrane dγ = 1 γ(Rγ 2 − Rγ 1), we get the Casimir force FC = 1 2zπzlz−1 � γ Rγ 2 − Rγ 1 �z+1 ζR(−z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='15) The Riemann ζ–function is a global factor of the energy and force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Here, two cases for integer values of z can occur: first, if z is an even number, the energy and force are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Second, if z is an odd number, the Casimir force can be either attractive or repulsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' When the separation dγ tends to infinity the Casimir force tends to zero for all γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='1 (a), we consider the globally flat case γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The force as a function of distance decays as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The unusual aspect is the change in the orientation of the force each time we fixed different z values, such as it is presented in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This is due directly to the Riemann ζ–function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' For the case where z takes on even values, the force is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='1 (b), the force tends to zero for different values of the parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Here, we note that the magnitude of the force is modified, that is, the force will be stronger (for a cone with deficit angle) or weaker (for a cone with excess angle) depending on the γ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (a) (b) Figure 1: (a): Casimir force versus separation dγ, considering γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (b): Casimir force versus separation dγ, considering z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 4 Casimir force at finite temperature In order to study the effect of temperature, it is convenient to use the path integral approach considering imaginary time, which is associated to finite temperature in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The path integral for the scalar field is given by Z = � Dφ exp (S(φ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='1) The effective action associated to operator O is Γ = − ln(Z) = 1 2 ln det[(−∂2 τ + P)/ϵ] = 1 2 ln det[O/ϵ] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2) where ϵ is an arbitrary parameter with mass dimension, introduced to render the ζ–function dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Eventually, the ζ–function will be independent of this parameter ϵ hence we say ϵ = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Therefore, the Casimir energy is given by EC = ∂ ∂ξΓ = −1 2 ∂ ∂ξ � d dsζO(s) ����� s=0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3) where ξ = 1/T is the inverse of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The eigenvalues problem of the operator O is expressed by (−∂2 τ + (−1)zl2(z−1)∆z)φ = ωφ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='4) where τ ∈ C and it has periodic border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' We propose the following solution to the scalar field φm,n(τ, xi) = 1 ξ e 2πim ξ τϕn(xi) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5) Fc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='15 上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10 上 Z=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='05 dy 4 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='15 Z=1 Z=5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='20Fc 8 10 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5 y=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='05 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='20where the eigenvalues associated to the time derivative are ωm = 2πm ξ , and the spatial eigenvalues come from the asymptotic behavior of the functions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Then, the ζ–function associated with the operator O is give by ζO(s) = ∞ � m=−∞ ∞ � n=1 ��2πm ξ �2 + l2(z−1) � nπγ Rγ 2 − Rγ 1 �2z�−s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='6) We use the integral representation ζ–function to rewrite the spectral function as ζO(s) = 1 Γ (s) � ∞ 0 dt ts−1 ∞ � m=−∞ ∞ � n=1 exp � −t ��2πm ξ �2 + l2(z−1) � nπγ Rγ 2 − Rγ 1 �2z�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='7) A suitable representation is obtained by using the Poisson resummation [6] ζO(s) = ξ √ 4π Γ (s − 1/2) Γ (s) � l2(z−1) � πγ Rγ 2 − Rγ 1 �2z�1/2−s ζR (z(2s − 1)) + ξ √πΓ(s) ∞ � n,m=1 � ∞ 0 dt ts−3/2 exp � −ξ2m2 4t − tl2(z−1) � nπγ Rγ 2 − Rγ 1 �2z� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='8) We can reduce the ζ–function by introducing the modified Bessel function through the following change of variables y = l2(z−1) � nπγ Rγ 2 − Rγ 1 �2z t , z = mξlz−1 � nπγ Rγ 2 − Rγ 1 �z , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='9) then, after performing the change, we have ζO(s) = ξ √ 4π Γ (s − 1/2) Γ (s) � l2(z−1) � πγ Rγ 2 − Rγ 1 �2z� 1 2 −s ζR (z(2s − 1)) + 2ξ √πΓ(s) ∞ � n,m=1 ��2lz−1 mξ � nπγ Rγ 2 − Rγ 1 �z� 1 2 −s K 1 2 −s � ξmlz−1 � nπγ Rγ 2 − Rγ 1 �z�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='10) Now, by expanding the derivative of the spectral function around s = 0, we have d dsζO(s) ���� s=0 = −ξlz−1 � πγ Rγ 2 − Rγ 1 �z ζR(−z) + 2 ∞ � m,n=1 1 m exp � −ξmlz−1 � nπγ Rγ 2 − Rγ 1 �z� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='11) The sum over m can be explicitly performed using a geometric serie, then the Casimir energy is EC = lz−1 � πγ Rγ 2 − Rγ 1 �z � �1 2ζR(−z) + ∞ � n=1 nz exp � ξlz−1 � nπγ Rγ 2−Rγ 1 �z� − 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='12) In the case of γ = 1, we recover the result for globally flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' By taking the derivative with respect to the separation dγ, we obtain the Casimir force FC = zπzlz−1 � γ Rγ 2 − Rγ 1 �z+1 � � � 1 2ζR(−z) + ∞ � n=1 nz exp � lz−1ξ � nπγ Rγ 2−Rγ 1 �z� − 1 − ξlz−1πz � γ Rγ 2 − Rγ 1 �z ∞ � n=1 n2z exp � ξlz−1 � nπγ Rγ 2−Rγ 1 �z� � exp � ξlz−1 � nπγ Rγ 2−Rγ 1 �z� − 1 �2 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='13) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2 (a) the behavior of the force as a function of γ changes in magnitude for different separation distances by considering a fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' When the separation distance increases, the force decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The exponential factor γ deter- mines how the force decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2 (b) we show how the force varies with respect to γ, for different temperature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' At high temperatures, the magnitude of the force is greater, and the decay is faster than in the case of nontemperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' It is important to note that the temperature is independent of the factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (a) (b) Figure 2: (a): Casimir force versus γ, T = 100 and different separation dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (b): Casimir force versus γ, for z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The solid, dashed and dotted curves are T = 100, 120, 140, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='3 (a), unlike what occurs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 1 (a), we can see that for a certain finite temperature, and γ = 1, the contribution of the sums in n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='13) are completely relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The orientation of the force for z = 1, 2, 3 is the same, that is, Fc 上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='0 1000 2000 3000 4000 5000Fc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='0 1000 2000 3000 4000 5000 6000there is no change in the sign of the force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In addition, for even values of z, there is a contribution to the Casimir force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 3 (b), we show how the force varies as a function of distance for three different values of γ by considering a constant temperature and anisotropic factor z = 1, such as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The force decays to a finite value for different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This behavior was expected for temperatures other than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The magnitude of the force is stronger with a deficit angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (a) (b) Figure 3: (a): Casimir force versus separation dγ, T = 100 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' (b): Casimir force versus separation dγ, z = 1, and T = 100 temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 5 Conclusions We study the case of a membrane embedded in a flat cone with a deficit or excess angle generated by a massive particle located at the origin of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' This cone is an exact solution of the Hoˇrava theory in 2 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The vacuum fluctuations are described by a scalar field, which is expressed using the Hoˇrava-Lifshitz theory, and satisfies Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' We have derived the Casimir energy and force of this particular manifold at both zero and finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' At zero temperature, the force is multiplied by a global Riemann ζ–function that depends on the anisotropic scaling factor z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' If z is even, the energy and force are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' For the special case of z = 3, the energy is positive, leading to a repulsive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' When temperature is taken into account, additional terms appear in the Casimir effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' If z is even, the Casimir energy and force are nonzero, and specifically, for z = 1, 2, 3, the force is attractive due to thermal influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The factor γ plays a significant role in the Casimir effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' When the cone has a deficit angle (γ < 1), the magnitude of the force is greater than in the case of a flat space or excess angle (γ ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' The results are consistent with those found in the literature about the Casimir effect in Lorentz-violating theories: the Casimir effect depends on the anisotropic scaling z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Furthermore, the topology has a strong influence on the decay of the Fc dy 2 3 4 5 6 7 8 Z=1 1000 2000 1 3000 Z=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Z=3 4000 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' 5000Fc L dy 2 3 4 6 8 1000 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5 2000 y=1 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='5 3000 4000 5000energy and force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Therefore, the Casimir effect in Hoˇrava-Lifshitz gravity theories will not only depend on boundary conditions, but also on the spatial configuration determined by the γ parameter, that is, by the presence of a massive point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' Acknowledgements C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' is partially supported by Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' CONICYT PFCHA/DOCTORADO BE- CAS CHILE /2019 – 21190960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' is a graduate student in the “Doctorado en F´ısica Menci´on F´ısica-Matem´atica” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' program at the Universidad de Antofa- gasta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE3T4oBgHgl3EQfgwr4/content/2301.04566v1.pdf'} diff --git a/jtE2T4oBgHgl3EQfdQd0/content/tmp_files/load_file.txt b/jtE2T4oBgHgl3EQfdQd0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0d6e5efe88c7a307c836930d6a2f4a807e5b592 --- /dev/null +++ b/jtE2T4oBgHgl3EQfdQd0/content/tmp_files/load_file.txt @@ -0,0 +1,1050 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf,len=1049 +page_content='1 RedMule: A Mixed-Precision Matrix-Matrix Operation Engine for Flexible and Energy-Efficient On-Chip Linear Algebra and TinyML Training Acceleration Yvan Tortorella∗, Luca Bertaccini†, Luca Benini∗†, Davide Rossi∗, and Francesco Conti∗ ∗Department of Electrical, Electronic and Information Engineering (DEI), University of Bologna, Italy †IIS Integrated Systems Laboratory, ETH Zurich, Switzerland Abstract—The increasing interest in TinyML, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=', near-sensor machine learning on power budgets of a few tens of mW, is currently pushing toward enabling TinyML-class training as opposed to inference only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Current training algorithms, based on various forms of error and gradient backpropagation, rely on floating-point matrix operations to meet the precision and dynamic range requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' So far, the energy and power cost of these operations has been considered too high for TinyML scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This paper addresses the open challenge of near- sensor training on a few mW power budget and presents RedMulE - Reduced-Precision Matrix Multiplication Engine, a low-power specialized accelerator conceived for multi-precision floating-point General Matrix-Matrix Operations (GEMM-Ops) acceleration, supporting FP16, as well as hybrid FP8 formats, with {sign, exponent, mantissa} = ({1, 4, 3}, {1, 5, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We inte- grate RedMule into a Parallel Ultra-Low-Power (PULP) cluster containing eight energy-efficient RISC-V cores sharing a tightly- coupled data memory and implement the resulting system in a 22 nm technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' At its best efficiency point (@ 470 MHz, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 V), the RedMulE-augmented PULP cluster achieves 755 GFLOPS/W and 920 GFLOPS/W during regular General Matrix- Matrix Multiplication (GEMM), and up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='19 TFLOPS/W and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='67 TFLOPS/W when executing GEMM-Ops, respectively, for FP16 and FP8 input/output tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In its best performance point (@ 613 MHz, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V), RedMulE achieves up to 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS and 117 GFLOPS for FP16 and FP8, respectively, with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% utilization of the array of Computing Elements and consuming less than 60 mW on average, thus enabling on-device training of deep learning models in TinyML application scenarios while retaining the flexibility to tackle other classes of common linear algebra problems efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Index Terms—General Matrix-Matrix Multiplication, General Matrix-Matrix Operations, Hardware Accelerator, Embedded- Systems, Online-Learning, TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' INTRODUCTION In the last few years, the number of Internet of Things (IoT) devices connected and executing Machine Learning (ML) and, in particular, Deep Learning (DL) based algorithms such as Deep Neural Networks (DNNs) increased considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To re- duce the amount of data sent over the network, improve energy efficiency, and prevent network congestion, the computation has been moved increasingly from data centers to energy-efficient IoT end-nodes with low power budgets (a few mW average, a hundred mW peak) [1], giving rise to the Tiny-ML field of research and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Extreme-edge applications like training and inference of Neural Networks (NNs), graph analysis and manipulation [2], [3], short-distance problems [4], and model-based control rely on General Matrix-Matrix Multiplications (GEMMs) or Gen- eral Matrix-Matrix Operations (GEMM-Ops) as the most sig- nificant kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' GEMM-Ops are operations that share the same structure of a GEMM but replace the canonical multiply/add with other mapping and reduction operations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Due to the similarity of these computational patterns, it has recently been proposed [6] to augment TensorCores with GEMM-Ops support, thereby extending their acceleration capabilities to a broader class of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' There is not yet an equal contribution targeting ultra-low-power embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In desktop, mobile, and data center computing, single and double-precision Floating-Point (FP) operations are typically employed for DL and linear algebra applications, providing high accuracy at an acceptable area and energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, on embedded devices, power and area constraints are much tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Recently, a significant effort has gone into adapting linear algebra-based algorithms as well as online learning [7] to low-precision formats, such as FP16 [8], [9] and FP8 [10], [11], while incurring in little accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' These algorith- mic advancements enabled performance and energy efficiency gains [12], [13], opening the way for deploying continual learn- ing and adaptation of DL models on extreme-edge computing systems such as smart wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, the compu- tational capabilities of microcontroller units (MCUs), typically used in these devices, are minimal, especially concerning the execution of FP operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this paper, we present RedMulE (Reduced-precision ma- trix Multiplication Engine), a TinyML-class fully parametric open-source hardware accelerator designed to support on-chip mixed FP precision (FP8, FP16) linear algebra within RISC-V- based Parallel Ultra-Low-Power (PULP) [14] clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Since GEMM is commonly known to be the key kernel behind DL and ML training algorithms, RedMulE enables the de- ployment of on-chip learning and adaptation capabilities while efficiently supporting GEMM-Ops, on ultra-low-power System arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='03904v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='AR] 10 Jan 2023 2 on-Chips (SoCs) suitable for TinyML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We pro- totyped our design within an 8-core PULP cluster in 22 nm CMOS technology, instantiating a RedMulE instance with 48 internal Computing Elements (CEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE occupies only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='15 mm2, accounting for 24% of the entire cluster area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' It achieves up to 15× speedup during regular FP16 GEMM and up to 62× during GEMM-Ops with respect to parallel execution on the RISC-V cores, reaching up to 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% CEs utilization) at 613 MHZ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In its best efficiency point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 470 MHz at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 V, RedMulE achieves up to 772 GFLOPS/W and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='19 TFLOPS/W energy efficiency for GEMM and GEMM-Ops respectively, while reaching 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 GFLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' When used with FP8 input/output tensors rep- resentation, a 96 CEs RedMulE implementation reaches up to 117 GFLOPS at 613 MHz, achieving up to 920 GFLOPS/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RELATED WORK The strong interest in executing linear algebra-based al- gorithms like inference and training of NNs led to the de- velopment of various hardware platforms specialized in this task, spanning from data-centers computing systems to ultra- low-power embedded platforms [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' NVIDIA’s recent Hopper H100 [16] Graphic Processing Unit (GPU) is the most repre- sentative example of data-center computing platform for DL tasks like inference and training of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The H100 achieves 1978 TFLOPS at 700 W power consumption and can be used to train huge NN models like transformers by using narrow FP8 formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, enabling the execution of DL-based algorithms on ultra-low-power TinyML SoCs for extreme- edge devices such as smart wearable systems is challenging due to the strict power, energy, and cost constraints imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Extreme-edge inference is achievable in practical cases since it can be performed employing low-precision integer arithmetic, which reduces the model’s memory footprint and increases the energy efficiency of the underlying architecture with a limited accuracy loss [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the contrary, extreme-edge NNs training faces large memory requirements and the need for FP calculations, which typically leads to power envelopes exceeding the TinyML constraints [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this section, we focus on embedded platforms emphasizing edge training at moderate power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Inference Accelerators Hardware accelerators specialized for low-power DL infer- ence provide attractive alternatives to software-based execu- tions [18], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Diana [21], a low-power NN SoC, features a digital NN inference accelerator and an analog in-memory- computing core integrated within a shared memory subsystem working only with narrow integer formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' DNPU [22] is a fully-digital energy-efficient DL processor for convolutional and recursive NN inference acceleration designed in 65 nm technology and based on a heterogeneous architecture support- ing 16-bit fixed-point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Gemmini [23] is a 16 × 16 systolic accelerator designed for inference of deep NNs with 8-bit multiply-accumulate units with runtime-programmable weight stationary and output stationary dataflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On-Device Learning On-device learning is an emerging and open challenge concerning training DL models on ultra-low-power general- purpose microcontrollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To reach this aim, many works inves- tigated algorithms like direct feedback alignment or equilibrium propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, such methods have been demonstrated to be less effective than the classical backpropagation method due to severe convergence difficulties [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' TinyOL [25] and [26] focus on training NNs using the low-budget Arduino Nano mi- crocontroller based on Cortex-M core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, PULP Trainlib [27], [28], and [29] are all examples of approaches to enable on-device learning and adaptation on RISC-V multi-core PULP clusters like Vega [30], that provide mixed FP precision capabilities, spanning from IEEE 754 Standard FP32 and FP16 to bfloat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, the low speed and number of available floating point units typical of ultra-low-power microcontrollers limit the performance of these libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Training Accelerators To address the limited training performance achievable by software libraries running on low-power processors, several researchers turned to hardware acceleration [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Cambricon-Q [31] is a training-oriented chip for high accu- racy and energy efficiency based on 8-bit fixed-point arith- metic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, many common training algorithms require floating-point operations to ensure convergence [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Most training-oriented chips employing FP arithmetic are all char- acterized by power envelopes unsuitable for extreme-edge ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' IBM proposes [33], [34], an AI computing platform featuring 8 × 8 mixed-precision engines supporting FP16 and hybrid FP8 training, while [35] support only FP16 and FP32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Similarly, LNPU [36] supports mixed 8-bit and 16-bit FP precision for on-chip training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' While these chips consume significantly less power than data-center GPUs during NN training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' a few Watts as opposed to hundreds of Watts), they still do not meet the tens of mW power constraints of TinyML devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Recently, a few training-oriented SoCs that fit the power budget of extreme-edge applications have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' T- PIM [37] is a processing-in-memory accelerator in 28 nm tech- nology for on-device learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' It reaches up to 250 GOPS/W during training with 0% of sparsity and within a power envelope of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='23 mW at 280 MHz operating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, T- PIM and all the recently proposed PIM approaches do not support FP computations and are not suitable for standard back- propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To support NNs training at reduced power budgets, many training-oriented chips extensively employ pruning to increase sparsity during training [38], lacking generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For example, TSUNAMI [39] and Trainer [40] are both accelerators designed for extreme-edge NN inference and training, meeting the TinyML power constraints by employing pruning and zero skipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' [41] propose a reconfigurable accelera- tor for dense-sparse matrix multiplications for mixed-precision computations, suitable for training-oriented applications since it features FP16 multiplications and FP32 accumulations with low area occupation and high energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' such an 3 TABLE I: Set of General Matrix-Matrix Operations supported by RedMulE Z = (X ◦ W) ⋆ Y Group Kernel ⋆ Res Matmul × + Z = (X × W) + Y Group 1 Maximum Critical Path + max Z = max[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (X + W)] All-Pairs Shortest Paths + min Z = min[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (X + W)] Maximum Reliability Path × max Z = max[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (X × W)] Minimum Reliability Path × min Z = min[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (X × W)] Group 2 Minimum Spanning Tree max min Z = min[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' max(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' W)] Maximym Capacity Path min max Z = max[Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' min(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' W)] accelerator is not parametric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' thus not allowing a fast scale-up at design time when higher performance is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In addition, its integration into a real system has not been evaluated, and it does not support compressed FP8 input/output tensors, which allows for training larger NN models on edge devices where the memory resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' GEMM-Ops Chips All examples of training and inference-oriented chips men- tioned so far target only the most common DL operations (such as matrix multiplications and convolutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, a large set of kernels share the same computational structure as GEMM but do not rely on multiplication and addition as elementary operations, falling into the GEMM-Ops scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Graph analytics, such as breadth-first search [2], [3], short-distance problems [4] that are commonly used for path planning optimization in embedded drones navigation [42], and minimum spanning tree, used for computer vision [43], are examples of applications that make use of GEMM-Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' SIMD2 [6] addresses this issue by building functional units for GEMM-Op acceleration on top of NVIDIA Streaming Multiprocessor architecture, resembling the TensorCores structure and providing dedicated ISA extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The design is implemented in 45 nm technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Adding all the SIMD2 extensions to the baseline matrix multiplication unit results in up to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8× speedup with respect to executing the same kernel on CUDA cores at the cost of 69% of area overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this paper, we propose an extended version of Red- MulE [44] with the following unique combination of features: An array of Floating-Point Units-based Computing Ele- ments (CEs) for efficient training and inference of general DL models on embedded SoCs with additional support for reduced bit-width FP computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We tightly couple RedMulE with a parallel cluster of RISC-V processors to achieve maximum flexibility in implementing complex training algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Supports for GEMM-Ops with a low area overhead (16%) with respect to a GEMM-only implementation to address a wider spectrum of applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' X W + x = Y Z = + x N N N N L L H a) X W Y Z = + x x + = N N L H L H L H b) 1 1 1 1 1 1 1 1 1 1 1 L H xN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1: Execution of a GEMM through a) scalar dot product microkernel and b) block-dot product (or outer product) micro- kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' A fully-parametric design that allows the instantiation of a wide range of CEs arrays, internal buffers and memory interface configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Generalized Matrix-Matrix Operations In this work, we define Generalized Matrix-Matrix Op- erations (GEMM-Ops) as all the operations of the kind f2(Y, f1(X, W)), in particular they can be expressed as: Z = (X ◦ W) ⋆ Y (1) where ◦ corresponds to f1() and ⋆ corresponds to f2().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Table I shows some examples of GEMM-Ops, divided into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Group 1 includes all the GEMM-Ops where the ◦ operator can be of the +/× kind while ⋆ can be min/max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Group 2 contains the GEMM-Ops kernels where the ◦ operator also belongs to the min/max kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' X is a matrix of size M × N, W is a matrix of size N × K, while Z and Y have size M × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The similarity of GEMMs and GEMM-Ops makes matrix computing units good candidates to be extended for supporting GEMM-Ops, extending their flexibility to accelerate general- ized parallel algebraic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This class of algorithms is also well-suited for ML applications since matrices are the baseline structure of all DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To this purpose, it is essential to note that the structure of Equation 1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' As a consequence, for ML applications, there is no need to identify X or W as input or weight matrices because their role can be flexibly exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Asymptotic Optimality of Linear Algebra Acceleration Strategies Memory load/store operations enlarge the gap between the- oretical and practical performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Therefore, 4 RedMulE RISCY 0 RISCY 7 iCACHE SHARED PERIPH INTERCO CLUSTER AXI BUS DMAC HW SYNC TCDM 0 TCDM 1 TCDM 15 TCDM 14 TCDM 2 TCDM 13 LOG INTC HWPE INTC HCI CTRL 32b 32b 32b 288b STREAMER 128b DataMover CTRL STREAMER 32b TCDM 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2: PULP cluster architecture with HWPEs integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' maximizing the number of operations performed per memory access, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' the arithmetic intensity, is the key to an efficient design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' As analyzed by Pedram [45], scalar dot products and vector units do not guarantee the best trade-off between the number of operations performed per memory load/store access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1a, a simple scalar dot product that operates on a N-dimensional array performs 2 × N operations (N multiplications + N additions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The memory operations performed in this kernel are N loads of X, N loads of W, one load of Y and one store of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The resultant arithmetic intensity is: Intensity 1D = OPs LD/ST = 2N 2N + 2 ∼ 1, (N → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (2) 2-Dimensional L×H arrays exploit block-dot products (outer product) microkernels to perform GEMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Let us consider an L × H 2D array that can operate on L × 1 and 1 × H vectors, each made of N elements, like those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The operations performed on the two vectors are 2×L×H, repeated N times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The resulting load/store operations are L × N loads of X, H ×N loads of W, L×H loads of Y and L×H stores of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' With these changes, Equation 2 becomes: OPs LD/ST = 2LHN (L + H)N + 2LH ∼ 2LH L + H , (N → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' (3) Equation 3 shows that if L = H, the number of operations is quadratic with the size of the 2-D array, while the number of memory accesses remains linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This demonstrates that 2- dimensional arrays are more efficient with respect to scalar or vector units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Thus, we will exploit the outer-product approach for the RedMulE design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' ARCHITECTURE In this section, we describe the PULP cluster, the hardware template we rely upon, and the RedMulE micro-architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' PULP Cluster and RedMulE In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2, we show the architecture of a PULP cluster, a multi-core architecture that features a parametric number (2–16) of 32-bit RISC-V general-purpose cores featuring a partially shared, partially private instruction cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this specific work, we focus on a PULP cluster containing 8 RISC-V cores, equipped with 128 kB of Tightly-Coupled Data Memory (TCDM) split among 16 banks for word-level interleaving with a low level of contention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The PULP cluster also features an event unit for flexible internal synchronization and a dedicated Direct Memory Access Controller (DMAC) to efficiently move data between the TCDM and external memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' A peripheral interconnect allows the RISC-V cores to program the on-board peripherals (like the DMAC), and an AXI4 full cross-bar inter- connect allows communications with the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The capabilities of the PULP cluster can be further enhanced by integrating application-specific hardware accelerators called Hardware Processing Engines (HWPEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' HWPEs are software programmed by the RISC-V cores through the peripheral interconnect and share the TCDM with the RISC-V cores and the DMAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this sense, the HWPEs are tightly-coupled with the cluster cores [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The cores, the DMAC, and the accelerators access the shared TCDM through a single-cycle latency Heterogeneous Cluster Interconnect (HCI) [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Such interconnect features a logarithmic branch that allows all-to- all single-cycle accesses from 32-bit master ports, like those of the cores or the DMAC, to each of the word-interleaved memory banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Conflicts are managed by granting only one initiator per bank with a round-robin scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The other branch is the shallow branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' It features a single n-bit parametric port, routed to adjacent 32-bit memory banks treated like a single wider bank without arbitration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This branch allows for simple integration of tightly-coupled accelerators like HWPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The bitwidth of the shallow branch port can be tuned to the HWPE requirements through a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The TCDM banks are connected to the two HCI branches through a set of multiplexers, which grant access to one branch or the other according to a configurable starvation-free rotation scheme, allocating a configurable maximum of K < N consecutive cycles to the HWPE over a period of N cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' During the execution of NNs workloads, particularly during inference and training, on-the-fly data marshalling operations are known to reduce performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For this reason, our PULP cluster features a DataMover [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The DataMover is a tiny accelerator capable of transposing 3-dimensional tensors stored in the TCDM, with 33% less time than eight RISC-V cores and up to 50× increased energy efficiency (the lower the precision of chunks to transpose the more significant the advantages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The accelerator works with a configurable data element bitwidth, from 32-bit down to 1-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE 1) Global Architecture: RedMulE is a domain-specific pro- cessor designed to accelerate GEMM-Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Its architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The core of RedMulE is the Datapath, a 2- Dimensional array of CEs interconnected as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The CEs are organized in L rows, each made of H columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Within each row, a number of H CEs are cascaded so that each CE computing an intermediate product will pass its result to the next CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The partial product computed by each row’s last CE is fed back as accumulation input of the same row’s first CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The 5 RedMulE Parameters L # of rows of CEs H # of columns of CEs (# of CEs per row) P # of pipeline stages per CE CE 0,0 X W 0 Z ROW 0 FNCOMP y x w z Computing Element (CE) COLUMN 0 COLUMN 1 c) a) L H clk_1 clk_2 en CLOCK GATING FMA !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='FMA TCDM RDATA STREAMER INPUT REDMULE CAST HFP8 FP16 STREAMER OUTPUT TCDM DATA SEL = (CAST) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1 : 0 1 0 1 0 REDMULE CAST HFP8 FP16 d) X FIFO W FIFO Z FIFO X-BUFFER DATAPATH W-BUFFER Y/Z-BUFFER CONTROLLER PERIPH INTERCO W-BUFFER 256b 256b X W X W X W CE 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 CE 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 CE 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 COLUMN 2 COLUMN 3 CE 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='0 X 0 Z Y ROW 1 X X X Y CE 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 CE 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 CE 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='0 X 0 Z Y ROW 11 X X X CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 Y FIFO 256b 256b RedMulE TCDM 288b b) STREAMER SCHEDULER Datapath RedMulE Cast Pipe COMB Pipe Pipe FMA COMB Pipe COMB Pipe Pipe FNCOMP COMB REDMULE CAST 288b CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='0 CE 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='0 CE 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3: a) RedMulE internal architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' b) RedMulE Datapath microarchitecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' c) RedMulE CE microarchitecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' d) RedMulE Cast module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' e) Table with RedMulE design-time available parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE Datapath features a design-time configurable number of internal CEs, pipeline registers (P) for each CE, and internal computing precision (FP bitwidth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' All RedMulE’s parameters are tunable at design time and are resumed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To feed the Datapath with data, RedMulE includes the Streamer, following the HWPE design strategy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The Streamer is a specialized memory access unit that connects RedMulE to the HCI shallow branch through a single wide port of parametric size (multiple of 32-bit), used for load and store operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The incoming stream from the HCI is propagated to a single input-multiple output dispatcher that forwards the valid only to the selected output channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' simultaneously, each output channel propagates the incoming stream from the HCI to the accelerator input ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, the streams produced by RedMulE are propagated to the HCI interface during write operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The Streamer is connected to three internal buffers: an X- Buffer that changes all the L inputs of a column once every H ×(P +1) cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' a W-Buffer made of H shift registers, each broadcasting a new W-element to all the L CEs of a column every cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' a Z-Buffer that buffers the computed Z-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The same buffer is used to pre-load Y-elements and push them into the Datapath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This solution saves area and power in the accelerator since there is no need for a separated buffer to store Y bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The control side of the accelerator is divided into two sub- modules, namely Scheduler and the Controller, that contain the register file, accessed by the cores to program the accelerator and cooperate to regulate the accelerator execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2) Computing Element Microarchitecture: The micro- architecture of each CE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The CE is divided into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The first stage is dedicated to the ◦ operation selection and contains one Fused Multiply-Add (FMA) and one 1https://hwpe-doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='rtfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='io Floating-Point Non-Computational Operations (FNCOMP) like comparisons, which implements FP MIN/MAX operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We adapted the FMA and FNCOMP modules from the open-source FPnew trans-precision Floating-Point Unit (FPU) [48], so that their internal pipeline registers could support backpressure coming from memory stalls during RedMulE’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' X and W elements are propagated into both the FMA and the FNCOMP modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Depending on the desired ◦ operation, a multiplexer selects the result of either the FMA or the FNCOMP, while the clock gating module shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3b freezes the input operands of one module or the other so that there is no switching activity in the unused module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The Y element is propagated to the FMA in the first stage and is also directly sent to the input of the second stage, containing a fully combinational FNCOMP module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This architectural solution guarantees the execution of all the operations listed in Table I with a compact architectural implementation, in which we duplicate just what is strictly needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3) Casting Module: Hybrid FP8 precision formats can be used as an efficient compression scheme to enable DL inference and training on extreme-edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Hybrid FP8 precision means that the {sign, exponent, mantissa} structure used to represent the tensors can be either {1, 5, 2} or {1, 4, 3}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The former format is best suited for backward propagation of gradients, as it provides a larger dynamic range but a lower ac- curacy, while the latter is a better fit for forward propagation of activations thanks to the larger mantissa [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' While 8-bit representation works for data compression, it could severely impact accuracy due to reduced-precision accumulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To support this use case, RedMulE works internally with fixed FP16 precision but still accepts compressed FP8 formats as inputs and is capable of generating FP8 compressed output tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To do this, RedMulE is augmented with a dedicated 2Also called E4M3 and E5M2 by NVIDIA [https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='com/mkhbxj3v] 6 0 Store Row_0 Col_0 X0,0 X0,4 X0,8 W0,0 W0,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} 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K N M N L Hx(P+1) Col_1 X0,1 X0,5 X0,9 4 x 0 W1,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='. W1,15 W1,1 W5,0 Col_2 CE 03 Col_3 8 x 0 W2,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='. 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CE 00 CE 01 CE 02 CE 03 Time Column Index 0 1 2 3 Partial Result a) Z W X Rw Rx Vz b) c) d) Load/Store Memory Access Schedule Broadcasted Time 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b 16b Broadcasted Broadcasted Broadcasted CE 02 CE 01 CE 00 Y0,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Y0,15 16b Ry L Hx(P+1) Y M K Accumulate Accumulate = 1 0 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4: a) GEMM-Op execution displayed on matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' b) Row of CEs within RedMulE Datapath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' c) Memory access schedule in load/store mode described in terms of R (Ready) and V (Valid) handshake signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' d) Pipeline evolution within a row of ces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' casting module placed between the Streamer and the HCI interface, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The cast module contains two FP cast units: the input cast unit is used to cast 8-bit FP incoming stream into 16-bit FP to feed the accelerator so that the CEs in the Datapath can operate on larger precision, guar- anteeing enough accuracy during intermediate accumulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' After the computation, the output cast unit can be used to convert the 16-bit FP results produced by RedMulE to 8-bit encoded outgoing stream before writing it to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The cast units can be excluded from the path if the input tensors are represented with 16-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For DL use cases only, RedMulE can also be instantiated at design time so that it can only load and store HFP8 operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this use-case, the input and output tensors represented with 8-bit formats allow to read or write from and to the memory twice the number of elements while keeping the same memory bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Consequently, this allows for doubling the number of CEs inside each row, doubling RedMulE’s performance with respect to the 16-bit inputs case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE Computational Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4a shows how RedMulE performs a GEMM-Op visual- ising it on the computed matrices, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4d show the detailed sequence of the operations within a row of CEs providing an example of GEMM execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For this discussion, let us focus on a RedMulE implementation that features L = 12, H = 4, and P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The RedMulE operation starts by pre-loading the Z-Buffer with L rows from the Y- matrix, each row made of H × (P + 1) = 16 FP16 elements (256-bit memory width/16-bit internal precision), namely y0,0 y0,15 for Row 0, y1,0 - y1,15 for Row 1, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Afterwards, RedMulE pre-loads the X-Buffer as well, following the same pattern, and then loads a set of H × (P + 1) = 16 W-elements (w0,0 - w0,15) inside the first shift register of the W-buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Each W-element is broadcasted to all the L CEs in the first Datpath column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' While W-elements are broadcasted, the Z-Buffer pushes Y-elements in the CEs array cycle-by- cycle to perform the ⋆ operation during the execution of the ◦ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' After P + 1 cycles, each of the L CEs in the first column forwards its computed partial result to the neighbour CE in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The accelerator loads another set of H × (P + 1) W-elements (w1,0 - w1,15) to broadcast them to all the CEs in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Once all the H CEs of a row have completed their computations, calculating a subset of H × (P + 1) row-column intermediate results, RedMulE activates its feedback (accumulate = 1) to provide the intermediate results to the accumulation input of the first CEs of the given row, then reiterating the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Immediately after, the Streamer reloads the next Y-submatrix in the Z-Buffer so that it will be ready for the next calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' During the Z-Buffer reload operation, the X-Buffer provides a new X-operand to the first column of CEs, and a new set of H ×(P +1) W-elements is reloaded in the first W shift register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' After (P + 1) cycles, all the L CEs of the first column produce a new partial product and provide it to the CEs in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The X-Buffer provides a new X-operand at the input of the second column of CEs, and the W-Buffer loads a new set of H × (P + 1) W- elements in the second W shift register for broadcasting, and the computation continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig 4d shows the detailed sequence of data within the pipeline of a row of CEs from the beginning of a GEMM operation until the moment of the reuse of the partial results (accumulate = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To guarantee a continuous data flow in the accelerator, the W-buffer accesses the memory once every (P + 1)-cycles to load a new set of H ×(P +1) W-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Once the X-Buffer and the Z-Buffer are empty, RedMulE reuses the Streamer port to load the X and Y-operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Such operation is made by interleaving the memory accesses to X or Y matrices between 7 RedMulE Baseline (8 Cores) Ideal (32 MAC/cycle) a) b) c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5: RedMulE benchmarking with comparison with software executed on 8 RISC-V cores: a) Synthetic GEMM execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' b) ResNet8 execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' c) GEMM-Ops execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' two adjacent W-matrix accesses until the complete fulfilment of the X and Z buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig 4c shows how the memory accesses to different matrices are interleaved, describing the memory accesses in terms of Ready (R) and Valid (V) handshake signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The Streamer load and store units fully support backpressure through a mechanism based on R/V handshake signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Such a mechanism fully decouples the memory access and data consumption/production from the Datapath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The V signals for loads and the R signals for stores are generated within the Streamer itself depending only on memory stalls, which can be amortized by the presence of FIFO elements, and not on the actual usage from the Datapath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, the Datapath uses the R signal of loads and the V signal of stores, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4c, to control the order of memory accesses interleaving them so that a continuous dataflow can be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This choice is made to maximize the memory port utilization since having a single memory port also helps reduce the overall streamer area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' After the conclusion of an entire row-column operation, the Z-Buffer buffers the final sub-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Afterwards, store oper- ations are interleaved between two adjacent W load accesses until the Z-Buffer is empty and can be reloaded with Y- elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' With this approach, RedMulE optimizes the band- width utilization using a single wide memory port and achieves up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% CEs utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' IMPLEMENTATION AND MEASUREMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Experimental Setup We focus our experiments on a RedMulE12x4 instance with H = 4, L = 12, P = 3, resulting in 48 CEs and a 288-bit wide HCI port, for 256-bit + 32-bit non-word-aligned accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We also address a RedMulE12x8 since, as described in Section V-B3, it uses the same memory interface with twice the number of CEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Our experiments target GlobalFoundries 22 nm technology using Synopsys Design Compiler for synthesis (slow corner at ftarg = 250 MHz, VDD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='59 V, T = 125 °C) and Cadence Innovus for full-cluster Place&Route in the same operating point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE’s timing analysis and power extraction were made using Prime Time with 100% annotated switching activity from post-layout simulation in typical corner at 25 °C, targeting two operating points: 470 MHz at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 V for high energy efficiency and 613 MHz at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V for high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Performance Evaluation 1) GEMM Performance Evaluation: We use square and rectangular matrices as a synthetic benchmark to evaluate RedMulE’s computation latency in cycles against the SW execution on 8 parallel RISC-V cores sharing 4 FPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the given benchmark, RedMulE reaches a peak throughput of more than 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4 OP/cycle, where we count both ⋆ and ◦ as one ”OP”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' for a regular GEMM we count 1 MAC = 2 OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE achieves up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% of CEs utilization on 96×96 FP16 matrices (55 kB memory occupation), leading to 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS at 613 MHz with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='80 V supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5a shows the number of computing cycles required to compute various matrices during parallel FP16 software executed on 8 RISC-V cores and compares them on RedMulE, showing that it reaches 15× average speedup over the software on large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' This performance increase with respect to the software counterpart settles around 13× with larger matrices since also the software execution becomes more efficient in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We also consider the acceleration of a small 8 × 8 × 8 case, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5a in which the accelerator is under-utilized, but it still introduces 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5× speedup over the software parallel execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2) FP16 Network Training: To further evaluate Red- MulE performance on a real-case NN training, our target is TinyMLPerf [49], and in particular, we focused on the ResNet [50] example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For the software infrastructure, we rely on the pulp-TrainLib [27], and we compared RedMulE with a software baseline executed on 8 RISC-V cores sharing 4 FPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The library takes into consideration all the training steps for the calculation of the gradients and backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5b shows the execution of a single step in the ResNet8 network when using 8 RISC-V cores in parallel and when using RedMulE for the matrix multiplication execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE accelerates the matrix multiplication execution of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6× with respect to the parallel RISC-V execution in SW, speeding up the entire single step of the ResNet8 of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE keeps its utilization constant at 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1% (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 MAC/cycle) with the only exceptions in the first and the last layers where it drops to 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2% (44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7 MAC/cycle) and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3% (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 MAC/cycle) due to leftovers that do not allow to exploit the full potential of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5b, it is also evident that the data reorganization during the Im2Col accounts for approximately 3 Millions computing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' To solve this problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' we aug- ment RedMulE’s operation with the support of the DataMover GEMM-Ops Speedup Over SW Execution RedMulE Cycles sGroup 1Group 2 62 x 1000000 59 x 47 x 100000 COMPUTING CYCLES 25 x 10000 1000 100 10 1 8x8x8 12x12x12 32x32x32 64x64x64 MATRIX SIZESResNet 8 FP16 - RISC-V Cores vs RedMulE Single Training Step RedMulE+Datamover RedMulE+Datamover (FP8) RISC-V RedMulE 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 x COMPUTING CYCLES 12000000 6 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 10000000 ↑ 8000000 3 6000000 9 15 4000000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 x 2000000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 x 0 DMA Im2Col Im2Col Other Matmul TRAINING Transfers (Core) (Padding) Marshalling STEP RESNET PHASEHW vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=" sW Execution XS'ET 1E+6 S x9's 1 x8'21 ycl 1E+5 1 1E+4 Computing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5 1E+3 1E+2 1E+1 1E+0 24x48x64 24x32x32 96x96x96 6 9 Computed MACs8 b) a) c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 6: a) RedMulE area breakdown with a focus on the datapath, b) RedMulE power distribution, c) PULP power distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' engine, halving the number of computing cycles required to perform the two Im2Col operations and thus speeding up the overall training step execution up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' As all the devices included in the PULP cluster (RISC-V cores and accelerators) are designed for synergistic cooperation and share the memory, the heterogeneity of the architecture can be efficiently and fully exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 3) HFP8 Network Training: For the same training example, we consider a RedMulE12x8 instance used to train the ResNet network encoded on 8-bit FP inputs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For the RedMulE12x4 we considered until now, the memory port of the Streamer is 288-bit wide, meaning a 256-bit memory port with non- word aligned memory accesses capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this configuration, RedMulE12x4 can load 16 × FP16 elements at a time that are used to fill the pipeline during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Having H = 4 columns, the pipeline stages within each row are calculated as H ×(P +1), where P = 3 in this implementation, resulting in 16 pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Considering a fixed 8-bit input encoding, with the same 288-bit memory port, RedMulE can access up to 32×FP8 elements at a time, meaning that we can implement a RedMule12x8 instance maintaining a 288-bit memory interface and obtaining 32 pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We show how the ResNet8 training can benefit from this configuration in the green bar of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Matrix multiplication execution can be accelerated up to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5×, resulting in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5× speed-up over the entire training step execution, with 97% utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 4) GEMM-Ops Performance Evaluation: To evaluate the GEMM-Ops performance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 5c we compare the RedMulE GEMM-Ops execution against parallel SW execution on the RISC-V cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE always takes the same number of computing cycles to perform any of the supported GEMM- Ops, while the parallel execution on the general-purpose cores changes depending on the executed kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' All the kernels belonging to Group 1 (see Table I), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' for which ◦ operation is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 7: RedMulE area sweep with several sizes of H and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' +/× and ⋆ is max/min, require the same number of computing cycles when executed on the cores, while up to 47× speedup can be achieved when leveraging RedMulE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' When also ⋆ are of the max/min kind, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Group 2, the execution overhead for the general-purpose cores is even higher, and RedMulE can accelerate such kernels up to 62×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE Area 1) Area Breakdown analysis: RedMulE12x4 occupies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='15 mm2, corresponding to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8% of the entire PULP cluster area (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='64 mm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 6a shows the breakdown of the RedMulE area, where the cast units account for the 7% to the overall accelerator area, and the FMA units account for the 72%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The support for GEMM-Ops, namely the introduction of the two FNCOMP modules and the operation selection logic, introduces an overhead of just the 16% over the entire accelerator area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The 13% of this overhead resides in the first stage FNCOMP and is dominated by the pipeline introduced to match the number of cycles required by an FMA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The second stage FNCOMP is fully combinational and accounts only for 3% of the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2) RedMulE Area Sweep: We studied the area overhead introduced when changing the number of CEs within RedMulE, fixing the CEs’ internal pipeline stages to P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 7 shows that RedMulE’s area occupation becomes comparable to the area of the entire PULP cluster when it contains 256 CEs, corresponding to a RedMulE16x16 instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, the area of RedMulE32x32 is 4× larger than the entire PULP cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 7 shows that changing the shape of the Datapath also affects the size of the Streamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In particular, for each CE that is added to a row of the Datapath (or equivalently, changing the H parameter), other P + 1 pipeline registers are added within each Datapath row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The consequence is that the number of elements needed to keep a high CEs utilization increases by P + 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Keeping P = 3 as an example, increasing the H parameter by 1 requires the Streamer to provide P +1(= 4) additional FP16 elements to the Datapath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The consequence is that the streamer port must be enlarged of 64-bit (= 4 × 16-bit), limiting the integration of RedMulE in the PULP cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2%3% 10% 5% 4% 76%RedMulE Area Sweep Control I Cast Streamer Buffers Datapath 10000 9000 8000 7000 [kGE] 6000 5000 Area 4000 3000 PULP CLUSTER 2000 1000 0 8x4 12x4 12x8 16x16 32x32 RedMulE Instance (LxH)9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 8: Energy efficiency of RedMulE compared with the SW baseline executed on 8 RISC-V cores with 4 shared FPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE Power At a cluster level, the power consumption in the efficiency point amounts to 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 mW during GEMM operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The RedMulE contribution dominates the power envelope account- ing for 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8% of the overall consumption, while the TCDM banks and the HCI interconnect contribution is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this operating point, we reach a cluster peak energy efficiency of 755 GFLOPS/W during GEMM execution, corresponding to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5× higher energy efficiency with respect to the software baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' During the execution of the algorithms belonging to GEMM-Ops’ Group 1 on RedMulE, the cluster-level power dissipation reaches 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 mW, leading to 842 GFLOPS/W, which is 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2× higher than SW execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, during the execution of the algorithms in GEMM-Ops’ Group 2, the power consumption is further reduced to 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 mW resulting in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='19 TFLOPS/W, thus 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2× more efficient than software execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Figure 8 compares the energy efficiency of RedMulE with the software baseline executed on 8 RISC-V cores with 4 shared FPUs on FP16 elements during the execution of GEMM, GEMM-Ops’ Group 1 and GEMM-Ops’ Group 2 kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Figure 6b and Figure 6c show respectively the power breakdown for RedMulE, where most of the power is consumed by the Datapath, and the PULP cluster during a GEMM operation, where the majority of the power is consumed by RedMulE and by the TCDM banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' COMPARISON WITH THE STATE-OF-THE-ART Table II resumes the comparison of our work with different State of the Art (SoA) architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We compare our work with GPU architectures, in particular with NVIDIA H100 containing TensorCores, that guarantee up to 989 TFLOPS of performance in FP16 and 1979 TFLOPS in FP8, meaning 17000× than our work, but at the cost of 700 W power consumption and 814 mm2, 12000× more power- consuming and 1300× larger than our work – representing an unfeasible solution for an IoT end-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' While RedMulE targets primarily training, it is also usable for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' For this reason, we include in our comparison some inference-oriented chips, like DNPU [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' DNPU’s per- formance is just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9× higher than our cluster, although DNPU contains 16× the number of CEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Moreover, DNPU features Technology GF22FDX Cluster Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='64 mm2 (48 CEs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='73 mm2 (96 CEs) Cluster SRAM 128 kB RedMulE Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='15 mm2 (48 CEs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='24 mm2 (96 CEs) RedMulE Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 117 GFLOPS (FP8) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS (FP16) Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Range 470 - 613 MHz VDD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V Power Cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 - 193 mW GEMM En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 920 GFLOPS/W (FP8) 775 GFLOPS/W (FP16) GEMM-Ops En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 TFLOPS/W (Group 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='67 TFLOPS/W (Group 2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 9: Area breakdown of the PULP cluster, layout and resume table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7× higher efficiency than RedMulE but is designed to work with fixed-point precision only, which helps increase energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We also compared our work with Diana [21] and Gemmini [23], being designed in the same technology node of RedMulE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The former achieves 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5% less performance than RedMulE12x8 and 12% less performance than RedMulE12x4 in the energy efficient mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Diana’s power consumption in efficiency mode is much lower than our design, but if we scale down the frequency to 50 MHz as they do, our PULP cluster with RedMulE12x4 consumes just 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The sig- nificant increase in Diana’s energy efficiency is justified as it uses only 8-bit integer arithmetic, which helps reduce power consumption and increases energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the contrary, Gemmini features one order of magnitude less energy efficiency than RedMulE12x4 despite it features 5× the number of CEs and works with 8-bit integer format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We also compared our design with other platforms specif- ically designed for on-chip training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' IBM [33] demonstrated a 4-core AI chip in 7 nm technology which is just 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4× more energy-efficient, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2× larger, and 74× more power- consuming than our PULP cluster with RedMulE12x4, despite the technology scaling and the reduced operating voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' IBM also proposes a chip [35], with more than 1 W of power con- sumption, which is not acceptable for extreme-edge training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On the other hand, LNPU [36] is an extreme-edge processor that features a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='67× higher power envelope than RedMulE12x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Vega is a valid candidate for on-chip embedded training, but RedMulE12x4 achieves 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8× higher performance and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2× higher energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Cambricon-Q [31] is designed in 45 nm and is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9× more energy-efficient than our design but makes use of narrow 8-bit fixed-point arithmetic, while generally available learning algorithms based on backpropaga- tion strictly require FP range and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Cambricon-Q is also 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7× more power-hungry than our design, therefore not suitable for TinyML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Similar considerations hold for T-PIM [37], a training chip designed in 28 nm technology that features an in-memory computing core for high energy efficiency but only works with 16-bit integer precision, not satisfying the precision requirements to enable on-chip training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' GEMM-Ops Energy Efficiency Over SW Execution Group 1 Group 2 GEMM RedMulE GOPS/W 1000 ENERGY EFFICIENCY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' [GFLOPS/W] 100 10 1 8x8x8 12x12x12 32x32x32 64x64x64 MATRIX SIZES3% 24% 7% 19% 22% 19%:10 TABLE II: State of the art comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' First line = Best Efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Second line = Peak Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 1 MAC = 2 OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Category Design Tech nm Area mm2 Freq MHz Volt V Power mW Perf GOPS Energy Eff GOPS/W CEs Precision GPU NVIDIA H100 [16] 4 814 1830 700000 1978900 989400 2827 1413 528 FP8 FP16 SIMD2 [6] 45 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 4190 INT16 Inference Chips DNPU [22] 65 16 50 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 279 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 279 2100 1000 768 INT16 Diana [21] 22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='91 50 280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='96 129 40 224 4040 1740 256 INT8 Gemmini [23] 22 16 700 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='91 70 50 256 INT8 Training Chips 4-core IBM [33] 7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 1000 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='75 4400 13000 8000 12800 1800 980 4096 FP16 LNPU [36] 65 16 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 367 600 300 1630 817 768 FP8 FP16 Oh, IBM [35] 14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 1000 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='62 1428 2727 2000 3000 1400 1100 128 FP32 FP16 T-PIM [37] 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='04 50 280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='25 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 43 7590 840 INT16 TSUNAMI [39] 65 16 50 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 45 419 612 3420 1480 2048 FP8 306 1710 740 1024 FP16 Trainer [40] 28 21 40 440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='56 1 23 363 900 4280 8192 FP8 450 2140 4096 FP16 Cambricon-Q [31] 45 888 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 1030 2000 2240 1024 INT8 Vega [30] 22 12 450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 250 180 4 FP16 Mat-Mul Anders [41] 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='024 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 1090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='023 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='068 34 2970 420 16 FP16 GEMM This Work RedMulE12x4 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='64 470 613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='3 116 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 775 506 48 FP16 Group 1 GEMM-Ops 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 103 842 576 Group 2 GEMM-Ops 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 1193 819 GEMM This Work RedMulE12x8 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='73 470 613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 193 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='7 117 920 608 96 FP8 Group 1 GEMM-Ops 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2 168 1052 694 Group 2 GEMM-Ops 54 104 1666 1123 TSUNAMI [39] and Trainer [40] are conceived for energy- efficient embedded training and extensively use pruning and sparse matrices generation to increase energy efficiency and reduce the number of required MAC operations during training with zero-skipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We compare with the results they pro- vide during dense calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In their best efficiency points, TSUNAMI and Trainer’s power consumption is comparable to RedMulE’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' However, those points correspond to 50 MHz and 40 MHz for TSUNAMI and Trainer, while RedMule is evaluated at 470 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Therefore, RedMulE would consume approximately one order of magnitude less power at compa- rable frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' TSUNAMI and Trainer reach up to 5× and 8× higher performance, respectively, since they feature 21× and 85× the number of CEs, but feature a much lower CEs’ utilization than our RedMulE (75% TSUNAMI and only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5% Trainer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' The systolic architecture of RedMulE enables, in principle, almost arbitrary architecture scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Assuming linear performance, area, and power ratio, scaling to 1024 or 4096 CEs (21× and 85× larger than RedMulE12x4), our utilization would still be 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4%, leading to higher overall performance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='25 TFLOPS and 5 TFLOPS respectively) once accounting overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We compare RedMulE12x4 with Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' [41], proposing a hardware accelerator for matrix multiplications in 14 nm tech- nology that targets TinyML learning and inference applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' It reaches a peak energy efficiency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='97 TFLOPS/W in FP16 precision, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='83× higher than RedMulE12x4, but only when operating at near-threshold voltage (260 mV) and extremely reduced frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In that operating point, their design is 659× less performant than RedMulE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Anders’ peak performance is obtained at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9 V and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='09 GHz, leading to 420 GFLOPS/W and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% MAC units utilization, similarly to RedMulE’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In 22 nm technology, at 613 MHz frequency and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V, RedMulE12x4 reaches 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='72× better than Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=', with a 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5% higher energy efficiency of 506 GFLOPS/W on FP16 GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We also compared RedMulE with SIMD2 [6], the only other design that features GEMM-Ops extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Even though SIMD2 works only with integer precision and cannot thus target on-chip training, it features 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='1× higher power consumption than RedMulE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In their case, the authors also claim that the area overhead to build GEMM-Ops extensions on top of 11 NVIDIA Streaming Multiprocessor accounted for 69%, while in RedMulE, the area overhead introduced by GEMM-Ops extension is just 16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' CONCLUSION In this paper, we presented RedMulE - Reduced-Precision Matrix Multiplication Engine, a fully-parametric open-source cluster-coupled accelerator enabling TinyML training on ultra- low-power devices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' near-sensor training on a few tens of mW of power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE is conceived for FP16 GEMM-Ops computation, and supports compressed FP8 inputs while also efficiently accelerating a wide range of operations that share the same structure of a GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE allows the instantiation of a wide range of Floating-Point Units- based Computing Elements (CEs), internal buffers, and memory interface configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' We integrated an instance of RedMulE, containing a 12×4 array of CEs into an ultra-low-power cluster containing 8 RISC-V cores, and implemented the resulting system in a 22 nm technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' RedMulE achieves 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='4% CEs utilization and an average 15× speedup during simple GEMM execution with respect to a parallel software baseline running on the eight cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' It occupies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='15 mm2 accounting for only 24% of the cluster area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' During GEMM-Ops execution, the performance speedup introduced by RedMulE over the RISC-V cores reaches up to 62×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In its best performance point (at 613 MHz, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 V), RedMulE achieves 506 GFLOPS/W @ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5 GFLOPS when executing FP GEMM kernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' while, in its best efficiency point (at 470 MHz, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='65 V), it reaches 775 GFLOPS/W @ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='8 GFLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' On a real example of NN training, RedMulE accelerates the matrix multiplication by up to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='6× and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='5× when the input tensors are represented with 16-bit and 8-bit respectively, accelerating the whole training step of ResNet8 by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='9× and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by Thales Alenia 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“Deep residual learning for image recognition,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Yvan Tortorella received his Master’s Degree in Electronic Engineering in October 2021 from the University of Bologna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He is currently pursuing a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' in Digital Systems Design in the group of Professor Luca Benini at the Department of Electrical and Information Engineering (DEI) of the University of Bologna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' His research interests include the design of PULP (Parallel Ultra-Low Power)-based hardware accelerators for ultra-low power Machine Learning and the design of RISC-V-based computer architec- tures for satellite applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Luca Bertaccini received the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' degree in Elec- tronic Engineering from the University of Bologna in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He is currently pursuing a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' degree at ETH Z¨urich in the Digital Circuits and Systems group led by Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Luca Benini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' His research inter- ests include heterogeneous systems-on-chip, energy- efficient hardware accelerators, computer arithmetic, and transprecision computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He received the 2021 IEEE ASAP Best Paper Honorable Mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Luca Benini holds the chair of digital Circuits and systems at ETHZ and is Full Professor at the Universit`a di Bologna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He received a PhD from Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Benini’s research interests are in energy-efficient parallel computing systems, smart sensing micro-systems and machine learning hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He has published more than 1000 peer- reviewed papers and five books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He is a Fellow of the IEEE, of the ACM and a member of the Academia Europaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He received the IEEE Mac Van Valkenburg award in 2016 and the ACM/IEEE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Richard Newton Award in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Davide Rossi received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' degree from the Uni- versity of Bologna, Bologna, Italy, in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He has been a Post-Doctoral Researcher with the Department of Electrical, Electronic and Information Engineering “Guglielmo Marconi,” University of Bologna, since 2015, where he is currently an Assistant Professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' His research interests focus on energy-efficient digital architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' In this field, he has published more than 100 papers in international peer-reviewed conferences and journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He is recipient of Donald O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' Pederson Best Paper Award 2018, 2020 IEEE TCAS Darlington Best Paper Award, 2020 IEEE TVLSI Prize Paper Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' 13 Francesco Conti received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' degree in elec- tronic engineering from the University of Bologna, Italy, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' He is currently an Assistant Professor in the DEI Department of the University of Bologna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' From 2016 to 2020, he held a research grant in the DEI department of University of Bologna and a position as postdoctoral researcher at the Integrated Systems Laboratory of ETH Zurich in the Digital Systems group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' His research focuses on the devel- opment of deep learning based intelligence on top of ultra-low power, ultra-energy efficient programmable Systems-on-Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} +page_content=' His research work has resulted in more than 40 publications in international conferences and journals and has been awarded several times, including the 2020 IEEE TCAS-I Darlington Best Paper Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfdQd0/content/2301.03904v1.pdf'} diff --git a/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/2301.05219v1.pdf.txt b/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/2301.05219v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2bbcda91e433600083c7f1182bbeedee36bd6ba --- /dev/null +++ b/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/2301.05219v1.pdf.txt @@ -0,0 +1,2059 @@ +Why is the State of Neural Network Pruning so Confusing? On the Fairness, +Comparison Setup, and Trainability in Network Pruning +Huan Wang† +Can Qin +Yue Bai +Yun Fu +Northeastern University, Boston, USA +Abstract +The state of neural network pruning has been noticed to +be unclear and even confusing for a while, largely due to “a +lack of standardized benchmarks and metrics” [3]. To stan- +dardize benchmarks, first, we need to answer: what kind of +comparison setup is considered fair? This basic yet crucial +question has barely been clarified in the community, unfor- +tunately. Meanwhile, we observe several papers have used +(severely) sub-optimal hyper-parameters in pruning exper- +iments, while the reason behind them is also elusive. These +sub-optimal hyper-parameters further exacerbate the dis- +torted benchmarks, rendering the state of neural network +pruning even more obscure. +Two mysteries in pruning represent such a confusing sta- +tus: the performance-boosting effect of a larger finetuning +learning rate, and the no-value argument of inheriting pre- +trained weights in filter pruning. +In this work, we attempt to explain the confusing state of +network pruning by demystifying the two mysteries. Specif- +ically, (1) we first clarify the fairness principle in prun- +ing experiments and summarize the widely-used compar- +ison setups; (2) then we unveil the two pruning myster- +ies and point out the central role of network trainability, +which has not been well recognized so far; (3) finally, we +conclude the paper and give some concrete suggestions re- +garding how to calibrate the pruning benchmarks in the fu- +ture. Code: https://github.com/mingsun-tse/why-the-state- +of-pruning-so-confusing. +1. Introduction +The past decade has witnessed the great success of deep +learning, empowered by deep neural networks [38,68]. The +success comes at the cost of over-parameterization [5, 12, +26, 31, 32, 36, 61, 62, 66, 69, 75], causing prohibitive model +footprint, slow inference/training speed, and rapid-growing +energy consumption. Neural network pruning, an old model +parameter reduction technique that used to be employed for +†Corresponding author: wang.huan@northeastern.edu +Table 1. Top-1 accuracy (%) benchmark of filter pruning with +ResNet50 [26] on ImageNet [10]. Simply by using a better fine- +tuning LR schedule, we manage to revive a 6-year-ago baseline +filter pruning method, L1-norm pruning [42], making it match or +beat many filter pruning papers published in recent top-tier venues. +Note, we achieve this simply by using the common step-decay LR +schedule, 90-epoch finetuning, and standard data augmentation, +no any advanced training recipe (like cosine annealing LR) used. +This paper studies the reasons and lessons behind this pretty con- +founding benchmark situation in filter pruning. +Method +Pruned acc. (%) Speedup +SFP [28] IJCAI’18 +74.61 +1.72× +DCP [88] NeurIPS’18 +74.95 +2.25× +GAL-0.5 [47] CVPR’19 +71.95 +1.76× +Taylor-FO [55] CVPR’19 +74.50 +1.82× +CCP-AC [60] ICML’19 +75.32 +2.18× +ProvableFP [45] ICLR’20 +75.21 +1.43× +HRank [46] CVPR’20 +74.98 +1.78× +GReg-1 [79] ICLR’21 +75.16 +2.31× +GReg-2 [79] ICLR’21 +75.36 +2.31× +CC [44] CVPR’21 +75.59 +2.12× +L1-norm [42] ICLR’17 (our reimpl.) +75.24 +2.31× +GAL-1 [47] CVPR’19 +69.88 +2.59× +Factorized [43] CVPR’19 +74.55 +2.33× +LFPC [27] CVPR’20 +74.46 +2.55× +GReg-1 [79] ICLR’21 +74.85 +2.56× +GReg-2 [79] ICLR’21 +74.93 +2.56× +CC [44] CVPR’21 +74.54 +2.68× +L1-norm [42] ICLR’17 (our reimpl.) +74.77 +2.56× +improving generalization [2, 6, 64], now is mostly used for +model size compression or/and speed acceleration [7,8,11, +20,30,73]. +The prevailing pipeline of pruning comprises three steps: +1) pretraining: train a dense model; 2) pruning: prune the +dense model based on certain rules; 3) finetuning: retrain +the pruned model to regain performance. Most existing re- +search focuses on the second step, seeking better criteria +to remove unimportant weights so as to incur as less per- +formance degradation as possible. This 3-step pipeline has +been practiced for more than 30 years [2,6,39,57] and is still +extensively adopted in today’s pruning methods [30,73]. +Despite the long history of network pruning, recent +progress seems a little bit unclear. As we quote from the +1 +arXiv:2301.05219v1 [cs.CV] 12 Jan 2023 + +abstract of a recent paper [3], “After aggregating results +across 81 papers and pruning hundreds of models in con- +trolled conditions, our clearest finding is that the community +suffers from a lack of standardized benchmarks and met- +rics. This deficiency is substantial enough that it is hard to +compare pruning techniques to one another or determine +how much progress the field has made over the past three +decades”. We also sense this kind of confusion. Particu- +larly, two mysteries in the area represent such confusion: +• Mystery 1 (M1): The performance-boosting effect +of a larger finetuning learning rate (LR). It was +broadly believed that finetuning the pruned model +should use a small LR, e.g., in the famous L1-norm +pruning [42], finetuning LR 0.001 is used for the Im- +ageNet experiments. +However, many other papers +choose a larger LR, e.g., 0.01, which delivers signifi- +cantly better performance than 0.001. For a pretty long +time, pruning papers do not officially investigate this +critical performance-boosting effect of a larger finetun- +ing LR (although they may have already used it in their +experiments; see Tab. 3), until this paper [37]. [37] +formally studies how different finetuning LR sched- +ules affect the final performance. They find random +pruning and magnitude-pruning (the two most basic +pruning methods) armed with a good finetuning LR +schedule (CLR, or Cyclic Learning Rate Restarting) +can counter-intuitively rival or even surpass other more +sophisticated pruning algorithms. Unfortunately, they +do not give explanations for this phenomenon except +for calling upon comparing pruning algorithms in the +same retraining configurations. +• Mystery 2 (M2): The value of network pruning. The +value of network pruning seems unquestionable given +the development history of over 30 years. However, +two papers [9, 50] empirically find that training the +pruned model from scratch can match pruning a pre- +trained model, thus radically challenging the necessity +of pretraining a big model first in the conventional 3- +step pruning pipeline. +Tab. 1 shows a concrete example that M1 makes the prun- +ing benchmarks unclear. +After using an improved fine- +tuning LR schedule (see Tab. 3), we make L1-norm prun- +ing [42], which is broadly considered the most basic base- +line approach, match or beat many top-performing pruning +methods published in top-tier conferences in the past sev- +eral years. Such a situation really bewilders us, especially +for those not in this area trying to borrow the most advanced +pruning methods in their projects – “Has the area really de- +veloped in the past several years?” they might question. +This paper is meant to unveil these mysteries. Over this +process, we hope to offer some helpful thoughts about why +we have run into the current chaotic benchmark situation +and how we can avoid it in the future. +Specifically, when we try to unveil the mysteries, only +to find there are various comparison setups enforced in the +area, many of the conclusions actually hinge on which com- +parison setup is used. To decide which comparison setup is +more trustworthy, we have to be clear with the fairness prin- +ciple in pruning experiments first (i.e., what kind of compar- +ison setup is considered fair?). After we sort out the fairness +principle and comparison setups, we empirically examine +the two mysteries and find M2 reduces to M1. When exam- +ining M1, we find the role of network trainability in network +pruning, through which we can easily explain M1. +Therefore, our investigation path and the contributions in +this paper can be summarized as follows. +• We first clarify the fairness principle in pruning exper- +iments and summarize the outstanding comparison se- +tups (Sec. 3), which have been unclear in the literature +for a long time. +• Then, we start to unveil M1 [37] and M2 [50]. +As +we shall show (Sec. 4), the conclusion of M2 actu- +ally varies up to which comparisons setup is used: If +a larger finetuning LR is allowed to be used, the no- +value-of-pruning argument cannot hold; otherwise, it +mostly holds. Thus, to unveil M2, we have to unveil +M1 first. +• Next, we focus on unveiling M1. We introduce the +perspective of network trainability to diagnose prun- +ing and clearly explain why the finetuning LR has a +significant impact on the final performance (Sec. 5) – +to our best knowledge, we are the first to clarify this +mystery in the area. +• Finally, we summarize the major reasons that have led +to the confusing benchmark situation of network prun- +ing up to now, and give some concrete suggestions +about how to avoid it in the future (Sec. 6). +2. Prerequisites +2.1. Taxonomy of Pruning and Related Work +A pruning algorithm has and only has five exclusive +aspects to be specified: (1) Base model (when to prune): +is pruning conducted on a pretrained model or a random +model? (2) Sparsity granularity (what to prune): what is the +smallest weight group in pruning? (3) Pruning ratio (how +many to prune): how many weights are to be pruned? (4) +Pruning criterion (by what to prune): what measure is used +to select the important weights (i.e., those to be kept) vs. the +unimportant weights (i.e., those to be pruned)? (5) Pruning +schedule (how to schedule): How is the sparsity scheduled +over the pruning process? +2 + +These five mutually orthogonal questions can shatter +most (if not all) pruning papers in the literature. +Re- +searchers typically use (1), (2), and (4) to classify different +pruning methods. We give the major backgrounds in these +three axes. +Pruning after training +vs. pruning at initialization. +Pruning has been mostly conducted on a pretrained model +over the past 30 years, which is thus called pruning af- +ter training or post-training pruning. +This fashion has +been the unquestioned norm until (at least) two papers +in 2019, SNIP [41] and LTH [17]. +They argue prun- +ing can be conducted on randomly initialized models and +can achieve promising performance (allegedly matching +the dense counterpart), too. This new fashion of pruning +is called pruning at initialization (PaI). Existing PaI ap- +proaches mainly include [19, 40, 41, 63, 77] and the series +of lottery ticket hypothesis [17,18]. PaI is not very relevant +to this work because the benchmarking chaos and the mys- +teries are mostly discussed in the PaT context, so here we +would not discuss in length the specific PaI techniques. In- +terested readers may refer to [80] for a comprehensive sum- +mary. +Sparsity structure: structured pruning vs. unstructured +pruning. If the smallest weight group in pruning is a single +weight element, this kind of pruning is called unstructured +(or fine-grained) pruning [17,22,23], because the resulting +zero-weight (i.e., pruned-weight) locations are typically ir- +regular (if no extra regularization is enforced). If the small- +est weight group in pruning presents some structure, this +kind of pruning is called structured (or structural/coarse- +grained) pruning [29, 42, 53, 82]. +In the area, structured +pruning typically narrowly refers to filter pruning or channel +pruning if not explained otherwise. Structured pruning ben- +efits more acceleration because the regular sparsity pattern +is more hardware-friendly; meanwhile, the regularity im- +poses more constraints on the network, so given the same +sparsity level, structured pruning typically underperforms +unstructured pruning. +Note, the definition of “structured” sparsity is severely +hardware-dependent and thus can vary as the hardware +condition changes. E.g., the N:M sparsity1 pioneered by +NVIDIA Ampere architecture was considered as unstruc- +tured sparsity, but since NVIDIA has launched new library +support to exploit such kind of sparsity for acceleration, +now it can be considered as structured sparsity (called fine- +grained structured sparsity [58,86]), too. +This paper focuses on filter pruning for now because the +two aforementioned mysteries (the effect of finetuning LR +and the value of network pruning) are mainly discussed in +this context. +1https://developer.nvidia.com/blog/accelerating-inference-with-sparsity- +using-ampere-and-tensorrt/ +Pruning criterion: Importance-based vs. regularization +based. The former prunes weights based on some estab- +lished importance criteria, such as magnitude (for unstruc- +tured pruning) [22,23] or L1-norm (for filter pruning) [42], +saliency based on 2nd-order gradients (e.g., Hessian or +Fisher) [24,39,70,74,76]. The latter adds a penalty term to +the objective function, drives unimportant weights toward +zero, then removes those with the smallest magnitude. Two +notable points: (1) Even in a regularization-based prun- +ing method, after the regularization process, the weights +are still removed by a certain importance (typically mag- +nitude). Namely, regularization-based pruning inherently +embeds importance-based pruning. (2) The two paradigms +are not exclusive; they can be employed simultaneously. +E.g., [13, 79, 81] select unimportant weights by magnitude +while also employing the regularization to penalize weights. +Finally, for more comprehensive literature on network +pruning, we refer interested readers to several surveys: an +early one [64], some recent surveys of pruning alone [3,20, +30] or pruning as a sub-topic under the general umbrella of +model compression and acceleration [7,8,11,73]. +Most relevant papers. The most relevant works to this pa- +per are +• [3], which is the first to systematically report the frus- +trating state of network pruning benchmarks, identify +some causes (such as the lack of standard benchmarks +and metrics in pruning), and give concrete remedies. +However, they do not go deeper and analyze why the +standard benchmarks are hard to achieve. Our paper +succeeds [3] and will elaborate more on this aspect and +point out the central role of trainability within. +• +[37], +which officially reports the performance- +boosting effect of a larger finetuning LR and calls upon +comparing pruning algorithms under the same retrain- +ing configurations. [37] is actually motivated by [65], +which proposes LR rewinding vs. the weight rewind- +ing [18] proposed for finding winning ticket on large- +scale networks and datasets. [65] virtually takes ad- +vantage of the performance-boosting effect of a larger +finetuning LR, yet we do not know if they were aware +at that point that the performance boosting is not be- +cause of the magic rewound LR schedule but sim- +ply because of a larger finetuning LR (i.e., any ap- +propriately larger LR would do, even not a rewound +one), as later clarified by [37]; so we tentatively con- +sider [37] as the first work to systematically report the +performance-boosting effect of a larger finetuning LR. +• [50], which brings forward the argument regarding the +value of network filter pruning against scratch training. +We will re-evaluate the major claim (scratch training +3 + +M1: +A +larger +finetuning +LR +can significantly +“boost” the final +performance. +🔍 Unveil M1: Network trainability is damaged +by the sparsifying action, slowing down the +finetuning optimization; larger LR makes the +model converge faster, thus better performance +is observed earlier. The performance is not really +“boosted”. +Fairness principle: The performance advantage of a pruning paper should originate from the pruning method stage, not pretraining or finetuning. +In terms of strictness, S1 < S2 < S3 < S4. S2 (the most popular one) cannot ensure fairness. Comparison setup ≥S3 is suggested to ensure fairness. +Pretraining +(big dense) +Random model +(big dense) +Base model +(big dense) +Pruning +method +Pruned model +(small dense) +Final model +(small dense) +Finetuning +(small dense) +Sparsifying action +Dataset +Network structure +Total sparsity/speedup +Given the same +S1: +M2: Training the pruned model from scratch with +adjusted (typically prolonged) epochs can produce +the final model with similar performance. +Training with adjusted epochs +(small dense) +Pruned model +(small dense) +Final model +(small dense) +S2: S1 + Keep +the same base +model (M1 +arises at this +setup) +S3: S2 + +Keep the +same +finetuning +S4: S3 + Keep the same pruning cost +M2 holds under S3, +not hold under S2. To +unveil M2, we need to +unveil M1 first. +S1-S4: +Four +pruning +comparison setups +M1-M2: Two mysteries +in filter pruning +Scratch training pipeline +Pruning pipeline +Figure 1. Overview of this paper. We are motivated by unveiling two mysteries (M1, M2) in filter pruning, which represent the confusing +pruning benchmark situation. We first clarify the fairness principle and summarize outstanding comparison setups to lay down the discus- +sion foundation (the notation “S1 < S2” means S1 is less strict than S2; others can be inferred likewise). Then we start to unveil M1 and +M2. M2 will be shown to reduce to M1, actually. To unveil M1, we introduce network trainability as an effective perspective to demystify +M1. The unawareness of the role of network trainability in pruning has actually led to several sub-optimal hyper-parameter settings, which +exacerbates the chaotic benchmark status. We finally give some concrete suggestions to calibrate the pruning benchmarks. +can match filter pruning) of this paper under our more +strictly controlled comparison setups. +2.2. Terminology Clarification +First, we make some critical concepts clear to lay down +the common ground for discussion. +Although they are +pretty simple concepts, misinterpreting them will twist our +discussions. +• Pruning pipeline vs. pruning method vs. sparsifying +action. Some papers refer to pruning as the whole +pruning algorithm, i.e., the 2nd step in the pruning +pipeline; while others may mean a pruning paper or +the instant sparsifying action. We realize such a vague +conception definition is one reason causing confusion, +so we make them exact here. We use pruning pipeline +to mean all the three steps in a pruning paper. We can +consider pruning pipeline to be interchangeable with +a pruning paper. +Then, we use pruning method = +pruning algorithm to mean the 2nd step of the prun- +ing pipeline. Finally, the instant pruning action (i.e., +zeroing out weights or physically taking away weights +from a network) is referred to as sparsifying action. To +summarize, in terms of concept scope, pruning paper = +pruning pipeline = pruning > pruning method = prun- +ing algorithm > sparsifying action2. +2The notation “pruning paper = pruning pipeline > pruning method” +means, in this paper we consider pruning paper interchangeable with +pruning pipeline, which includes pruning method as one part. +• Training from scratch. +Training from scratch = +scratch training, means to train a randomly initialized +model to convergence. Scratch training of a pruned +model means, we already know the network architec- +ture of the pruned model; the weights are randomly ini- +tialized; train this network from scratch using the same +training recipe as training the dense model. Notably, +for filter pruning, when the architecture of the pruned +model is known, the model should be implemented as +a small-dense model, not a large-sparse model (with +structural masks). +The reason is, the widely-used +parameter initialization schemes (e.g., He initializa- +tion [25], the default initialization scheme for CONV +and Linear layers in PyTorch [59]) depend on the pa- +rameter shape. The large-sparse implementation is not +equivalent to (often underperforms) the small-dense +implementation. For unstructured pruning, the stan- +dard implementation scheme is large-sparse weights +(with unstructured masks) [50,79]. +• Finetuning. After the sparsifying action action, the +subsequent training process is called finetuning or re- +training. +We notice the community seems to have +different interpretations about these two terms. E.g., +in [37], finetuning is a sub-concept of retraining, +specifically meaning retraining with the last (small- +est) learning rate of original training3; while many +3 [37] attributes this term usage to [23, 42, 48]. We double-checked these +4 + +more papers [50,56,76,79,82] consider finetuning the +same as retraining, meaning the 3rd step of the prun- +ing pipeline. In this paper, we take the more com- +mon stance: considering finetuning and retraining in- +terchangeable, and in the end of this paper, we will +show the term “finetuning” should be deprecated in fa- +vor of “retraining”. +• Scratch-E, Scratch-B. These two terms are from [42], +denoting two scratch training schemes. “E” is short for +epochs, “B” short for (computation) budget. In prac- +tice, [42] uses FLOPs as an approximation for the com- +putation budget. In Scratch-E, the point is to maintain +the same total epochs when comparing scratch train- +ing to pruning. +In Scratch-B, the point is to main- +tain the same total FLOPs when comparing scratch +training to pruning. +Here is a concrete example of +Scratch-B: a dense model has FLOPs F1; pretrain- +ing the dense model takes K1 epochs; it is pruned by +L1-norm pruning, giving a pruned model with FLOPs +F2; the finetuning takes another K2 epochs; then the +scratch training should take (K1F1 + K2F2)/F2 = +K1(F1/F2)+K2 epochs. The ratio F1/F2 is typically +called speedup in network pruning. +• Value of network pruning. +This term comes +from [42]. The conclusion of [42] is that scratch train- +ing can match the performance of the 3-step prun- +ing pipeline if Scratch-B is adopted, for filter pruning. +Therefore, they argue there is no value for filter prun- +ing algorithms that use predefined layerwise pruning +ratios. For the filter pruning algorithms that do not +use predefined layerwise pruning ratios, their role is +to decide the favorable network architectures, akin to +NAS [15,89]. As for unstructured pruning, [42] shows +scratch training cannot match pruning. Therefore, rig- +orously, the argument about the value of network prun- +ing means the value of inheriting pretrained weights in +filter pruning with predefined layerwise pruning ratios. +We will use the short notion, value of network pruning, +without mentioning its much richer context. +3. Fairness and Comparison Setups in Pruning +3.1. Fairness Principle in Network Pruning +This section is prepared for the next section, Sec. 3.2, +where we will summarize the major comparison setups in +pruning. As we shall see in the experiments, a pruning al- +gorithm A can be better than B under one comparison setup, +while worse than or on par with B under another comparison +setup. To decide which comparison is more trustworthy, we +three papers and found they do not evidently have the inclination to mean +finetuning as one particular type of retraining. +have to evaluate which comparison setup is fairer. Thence +comes the necessity of a clear fairness principle in pruning. +Fairness Principle. The performance advantage of a prun- +ing paper should be attributed to the pruning method stage. +As aforementioned (Sec. 2), a pruning algorithm exclu- +sively has five aspects [30, 80]: base model, sparsity gran- +ularity, pruning ratio, pruning criterion, and pruning sched- +ule. +Excluding the base model and sparsity granularity +axes4, therefore, when we say a pruning method is bet- +ter than another one, the performance advantage should +be attributed to at least one of the three aspects; namely, +a better pruning ratio scheme, or/and a better pruning +criterion, or/and a better pruning schedule. Otherwise, it +means the performance advantage comes from some outside +factors other than the pruning algorithm itself – in this case, +attributing the performance credit to the pruning method +would be an unjustified claim, potentially leading to an un- +fair comparison. +Per this fairness principle, clearly, we should keep the +same base model and the finetuning process (i.e., the 1st +and 3rd steps) in the pruning pipeline. Next, we elaborate +on the outstanding comparison setups in network pruning +and examine their fairness. +3.2. Comparison Setups in Network Pruning +In the literature, we find there are at least the following +four groups of comparison setups, summarized in Tab. 2. +Note, in the following discussion, we consider different +pruning algorithms (and scratch training) that remove or +zero out unimportant weights only once. Namely, we do +not consider iterative pruning for now, and we will discuss +how the conclusions can carry over to iterative pruning. +Historical contexts of Tab. 2. In the following paragraphs, +we briefly go through the historical context of the different +comparison setups in Tab. 2. +S1. Obviously, the S1 setup is the most basic one, and +also the earliest one. It simply compares performance re- +gardless of many factors, such as training epochs and even +the base model. This setup is adopted by early pruning pa- +pers, especially those using Caffe [33], e.g., [22,23,29,82]. +Such a comparison setup does not even demand the same +base model, which we may consider problematic today; yet +back in those days, they had their own reasons – before +the deep learning (DL) community had mature DL plat- +forms/tools/computation power as we have today, these pa- +pers usually trained their own base models. Consequently, +due to different implementations, their base models do not +have the same (or even close) accuracy, e.g., the VGG16 +4Most pruning works do not consider the performance gain due to a better +base model or sparsity granularity as a valid advantage over other meth- +ods, because a pruning method can be easily applied to different base +models and sparsity granularities (e.g., SSL [82]) although the paper may +only focus on one kind. +5 + +Table 2. Summary of popular comparison setups in pruning pa- +pers. It is helpful to review them along with the 3-step pruning +pipeline: pretraining (output: base model) ⇒ pruning (output: +pruned model) ⇒ finetuning (output: final small model). In terms +of strictness, S1 < S2 < S3.1 < S3.2 < S4.1 < S4.2 (the +notation “S1 < S2” means S1 is less strict than S2; others can +be inferred likewise). Most existing pruning papers follow the S2 +comparison setup. +No. +Comparison setups +S1 +Compare performance or performance drop on the same +dataset and network at the same compression or speedup rate +S2 ++Same base model +S3.1 +Same base model ++Same finetuning epochs +S3.2 +Same base model ++Same finetuning LR schedule +S4.1 ++Same base model ++Same finetuning LR schedule ++Same pruning epochs +S4.2 ++Same base model ++Same finetuning LR schedule ++Same pruning LR schedule +SX-A +Same epochs of “pretraining + pruning + finetuning” +SX-B +Same FLOPs of “pretraining + pruning + finetuning” +base model reported by ThiNet [52] has top-5 accuracy +88.44% while CP [29], a concurrent work with ThiNet, re- +ported 89.9% (for those who are not familiar with these +numbers, 1.5% top-5 accuracy is a very significant gap for +ImageNet-1K classification). +As a remedy, to make the results comparable, many pa- +pers report the relative performance drop, namely, base +model accuracy minus final model accuracy. Such an idea +is still broadly practiced at present [50,79], esp. when com- +paring methods that are implemented under quite different +conditions. +S2. Later, as the DL community develops, more DL +platforms e.g., PyTorch [59] and TensorFlow [1] mature. +There is usually a well-accepted model zoo (such as torchvi- +sion models5) for others to use. +As a result, more and +more pruning papers adopt them as the base models, such +as [42, 79], which has become the mainstream practice at +present. Thus, the S2 comparison setup arises. +At this stage, few researchers have noticed the impor- +tance of finetuning. This makes sense since, in the pruning +pipeline, only the pruning method part (i.e., the 2nd step) is +regarded as the central one. The finetuning process is often +so downplayed that many papers do not even clearly report +the hyper-parameters, as also noted by [3]. +S3. Later, in the endless pursuit of higher and higher per- +formance, there is a clear trend that the finetuning epochs +become longer and longer (see Tab. 3 for an incomplete +summary). This in effect renders the comparison more and +more unfair. Besides, the finetuning LR has been noticed +5https://pytorch.org/vision/stable/models.html +Table 3. +Summary of finetuning epochs and LR schedules of +many filter pruning papers published in recent top-tier venues, +with ResNets [26]. The default dataset is ImageNet [10]; other +datasets are explicitly pointed out. +Method +#Epochs +LR schedule +SSL [82]NeurIPS’16 (CIFAR10) +– +0.01 +L1-norm [42]ICLR’17 +20 +0.001, fixed +DCP [88]NeurIPS’18 +60 +0.01, step (36/48/54) +GAL-0.5/1 [47]CVPR’19 +30 +0.01, step decay (10/20) +Taylor-FO [55]CVPR’19 +∼25 +0.01, step decay (10/20) +Factorized [43]CVPR’19 +90 +0.01, step decay (30/60) +CCP-AC [60]ICML’19 +100 +0.001, step decay (30/60/90) +HRank [46] CVPR’20 +30×#layers +0.01, step decay (10/20) +GReg-1/2 [79] ICLR’21 +90 +0.01, step decay (30/60/75) +ResRep [14] ICCV’21 +180 +0.01, cosine annealing +L1-norm [42] ICLR’17 (our reimpl.) +90 +0.01, step decay (30/60/75) +to have a significant impact on the final performance, as +formally studied by [37] (although [37] is the first one to +formally study this phenomenon, a larger finetuning LR has +been employed by many papers even before). Because of +these, the finetuning process must be taken into account to +maintain fairness. +The most exact way to rule out the impact of finetuning +is to use exactly the same finetuning process – the same LR +schedule (including the same epochs; we omit the hyper- +parameters, like weight decay, momentum, etc., and assume +they are maintained the same), i.e., the S3.2 in Tab. 2. +However, due to various objective or subjective reasons +(e.g., prior papers may not release their finetuning details, +making the follow-ups unable to reproduce the same fine- +tuning), S3.2 is often impractical. Thence comes a weaker +setup S3.1, which only keeps the same epochs of finetun- +ing. It is allowed to use different finetuning LR schedules +(e.g., different initial LR) – this is where M1, the mystery of +the finetuning LR effect, arises. +Several papers (such as [78, 79, 84, 85]) have abla- +tive analysis experiments on small-scale datasets (e.g., CI- +FAR10 [35]) using the setup S3.2, while the main bench- +mark experiments (e.g., with ResNet-50 on ImageNet) us- +ing S2. The primary reason is that, following the setup +S3.2 means re-running the experiments for other compari- +son methods in the large-scale benchmarks, which is usually +impractical (too costly) or even impossible (e.g., the com- +parison methods do not release usable code). +S4. The S3.2 is still not the most strictly fair setup +since it does not consider the cost (measured by the num- +ber of epochs) of the pruning method. For one-shot prun- +ing (such as L1-norm pruning [42]), the cost of pruning +is zero; while for a regularization-based method (such as +GReg [79]), it may take another few epochs for regular- +ized training. Considering these cases, S4.2 comes out: +it builds upon S3.2 and demands the same LR schedule +for the pruning algorithm – as far as we know, this is the +most strict comparison setup. In practice, again, for various +reasons, we may not know the LR schedule of a pruning +6 + +algorithm. Then, S4.2 degrades to S4.1, which only de- +mands the same epochs. +SX. In setups S2 to S4.2, when comparing pruning to +scratch training in obtaining the same pruned (small) model, +the scratch training employs the same training recipe of ob- +taining the base (big) model. [50] challenges this practice. +They argue, the scratch training scheme spends less cost +than pruning, so the comparison is unfair. As a remedy, +they propose to take into account the cost of the pretrain- +ing stage, which gives the SX-A and SX-B setups. About +the cost, one way to measure it is to use the number epochs +(hence the SX-A); another is to consider the same computa- +tion and they approximate computation with FLOPs (hence +the SX-B). +It is hard to say if considering the cost of the pretraining +stage is really necessary and practical. Advocates of the +older practice may list reasons, e.g., pretrained models often +exist already (like those pretrained on ImageNet [10] and +shared on HuggingFace6), so we do not need to consider the +cost of pretraining. However, advocates of SX may argue +that not all pretrained models are available; for many tasks, +we still need to train the pretrained models first, so the cost +of scratch training should be considered. +We have no inclination here regarding which one is more +correct. We make two points that we are fairly certain about: +(1) In the pruning literature, most papers still follow the +older practice when reporting the scratch training results of +the pruned model. (2) Given the recent rise of foundation +models [4] (e.g., Bert [12], GPT-3 [5], CLIP [61], diffu- +sion models [66,71]), common researchers barely have the +resources to train a model from scratch, so pruning would +inevitably be conducted on the pretrained model, probably, +for those big models. +What comparison setup is mostly used now? +Unfor- +tunately, S2 is the most prevailing comparison setup at +present [3]. This setup ignores at least one important fac- +tor that, we now know [37], has a significant impact on the +final performance: the finetuning LR schedule. +In the following sections, we start our empirical investi- +gation of unveiling M1 and M2. We study M2 first and then +M1, because the conclusion about M2 actually depends on +M1, as we are about to show. +4. Reexamining the Value of Pruning +The rethinking paper [50] presents many valuable +thoughts regarding the value of the 3-step pruning pipeline +against scratch training. However, there are a few potential +concerns in their experiments that may shake the validity of +their conclusion. First, they directly cite the results of a few +pruning papers and compare the relative performance drop. +6https://huggingface.co/ +Because of the stark differences between different DL plat- +forms, such a comparison (e.g., comparing methods that use +different base models) may not be convincing enough for +rigorous analysis. Second, when reproducing the L1-norm +pruning [42], they use fixed LR 0.001 and 20 epochs, fol- +lowing [42], for the finetuning stage, which is now known +to be severely sub-optimal (see Tab. 4, a larger finetuning +LR 0.01 can significantly boost performance). +It is thereby of interest whether the no-value-of-pruning +argument would change if the comparison is conducted un- +der a strictly controlled condition and a better finetuning +LR is employed. This section attempts to answer this ques- +tion. Three comparison setups (SX-A, SX-B, and S4.2) +are considered since they are the most strict setups up to +date. +Pruning method. We choose L1-norm pruning [42] be- +cause it is the most representative pruning method and easy +to control at a strict comparison setup. Specifically, L1- +norm pruning prunes the filters of a pretrained model with +the smallest L1-norms to a predefined pruning ratio. After +pruning, the pruned model is finetuned for a few epochs +to regain performance. +Other pruning methods, such as +regularization-based methods (e.g., [49, 79, 82]), introduce +many factors that are hard to control for rigorous analysis, +so we do not adopt them here. We will discuss how the +findings can generalize to those cases later. +Networks and datasets. The network used for analysis is +ResNet34 [26], following [42]. For standard benchmarks +(e.g., Tab. 1), we use ResNet50 [26] because it is one of the +most representative benchmark networks in filter pruning. +The datasets are ImageNet100 and the full ImageNet [10]. +ImageNet100 is a randomly drawn 100-class subset of Im- +ageNet. We use it for faster analysis given our limited re- +source budget. The full ImageNet is used for benchmarks. +Implementation details of pruning. For analysis, pruning +is conducted on the 1st CONV layer (the 2nd CONVs are +not pruned, following L1-norm pruning [42]) in all resid- +ual blocks of ResNet34. The first CONV and all FC layers +are spared, also following the common practice [20,79,87]. +Uniform layerwise pruning ratio is employed (which usu- +ally under-performs a tuned non-uniform layerwise pruning +ratio scheme; but since this paper does not target the best +performance but explanation, we adopt it for easy analysis). +We conduct pruning at a wide sparsity spectrum (10% to +95%) in the hopes of comprehensive coverage. +One table to show them all. The results are presented in +Tab. 4. Before we present the analyses, we introduce a no- +tion, pruning epoch, which is defined as the epoch when the +sparsifying action is physically enforced. E.g., if a model +is trained for 30 epochs and then the sparsifying action is +enforced, the pruning epoch is 30. We observe: +(1) For the S4.2 setup (rows marked by +), we are not +7 + +Table 4. Top-1 accuracy (%) comparison between L1-norm pruning [42] and training from scratch with ResNet34 on ImageNet100. Each +result is averaged by at least three random runs. The learning rate (LR) schedule of scratch training is: Initial LR 0.1, decayed at epoch +30/60/90/105 by multiplier 0.1, total: 120 epochs (top-1 accuracy of dense ResNet34: 84.56%, FLOPs: 3.66G). “P30F90, 1e-1” means the +model is pruned at epoch 30 and finetuned for another 90 epochs with initial finetune LR 1e-1 (please refer to our supplementary material +for the detailed LR schedule); the others can be inferred likewise. The best result within each comparison setup is highlighted in bold. +Pruning ratio +10% +30% +50% +70% +90% +95% +FLOPs (G, speedup: k×) +3.30 (1.11×) +2.59 (1.41×) +1.90 (1.93×) +1.19 (3.09×) +0.48 (7.68×) +0.30 (12.06×) +Scratch training +83.68±0.38 +83.31±0.13 +82.90±0.16 +82.45±0.13 +79.37±0.76 +76.67±0.90 +L1-norm (P15F105, 1e-1) +83.95±0.17 +84.01±0.23 +83.87±0.44 +82.93±0.10 +79.86±0.11 +77.41±0.11 +L1-norm (P30F90, 1e-2) +83.88±0.07 +84.00±0.22 +83.29±0.14 +82.61±0.07 +80.41±0.32 +77.64±0.39 +L1-norm (P45F75, 1e-2) +83.56±0.03 +83.95±0.14 +83.28±0.08 +82.47±0.12 +79.88±0.10 +76.17±0.21 +L1-norm (P60F60, 1e-3) +84.21±0.07 +83.87±0.09 +82.90±0.10 +81.24±0.17 +77.29±0.05 +70.53±0.37 +L1-norm (P75F45, 1e-3) +84.24±0.04 +83.47±0.12 +82.45±0.14 +80.81±0.09 +73.94±0.24 +64.98±0.31 +L1-norm (P90F30, 1e-4) +84.09±0.07 +82.47±0.02 +79.70±0.00 +74.87±0.19 +49.23±0.21 +29.89±0.26 +L1-norm (P30F90, 1e-1) +85.27±0.13 +85.37±0.19 +85.48±0.18 +83.83±0.17 +81.56±0.29 +79.57±0.15 +L1-norm (P60F60, 1e-2) +83.72±0.14 +83.88±0.07 +83.67±0.11 +82.96±0.23 +80.78±0.23 +77.81±0.25 +L1-norm (P90F30, 1e-2) +83.91±0.08 +84.02±0.20 +83.41±0.15 +82.91±0.12 +79.43±0.07 +75.20±0.23 +L1-norm (P30/kF90, 1e-1) +85.45±0.24 +85.06±0.24 +84.85±0.31 +83.64±0.09 +79.65±0.31 +75.79±0.28 +L1-norm (P30/kF90, 1e-2) +83.40±0.04 +82.69±0.27 +82.16±0.03 +79.97±0.16 +74.76±0.24 +70.61±0.52 +Under comparison setup S4.2 (same overall LR schedule), +Under comparison setup SX-A (same total epochs; finetuning LR +increased), +Under comparison setup SX-B (same total FLOPs). +allowed to change the LR schedule. +The only thing we +can change is the pruning epoch. As seen, the best prun- +ing epoch varies w.r.t. the sparsity level – at a small pruning +ratio, different pruning epochs give a similar performance; +while as the pruning ratio arises, the performance becomes +more sensitive to the pruning epoch, e.g., for pruning ratio +95%, P90F30, 1e-4 severely underperforms P30F90, +1e-2. Notably, a clear trend is, when the pruning ratio is +large (70% to 95%), it is better to have a smaller pruning +epoch. +Under this setup, only at pruning ratios of 30%-70%, +pruning surpasses scratch training by a statistically signifi- +cant gap. Therefore, we can only say pruning has a marginal +advantage over scratch training here. +(2) Then we look at the setup SX-A (rows marked by +). +Under this setup, we are allowed to adjust the finetuning LR +as long as the total epochs are kept the same. We increase +the initial finetuning LR. As seen, it significantly improves +the accuracies, e.g., (P30F90, 1e-1) improves the accu- +racy by nearly 2% at pruning ratio 95%, against (P30F90, +1e-2). This is the performance-boosting effect aforemen- +tioned [37]. We also apply the larger LR trick to another +two settings P60F60 and P90F30. In all of them, we see +a larger finetuning LR improves performance by an obvious +margin. +Now, the gap between pruning and scratch training be- +comes much more significant. Pruning is more surely valu- +able under this setup. +(3) Next, we use the comparison setup SX-B (rows +marked by +), which maintains the total FLOPs. We ap- +ply this scheme to the best pruning setup P30F90, 1e-1 +in S4.1 in the hopes of better performances. Since the +dense model is trained for 30 epochs, to compensate for +the FLOPs, the pruning epoch should be squeezed by the +speedup ratio k. E.g., for pruning ratio 10%, the speedup +ratio is 1.11, then the pruning epoch should be adjusted to +30/k ≈ 27. +As seen, the squeezing of the pruning epoch does close +the gap between pruning and scratch training: At pruning +ratios of 10% to 70%, pruning is still better; while for 90% +and 95%, pruning only matches or underperforms scratch +training – this is a concrete example that we do not have a +once-for-all answer to questions like “is pruning better than +scratch training?” +We also try a smaller finetuning LR in this setup, as +shown in the row (P30/kF90, 1e-2). The LR effect also +translates to this case – a smaller finetuning LR degrades the +performance. +Short summary. As seen, the argument about the value +of network pruning severely hinges on which comparison +setup is employed and the pruning ratio. +For the setup +SX-A, where pruning outperforms scratch training obvi- +ously, the advantage comes from a better finetuning LR. +Yet, we are not sure if such better LR schedules also exist +for the scratch training; if so, scratch training can be fur- +ther boosted, too – as such, this kind of “competition” can +be endless. There are two kinds of attitudes toward this sit- +uation: (1) Do not consider the performance improvement +from a better finetuning LR as a fair/valid performance ad- +vantage as it is not from the pruning algorithm. (2) Still con- +sider it as a valid performance advantage but will meet the +“endless competition” challenge we just mentioned. The +community now is mostly using (2). We suggest using (1), +following our fairness definition clarified in Sec. 3. +8 + +Despite many uncertainties, we are certain about one +thing from Tab. 4: Whichever setup is favored, the fine- +tuning LR holds a critical role in performance. Even for +the comparisons setup S4.2, where the finetuning LR does +not change, by changing the pruning epoch, implicitly, we +change the finetuning LR, and it has been shown very per- +tinent to the final performance as well. In this sense, the +two mysteries of pruning actually boil down to one (M1): +Why does finetuning LR have such a great impact on the +performance? +This is the next question we would like to answer. LR, +(arguably) as the most influential hyper-parameter in train- +ing neural networks, has a significant impact on perfor- +mance – this is definitely not surprising; what is really sur- +prising might be, why the prior pruning works (e.g., the +original L1-norm pruning [42] adopts LR 0.001 in finetun- +ing for their ImageNet experiments) did not realize that such +a simple “trick” is so important to performance? This ques- +tion is also worth our thinking. +5. Trainability in Network Pruning +5.1. Background of Trainability +Trainability, by its name, means the ability (easiness) of +training a neural network, i.e., the optimization speed (note, +speed is not equal to quality, so we may see a network with +good trainability turns out to have a bad generalization abil- +ity eventually). +Notably, essentially, the role of a pruning method is to +provide the initial weights for the later finetuning process, +that is, pruning is essentially a kind of initialization. In stark +contrast to the broad awareness that initialization is very +critical to neural network training [21,25,34,54,72], the ini- +tialization role of pruning has received negligible research +attention, however. Trainability is also mostly studied for +random initialization [67,83]. +A few recent works marry it with network pruning in +some other similar forms like signal propagation [40] and +gradient flows [77] (a good signal propagation or strong +gradient flow usually suggests a good trainability). These +works are inspiring, while they mostly stay in the domain +of pruning at initialization (PaI). Few attempts before, to +our best knowledge, tried to utilize the notion of trainability +to examine pruning after training (PaT), at least, for the two +mysteries we study here. This paper is meant to bridge this +gap. The major difference between PaI and PaT is whether +using a pretrained model as base. Such a context is essential +to this paper since the above two mysteries are both brought +forward in the context of PaT. +Trainability accuracy. Literally, a bad trainability implies +the training is hard and the training performance will arise +slowly. Per this idea, there is a straightforward metric to +measure trainability – we introduce trainability accuracy, +Table 5. Top-1 accuracy (%) comparison of different setups of +L1-norm pruning [42] with ResNet34 on ImageNet100. Pruning +ratio: 95%. TA: trainability accuracy (the metric used to measure +trainability; see Eq. (1)). This table shows, the performance gap +between a smaller LR and a larger LR is not fundamental. It can +be closed simply by training more epochs. The root cause that a +smaller LR appears to under-perform a larger LR is simply that +the model trained by the smaller LR does not fully converge. +Finetuning setup +Top-1 acc. (%) +TA (%) +P30F90, 1e-1 +79.57±0.15 +88.00 +P30F90, 1e-2 +77.64±0.39 +77.45 +P30F90, 1e-2 (+30 epochs) +79.12±0.19 +/ +P30F90, 1e-2 (+60 epochs) +79.59±0.25 +/ +P60F60, 1e-2 +77.81±0.25 +87.39 +P60F60, 1e-3 +70.53±0.37 +68.19 +P60F60, 1e-3 (+60 epochs) +75.71±0.09 +/ +P60F60, 1e-3 (+120 epochs) +77.17±0.13 +/ +P60F60, 1e-3 (+180 epochs) +77.33±0.09 +/ +P90F30, 1e-2 +75.20±0.23 +84.83 +P90F30, 1e-4 +29.89±0.26 +37.93 +P90F30, 1e-4 (+60 epochs) +60.69±0.17 +/ +P90F30, 1e-4 (+270 epochs) +70.78±0.16 +/ +P90F30, 1e-4 (+1485 epochs) +78.18 +/ +the average of the first N epochs, +T = 1 +N +N +� +i=1 +Acci. +(1) +Since the optimization speed depends on the LR used, when +we calculate trainability accuracy, we must ensure they are +under the same LR schedule. In this paper, we choose N +as the number of the 1st LR stage, which characterizes the +optimization speed in the early phase. +Next, we utilize trainability to explain the mysterious ef- +fect of the finetuning LR. +5.2. Examining the Effect of Finetuning LR +Two facts as foundation. We first lay down two facts as +the common ground for the discussion of this section. We +will show the mystery about the finetuning LR effect boils +down to these two simple facts. +First, pruning damages trainability. This is an intuitively +straightforward fact since pruning removes connections or +neurons, which virtually makes the network harder to train. +This fact holds for not only pruning a random network [40], +but also for pruning a pretrained model here. Moreover, +notably, more aggressive pruning leads to more damaged +trainability. +Second, a model of worse trainability will +need more effective updates to reach convergence. More +effective updates mean two cases: If LR is not changed, +more epochs are needed; if the number of epochs does +not change, a larger LR is needed. This is also easy to +understand since trainability measures the easiness of op- +timization; a bad trainability implies harder optimization +9 + +0 +10 +20 +30 +40 +50 +60 +Epoch +20 +40 +60 +80 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(a) P60F60, 1e-3 +0 +50 +100 +150 +200 +250 +Epoch +20 +40 +60 +80 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(b) P60F60, 1e-3 (+180 epochs) +Figure 2. Test accuracy vs. epoch during finetuning of the setting +P60F60, 1e-3 at pruning ratio 95% in Tab. 5. Red vertical +lines mark the epoch of decaying LR by 0.1. Particularly note be- +fore the 1st LR decay, the accuracy keeps arising in (a), implying +the 1st LR decay may be too early – this is confirmed in (b), where +the red cross marker (×) indicates the time point of the 1st LR de- +cay in (a). See more similar plots in our supplementary material. +literally, hence the more effective updates. Such observa- +tion has been made by some sparse training papers, e.g., +RigL [16] notes that “sparse training methods benefit sig- +nificantly from increased training steps”. +When we observe that a larger LR improves the final +test accuracy of the pruned model (e.g., Row P30F90, +1e-1 vs. Row P30F90, 1e-2 in Tab. 4), it is worthwhile +to differentiate two subtle yet distinct possibilities: +• A larger LR helps the pruned model reach a solution +that the smaller LR cannot reach, i.e., a larger LR help +the model located into a better local minimum basin. +• The smaller LR can also help the model reach the so- +lution as the larger LR does; just, the larger LR helps +the model get there faster. +The former implies the performance-boosting effect of a +larger LR is fundamental; while the latter implies there is +no fundamental gap between the two solutions; it is only an +issue of optimization speed. +Let’s analyze a concrete example of P60F60 in Tab. 2. +For pruning ratio 95% (we use this for example because at +larger sparsity, the performance boosting effect is most pro- +nounced), using 1e-2 vs. 1e-3 improves the test accuracy +from 70.53 to 76.74, a very significant jump. This improve- +ment also translates to the rows of P30F90 and P90F30. +However, in Tab. 2, we note the performance improve- +ment coincides with trainability accuracy improvement. We +were wondering if the performance improvement is actually +due to a better trainability. +Fig. 2(a) plots the test accuracy during the finetuning of +P60F60, 1e-3. We notice before the 1st LR decay at +epoch 30, the accuracy keeps arising even at epoch 30. This +triggers a question: usually, we decay LR when the accu- +racy saturates; now, when the accuracy is still steadily ris- +ing, is the LR decayed too early? This question matters be- +cause if the LR decays too early, the model is forced to sta- +bilize due to the small step size and insufficient updates, not +because it gets close to the local minimum, i.e., the model +may not converge at all. +To verify this, we extend the epochs of the LR 0.001 +phase by 60/120/180 epochs. See the results in Tab. 5 (note +the rows P60F60, 1e-3 (+60/120/180 epochs)) +and Fig. 2(b). Now, the model finetuned by LR 1e-3 can +reach 77.33, very close to 77.81 reached by LR 1e-2. The +test accuracy plot in Fig. 2(b) also confirms that the seem- +ing underperformance of LR 1e-3 is due to insufficient +epochs – namely, the advantage of a larger LR 0.01 is +not some magic fundamental advantage, but a simple +consequence of faster optimization. +We also verify this on other cases (P30F90 and +P90F30) that the smaller LR “underperforms” the larger +LR. The results are also presented in Tab. 5. +In all of +these cases, given abundant epochs, the gap between the +larger LR and the smaller LR can be closed. Especially for +P90F30, the smaller LR 1e-4 can achieve a much better +result than LR 1e-2 (78.18 vs. 75.20). This strongly demon- +strates the smaller LR can also achieve what the larger LR +can do. +To summarize, our results suggest a larger LR does not +really “improve” the performance. +What really hap- +pens is, a larger LR accelerates the optimization pro- +cess, making the higher performance observed earlier. +In practice, when researchers tune different LR’s, they usu- +ally keep the total epochs fixed (for the sake of fairness). +Given the same total epochs, the pruned model using the +smaller finetuning LR does not fully converge, making the +performance appear “worse”. +Further remarks. It is worthwhile to note that such an ex- +perimenting trap is so covert if we are unaware of the dam- +aged trainability issue in pruning. We may never realize that +the epochs should be increased properly if a smaller finetun- +ing LR is used. What’s even trickier, we do not know how +many more epochs is the so-called proper – Tab. 5 is a living +example. For some cases (e.g., P30F90), 60 more epochs +is enough, while for others (e.g., P60F60, P90F30), 180 +epochs or more is not enough to bridge the performance +gap. Clearly, there is still much work to be done here toward +a more rigorous understanding of the influence of damaged +10 + +trainability on pruning. +Retrospective remarks and the lessons. It is worthwhile +to ponder why [42] employed a seriously sub-optimal fine- +tuning LR scheme. +This, we conceive, may originate +from a long-standing misunderstanding in the area of net- +work pruning – many have believed that because pruning +is conducted onto a converged model, the retraining of the +pruned model needs not to be long and the LR should be +small to avoid destroying the knowledge the model has ac- +quired, e.g., in [65], the authors mentioned in their abstract +“The standard retraining technique, fine-tuning, trains the +unpruned weights from their final trained values using a +small fixed learning rate”, implying that such misconcep- +tion spreads so widely that it is taken for “standard”.7 +However, the results in Tab. 4 suggest, such thought only +holds for the cases of low pruning ratios. For a moderate or +large pruning ratio, this thought hardly holds. What was +neglected is that the sparsifying action damages network +trainability, slowing down the optimization. As a compen- +sation, it is supposed to use a larger LR to accelerate the +optimization, not a smaller LR; similarly, more epochs are +needed to compensate for the slow optimization. However, +the original L1-norm pruning [42] chose LR 0.001 and only +20 epochs for their ImageNet experiments, exactly the op- +posite of what is expected. This, we conceive, is the reason +that L1-norm pruning has been underrated for a long time. +Its real performance is actually pretty strong even compared +with recent top-performing approaches (see Tab. 1). +Similarly, based on what we just learned about the truth +of M1, if we examine the other filter pruning methods, e.g., +GAL [47] (see Tab. 3), its reported results are probably also +underrated, because it uses only 30 epochs for finetuning +and the model may well not expose its full potential, as a re- +sult of the immature convergence. This implies a pretty dis- +turbing concern – for many filter pruning papers, we have to +calibrate their results for fairness. Directly citing the num- +bers may (well) not show the real performance comparison. +6. Conclusion and Discussion +This paper attempts to figure out the confounding bench- +mark situation in filter pruning. Two particular mysteries +are explored, the performance-boosting effect of a larger +finetuning LR, the no-value-of-pruning argument. +We +present a clear fairness principle and sort out four groups +of popular comparison setups used by many pruning pa- +pers. Under a strictly controlled condition, we examine the +two mysteries and find they both boil down to the issue of +damaged network trainability. This issue was not well rec- +ognized by prior works, leading to (severely) sub-optimal +7Actually, the 3rd-step of the pruning pipeline is broadly referred to as +finetuning – this term per se already implies the inclination of using a +small LR. To rule out such conceptual bias, a more accurate way to phrase +the 3rd step in the pruning pipeline may be retraining the pruned model. +hyper-parameter settings, which ultimately exacerbates the +confounding benchmark situation in filter pruning now. We +hope this paper helps the community towards a clearer un- +derstanding of pruning and more reliable benchmarking of +it. +Takeaways and suggestions from this paper: +• Why is the state of neural network pruning so confus- +ing? Non-standard comparison setups (and its funda- +mental reason: unclear fairness principle) and the un- +awareness of the role of trainability are the two major +reasons. The latter further leads to sub-optimal hyper- +parameter settings, inherited by many follow-up pa- +pers, exacerbating the messy benchmark situation. +• As the area of network pruning develops, various com- +parison setups have appeared (see Tab. 2). Each has +its own historical context. +Unfortunately, the most +prevailing comparison setup now, setup S2 in Tab. 2, +cannot ensure fairness. We suggest using the setup +S3.2 or higher, i.e., maintaining the same base +model and the same finetuning process – Higher com- +parison setup means stricter experiment control, also +means more resources and efforts; so there would be +inevitably a trade-off between how fair we want to be +and how much we can invest in8. +• Reporting all the finetuning details (esp. the LR sched- +ule) is rather necessary and should be standardized. +• Filter pruning can beat scratch training or not, up to the +specific comparison setup and pruning ratio in consid- +eration. Given the recent rise of large foundation mod- +els, pruning may still follow the conventional 3-step +pipeline. +• The observation that a larger finetuning LR “improves” +pruning performance is largely a misinterpretation – +the performance is not “improved”; what really hap- +pens is that the good performance is observed ear- +lier because the larger LR accelerates the optimization. +The fundamental factor playing under the hood is the +network trainability damaged by the sparsifying action +(or zeroing out) action in pruning. +• The damaged network trainability was not well rec- +ognized by prior pruning works, resulting in severely +sub-optimal hyper-parameters, rendering the potential +of a baseline method, L1-norm pruning [42], underes- +timated for a long time. 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This is adopted by L1-norm prun- +ing [42] (and inherited by [50]), so in their finetuning, they +use the last-stage LR (i.e., 0.001) as finetuning LR and never +decay it. +How we decide the LR schedule for scratch train- +ing? +Per our empirical observations, +decaying the +LR from 1e-3 to 1e-4 can still improve the model by +around 0.5-0.8% top-1 accuracy. +Namely, 1e-3 is not +where the model finally converges. +Since we want to +compare models at their best potential, we add another +30 epochs and decay another two times to make sure +the model fully utilizes its potential – this gives us +the LR schedule for training a scratch model in Tab. 4: +0:0.1,30:0.01,60:0.001,90:1e-4,105:1e-5, +total epochs:120. +How we decide the LR schedule for finetuning? We abide +by the following rules to decide the finetuning LR schedule: +• Rule 1: When comparing scratch training to pruning +methods, the final LR should be the same – as we +mentioned, decaying LR from 1e-3 to 1e-4 on Ima- +geNet can see another 0.5-0.8% top-1 accuracy bump, +thus it would be unfair to compare a pruning method +whose model is finetuned to LR smaller than 1e-3 to a +scratch-training model whose smallest LR is only 1e-3. +The final LR for our ImageNet/ImageNet100 results, +as mentioned above, is 1e-5. +• Rule 2: Halving. This means a kind of epoch splitting +scheme: Given N total epochs, split half of them (i.e., +N/2 epochs) to the 1st LR, then split the half of the +left epochs (i.e., N/4) to the 2nd LR, etc. To our best +9https://github.com/pytorch/examples/tree/main/imagenet +14 + +knowledge, this scheme is due to (no later than) the pa- +per of ResNet [26] (see their CIFAR10 experiments). +• Rule 3: The epochs for each LR stage is no more than +30. This is mainly due to the design in the official Py- +Torch ImageNet example. We do not know why they +chose 30 historically, but since this example is exten- +sively followed as a baseline, we follow it too. +Based on these rules, given the total number of fine- +tuning +epochs, +we +can +exactly +derive +the +finetun- +ing LR used in the paper. +For example, in Tab. 4, +for +P30F90, +1e-1, +the finetuning LR schedule is: +0:1e-1,30:1e-2,60:1e-3,75:1e-4,83:1e-5; +for +P60F60, +1e-2, +the finetuning LR schedule is: +0:1e-2,30:1e-3,45:1e-4,53:1e-5. +Layerwise +pruning +ratio +for +the +experiments +of +ResNet50 on ImageNet. For the results of ResNet50 on +ImageNet we report in Tab. 1, its finetuning LR schedule +is: 0:1e-2,30:1e-3,60:1e-4,75:1e-5, total +epochs:90. As seen, this is never some heavily tuned +magic LR schedule; nevertheless, we use it to finetune +the pruned model by L1-norm pruning [42], only to find +the final performance actually can beat/match many top- +performing methods after ICLR’17. The layerwise prun- +ing ratios are borrowed from GReg [79] (as they released +their ratios; for many other papers, we only know the total +speedup, not aware of the detailed layerwise pruning ratios) +to keep a fair comparison with it – speedup 2.31×: [0, 0.60, +0.60, 0.60, 0.21, 0]; speedup 2.56×: [0, 0.74, 0.74, 0.60, +0.21, 0]. +Code references. We mainly refer to the following code +implementations in this work. They are all publicly avail- +able. +• Official PyTorch ImageNet example10; +• GReg-1/GReg-2 [79]11; +• Rethinking the value of network pruning [50]12. +B. Can the findings generalize to other pruning +methods than L1-norm pruning? +Pruning methods, +according to their methodology, +typically are classified into two groups, regularization- +based +(a.k.a. +penalty-based) +and +importance-based +(a.k.a. saliency-based), from a long time ago [64]. +Despite the different categorization, any pruning method +has a step to physically zero out the weights, i.e., the spar- +sifying action step, per the terminology in this paper. Typi- +10https://github.com/pytorch/examples/tree/master/imagenet +11https://github.com/MingSun-Tse/Regularization-Pruning +12https://github.com/Eric-mingjie/rethinking-network- +pruning/tree/master/imagenet/l1-norm-pruning +cally, this step is the magnitude pruning (or L1-norm prun- +ing when it comes to filter pruning). E.g., SSL [82] and +GReg [79] are two regularization-based pruning methods, +with different penalty terms proposed, yet both of them +have a step to physically zero out unimportant weights by +sorting their magnitude before finetuning. In other words, +regularization-based methods, although they are classified +into a different group from magnitude pruning (which +is importance-based), they essentially include magnitude +pruning as a part. +We have analyzed in the paper, the fundamental reason +that incurs damaged trainability is the sparsifying action ac- +tion in magnitude pruning. Therefore, any pruning method +that employs magnitude pruning as a part is subject to the +analyses of this paper – this means the discoveries of this +paper are generic. The attended broken trainability in these +methods should also lead to similar13 finetuning LR effect +to the L1-norm pruning case. +C. Can the finetuning LR effect generalize to +other LR schedules than the traditional +step decay? +In the paper, we explore the finetuning LR effect (a large +LR vs. a small LR,e.g., 0.01 vs. 0.001) to the final perfor- +mance using the conventional step decay LR schedule. It is +of interest if the effect can translate to other more advanced +LR schedules. +We consider Cosine Annealing LR schedule [51] here, +referring to the official PyTorch Cosine LR implementa- +tion14. When we switch from Step LR schedule to Cosine, +the initial LR and minimum LR are kept the same (namely, +the start point and end point of LR are the same; the only +difference is the scheduling in between). The scratch model +is trained for 200 epochs, initial LR 0.1, step decayed at +epoch 100 and 150 by multiplier 0.1 (referring to the orig- +inal ResNet CIFAR10 training recipe in the ResNet pa- +per [26]). For finetuning, the initial LR is 0.01 or 0.001, +the minimum LR 0.0001, total epochs 120. +Table 6. Effect of LR schedule of ResNet56 on CIFAR10. Base- +line accuracy 93.78%, Params: 0.85M, FLOPs: 0.25G. +Pruning ratio +0.3 +0.5 +0.7 +0.9 +Sparsity/Speedup +31.14%/1.45× 49.82%/1.99× 70.57%/3.59× 90.39%/11.41× +Scratch (Step LR) +93.16±0.16 +92.78±0.23 +92.11±0.12 +88.36±0.20 +Scratch (Cosine LR) +93.84±0.06 +93.20±0.31 +92.15±0.21 +88.17±0.43 +L1 [42] (Step LR 0.001) +93.43±0.06 +93.12±0.10 +91.77±0.11 +87.57±0.09 +L1 [42] (Step LR 0.01) +93.79±0.06 +93.51±0.07 +92.26±0.17 +86.75±0.31 +L1 [42] (Cosine LR 0.001) +93.48±0.04 +93.11±0.09 +91.65±0.11 +87.17±0.14 +L1 [42] (Cosine LR 0.01) +93.82±0.07 +93.74±0.06 +92.27±0.00 +86.90±0.20 +The results of ResNet56 (on CIFAR10) and VGG19 (on +CIFAR100) are presented in Tab. 6 and Tab. 7. As seen, +13This said, the severity of the trainability issue may vary up to specific +pruning methods. E.g., we have observed the GReg method [79] is less +bothered by such damaged trainability due to their growing regulariza- +tion design. +14https://pytorch.org/docs/stable/generated/torch.optim.lr scheduler.CosineAnnealingLR.html +15 + +Table 7. Effect of LR schedule of VGG19 on CIFAR100. Baseline +accuracy: 74.02%, Params: 20.08M, FLOPs: 0.80G. +Pruning ratio +0.3 +0.5 +0.7 +0.9 +Sparsity/Speedup +19.24%/1.23× 51.01%/1.97× 74.87%/3.60× 90.98%/8.84× +Scratch (Step LR) +72.84±0.25 +71.88±0.14 +70.79±0.08 +66.52±0.37 +Scratch (Cosine LR) +73.54±0.22 +71.87±0.09 +70.10±0.24 +65.92±0.10 +L1 [42] (Step LR 0.001) +73.67±0.05 +72.04±0.12 +70.21±0.02 +64.72±0.17 +L1 [42] (Step LR 0.01) +74.01±0.18 +73.01±0.22 +71.49±0.14 +66.05±0.04 +L1 [42] (Cosine LR 0.001) +73.69±0.08 +72.10±0.08 +69.96±0.09 +63.93±0.15 +L1 [42] (Cosine LR 0.01) +74.39±0.07 +73.51±0.18 +71.78±0.21 +65.70±0.11 +Table 8. Top-1 accuracy (%) comparison of different setups of +L1-norm pruning [42] with ResNet34 on ImageNet100. Prun- +ing ratio: 95%. This table shows, the performance gap between a +smaller LR and a larger LR is not fundamental. It can be closed +(or squeezed) simply by training more epochs. The root cause that +a smaller LR appears to under-perform a larger LR is simply that +the model trained by the smaller LR does not fully converge. +Finetuning setup +Top-1 acc. (%) Trainability acc. (%) +P90F30, 1e-2 +75.20±0.23 +84.83 +P90F30, 1e-4 +29.89±0.26 +37.93 +P90F30, 1e-4 (+60 epochs) +60.69±0.17 +/ +P90F30, 1e-4 (+270 epochs) +70.78±0.16 +/ +P90F30, 1e-4 (+1485 epochs) +78.18 +/ +the advantage of initial LR 0.01 over 0.001 does not only +appear with the Step LR schedule, but also appears with +the Cosine LR schedule in most cases (esp. for VGG19). +This implies the finetuning LR effect is generic, not limited +to one particular LR schedule, which further highlights the +importance of the topic we have been studying in the paper. +D. Additional results +1e-4 vs. 1e-2 for P90F30. In Tab. 5 of the paper, we men- +tion the performance gap between a small LR and a large +LR is not fundamental, but a simple consequence of conver- +gence speed under different LRs. When we add more fine- +tuning epochs, the performance gap can be closed fully or +by a large part for P30F90 and P60F60; while on P90F30, +the gap is still obvious even after we add 270 epochs. +Here we add even more, 1485 epochs, so that the num- +ber of the 1st LR stage is now 1500 epochs, exactly 100 +times of the 1st-LR-stage epochs (i.e., 15 epochs) when +using 1e-2 as initial LR. As we see, now LR 1e-4 can +achieve 78.18 top-1 accuracy, which is significantly better +than 75.20 achieved by LR 1e-2. This is yet another strong +piece of evidence to show that the seeming performance gap +between a large LR and a small LR is never a gap that can- +not be bridged, further confirming our opinion in the paper. +More learning curves. In Fig. 1 of the paper, we present +the learning curves for P60F60, 1e-3 without and with more +finetuning epochs, to show the underperformance of a small +LR is actually due to insufficient training. Here we present +more plots for P30F90, 1e-2 and P90F30, 1e-4 – see Fig. 3. +In both cases (note the red crosses × in (1.b) and (2.b)), +the performance can be boosted by adding more epochs to +the 1st LR stage, especially for the case of P90F30, 1e-4, +where the 1st LR decay is actually way too early. These +plots further confirm our opinion in the paper – the seem- +ing underperformance of a small finetuning LR is not some- +thing magic, but a simple consequence of slow convergence +(caused by the broken trainability, esp. at large pruning ra- +tios like 95%). +16 + +0 +20 +40 +60 +80 +Epoch +40 +60 +80 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(1.a) P30F90, 1e-2 +0 +25 +50 +75 +100 +125 +150 +Epoch +40 +60 +80 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(1.b) P30F90, 1e-2 (+60 epochs) +0 +5 +10 +15 +20 +25 +30 +Epoch +10 +20 +30 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(2.a) P90F30, 1e-4 +0 +250 +500 +750 +1000 +1250 +1500 +Epoch +20 +40 +60 +80 +Test accuracy (%) +ResNet34, ImageNet100, PR 95% +(2.b) P90F30, 1e-4 (+1485 epochs) +Figure 3. Test accuracy vs. epoch during finetuning of the setting +P30F90, 1e-2 and P90F30, 1e-4 at pruning ratio 95% in Tab. 5. +Red vertical lines mark the epoch of decaying LR by 0.1. Partic- +ularly note before the 1st LR decay, the accuracy keeps arising in +(a), implying the 1st LR decay may be too early – this is confirmed +in (b), where the red cross marker (×) indicates the time point of +the 1st LR decay in (a). +17 + diff --git a/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/load_file.txt b/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aac200558d12a5ec408b4c0775dc12a82d2ea7a7 --- /dev/null +++ b/kNE4T4oBgHgl3EQfsw2X/content/tmp_files/load_file.txt @@ -0,0 +1,1537 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf,len=1536 +page_content='Why is the State of Neural Network Pruning so Confusing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' On the Fairness, Comparison Setup, and Trainability in Network Pruning Huan Wang† Can Qin Yue Bai Yun Fu Northeastern University, Boston, USA Abstract The state of neural network pruning has been noticed to be unclear and even confusing for a while, largely due to “a lack of standardized benchmarks and metrics” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To stan- dardize benchmarks, first, we need to answer: what kind of comparison setup is considered fair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This basic yet crucial question has barely been clarified in the community, unfor- tunately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Meanwhile, we observe several papers have used (severely) sub-optimal hyper-parameters in pruning exper- iments, while the reason behind them is also elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' These sub-optimal hyper-parameters further exacerbate the dis- torted benchmarks, rendering the state of neural network pruning even more obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Two mysteries in pruning represent such a confusing sta- tus: the performance-boosting effect of a larger finetuning learning rate, and the no-value argument of inheriting pre- trained weights in filter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In this work, we attempt to explain the confusing state of network pruning by demystifying the two mysteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Specif- ically, (1) we first clarify the fairness principle in prun- ing experiments and summarize the widely-used compar- ison setups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) then we unveil the two pruning myster- ies and point out the central role of network trainability, which has not been well recognized so far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (3) finally, we conclude the paper and give some concrete suggestions re- garding how to calibrate the pruning benchmarks in the fu- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Code: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/mingsun-tse/why-the-state- of-pruning-so-confusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Introduction The past decade has witnessed the great success of deep learning, empowered by deep neural networks [38,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The success comes at the cost of over-parameterization [5, 12, 26, 31, 32, 36, 61, 62, 66, 69, 75], causing prohibitive model footprint, slow inference/training speed, and rapid-growing energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Neural network pruning, an old model parameter reduction technique that used to be employed for †Corresponding author: wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='huan@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='edu Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Top-1 accuracy (%) benchmark of filter pruning with ResNet50 [26] on ImageNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Simply by using a better fine- tuning LR schedule, we manage to revive a 6-year-ago baseline filter pruning method, L1-norm pruning [42], making it match or beat many filter pruning papers published in recent top-tier venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Note, we achieve this simply by using the common step-decay LR schedule, 90-epoch finetuning, and standard data augmentation, no any advanced training recipe (like cosine annealing LR) used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This paper studies the reasons and lessons behind this pretty con- founding benchmark situation in filter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Method Pruned acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (%) Speedup SFP [28] IJCAI’18 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='72× DCP [88] NeurIPS’18 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25× GAL-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5 [47] CVPR’19 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='76× Taylor-FO [55] CVPR’19 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='82× CCP-AC [60] ICML’19 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18× ProvableFP [45] ICLR’20 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='43× HRank [46] CVPR’20 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78× GReg-1 [79] ICLR’21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31× GReg-2 [79] ICLR’21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31× CC [44] CVPR’21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12× L1-norm [42] ICLR’17 (our reimpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=') 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31× GAL-1 [47] CVPR’19 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='59× Factorized [43] CVPR’19 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='33× LFPC [27] CVPR’20 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='55× GReg-1 [79] ICLR’21 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56× GReg-2 [79] ICLR’21 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56× CC [44] CVPR’21 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='68× L1-norm [42] ICLR’17 (our reimpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=') 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56× improving generalization [2, 6, 64], now is mostly used for model size compression or/and speed acceleration [7,8,11, 20,30,73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The prevailing pipeline of pruning comprises three steps: 1) pretraining: train a dense model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2) pruning: prune the dense model based on certain rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3) finetuning: retrain the pruned model to regain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Most existing re- search focuses on the second step, seeking better criteria to remove unimportant weights so as to incur as less per- formance degradation as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This 3-step pipeline has been practiced for more than 30 years [2,6,39,57] and is still extensively adopted in today’s pruning methods [30,73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Despite the long history of network pruning, recent progress seems a little bit unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As we quote from the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='05219v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='CV] 12 Jan 2023 abstract of a recent paper [3], “After aggregating results across 81 papers and pruning hundreds of models in con- trolled conditions, our clearest finding is that the community suffers from a lack of standardized benchmarks and met- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This deficiency is substantial enough that it is hard to compare pruning techniques to one another or determine how much progress the field has made over the past three decades”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We also sense this kind of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Particu- larly, two mysteries in the area represent such confusion: Mystery 1 (M1): The performance-boosting effect of a larger finetuning learning rate (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It was broadly believed that finetuning the pruned model should use a small LR, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', in the famous L1-norm pruning [42], finetuning LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 is used for the Im- ageNet experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, many other papers choose a larger LR, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, which delivers signifi- cantly better performance than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For a pretty long time, pruning papers do not officially investigate this critical performance-boosting effect of a larger finetun- ing LR (although they may have already used it in their experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3), until this paper [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [37] formally studies how different finetuning LR sched- ules affect the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' They find random pruning and magnitude-pruning (the two most basic pruning methods) armed with a good finetuning LR schedule (CLR, or Cyclic Learning Rate Restarting) can counter-intuitively rival or even surpass other more sophisticated pruning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Unfortunately, they do not give explanations for this phenomenon except for calling upon comparing pruning algorithms in the same retraining configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Mystery 2 (M2): The value of network pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The value of network pruning seems unquestionable given the development history of over 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, two papers [9, 50] empirically find that training the pruned model from scratch can match pruning a pre- trained model, thus radically challenging the necessity of pretraining a big model first in the conventional 3- step pruning pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1 shows a concrete example that M1 makes the prun- ing benchmarks unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' After using an improved fine- tuning LR schedule (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3), we make L1-norm prun- ing [42], which is broadly considered the most basic base- line approach, match or beat many top-performing pruning methods published in top-tier conferences in the past sev- eral years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Such a situation really bewilders us, especially for those not in this area trying to borrow the most advanced pruning methods in their projects – “Has the area really de- veloped in the past several years?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' they might question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This paper is meant to unveil these mysteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Over this process, we hope to offer some helpful thoughts about why we have run into the current chaotic benchmark situation and how we can avoid it in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Specifically, when we try to unveil the mysteries, only to find there are various comparison setups enforced in the area, many of the conclusions actually hinge on which com- parison setup is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To decide which comparison setup is more trustworthy, we have to be clear with the fairness prin- ciple in pruning experiments first (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', what kind of compar- ison setup is considered fair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' After we sort out the fairness principle and comparison setups, we empirically examine the two mysteries and find M2 reduces to M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' When exam- ining M1, we find the role of network trainability in network pruning, through which we can easily explain M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, our investigation path and the contributions in this paper can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We first clarify the fairness principle in pruning exper- iments and summarize the outstanding comparison se- tups (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3), which have been unclear in the literature for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Then, we start to unveil M1 [37] and M2 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As we shall show (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4), the conclusion of M2 actu- ally varies up to which comparisons setup is used: If a larger finetuning LR is allowed to be used, the no- value-of-pruning argument cannot hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' otherwise, it mostly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Thus, to unveil M2, we have to unveil M1 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Next, we focus on unveiling M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We introduce the perspective of network trainability to diagnose prun- ing and clearly explain why the finetuning LR has a significant impact on the final performance (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5) – to our best knowledge, we are the first to clarify this mystery in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finally, we summarize the major reasons that have led to the confusing benchmark situation of network prun- ing up to now, and give some concrete suggestions about how to avoid it in the future (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Prerequisites 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Taxonomy of Pruning and Related Work A pruning algorithm has and only has five exclusive aspects to be specified: (1) Base model (when to prune): is pruning conducted on a pretrained model or a random model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) Sparsity granularity (what to prune): what is the smallest weight group in pruning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (3) Pruning ratio (how many to prune): how many weights are to be pruned?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (4) Pruning criterion (by what to prune): what measure is used to select the important weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', those to be kept) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the unimportant weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', those to be pruned)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (5) Pruning schedule (how to schedule): How is the sparsity scheduled over the pruning process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2 These five mutually orthogonal questions can shatter most (if not all) pruning papers in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Re- searchers typically use (1), (2), and (4) to classify different pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We give the major backgrounds in these three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning after training vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' pruning at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning has been mostly conducted on a pretrained model over the past 30 years, which is thus called pruning af- ter training or post-training pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This fashion has been the unquestioned norm until (at least) two papers in 2019, SNIP [41] and LTH [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' They argue prun- ing can be conducted on randomly initialized models and can achieve promising performance (allegedly matching the dense counterpart), too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This new fashion of pruning is called pruning at initialization (PaI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Existing PaI ap- proaches mainly include [19, 40, 41, 63, 77] and the series of lottery ticket hypothesis [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' PaI is not very relevant to this work because the benchmarking chaos and the mys- teries are mostly discussed in the PaT context, so here we would not discuss in length the specific PaI techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In- terested readers may refer to [80] for a comprehensive sum- mary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Sparsity structure: structured pruning vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' unstructured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' If the smallest weight group in pruning is a single weight element, this kind of pruning is called unstructured (or fine-grained) pruning [17,22,23], because the resulting zero-weight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', pruned-weight) locations are typically ir- regular (if no extra regularization is enforced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' If the small- est weight group in pruning presents some structure, this kind of pruning is called structured (or structural/coarse- grained) pruning [29, 42, 53, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In the area, structured pruning typically narrowly refers to filter pruning or channel pruning if not explained otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Structured pruning ben- efits more acceleration because the regular sparsity pattern is more hardware-friendly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' meanwhile, the regularity im- poses more constraints on the network, so given the same sparsity level, structured pruning typically underperforms unstructured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Note, the definition of “structured” sparsity is severely hardware-dependent and thus can vary as the hardware condition changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the N:M sparsity1 pioneered by NVIDIA Ampere architecture was considered as unstruc- tured sparsity, but since NVIDIA has launched new library support to exploit such kind of sparsity for acceleration, now it can be considered as structured sparsity (called fine- grained structured sparsity [58,86]), too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This paper focuses on filter pruning for now because the two aforementioned mysteries (the effect of finetuning LR and the value of network pruning) are mainly discussed in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/blog/accelerating-inference-with-sparsity- using-ampere-and-tensorrt/ Pruning criterion: Importance-based vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' regularization based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The former prunes weights based on some estab- lished importance criteria, such as magnitude (for unstruc- tured pruning) [22,23] or L1-norm (for filter pruning) [42], saliency based on 2nd-order gradients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', Hessian or Fisher) [24,39,70,74,76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The latter adds a penalty term to the objective function, drives unimportant weights toward zero, then removes those with the smallest magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Two notable points: (1) Even in a regularization-based prun- ing method, after the regularization process, the weights are still removed by a certain importance (typically mag- nitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Namely, regularization-based pruning inherently embeds importance-based pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) The two paradigms are not exclusive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' they can be employed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', [13, 79, 81] select unimportant weights by magnitude while also employing the regularization to penalize weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finally, for more comprehensive literature on network pruning, we refer interested readers to several surveys: an early one [64], some recent surveys of pruning alone [3,20, 30] or pruning as a sub-topic under the general umbrella of model compression and acceleration [7,8,11,73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Most relevant papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The most relevant works to this pa- per are [3], which is the first to systematically report the frus- trating state of network pruning benchmarks, identify some causes (such as the lack of standard benchmarks and metrics in pruning), and give concrete remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, they do not go deeper and analyze why the standard benchmarks are hard to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Our paper succeeds [3] and will elaborate more on this aspect and point out the central role of trainability within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [37], which officially reports the performance- boosting effect of a larger finetuning LR and calls upon comparing pruning algorithms under the same retrain- ing configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [37] is actually motivated by [65], which proposes LR rewinding vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the weight rewind- ing [18] proposed for finding winning ticket on large- scale networks and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [65] virtually takes ad- vantage of the performance-boosting effect of a larger finetuning LR, yet we do not know if they were aware at that point that the performance boosting is not be- cause of the magic rewound LR schedule but sim- ply because of a larger finetuning LR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', any ap- propriately larger LR would do, even not a rewound one), as later clarified by [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' so we tentatively con- sider [37] as the first work to systematically report the performance-boosting effect of a larger finetuning LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [50], which brings forward the argument regarding the value of network filter pruning against scratch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We will re-evaluate the major claim (scratch training 3 M1: A larger finetuning LR can significantly “boost” the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 🔍 Unveil M1: Network trainability is damaged by the sparsifying action, slowing down the finetuning optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' larger LR makes the model converge faster, thus better performance is observed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The performance is not really “boosted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Fairness principle: The performance advantage of a pruning paper should originate from the pruning method stage, not pretraining or finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In terms of strictness, S1 < S2 < S3 < S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S2 (the most popular one) cannot ensure fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Comparison setup ≥S3 is suggested to ensure fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pretraining (big dense) Random model (big dense) Base model (big dense) Pruning method Pruned model (small dense) Final model (small dense) Finetuning (small dense) Sparsifying action Dataset Network structure Total sparsity/speedup Given the same S1: M2: Training the pruned model from scratch with adjusted (typically prolonged) epochs can produce the final model with similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Training with adjusted epochs (small dense) Pruned model (small dense) Final model (small dense) S2: S1 + Keep the same base model (M1 arises at this setup) S3: S2 + Keep the same finetuning S4: S3 + Keep the same pruning cost M2 holds under S3, not hold under S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To unveil M2, we need to unveil M1 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S1-S4: Four pruning comparison setups M1-M2: Two mysteries in filter pruning Scratch training pipeline Pruning pipeline Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Overview of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We are motivated by unveiling two mysteries (M1, M2) in filter pruning, which represent the confusing pruning benchmark situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We first clarify the fairness principle and summarize outstanding comparison setups to lay down the discus- sion foundation (the notation “S1 < S2” means S1 is less strict than S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' others can be inferred likewise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Then we start to unveil M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' M2 will be shown to reduce to M1, actually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To unveil M1, we introduce network trainability as an effective perspective to demystify M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The unawareness of the role of network trainability in pruning has actually led to several sub-optimal hyper-parameter settings, which exacerbates the chaotic benchmark status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We finally give some concrete suggestions to calibrate the pruning benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' can match filter pruning) of this paper under our more strictly controlled comparison setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Terminology Clarification First, we make some critical concepts clear to lay down the common ground for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Although they are pretty simple concepts, misinterpreting them will twist our discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning pipeline vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' pruning method vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' sparsifying action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Some papers refer to pruning as the whole pruning algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the 2nd step in the pruning pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while others may mean a pruning paper or the instant sparsifying action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We realize such a vague conception definition is one reason causing confusion, so we make them exact here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We use pruning pipeline to mean all the three steps in a pruning paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We can consider pruning pipeline to be interchangeable with a pruning paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Then, we use pruning method = pruning algorithm to mean the 2nd step of the prun- ing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finally, the instant pruning action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', zeroing out weights or physically taking away weights from a network) is referred to as sparsifying action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To summarize, in terms of concept scope, pruning paper = pruning pipeline = pruning > pruning method = prun- ing algorithm > sparsifying action2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2The notation “pruning paper = pruning pipeline > pruning method” means, in this paper we consider pruning paper interchangeable with pruning pipeline, which includes pruning method as one part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Training from scratch = scratch training, means to train a randomly initialized model to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Scratch training of a pruned model means, we already know the network architec- ture of the pruned model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the weights are randomly ini- tialized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' train this network from scratch using the same training recipe as training the dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Notably, for filter pruning, when the architecture of the pruned model is known, the model should be implemented as a small-dense model, not a large-sparse model (with structural masks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The reason is, the widely-used parameter initialization schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', He initializa- tion [25], the default initialization scheme for CONV and Linear layers in PyTorch [59]) depend on the pa- rameter shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The large-sparse implementation is not equivalent to (often underperforms) the small-dense implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For unstructured pruning, the stan- dard implementation scheme is large-sparse weights (with unstructured masks) [50,79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' After the sparsifying action action, the subsequent training process is called finetuning or re- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We notice the community seems to have different interpretations about these two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', in [37], finetuning is a sub-concept of retraining, specifically meaning retraining with the last (small- est) learning rate of original training3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while many 3 [37] attributes this term usage to [23, 42, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We double-checked these 4 more papers [50,56,76,79,82] consider finetuning the same as retraining, meaning the 3rd step of the prun- ing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In this paper, we take the more com- mon stance: considering finetuning and retraining in- terchangeable, and in the end of this paper, we will show the term “finetuning” should be deprecated in fa- vor of “retraining”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Scratch-E, Scratch-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' These two terms are from [42], denoting two scratch training schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' “E” is short for epochs, “B” short for (computation) budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In prac- tice, [42] uses FLOPs as an approximation for the com- putation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In Scratch-E, the point is to maintain the same total epochs when comparing scratch train- ing to pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In Scratch-B, the point is to main- tain the same total FLOPs when comparing scratch training to pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Here is a concrete example of Scratch-B: a dense model has FLOPs F1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' pretrain- ing the dense model takes K1 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' it is pruned by L1-norm pruning, giving a pruned model with FLOPs F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the finetuning takes another K2 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' then the scratch training should take (K1F1 + K2F2)/F2 = K1(F1/F2)+K2 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The ratio F1/F2 is typically called speedup in network pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Value of network pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This term comes from [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The conclusion of [42] is that scratch train- ing can match the performance of the 3-step prun- ing pipeline if Scratch-B is adopted, for filter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, they argue there is no value for filter prun- ing algorithms that use predefined layerwise pruning ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For the filter pruning algorithms that do not use predefined layerwise pruning ratios, their role is to decide the favorable network architectures, akin to NAS [15,89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As for unstructured pruning, [42] shows scratch training cannot match pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, rig- orously, the argument about the value of network prun- ing means the value of inheriting pretrained weights in filter pruning with predefined layerwise pruning ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We will use the short notion, value of network pruning, without mentioning its much richer context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Fairness and Comparison Setups in Pruning 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Fairness Principle in Network Pruning This section is prepared for the next section, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2, where we will summarize the major comparison setups in pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As we shall see in the experiments, a pruning al- gorithm A can be better than B under one comparison setup, while worse than or on par with B under another comparison setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To decide which comparison is more trustworthy, we three papers and found they do not evidently have the inclination to mean finetuning as one particular type of retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' have to evaluate which comparison setup is fairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Thence comes the necessity of a clear fairness principle in pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Fairness Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The performance advantage of a prun- ing paper should be attributed to the pruning method stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As aforementioned (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2), a pruning algorithm exclu- sively has five aspects [30, 80]: base model, sparsity gran- ularity, pruning ratio, pruning criterion, and pruning sched- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Excluding the base model and sparsity granularity axes4, therefore, when we say a pruning method is bet- ter than another one, the performance advantage should be attributed to at least one of the three aspects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' namely, a better pruning ratio scheme, or/and a better pruning criterion, or/and a better pruning schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Otherwise, it means the performance advantage comes from some outside factors other than the pruning algorithm itself – in this case, attributing the performance credit to the pruning method would be an unjustified claim, potentially leading to an un- fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Per this fairness principle, clearly, we should keep the same base model and the finetuning process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the 1st and 3rd steps) in the pruning pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Next, we elaborate on the outstanding comparison setups in network pruning and examine their fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Comparison Setups in Network Pruning In the literature, we find there are at least the following four groups of comparison setups, summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Note, in the following discussion, we consider different pruning algorithms (and scratch training) that remove or zero out unimportant weights only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Namely, we do not consider iterative pruning for now, and we will discuss how the conclusions can carry over to iterative pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Historical contexts of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In the following paragraphs, we briefly go through the historical context of the different comparison setups in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Obviously, the S1 setup is the most basic one, and also the earliest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It simply compares performance re- gardless of many factors, such as training epochs and even the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This setup is adopted by early pruning pa- pers, especially those using Caffe [33], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', [22,23,29,82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Such a comparison setup does not even demand the same base model, which we may consider problematic today;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' yet back in those days, they had their own reasons – before the deep learning (DL) community had mature DL plat- forms/tools/computation power as we have today, these pa- pers usually trained their own base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Consequently, due to different implementations, their base models do not have the same (or even close) accuracy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the VGG16 4Most pruning works do not consider the performance gain due to a better base model or sparsity granularity as a valid advantage over other meth- ods, because a pruning method can be easily applied to different base models and sparsity granularities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', SSL [82]) although the paper may only focus on one kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Summary of popular comparison setups in pruning pa- pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is helpful to review them along with the 3-step pruning pipeline: pretraining (output: base model) ⇒ pruning (output: pruned model) ⇒ finetuning (output: final small model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In terms of strictness, S1 < S2 < S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 < S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 < S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 < S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 (the notation “S1 < S2” means S1 is less strict than S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' others can be inferred likewise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Most existing pruning papers follow the S2 comparison setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Comparison setups S1 Compare performance or performance drop on the same dataset and network at the same compression or speedup rate S2 +Same base model S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 +Same base model +Same finetuning epochs S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 +Same base model +Same finetuning LR schedule S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 +Same base model +Same finetuning LR schedule +Same pruning epochs S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 +Same base model +Same finetuning LR schedule +Same pruning LR schedule SX-A +Same epochs of “pretraining + pruning + finetuning” SX-B +Same FLOPs of “pretraining + pruning + finetuning” base model reported by ThiNet [52] has top-5 accuracy 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='44% while CP [29], a concurrent work with ThiNet, re- ported 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='9% (for those who are not familiar with these numbers, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5% top-5 accuracy is a very significant gap for ImageNet-1K classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As a remedy, to make the results comparable, many pa- pers report the relative performance drop, namely, base model accuracy minus final model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Such an idea is still broadly practiced at present [50,79], esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' when com- paring methods that are implemented under quite different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Later, as the DL community develops, more DL platforms e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', PyTorch [59] and TensorFlow [1] mature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' There is usually a well-accepted model zoo (such as torchvi- sion models5) for others to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As a result, more and more pruning papers adopt them as the base models, such as [42, 79], which has become the mainstream practice at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Thus, the S2 comparison setup arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' At this stage, few researchers have noticed the impor- tance of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This makes sense since, in the pruning pipeline, only the pruning method part (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the 2nd step) is regarded as the central one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The finetuning process is often so downplayed that many papers do not even clearly report the hyper-parameters, as also noted by [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Later, in the endless pursuit of higher and higher per- formance, there is a clear trend that the finetuning epochs become longer and longer (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3 for an incomplete summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This in effect renders the comparison more and more unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Besides, the finetuning LR has been noticed 5https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='org/vision/stable/models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='html Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Summary of finetuning epochs and LR schedules of many filter pruning papers published in recent top-tier venues, with ResNets [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The default dataset is ImageNet [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' other datasets are explicitly pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Method #Epochs LR schedule SSL [82]NeurIPS’16 (CIFAR10) – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 L1-norm [42]ICLR’17 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001, fixed DCP [88]NeurIPS’18 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step (36/48/54) GAL-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5/1 [47]CVPR’19 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (10/20) Taylor-FO [55]CVPR’19 ∼25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (10/20) Factorized [43]CVPR’19 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (30/60) CCP-AC [60]ICML’19 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001, step decay (30/60/90) HRank [46] CVPR’20 30×#layers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (10/20) GReg-1/2 [79] ICLR’21 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (30/60/75) ResRep [14] ICCV’21 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, cosine annealing L1-norm [42] ICLR’17 (our reimpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=') 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01, step decay (30/60/75) to have a significant impact on the final performance, as formally studied by [37] (although [37] is the first one to formally study this phenomenon, a larger finetuning LR has been employed by many papers even before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Because of these, the finetuning process must be taken into account to maintain fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The most exact way to rule out the impact of finetuning is to use exactly the same finetuning process – the same LR schedule (including the same epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' we omit the hyper- parameters, like weight decay, momentum, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', and assume they are maintained the same), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, due to various objective or subjective reasons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', prior papers may not release their finetuning details, making the follow-ups unable to reproduce the same fine- tuning), S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 is often impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Thence comes a weaker setup S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1, which only keeps the same epochs of finetun- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is allowed to use different finetuning LR schedules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', different initial LR) – this is where M1, the mystery of the finetuning LR effect, arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Several papers (such as [78, 79, 84, 85]) have abla- tive analysis experiments on small-scale datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', CI- FAR10 [35]) using the setup S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2, while the main bench- mark experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', with ResNet-50 on ImageNet) us- ing S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The primary reason is that, following the setup S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 means re-running the experiments for other compari- son methods in the large-scale benchmarks, which is usually impractical (too costly) or even impossible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the com- parison methods do not release usable code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 is still not the most strictly fair setup since it does not consider the cost (measured by the num- ber of epochs) of the pruning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For one-shot prun- ing (such as L1-norm pruning [42]), the cost of pruning is zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while for a regularization-based method (such as GReg [79]), it may take another few epochs for regular- ized training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Considering these cases, S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 comes out: it builds upon S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 and demands the same LR schedule for the pruning algorithm – as far as we know, this is the most strict comparison setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In practice, again, for various reasons, we may not know the LR schedule of a pruning 6 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Then, S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 degrades to S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1, which only de- mands the same epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In setups S2 to S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2, when comparing pruning to scratch training in obtaining the same pruned (small) model, the scratch training employs the same training recipe of ob- taining the base (big) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' [50] challenges this practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' They argue, the scratch training scheme spends less cost than pruning, so the comparison is unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As a remedy, they propose to take into account the cost of the pretrain- ing stage, which gives the SX-A and SX-B setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' About the cost, one way to measure it is to use the number epochs (hence the SX-A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' another is to consider the same computa- tion and they approximate computation with FLOPs (hence the SX-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is hard to say if considering the cost of the pretraining stage is really necessary and practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Advocates of the older practice may list reasons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', pretrained models often exist already (like those pretrained on ImageNet [10] and shared on HuggingFace6), so we do not need to consider the cost of pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, advocates of SX may argue that not all pretrained models are available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' for many tasks, we still need to train the pretrained models first, so the cost of scratch training should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We have no inclination here regarding which one is more correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We make two points that we are fairly certain about: (1) In the pruning literature, most papers still follow the older practice when reporting the scratch training results of the pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) Given the recent rise of foundation models [4] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', Bert [12], GPT-3 [5], CLIP [61], diffu- sion models [66,71]), common researchers barely have the resources to train a model from scratch, so pruning would inevitably be conducted on the pretrained model, probably, for those big models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' What comparison setup is mostly used now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Unfor- tunately, S2 is the most prevailing comparison setup at present [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This setup ignores at least one important fac- tor that, we now know [37], has a significant impact on the final performance: the finetuning LR schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In the following sections, we start our empirical investi- gation of unveiling M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We study M2 first and then M1, because the conclusion about M2 actually depends on M1, as we are about to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Reexamining the Value of Pruning The rethinking paper [50] presents many valuable thoughts regarding the value of the 3-step pruning pipeline against scratch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, there are a few potential concerns in their experiments that may shake the validity of their conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' First, they directly cite the results of a few pruning papers and compare the relative performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='co/ Because of the stark differences between different DL plat- forms, such a comparison (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', comparing methods that use different base models) may not be convincing enough for rigorous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Second, when reproducing the L1-norm pruning [42], they use fixed LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 and 20 epochs, fol- lowing [42], for the finetuning stage, which is now known to be severely sub-optimal (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4, a larger finetuning LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 can significantly boost performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is thereby of interest whether the no-value-of-pruning argument would change if the comparison is conducted un- der a strictly controlled condition and a better finetuning LR is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This section attempts to answer this ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Three comparison setups (SX-A, SX-B, and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2) are considered since they are the most strict setups up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We choose L1-norm pruning [42] be- cause it is the most representative pruning method and easy to control at a strict comparison setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Specifically, L1- norm pruning prunes the filters of a pretrained model with the smallest L1-norms to a predefined pruning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' After pruning, the pruned model is finetuned for a few epochs to regain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Other pruning methods, such as regularization-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', [49, 79, 82]), introduce many factors that are hard to control for rigorous analysis, so we do not adopt them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We will discuss how the findings can generalize to those cases later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Networks and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The network used for analysis is ResNet34 [26], following [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For standard benchmarks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1), we use ResNet50 [26] because it is one of the most representative benchmark networks in filter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The datasets are ImageNet100 and the full ImageNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' ImageNet100 is a randomly drawn 100-class subset of Im- ageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We use it for faster analysis given our limited re- source budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The full ImageNet is used for benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Implementation details of pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For analysis, pruning is conducted on the 1st CONV layer (the 2nd CONVs are not pruned, following L1-norm pruning [42]) in all resid- ual blocks of ResNet34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The first CONV and all FC layers are spared, also following the common practice [20,79,87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Uniform layerwise pruning ratio is employed (which usu- ally under-performs a tuned non-uniform layerwise pruning ratio scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' but since this paper does not target the best performance but explanation, we adopt it for easy analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We conduct pruning at a wide sparsity spectrum (10% to 95%) in the hopes of comprehensive coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' One table to show them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The results are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Before we present the analyses, we introduce a no- tion, pruning epoch, which is defined as the epoch when the sparsifying action is physically enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', if a model is trained for 30 epochs and then the sparsifying action is enforced, the pruning epoch is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We observe: (1) For the S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 setup (rows marked by ), we are not 7 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Top-1 accuracy (%) comparison between L1-norm pruning [42] and training from scratch with ResNet34 on ImageNet100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Each result is averaged by at least three random runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The learning rate (LR) schedule of scratch training is: Initial LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1, decayed at epoch 30/60/90/105 by multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1, total: 120 epochs (top-1 accuracy of dense ResNet34: 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56%, FLOPs: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='66G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' “P30F90, 1e-1” means the model is pruned at epoch 30 and finetuned for another 90 epochs with initial finetune LR 1e-1 (please refer to our supplementary material for the detailed LR schedule);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the others can be inferred likewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The best result within each comparison setup is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning ratio 10% 30% 50% 70% 90% 95% FLOPs (G, speedup: k×) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='30 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11×) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='59 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='41×) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='90 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93×) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='19 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09×) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='48 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='68×) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='30 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='06×) Scratch training 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='38 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='13 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='13 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='76 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='90 L1-norm (P15F105, 1e-1) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='44 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11 L1-norm (P30F90, 1e-2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='32 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='39 L1-norm (P45F75, 1e-2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='03 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='08 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21 L1-norm (P60F60, 1e-3) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='05 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='37 L1-norm (P75F45, 1e-3) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='04 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31 L1-norm (P90F30, 1e-4) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 82.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25 L1-norm (P90F30, 1e-2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='91±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='28 L1-norm (P30/kF90, 1e-2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='04 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='27 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='03 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='52 Under comparison setup S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 (same overall LR schedule), Under comparison setup SX-A (same total epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' finetuning LR increased), Under comparison setup SX-B (same total FLOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' allowed to change the LR schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The only thing we can change is the pruning epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, the best prun- ing epoch varies w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the sparsity level – at a small pruning ratio, different pruning epochs give a similar performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while as the pruning ratio arises, the performance becomes more sensitive to the pruning epoch, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', for pruning ratio 95%, P90F30, 1e-4 severely underperforms P30F90, 1e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Notably, a clear trend is, when the pruning ratio is large (70% to 95%), it is better to have a smaller pruning epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Under this setup, only at pruning ratios of 30%-70%, pruning surpasses scratch training by a statistically signifi- cant gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, we can only say pruning has a marginal advantage over scratch training here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) Then we look at the setup SX-A (rows marked by ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Under this setup, we are allowed to adjust the finetuning LR as long as the total epochs are kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We increase the initial finetuning LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, it significantly improves the accuracies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', (P30F90, 1e-1) improves the accu- racy by nearly 2% at pruning ratio 95%, against (P30F90, 1e-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is the performance-boosting effect aforemen- tioned [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We also apply the larger LR trick to another two settings P60F60 and P90F30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In all of them, we see a larger finetuning LR improves performance by an obvious margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Now, the gap between pruning and scratch training be- comes much more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning is more surely valu- able under this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (3) Next, we use the comparison setup SX-B (rows marked by ), which maintains the total FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We ap- ply this scheme to the best pruning setup P30F90, 1e-1 in S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 in the hopes of better performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Since the dense model is trained for 30 epochs, to compensate for the FLOPs, the pruning epoch should be squeezed by the speedup ratio k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', for pruning ratio 10%, the speedup ratio is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11, then the pruning epoch should be adjusted to 30/k ≈ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, the squeezing of the pruning epoch does close the gap between pruning and scratch training: At pruning ratios of 10% to 70%, pruning is still better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while for 90% and 95%, pruning only matches or underperforms scratch training – this is a concrete example that we do not have a once-for-all answer to questions like “is pruning better than scratch training?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We also try a smaller finetuning LR in this setup, as shown in the row (P30/kF90, 1e-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The LR effect also translates to this case – a smaller finetuning LR degrades the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Short summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, the argument about the value of network pruning severely hinges on which comparison setup is employed and the pruning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For the setup SX-A, where pruning outperforms scratch training obvi- ously, the advantage comes from a better finetuning LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Yet, we are not sure if such better LR schedules also exist for the scratch training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' if so, scratch training can be fur- ther boosted, too – as such, this kind of “competition” can be endless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' There are two kinds of attitudes toward this sit- uation: (1) Do not consider the performance improvement from a better finetuning LR as a fair/valid performance ad- vantage as it is not from the pruning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (2) Still con- sider it as a valid performance advantage but will meet the “endless competition” challenge we just mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The community now is mostly using (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We suggest using (1), following our fairness definition clarified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 8 Despite many uncertainties, we are certain about one thing from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4: Whichever setup is favored, the fine- tuning LR holds a critical role in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Even for the comparisons setup S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2, where the finetuning LR does not change, by changing the pruning epoch, implicitly, we change the finetuning LR, and it has been shown very per- tinent to the final performance as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In this sense, the two mysteries of pruning actually boil down to one (M1): Why does finetuning LR have such a great impact on the performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is the next question we would like to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' LR, (arguably) as the most influential hyper-parameter in train- ing neural networks, has a significant impact on perfor- mance – this is definitely not surprising;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' what is really sur- prising might be, why the prior pruning works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the original L1-norm pruning [42] adopts LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 in finetun- ing for their ImageNet experiments) did not realize that such a simple “trick” is so important to performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This ques- tion is also worth our thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Trainability in Network Pruning 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Background of Trainability Trainability, by its name, means the ability (easiness) of training a neural network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the optimization speed (note, speed is not equal to quality, so we may see a network with good trainability turns out to have a bad generalization abil- ity eventually).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Notably, essentially, the role of a pruning method is to provide the initial weights for the later finetuning process, that is, pruning is essentially a kind of initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In stark contrast to the broad awareness that initialization is very critical to neural network training [21,25,34,54,72], the ini- tialization role of pruning has received negligible research attention, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Trainability is also mostly studied for random initialization [67,83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' A few recent works marry it with network pruning in some other similar forms like signal propagation [40] and gradient flows [77] (a good signal propagation or strong gradient flow usually suggests a good trainability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' These works are inspiring, while they mostly stay in the domain of pruning at initialization (PaI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Few attempts before, to our best knowledge, tried to utilize the notion of trainability to examine pruning after training (PaT), at least, for the two mysteries we study here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This paper is meant to bridge this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The major difference between PaI and PaT is whether using a pretrained model as base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Such a context is essential to this paper since the above two mysteries are both brought forward in the context of PaT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Trainability accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Literally, a bad trainability implies the training is hard and the training performance will arise slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Per this idea, there is a straightforward metric to measure trainability – we introduce trainability accuracy, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Top-1 accuracy (%) comparison of different setups of L1-norm pruning [42] with ResNet34 on ImageNet100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning ratio: 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' TA: trainability accuracy (the metric used to measure trainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This table shows, the performance gap between a smaller LR and a larger LR is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It can be closed simply by training more epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The root cause that a smaller LR appears to under-perform a larger LR is simply that the model trained by the smaller LR does not fully converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finetuning setup Top-1 acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (%) TA (%) P30F90, 1e-1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='15 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='00 P30F90, 1e-2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='39 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='45 P30F90, 1e-2 (+30 epochs) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='19 / P30F90, 1e-2 (+60 epochs) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25 / P60F60, 1e-2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='39 P60F60, 1e-3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='37 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='19 P60F60, 1e-3 (+60 epochs) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 / P60F60, 1e-3 (+120 epochs) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='13 / P60F60, 1e-3 (+180 epochs) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 / P90F30, 1e-2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='83 P90F30, 1e-4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='26 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93 P90F30, 1e-4 (+60 epochs) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 / P90F30, 1e-4 (+270 epochs) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 / P90F30, 1e-4 (+1485 epochs) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 / the average of the first N epochs, T = 1 N N � i=1 Acci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (1) Since the optimization speed depends on the LR used, when we calculate trainability accuracy, we must ensure they are under the same LR schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In this paper, we choose N as the number of the 1st LR stage, which characterizes the optimization speed in the early phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Next, we utilize trainability to explain the mysterious ef- fect of the finetuning LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Examining the Effect of Finetuning LR Two facts as foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We first lay down two facts as the common ground for the discussion of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We will show the mystery about the finetuning LR effect boils down to these two simple facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' First, pruning damages trainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is an intuitively straightforward fact since pruning removes connections or neurons, which virtually makes the network harder to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This fact holds for not only pruning a random network [40], but also for pruning a pretrained model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Moreover, notably, more aggressive pruning leads to more damaged trainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Second, a model of worse trainability will need more effective updates to reach convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' More effective updates mean two cases: If LR is not changed, more epochs are needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' if the number of epochs does not change, a larger LR is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is also easy to understand since trainability measures the easiness of op- timization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' a bad trainability implies harder optimization 9 0 10 20 30 40 50 60 Epoch 20 40 60 80 Test accuracy (%) ResNet34, ImageNet100, PR 95% (a) P60F60, 1e-3 0 50 100 150 200 250 Epoch 20 40 60 80 Test accuracy (%) ResNet34, ImageNet100, PR 95% (b) P60F60, 1e-3 (+180 epochs) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' epoch during finetuning of the setting P60F60, 1e-3 at pruning ratio 95% in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Red vertical lines mark the epoch of decaying LR by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Particularly note be- fore the 1st LR decay, the accuracy keeps arising in (a), implying the 1st LR decay may be too early – this is confirmed in (b), where the red cross marker (×) indicates the time point of the 1st LR de- cay in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' See more similar plots in our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' literally, hence the more effective updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Such observa- tion has been made by some sparse training papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', RigL [16] notes that “sparse training methods benefit sig- nificantly from increased training steps”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' When we observe that a larger LR improves the final test accuracy of the pruned model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', Row P30F90, 1e-1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Row P30F90, 1e-2 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4), it is worthwhile to differentiate two subtle yet distinct possibilities: A larger LR helps the pruned model reach a solution that the smaller LR cannot reach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', a larger LR help the model located into a better local minimum basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The smaller LR can also help the model reach the so- lution as the larger LR does;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' just, the larger LR helps the model get there faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The former implies the performance-boosting effect of a larger LR is fundamental;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while the latter implies there is no fundamental gap between the two solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' it is only an issue of optimization speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Let’s analyze a concrete example of P60F60 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For pruning ratio 95% (we use this for example because at larger sparsity, the performance boosting effect is most pro- nounced), using 1e-2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1e-3 improves the test accuracy from 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='53 to 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='74, a very significant jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This improve- ment also translates to the rows of P30F90 and P90F30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2, we note the performance improve- ment coincides with trainability accuracy improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We were wondering if the performance improvement is actually due to a better trainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2(a) plots the test accuracy during the finetuning of P60F60, 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We notice before the 1st LR decay at epoch 30, the accuracy keeps arising even at epoch 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This triggers a question: usually, we decay LR when the accu- racy saturates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' now, when the accuracy is still steadily ris- ing, is the LR decayed too early?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This question matters be- cause if the LR decays too early, the model is forced to sta- bilize due to the small step size and insufficient updates, not because it gets close to the local minimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the model may not converge at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To verify this, we extend the epochs of the LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 phase by 60/120/180 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' See the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5 (note the rows P60F60, 1e-3 (+60/120/180 epochs)) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Now, the model finetuned by LR 1e-3 can reach 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='33, very close to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='81 reached by LR 1e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The test accuracy plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2(b) also confirms that the seem- ing underperformance of LR 1e-3 is due to insufficient epochs – namely, the advantage of a larger LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 is not some magic fundamental advantage, but a simple consequence of faster optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We also verify this on other cases (P30F90 and P90F30) that the smaller LR “underperforms” the larger LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The results are also presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In all of these cases, given abundant epochs, the gap between the larger LR and the smaller LR can be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Especially for P90F30, the smaller LR 1e-4 can achieve a much better result than LR 1e-2 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This strongly demon- strates the smaller LR can also achieve what the larger LR can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To summarize, our results suggest a larger LR does not really “improve” the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' What really hap- pens is, a larger LR accelerates the optimization pro- cess, making the higher performance observed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In practice, when researchers tune different LR’s, they usu- ally keep the total epochs fixed (for the sake of fairness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Given the same total epochs, the pruned model using the smaller finetuning LR does not fully converge, making the performance appear “worse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Further remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is worthwhile to note that such an ex- perimenting trap is so covert if we are unaware of the dam- aged trainability issue in pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We may never realize that the epochs should be increased properly if a smaller finetun- ing LR is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' What’s even trickier, we do not know how many more epochs is the so-called proper – Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5 is a living example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', P30F90), 60 more epochs is enough, while for others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', P60F60, P90F30), 180 epochs or more is not enough to bridge the performance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Clearly, there is still much work to be done here toward a more rigorous understanding of the influence of damaged 10 trainability on pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Retrospective remarks and the lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is worthwhile to ponder why [42] employed a seriously sub-optimal fine- tuning LR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This, we conceive, may originate from a long-standing misunderstanding in the area of net- work pruning – many have believed that because pruning is conducted onto a converged model, the retraining of the pruned model needs not to be long and the LR should be small to avoid destroying the knowledge the model has ac- quired, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', in [65], the authors mentioned in their abstract “The standard retraining technique, fine-tuning, trains the unpruned weights from their final trained values using a small fixed learning rate”, implying that such misconcep- tion spreads so widely that it is taken for “standard”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='7 However, the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4 suggest, such thought only holds for the cases of low pruning ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For a moderate or large pruning ratio, this thought hardly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' What was neglected is that the sparsifying action damages network trainability, slowing down the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As a compen- sation, it is supposed to use a larger LR to accelerate the optimization, not a smaller LR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' similarly, more epochs are needed to compensate for the slow optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' However, the original L1-norm pruning [42] chose LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 and only 20 epochs for their ImageNet experiments, exactly the op- posite of what is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This, we conceive, is the reason that L1-norm pruning has been underrated for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Its real performance is actually pretty strong even compared with recent top-performing approaches (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Similarly, based on what we just learned about the truth of M1, if we examine the other filter pruning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', GAL [47] (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3), its reported results are probably also underrated, because it uses only 30 epochs for finetuning and the model may well not expose its full potential, as a re- sult of the immature convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This implies a pretty dis- turbing concern – for many filter pruning papers, we have to calibrate their results for fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Directly citing the num- bers may (well) not show the real performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Conclusion and Discussion This paper attempts to figure out the confounding bench- mark situation in filter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Two particular mysteries are explored, the performance-boosting effect of a larger finetuning LR, the no-value-of-pruning argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We present a clear fairness principle and sort out four groups of popular comparison setups used by many pruning pa- pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Under a strictly controlled condition, we examine the two mysteries and find they both boil down to the issue of damaged network trainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This issue was not well rec- ognized by prior works, leading to (severely) sub-optimal 7Actually, the 3rd-step of the pruning pipeline is broadly referred to as finetuning – this term per se already implies the inclination of using a small LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To rule out such conceptual bias, a more accurate way to phrase the 3rd step in the pruning pipeline may be retraining the pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' hyper-parameter settings, which ultimately exacerbates the confounding benchmark situation in filter pruning now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We hope this paper helps the community towards a clearer un- derstanding of pruning and more reliable benchmarking of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Takeaways and suggestions from this paper: Why is the state of neural network pruning so confus- ing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Non-standard comparison setups (and its funda- mental reason: unclear fairness principle) and the un- awareness of the role of trainability are the two major reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The latter further leads to sub-optimal hyper- parameter settings, inherited by many follow-up pa- pers, exacerbating the messy benchmark situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As the area of network pruning develops, various com- parison setups have appeared (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Each has its own historical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Unfortunately, the most prevailing comparison setup now, setup S2 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 2, cannot ensure fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We suggest using the setup S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2 or higher, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', maintaining the same base model and the same finetuning process – Higher com- parison setup means stricter experiment control, also means more resources and efforts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' so there would be inevitably a trade-off between how fair we want to be and how much we can invest in8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Reporting all the finetuning details (esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the LR sched- ule) is rather necessary and should be standardized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Filter pruning can beat scratch training or not, up to the specific comparison setup and pruning ratio in consid- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Given the recent rise of large foundation mod- els, pruning may still follow the conventional 3-step pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The observation that a larger finetuning LR “improves” pruning performance is largely a misinterpretation – the performance is not “improved”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' what really hap- pens is that the good performance is observed ear- lier because the larger LR accelerates the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The fundamental factor playing under the hood is the network trainability damaged by the sparsifying action (or zeroing out) action in pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The damaged network trainability was not well rec- ognized by prior pruning works, resulting in severely sub-optimal hyper-parameters, rendering the potential of a baseline method, L1-norm pruning [42], underes- timated for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This fact may spur us to re- evaluate the actual efficacy of (so) many sophisticated pruning methods against the simple L1-norm pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 8If we look back at the historical contexts of different comparison setups (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='2), sometimes, the unfairness is not because we do not want, but because we cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 11 The term finetuning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the 3rd step in the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Eigendamage: Structured pruning in the kronecker- factored eigenbasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICML, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3, 5 [77] Chaoqi Wang, Guodong Zhang, and Roger Grosse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Picking winning tickets before training by preserving gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3, 9 [78] Huan Wang and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Trainability preserving neural struc- tured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12534, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6 [79] Huan Wang, Can Qin, Yulun Zhang, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Neural pruning via growing regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1, 3, 4, 5, 6, 7, 15 [80] Huan Wang, Can Qin, Yulun Zhang, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Recent ad- vances on neural network pruning at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In IJCAI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3, 5 [81] Huan Wang, Qiming Zhang, Yuehai Wang, Lu Yu, and Haoji Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Structured pruning for efficient convnets via incremental regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In IJCNN, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3 [82] Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Learning structured sparsity in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In NeurIPS, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3, 5, 6, 7, 15 [83] Lechao Xiao, Yasaman Bahri, Jascha Sohl-Dickstein, Samuel Schoenholz, and Jeffrey Pennington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Dynami- cal isometry and a mean field theory of cnns: How to train 10,000-layer vanilla convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICML, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 9 [84] Yulun Zhang, Huan Wang, Can Qin, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Aligned structured sparsity learning for efficient image super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6 [85] Yulun Zhang, Huan Wang, Can Qin, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Learn- ing efficient image super-resolution networks via structure- regularized pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6 [86] Aojun Zhou, Yukun Ma, Junnan Zhu, Jianbo Liu, Zhijie Zhang, Kun Yuan, Wenxiu Sun, and Hongsheng Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Learn- ing n: M fine-grained structured sparse neural networks from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3 [87] Michael H Zhu and Suyog Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To prune, or not to prune: Exploring the efficacy of pruning for model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR Workshop, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 7 [88] Zhuangwei Zhuang, Mingkui Tan, Bohan Zhuang, Jing Liu, Yong Guo, Qingyao Wu, Junzhou Huang, and Jinhui Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Discrimination-aware channel pruning for deep neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1, 6 [89] Barret Zoph and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Neural architecture search with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In ICLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Training setting summary In the paper, we evaluate on two datasets ImageNet [10] and ImageNet100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The latter is a small version (100 classes) of ImageNet for faster experimenting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We use Py- Torch [59] to implement all of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, we mainly refer to the official PyTorch ImageNet example9 for hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The default LR schedule in PyTorch ImageNet example is: 0:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1,30:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01,60:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001, total epochs:90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is adopted by L1-norm prun- ing [42] (and inherited by [50]), so in their finetuning, they use the last-stage LR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) as finetuning LR and never decay it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' How we decide the LR schedule for scratch train- ing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Per our empirical observations, decaying the LR from 1e-3 to 1e-4 can still improve the model by around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='8% top-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Namely, 1e-3 is not where the model finally converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Since we want to compare models at their best potential, we add another 30 epochs and decay another two times to make sure the model fully utilizes its potential – this gives us the LR schedule for training a scratch model in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4: 0:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1,30:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01,60:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001,90:1e-4,105:1e-5, total epochs:120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' How we decide the LR schedule for finetuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We abide by the following rules to decide the finetuning LR schedule: Rule 1: When comparing scratch training to pruning methods, the final LR should be the same – as we mentioned, decaying LR from 1e-3 to 1e-4 on Ima- geNet can see another 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='8% top-1 accuracy bump, thus it would be unfair to compare a pruning method whose model is finetuned to LR smaller than 1e-3 to a scratch-training model whose smallest LR is only 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The final LR for our ImageNet/ImageNet100 results, as mentioned above, is 1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Rule 2: Halving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This means a kind of epoch splitting scheme: Given N total epochs, split half of them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', N/2 epochs) to the 1st LR, then split the half of the left epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', N/4) to the 2nd LR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' To our best 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/pytorch/examples/tree/main/imagenet 14 knowledge, this scheme is due to (no later than) the pa- per of ResNet [26] (see their CIFAR10 experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Rule 3: The epochs for each LR stage is no more than 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is mainly due to the design in the official Py- Torch ImageNet example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We do not know why they chose 30 historically, but since this example is exten- sively followed as a baseline, we follow it too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Based on these rules, given the total number of fine- tuning epochs, we can exactly derive the finetun- ing LR used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For example, in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 4, for P30F90, 1e-1, the finetuning LR schedule is: 0:1e-1,30:1e-2,60:1e-3,75:1e-4,83:1e-5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' for P60F60, 1e-2, the finetuning LR schedule is: 0:1e-2,30:1e-3,45:1e-4,53:1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Layerwise pruning ratio for the experiments of ResNet50 on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For the results of ResNet50 on ImageNet we report in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1, its finetuning LR schedule is: 0:1e-2,30:1e-3,60:1e-4,75:1e-5, total epochs:90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, this is never some heavily tuned magic LR schedule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' nevertheless, we use it to finetune the pruned model by L1-norm pruning [42], only to find the final performance actually can beat/match many top- performing methods after ICLR’17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The layerwise prun- ing ratios are borrowed from GReg [79] (as they released their ratios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' for many other papers, we only know the total speedup, not aware of the detailed layerwise pruning ratios) to keep a fair comparison with it – speedup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31×: [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' speedup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='56×: [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='74, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='74, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Code references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We mainly refer to the following code implementations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' They are all publicly avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Official PyTorch ImageNet example10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' GReg-1/GReg-2 [79]11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Rethinking the value of network pruning [50]12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Can the findings generalize to other pruning methods than L1-norm pruning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning methods, according to their methodology, typically are classified into two groups, regularization- based (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' penalty-based) and importance-based (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' saliency-based), from a long time ago [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Despite the different categorization, any pruning method has a step to physically zero out the weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', the spar- sifying action step, per the terminology in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Typi- 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/pytorch/examples/tree/master/imagenet 11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/MingSun-Tse/Regularization-Pruning 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='com/Eric-mingjie/rethinking-network- pruning/tree/master/imagenet/l1-norm-pruning cally, this step is the magnitude pruning (or L1-norm prun- ing when it comes to filter pruning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', SSL [82] and GReg [79] are two regularization-based pruning methods, with different penalty terms proposed, yet both of them have a step to physically zero out unimportant weights by sorting their magnitude before finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In other words, regularization-based methods, although they are classified into a different group from magnitude pruning (which is importance-based), they essentially include magnitude pruning as a part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We have analyzed in the paper, the fundamental reason that incurs damaged trainability is the sparsifying action ac- tion in magnitude pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Therefore, any pruning method that employs magnitude pruning as a part is subject to the analyses of this paper – this means the discoveries of this paper are generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The attended broken trainability in these methods should also lead to similar13 finetuning LR effect to the L1-norm pruning case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Can the finetuning LR effect generalize to other LR schedules than the traditional step decay?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In the paper, we explore the finetuning LR effect (a large LR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' a small LR,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) to the final perfor- mance using the conventional step decay LR schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It is of interest if the effect can translate to other more advanced LR schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' We consider Cosine Annealing LR schedule [51] here, referring to the official PyTorch Cosine LR implementa- tion14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' When we switch from Step LR schedule to Cosine, the initial LR and minimum LR are kept the same (namely, the start point and end point of LR are the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' the only difference is the scheduling in between).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The scratch model is trained for 200 epochs, initial LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1, step decayed at epoch 100 and 150 by multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1 (referring to the orig- inal ResNet CIFAR10 training recipe in the ResNet pa- per [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' For finetuning, the initial LR is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001, the minimum LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='0001, total epochs 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Effect of LR schedule of ResNet56 on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Base- line accuracy 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78%, Params: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='85M, FLOPs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='9 Sparsity/Speedup 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='45× 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='82%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='99× 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='57%/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='59× 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='39%/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='41× Scratch (Step LR) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20 Scratch (Cosine LR) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='06 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='43 L1 [42] (Step LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='06 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 L1 [42] (Step LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='06 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='31 L1 [42] (Cosine LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='04 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 L1 [42] (Cosine LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='06 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='00 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20 The results of ResNet56 (on CIFAR10) and VGG19 (on CIFAR100) are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 6 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As seen, 13This said, the severity of the trainability issue may vary up to specific pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', we have observed the GReg method [79] is less bothered by such damaged trainability due to their growing regulariza- tion design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 14https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='lr scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='CosineAnnealingLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='html 15 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Effect of LR schedule of VGG19 on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Baseline accuracy: 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='02%, Params: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='08M, FLOPs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='80G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='9 Sparsity/Speedup 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23× 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='97× 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='87%/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='60× 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='98%/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='84× Scratch (Step LR) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='25 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='08 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='37 Scratch (Cosine LR) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='24 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10 L1 [42] (Step LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='05 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='12 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='02 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 L1 [42] (Step LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='14 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='04 L1 [42] (Cosine LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='08 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='08 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='09 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='15 L1 [42] (Cosine LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='07 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='21 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='11 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Top-1 accuracy (%) comparison of different setups of L1-norm pruning [42] with ResNet34 on ImageNet100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Prun- ing ratio: 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This table shows, the performance gap between a smaller LR and a larger LR is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' It can be closed (or squeezed) simply by training more epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' The root cause that a smaller LR appears to under-perform a larger LR is simply that the model trained by the smaller LR does not fully converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Finetuning setup Top-1 acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (%) Trainability acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' (%) P90F30, 1e-2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='23 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='83 P90F30, 1e-4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='26 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='93 P90F30, 1e-4 (+60 epochs) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='17 / P90F30, 1e-4 (+270 epochs) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='16 / P90F30, 1e-4 (+1485 epochs) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 / the advantage of initial LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='01 over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='001 does not only appear with the Step LR schedule, but also appears with the Cosine LR schedule in most cases (esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' for VGG19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This implies the finetuning LR effect is generic, not limited to one particular LR schedule, which further highlights the importance of the topic we have been studying in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Additional results 1e-4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1e-2 for P90F30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5 of the paper, we men- tion the performance gap between a small LR and a large LR is not fundamental, but a simple consequence of conver- gence speed under different LRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' When we add more fine- tuning epochs, the performance gap can be closed fully or by a large part for P30F90 and P60F60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' while on P90F30, the gap is still obvious even after we add 270 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Here we add even more, 1485 epochs, so that the num- ber of the 1st LR stage is now 1500 epochs, exactly 100 times of the 1st-LR-stage epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=', 15 epochs) when using 1e-2 as initial LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' As we see, now LR 1e-4 can achieve 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='18 top-1 accuracy, which is significantly better than 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='20 achieved by LR 1e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' This is yet another strong piece of evidence to show that the seeming performance gap between a large LR and a small LR is never a gap that can- not be bridged, further confirming our opinion in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' More learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 1 of the paper, we present the learning curves for P60F60, 1e-3 without and with more finetuning epochs, to show the underperformance of a small LR is actually due to insufficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Here we present more plots for P30F90, 1e-2 and P90F30, 1e-4 – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' In both cases (note the red crosses × in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='b) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='b)), the performance can be boosted by adding more epochs to the 1st LR stage, especially for the case of P90F30, 1e-4, where the 1st LR decay is actually way too early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' These plots further confirm our opinion in the paper – the seem- ing underperformance of a small finetuning LR is not some- thing magic, but a simple consequence of slow convergence (caused by the broken trainability, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' at large pruning ra- tios like 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 16 0 20 40 60 80 Epoch 40 60 80 Test accuracy (%) ResNet34, ImageNet100, PR 95% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='a) P30F90, 1e-2 0 25 50 75 100 125 150 Epoch 40 60 80 Test accuracy (%) ResNet34, ImageNet100, PR 95% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='b) P30F90, 1e-2 (+60 epochs) 0 5 10 15 20 25 30 Epoch 10 20 30 Test accuracy (%) ResNet34, ImageNet100, PR 95% (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='a) P90F30, 1e-4 0 250 500 750 1000 1250 1500 Epoch 20 40 60 80 Test accuracy (%) ResNet34, ImageNet100, PR 95% (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='b) P90F30, 1e-4 (+1485 epochs) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' epoch during finetuning of the setting P30F90, 1e-2 and P90F30, 1e-4 at pruning ratio 95% in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Red vertical lines mark the epoch of decaying LR by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' Partic- ularly note before the 1st LR decay, the accuracy keeps arising in (a), implying the 1st LR decay may be too early – this is confirmed in (b), where the red cross marker (×) indicates the time point of the 1st LR decay in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE4T4oBgHgl3EQfsw2X/content/2301.05219v1.pdf'} diff --git a/kNFIT4oBgHgl3EQfqyvG/content/2301.11329v1.pdf 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Elekes_Saur_2022_TauBooB +©ESO 2023 +January 13, 2023 +Space environment and magnetospheric Poynting fluxes of the +exoplanet τ Boötis b +F. Elekes1 and J. Saur1,2 +1 Institute of Geophysics and Meteorology, University of Cologne, Pohligstr. 3, D-50969 Köln +e-mail: f.elekes@uni-koeln.de +2 e-mail: jsaur@uni-koeln.de +Accepted: January 8, 2023 +ABSTRACT +Context. The first tentative detection of a magnetic field on the Hot Jupiter type exoplanet τ Boötis b was recently reported by Turner +et al. (2021). The magnetic field was inferred from observations of circularly-polarized radio emission obtained with the LOFAR +telescopes. The observed radio emission is possibly a consequence of the interaction of the surrounding stellar wind with the planet’s +magnetic field. +Aims. We aim to better understand the near space environment of τ Boötis b and to shed light on the structure and energetics of its +near-field interaction with the stellar wind. We are particularly interested in understanding the magnetospheric energy fluxes powered +by the star-planet interaction and in localizing the source region of possible auroral radio emission. +Methods. We perform magnetohydrodynamic simulations of the space environment around τ Boötis b and its interaction with the +stellar wind using the PLUTO code. We investigate the magnetospheric energy fluxes and effects of different magnetic field orienta- +tions in order to understand the physical processes which cause energy fluxes that may lead to the observed radio emission given the +proposed magnetic field strength in Turner et al. (2021). Furthermore we study the effect of various stellar wind properties, such as +density and pressure, on magnetospheric energy fluxes given the uncertainty of extrasolar stellar wind predictions. +Results. We find in our simulations that the interaction is most likely super–Alfvénic and energy fluxes generated by the stellar wind– +planet interaction are consistent with the observed radio powers. Magnetospheric Poynting fluxes are of the order of 1–8 ×1018 W for +hypothetical open, semi-open and closed magnetospheres. These Poynting fluxes are energetically consistent with the radio powers +in Turner et al. (2021) for a magnetospheric Poynting flux–to–radio efficiency > 10−3 when the magnetic fields of the planet and +star are aligned. In case of lower efficiency factors the magnetospheric radio emission scenario is according to the parameter space +modeled in this study not powerful enough. A sub–Alfvénic interaction with decreased stellar wind density could channel Poynting +fluxes on the order of 1018W towards the star. In case of a magnetic polarity reversal of the host star from an aligned to anti-aligned +field configuration, expected radio powers in the magnetospheric emission scenario fall below the observable threshold. Furthermore +we constrain the possible structure of the auroral oval to a narrow band near the open-closed field line boundary. Strongest emission +is likely to originate from the night side of the planet. More generally, we find that stellar wind variability in terms of density and +pressure does influence magnetospheric energy fluxes significantly for close-in magnetized exoplanets. +Key words. Magnetohydrodynamics (MHD) – Methods: numerical – Planet-star interactions – Planets and satellites: aurorae +1. Introduction +Recently, tentative measurements of auroral radio emission from +the hot Jupiter exoplanet τ Boötis b were obtained with the Low +Frequency Array (LOFAR) (Turner et al. 2021). These obser- +vations might be considered the strongest evidence so far of an +intrinsic magnetic field on a planet outside the solar system if the +emission indeed originates from the planets vicinity. Then they +also imply that τ Boötis b possess a magnetosphere which inter- +acts with its surrounding stellar wind. The radio observations by +Turner et al. (2021), if confirmed, thus help pave the way for the +field of extrasolar space physics. In this work we therefore use +properties from the observed radio signals to derive new con- +straints on the space environment around τ Boötis b. +The massive Hot Jupiter τ Boötis b (Butler et al. 1997) is a very +good candidate to remotely observe a powerful interaction of a +stellar wind with an exoplanet’s magnetosphere for several rea- +sons: With ∼16 pc, the τ Boötis system is relatively close to the +solar system. The planet orbits its host star τ Boötis A at a short +distance of 0.046 astronomical units (Butler et al. 1997). Addi- +tionally, its large mass (> 5MJupiter) may cause its exobase to +remain close to the planet, leading to a magnetosphere (MS) not +completely filled with dense plasma and thus allowing for ra- +dio emission to be produced efficiently and to escape the planets +vicinity (Weber et al. 2018; Daley-Yates & Stevens 2018). +The tentative radio measurements obtained with LOFAR com- +prise two signals allegedly from the vicinity of τ Boötis b (Turner +et al. 2021). The circularly polarized signals were detected in +the 21 – 30 MHz and 15 – 21 MHz frequency bands, respec- +tively. The emission possibly originates from gyrating, ener- +getic electrons precipitating towards the planetary polar regions, +emitting radio waves generated through the Electron Cyclotron +Maser Instability (ECMI), which is expected to be the domi- +nant mechanism for exoplanetary radio emission (Zarka 1998; +Treumann 2006). From these signals the planetary magnetic field +strength can be inferred directly since the emission frequency +corresponds to the local electron gyro-frequency. The existing +observations are consistent with the expectations on the emit- +ted power from the radio-magnetic Bode’s law (Zarka et al. +2001; Zarka 2007; Zarka et al. 2018), on the polarization (e.g. +Article number, page 1 of 19 +arXiv:2301.05015v1 [astro-ph.EP] 12 Jan 2023 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +circular polarization; Zarka 1998; Grießmeier et al. 2005) and +on the frequency (i.e. slightly above Earth’s ionospheric cutoff; +Grießmeier et al. 2007b, 2011; Griessmeier 2017). The mea- +sured radio signal however needs multi-site follow up observa- +tions preferably at various radio wavelengths to confirm and to +further constrain the magnetic field environment of τ Boötis b +(Turner et al. 2021). +In addition to radio emission (e.g. Grießmeier 2015; Farrell +et al. 1999; Zarka et al. 2001; Zarka 2007) other indirect meth- +ods have been proposed to detect and constrain the magnetic +fields of exoplanets. These are, for example, optical signatures +in the stellar chromosphere by means of Ca II H&K line excess +emission induced by star-planet interactions (SPI) (Cuntz et al. +2000; Cauley et al. 2019; Shkolnik et al. 2003, 2005, 2008), +and asymmetries in near-ultraviolet stellar light curves together +with UV absorption signatures caused by transiting planetary +bow shocks (Vidotto et al. 2010, 2011; Llama et al. 2011). The +SPI and transit related observations can lead to false-positives +(e.g. Turner et al. 2021; Kislyakova et al. 2016; Preusse et al. +2006; Lai et al. 2010; Kopp et al. 2011; Miller et al. 2012, +2015; Bisikalo et al. 2013; Alexander et al. 2016; Turner et al. +2016; Gurumath et al. 2018; Route 2019) due to sets of model +assumptions involved in the process. Radio observations in con- +trast, can directly constrain the magnetic field amplitude and are +therefore less susceptible to false–positives (Grießmeier 2015). +The success of radio observations has been demonstrated in the +past in the solar system. For example, Jupiter’s magnetic field +was discovered through radio observations (Franklin & Burke +1958) before spacecraft confirmed it with in-situ magnetometer +measurements. +Since τ Boötis b may be the first exoplanet with a directly +observed magnetic field it provides an unique opportunity +to constrain the space environment around this exoplanet. +However, various properties of τ Boötis b are unknowns such as +radius, size and extend of its atmosphere above the 1 bar level +as well as stellar wind parameters. τ Boötis A is a solar-like F7 +IV-V star (Gray et al. 2001), coronal temperature and pressure +might therefore be comparable to the sun. The coronal base +density and consequently the stellar wind mass loss rate poses +the most uncertain free parameter of previous studies of the +stellar wind from τ Boötis A (Vidotto et al. 2012; Nicholson +et al. 2016). Recently new constraints on stellar winds of M +dwarf stars were reported utilizing astrospherical absorption +signatures induced by the interaction of the stellar wind with the +interstellar medium (Wood et al. 2021). The question naturally +arises if stellar wind – planet interactions might as well produce +observable signatures capable to provide constrains on stellar +wind properties such as density (i.e. mass loss rate) or pressure +(i.e. temperature), which will be addressed in this paper. +The proximity of Hot Jupiters to their host stars can potentially +cause sub–Alfvénic star – planet interactions, which is expected +to produce observable signatures in the stellar (e.g. chromo- +spheric emission Ca II H & K line emission Shkolnik et al. 2003, +2008; Cauley et al. 2019) or the planetary atmosphere (e.g. au- +roral radio emission Cohen et al. 2018; Turnpenney et al. 2018; +Bastian et al. 2022; Kavanagh et al. 2021, 2022). Such magnetic +SPI in exoplanetary systems were excessively studied by means +of magnetohydrodynamic simulations (e.g. Preusse et al. 2006, +2007; Zhilkin & Bisikalo 2020; Varela et al. 2018, 2022), +partly with a focus on the far–field interaction incorporating +self–consistent stellar wind models (e.g. Strugarek et al. 2014, +2019a,b; Cohen et al. 2011; Cohen et al. 2014; Vidotto et al. +2015; Vidotto & Donati 2017). Modeling of super-Alfvénic +stellar wind – magnetosphere interactions was sparsely done to +our knowledge although a large number of close-in exoplanets +might be exposed to, at least temporarily, super-Alfvénic wind +conditions (Zhilkin & Bisikalo 2019) (i.e. orbiting outside the +Alfvén surface). In case of super-Alfvénic or, more precisely, +super-fast magnetosonic stellar wind flows a bow shock forms +upstream of the planetary magnetosphere because of the flow +being faster than the fastest magnetohydrodynamic wave mode. +In this case the planet does not interact with the star since waves +are not able to propagate upstream. This might be the case for τ +Boötis b, as the planet is likely exposed to a super–fast stellar +wind according to Nicholson et al. (2016). +The generation of radio emission from exoplanets, its properties +and dependence on stellar wind and planetary parameters was +studied intensively using numerical simulations (Nichols & +Milan 2016; Varela et al. 2016, 2018; Turnpenney et al. 2020; +Daley-Yates & Stevens 2018; Kavanagh et al. 2020) for other +or generic exoplanets. However, little to no emphasis was +given to studying the detailed spatial structure and energetics +of magnetospheric Poynting fluxes that ultimately deliver the +available electromagnetic energy capable of driving planetary +auroral emissions at various wavelengths. +In order to better understand the space environment around τ +Boötis b we perform magnetohydrodynamic (MHD) simulations +of the near space environment of τ Boötis b and its magnetic +field interacting with the surrounding stellar wind plasma using +the PLUTO code. The stellar wind model is based on wind sim- +ulations (Vidotto et al. 2012; Nicholson et al. 2016) driven by +magnetic surface maps derived from magnetic measurements of +τ Boötis A (Marsden et al. 2014; Mengel et al. 2016; Jeffers et al. +2018). The magnetic field estimate of the planet’s intrinsic field, +based on the tentative magnetic field strengths derived by Turner +et al. (2021), is used to model the planetary magnetosphere. We +specifically aim to better understand the magnetospheric energy +fluxes around τ Boötis b and of Hot Jupiter type exoplanets in +general that are exposed to similar stellar wind conditions. We +also address the question on how stellar wind variability in the +time-independent case affects magnetospheric Poynting fluxes +and therefore possible radio powers generated by the interaction. +The paper is structured in the following way: An overview +of the physical model to describe the plasma interaction τ +Boötis b with the surrounding stellar wind is given in Sect. 2. +The numerical setup is summarized in Sect. 2.1, details about +the stellar wind model can be found in Sect. 2.2. The τ Boötis +b model is described in Sect. 2.3. In the subsequent Sect. 3 +we show our results, starting with a general description of +the interaction in Sect. 3.1, followed by a study of the spatial +structure of Poynting fluxes in Sect. 3.2.1. Then we discuss +the energetics of the interaction in Sect. 3.2.2, where we also +compare possible radio emission output with the observations +by Turner et al. (2021). The results are followed by a discussion +about the role and importance of the stellar wind to power the +energy fluxes in the magnetosphere of the exoplanet in Sect. +4.1. At last, we discuss possible auroral radio emission and its +detectability in scope of stellar wind variability (Sect. 4.2). +2. Numerical simulation +In this section we introduce our physical model and the numerics +to describe the interaction of τ Boötis b and its intrinsic magnetic +field with its surrounding stellar wind. The MHD model together +with the numerical model and coordinate system are presented in +Sect. 2.1. We introduce the stellar wind that is included as bound- +ary condition for the plasma variables in Sect. 2.2 followed by +Article number, page 2 of 19 + +F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +the description of parametrizations of physical processes intro- +duced by the planet and its atmosphere in Sect. 2.3. +2.1. Method +We performed single-fluid ideal, non-resistive and non-viscous +MHD simulations using the open-source code PLUTO (v. 4.4) +in spherical coordinates (Mignone et al. 2007). The MHD equa- +tions to solve are +∂ρ +∂t + ∇ · �ρv� += +Pmn − Lmp +(1) +∂ρv +∂t + ∇ · +� +ρvv + p − BB + 1 +2 B2 +� += +−(Lmp + νnρ)v +(2) +∂Et +∂t + ∇ · �(Et + pt)v − B(v · B)� += +−1 +2(Lmp + νnρ)v2 +−3 +2(Lmp + νnρ) p +ρ ++3 +2(Pmn + νnρ)kBTn +mn +(3) +∂B +∂t − ∇ × [v × B] += +0 , +(4) +where ρv is the momentum density, v the velocity, ρ the mass +density, pt the total pressure (e.g. magnetic and thermal) and p +the thermal pressure. B is the magnetic flux density, −v×B in Eq. +4 is the electric field in the ideal limit with infinite electrical con- +ductivity. Et is the total energy density, Et = ρe+ρv2/2+B2/2µ0, +and e the specific internal energy. The system is closed by the +equation of state in the form p = ρe(γ − 1), where γ is the ratio +of specific heats for the adiabatic case. +As for magnetic diffusion we do not include a diffusion term in +the induction equation (Eq. 4) but point out that numerical dif- +fusion, especially for coarse grids such as in our simulation, in- +troduce numerical diffusion sufficient to allow for reconnection +(see Varela et al. 2018). To justify this assumption we performed +test simulations incorporating magnetic diffusion and found it to +not influence the results of this paper significantly (see appendix +B for a detailed discussion on this topic). +We include plasma production, P, and loss terms, L, (Eqs. 1-3) +to account for photo-ionization, dissociative recombination to- +gether with associated momentum and internal energy transfer +between neutral atmospheric and magnetospheric plasma parti- +cles as well as ion-neutral collisions. We note that the neutral +species is not simulated and altered by the interaction with the +ion species. Details on how plasma production and loss are mod- +eled can be found in Sect. 2.3. The mass of plasma particles is +denoted by mp and mn describes the mass of neutral particles. +We assume the plasma to completely consist of ionized hydro- +gen atoms, mp = mH+. The atmosphere only consists of neutral +molecular hydrogen, mn = mH2. +The conservative form of Eqs. 1 – 4 are integrated using a ap- +proximate hll-Riemann solver (Harten, Lax, Van Leer) with the +diffusive minmod limiter function. The ∇ · B = 0 condition was +ensured by the mixed hyperbolic–parabolic divergence cleaning +technique (Dedner et al. 2002; Mignone et al. 2010). +The spherical grid consists of 256 non–equidistant radial, 64 and +128 equally spaced angular grid cells in θ and φ dimension re- +spectively. The radial grid is divided into three regions. From 1 +to 1.2 planetary radii (Rp) the grid contains 10 uniform cells. Af- +ter that from 1.2 to 12 Rp the next 150 cells increase in size with +a factor of ∼ 1.01 per cell. The last 96 cells from 12 Rp towards +the outer boundary at 70 Rp increase gradually with a factor of +∼ 1.02. The positive x axis points parallel to the relative veloc- +ity v0 of the stellar wind in the frame of the planet. The stellar +wind magnetic field is assumed to be perpendicular to v0 and is +anti-parallel to the z axis. The y axis completes the right handed +coordinate system. Co-latitude θ is measured from the positive z- +axis, longitudes Φ are measured from the positive y axis within +the xy plane. The origin is located at the planetary center. We +run all simulations for approximately 3.6 h physical time until a +quasi steady-state is reached in the vicinity of the planet (r < 30). +Small fluctuations cannot be avoided although larger scale struc- +ture and dynamics within the MS remain already almost constant +after approximately 2 hours physical time. +2.2. Stellar wind model +The derived stellar wind parameters from Nicholson et al. (2016) +resemble those of the sun, such as the polytropic index, γ = 1.1 +(Van Doorsselaere et al. 2011), and the stellar coronal base tem- +perature, which is not well constrained by observations, is set +to 2 × 106 K as typical value for the solar coronae (Nicholson +et al. 2016; Vidotto et al. 2012; Van Doorsselaere et al. 2011). +The magnetic field of τ Boötis A was studied excessively dur- +ing several epochs and magnetic surface maps as well as several +magnetic polarity reversals were observed (Donati et al. 2008; +Fares et al. 2009, 2013). The coronal base density remains an ed- +ucated guess based on a comparison of emission measure (EM) +values obtained from X-Ray spectra of τ Boötis A (Vidotto et al. +2012; Maggio et al. 2011). Due to the uncertainty of the base +density estimate, different stellar wind densities will be investi- +gated separately in the scope of magnetospheric Poynting fluxes +and possible radio powers in Sects. 3.2.3 and 4.1. +The stellar wind is applied through constant in–flow boundary +conditions at the upstream hemisphere (Φ = 0 to 180◦). The +magnetic field is assumed to be perpendicular to the relative ve- +locity v0 of the wind (i.e. parallel to the negative z axis). The +in–flow velocity of the plasma, which we call the relative ve- +locity v0, is parallel to the x–axis and is composed of the radial +velocity of the wind vsw and the orbital velocity of the planet. +The adopted plasma parameters of the wind are summarized in +Table 1 which were averaged over the several epochs studied by +Nicholson et al. (2016). +2.3. τ Boötis b model +We assumed a radially symmetric neutral atmosphere with a +scale height of H = 4373 km. Thus, the scale height extends +over three radial grid cells and consequently the neutral atmo- +sphere is sufficiently resolved within the numerical grid. We as- +sume an atmosphere consisting of molecular hydrogen as it is, +followed by helium, the most abundant constituent of the Jovian +atmosphere (Atreya et al. 2003). The collisional cross-section is +assumed to be σin = 2 × 10−19 m2 for H+ – H2 collisions with +momentum transfer for low–eV relative velocities between the +colliding particles (Tabata & Shirai 2000). In our simulations +the collision frequency is νin ≈ 0.5 s−1, so that νin = ¯vσinn(r), +where ¯v ≈ v0 denotes a typical velocity in the system and nn(r) +is the atmosphere number density as function of radial distance +from the center, +nn(r) = nn,0 exp +�Rp − r +H +� +, +(5) +where nn,0 = 8 × 1012 m−3 is the surface number density. Based +on test studies, we found that for nn,0 ≈ 8 × 1012 m−3 the ion- +Article number, page 3 of 19 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +Table 1: Physical simulation parameters. Details of the stellar wind and planet model are discussed in Sects. 2.2 and 2.3. +Symbol +Value +Source +Note +τ Boötis b +Planet radius +Rp +72875 km +Wang & Ford (2011) +Theoretical +Orbital period +Porb +3.31d +Butler et al. (1997); Wang & Ford (2011) +Semi-major axis +a +0.046 au +Butler et al. (1997) +Atm. surface density +nn,0 +8×1012 m−3 +– +Atm. scale height +H +0.06 Rp +– +3 radial grid cells +Magn. flux density (eq) +Bp +455µT +Turner et al. (2021) +Average of observations 1,2 +Basic Stellar wind model +Therm. pressure +psw +2.5 × 10−5 Pa +Nicholson et al. (2016) +Ion density +nsw +1.4 × 1012 m−3 +Nicholson et al. (2016) +Velocity +vsw +224.5 km s−1 +Nicholson et al. (2016) +Stellar frame of reference +Relative velocity +v0 +270.98 km s−1 +– +v0 = +� +v2sw + 4π2 × a2/P2 +orb +Magnetic flux density +Bsw +2.715 µT +Nicholson et al. (2016) +Alfvén Mach number +MA +5.36 +Fast mode Mach number +M f +1.9 +Plasma beta +β +8.31 +neutral collisions nearly completely bring the incoming plasma +flow to a halt in the atmosphere. This results in plasma pile up in +form of a shell around the planet. Increasing the density would +thus not produce a larger interaction. +We use a simplified description of photo-ionization. We neglect +the shadow zone exerted by the planet’s body and parameterize +plasma production through photo-ionization using only the ra- +dial dependence of the neutral atmosphere density, +P(r) = νionnn(r) . +(6) +The radial symmetric ionization partially mimics some night +side ionization through electron impact ionization. For the +photo-ionization frequency of hydrogen exposed to a solar-like +UV radiation environment at a distance of approximately 0.046 +AU from the star we take the value from Kislyakova et al. (2014), +νion = 6 × 10−5 s−1. +Plasma loss is introduced through recombination of hydrogen +ions. The loss term therefore depends on the plasma density, +L(r, t) = αn(r, t)(n(r, t) − nsw) . +(7) +Plasma loss is switched off if the plasma density falls below +the background density (i.e. n(r, t) ≤ nsw) as stellar wind ions +and electrons recombine significantly slower due to the higher +electron temperatures in the stellar wind. Given an electron tem- +perature of roughly Te ≈ 7500 K for a Hot Jupiter exoplanet’s +ionosphere with semi-major axis of 0.046 AU around a sun-like +star derived by Koskinen et al. (2010) and using the formula of +Storey & Hummer (1995), +α = 4 × 10−12 +�300K +Te +�0.64 +cm3 s−1 , +(8) +we find the hydrogen ion recombination rate, α, to be 5.1×10−19 +m3 s−1. Further discussion about the underlying assumption +about our atmosphere model can be found in appendix D. +Recent tentative auroral radio measurements from τ Boötis b +give a first observational constraint on its magnetic field strength. +Turner et al. (2021) found the polar surface magnetic flux density +Bp to lie between 7.5 and 10.7 G for two right-handed circularly +polarized signals. We assume a dipole field and adopt the average +value of both Stokes V+ signals (Turner et al. 2021), Bp = 9.1 +G, for our simulations. Furthermore we study the effect of dipole +orientation on the stellar wind – planet interaction through simu- +lating an open (0◦ tilt), semi-open (90◦ tilt) and closed MS (180◦ +tilt), where the tilt is measured with respect to the negative z +axis. The various tilts are realized by rotating the stellar back- +ground magnetic field accordingly so that the planetary dipole +axis is always parallel to the z-axis. Given the strong magnetic +variability of τ Boötis A (e.g. several magnetic polarity reversals +were observed as well as a chromospheric activity cycle in terms +of S–indices of roughly 240 days Donati et al. 2008; Fares et al. +2009, 2013; Mengel et al. 2016; Mittag et al. 2017; Jeffers et al. +2018) we are also able to study the effect of the host star’s mag- +netic field topology on the stellar wind–planet interaction and +associated magnetospheric energy fluxes. +The magnetic field is implemented using the insulating– +boundary method by Duling et al. (2014) which ensures that +no radial electric currents exist within the insulating boundary, +which we assume to be the planet’s neutral atmosphere below its +ionosphere. +3. Results +In this section we first present results of our modeling which pro- +vides an overview of the space plasma environment of τ Boötis b +(Sect. 3.1). Then we study in detail the magnetospheric Poynting +fluxes in Sect. 3.2. +3.1. Structure of the interaction +The simulated plasma velocities and pressures according to the +basic model (Table 1) are displayed in Fig. 1 for the open +(θB = 0◦), semi-open MS (θB = 90◦) and closed MS (θB = 180◦) +case. The magnetic field tilt θB is the angle between the external +field (parallel to the z axis) and the planet’s magnetic moment. +We note that, due to the symmetries chosen in our model studies, +the stellar wind and intrinsic magnetic field are not inclined with +respect to the z-axis, therefore we also show projected field lines +(black solid lines) in the xz-plane. Color contours denote plasma +pressure in µPa (right colorbar). Arrows represent velocity com- +ponents, their magnitudes are color coded (left color bar). The +length of arrows indicate the magnitudes of the shown compo- +Article number, page 4 of 19 + +F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +Fig. 1: Velocity fields (colored arrows, left colorbars) and plasma pres- +sure (color contours, right colorbars) in the xz-plane for the open MS +(θB = 0◦, top), semi-open (θB = 90◦, middle) and closed (θB = 180◦, +bottom) MS case. Projected stellar wind magnetic field lines are indi- +cated as black solid lines within the xz-plane parallel to the ambient +magnetic field. Closed and open magnetospheric field lines are colored +in magenta. +nents. Spatial dimensions are given in units of planetary radii. +The intrinsic magnetic field and its corresponding MS poses an +obstacle to the stellar wind flow coming from negative x – di- +rection. The flow outside the MS is super-Alfvénic (MA = 5.36) +and super-fast magnetosonic (Mf = 1.6) (see Table 1), where +M f = v0/(v2 +A + c2 +s)1/2, with the sound speed cs = +� +γpsw/ρsw, +polytropic index γ = 1.1 (Nicholson et al. 2016) and Alfvén ve- +locity vA = Bsw/ √µ0ρsw. The super-fast interaction enforces a +bow shock to be formed roughly 5Rp in front of the planet fol- +lowed by a fairly thick magnetosheath. Since no wave is able +to propagate upstream, the stellar wind plasma is unperturbed +until the bow shock. The structure of the MS strongly depends +on the internal field orientation as visible in Fig. 1 with an in- +crease of overall MS size towards higher magnetic axis tilts. +For the open and semi-open MS (Fig. 1 top and middle plot re- +spectively) two magnetic lobes form, separated by a thin plasma +sheet, where open magnetic field lines connect to the stellar wind +field several planetary radii downstream (not shown in the plots). +The day side magnetopause, defined by the location of the last +closed field line, lies between 3 and 3.5 Rp, while the night side +magnetopause is located at roughly 5 Rp for the open and semi- +open MS respectively. The downstream side magnetopause is +very narrow in the z – direction as expected due to the mag- +netic field lines convected downstream together with the stel- +lar wind flow and due to the magnetic stresses stretching the +magnetic field. The closed MS case (Fig. 1, bottom plot) has a +night side magnetopause lying several planetary radii (∼ 17Rp) +downstream (not shown in the plots). While the day side mag- +netopause is controlled by the stellar wind thermal and magnetic +pressure balanced with those exerted by the planet’s surround- +ings, the night side MS is influenced by reconnection (i.e. the +merging of planetary with stellar wind field lines). Magnetic re- +connection is most efficient for a magnetic moment parallel to +the ambient field (here the z-axis), therefore the fraction of open +planetary field lines connected to the star decreases significantly +with an intrinsic field moment being directed anti-parallel to the +stellar field. As the stellar wind plasma primarily penetrates the +MS along magnetic field lines, the amount of plasma and ther- +mal pressure decreases as well with increasing magnetic axis tilt. +We note that, as visible in Fig. 1 (bottom), the magnetosphere is +completely closed. This is due to the perfect anti–parallel align- +ment of the planetary and stellar wind magnetic field. +Within the MS the flow velocity is strongly reduced and has +weak upstream components in the negative x direction due to +magnetic tension exerted on planetary field lines. Magnetic re- +connection takes place at the upstream and downstream side +where velocities, both within and outside the MS, are strongly +enhanced due to acceleration through released magnetic energy. +Velocities are slightly larger at the flanks of the MS compared +to the upstream side and exceed the initial stellar wind velocity +at the downstream side where stellar wind as well as planetary +field lines merge together again and accelerate the plasma. +Thermal pressures are strongly enhanced within the magne- +tosheath, where stellar wind plasma is decelerated abruptly and +compressed, so that kinetic energy is converted into heat. Plasma +may penetrate the MS along open magnetic field lines in the +polar cusps where pressure is enhanced as well. The cusps act +as channels for plasma transport into the MS. There is a trend +towards lower pressures in the cusps for increasing magnetic +axis tilt. This is directly connected to the amount of stellar wind +plasma advected towards the planet as the amount of injected +plasma is related to the ability of magnetic field lines to merge +with the ambient field. This becomes increasingly difficult for +planetary magnetic moments having components anti-parallel to +the ambient field, therefore the area fraction of open magnetic +field lines and thus the size of the plasma injection channel is +maximal for a completely open MS. Here pressures up to 160 +µPa can be reached while the closed MS case shows pressures +up to roughly 90 µPa. +Article number, page 5 of 19 + +400 +Open magnetosphere +160 +8 +350 +6 +140 +300 +4 +120 +250 +2 +100 +v [km/s] +[Ro] +[edn] +0 +200 +80 +N +d +-2 +150 +60 +-4 +100 +40 +-6 +-50 +20 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x [Ro]400 +Semi-open magnetosphere +160 +8 +350 +6 +140 +300 +4 +120 +250 +2 +100 +v [km/s] +[Ro] +[edr] +0 +200 +80 +N +p +-2 +150 +60 +-4 + 100 +40 +-6 +- 50 +20 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x [Ro]400 +Closed magnetosphere +160 +8 +350 +6 +140 +300 +4 +120 +250 +2 +100 +v [km/s] +[Ro] +[μPa] +0 +200 +80 +N +p +-2 +150 +60 +-4 +-100 +40 +-6 +50 +20 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x [Ro]A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +Fig. 2: Mercator projections of the Poynting flux (upper row), plasma velocity (middle row) and absolute values of Poynting flux components +parallel to the unperturbed planetary field plus small background field from stellar wind (bottom row). The results are shown at an altitude of +one planetary radius above the surface. The left column displays maps for the open MS case (θB = 0◦), the right column for the closed MS case +(θB = 180◦). Arrows indicate normalized angular components, color contours denote radial components. Red dashed lines indicate the location of +the OCFB. +3.2. Poynting fluxes and aurorae +We are interested in understanding the electromagnetic coupling +of the stellar wind with the magnetosphere of the exoplanet, its +atmosphere and ionosphere. The energy fluxes associated with +these electromagnetic coupling processes provide the energet- +ics for the auroral emission from the exoplanet’s magnetosphere +at radio and other wavelengths. Therefore we study the Poynt- +ing flux to calculate the maximum available electromagnetic en- +ergy fluxes. We describe the spatial structure of magnetospheric +Poynting fluxes in Sect. 3.2.1. Then we study the energetics of +the interaction and effects of stellar wind variability on magne- +tospheric energetics in the subsequent sections 3.2.2 and 3.2.3. +3.2.1. Spatial structure +We first describe the spatial structure of the Poynting fluxes and +plasma velocities within the MS as displayed in the top and mid- +dle row of Fig. 2 respectively. The plots show Mercator projec- +tions of the angular vector components over a spherical shell +with radius 2Rp. The angle θ denotes the co-latitude, and φ the +longitude. Arrows indicate the orientation of vectorial quanti- +ties and color contours their magnitudes. Positive values indicate +radial components pointing away from the planet. Red dashed +lines represent the open – closed planetary field line boundaries +(OCFB). Magnetic field lines with both foot points on the plan- +etary surface are closed field lines. Each field line having only +one foot point on the planet is an open field line. The OCFB sep- +arates areas with open from areas with closed field lines. Thus, +the OCFB also represents the magnetopause at that specific ra- +dial location. The open and closed MS cases are shown left and +right, respectively. +The Poynting vector S can be rewritten in the ideal MHD case +using the convective electrical field (e.g. Saur et al. 2013) +S = E × B +µ0 += −v × B × B +µ0 += B2 +µ0 +v⊥ , +(9) +which is bodily carried by the plasma flow perpendicular to +magnetic field lines, denoted by the perpendicular velocity v⊥. +The Poynting flux describes the transport of magnetic enthalpy, +which is a factor of two larger than the magnetic energy density +B2/2µ0 (e.g. Saur et al. 2013). In the remainder of this work +we mostly present Poynting fluxes, but need to consider the +factor of two when we compare magnetic energy densities with +Article number, page 6 of 19 + +Poynting flux S +0 +60 +25 +50 +40 +[W/m²] +75 +20 +100 +0 +S +125 +-20 +150 +175 +-40 +0 +50 +100 +150 +200 +250 +300 +350 +[。] ΦPoynting flux S +0 +25 +10 +50 +5 +75 +0 +100 +S +125 +150 +-10 +175 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma velocity v +0 +0.2 +25 +50 +0.1 +75 +[km/s] +0.0 +100 +> +125 +-0.1 +150 +175 +-0.2 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma velocity v +0 +0.2 +25 +50 +0.1 +75 +v [km/s] +0.0 +e +100 +125 +-0.1 +150 +175 +-0.2 +0 +50 +100 +150 +200 +250 +300 +350 +[。] ΦPoynting flux iS ·: eBo +0 +10 +25 +8 +50 +[W/m2 ] +75 +6 +eBo! +e +100 +4 +125 +S +150 +2 +175 +0 +0 +50 +100 +150 +200 +250 +300 +350 +[。] ΦPoynting flux IS ·eBo +0 +10 +25 +8 +50 - +[zw/M +75 +6 +S ebo! +100 - +4 +125 +S +2 +150 +175 +0 +0 +50 +100 +150 +200 +250 +300 +350 +[。] ΦF. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +thermal (i.e. internal energy) or kinetic energy densities based +on their flows. +For both, the open and closed MS case, flow velocities are +strongly reduced at the upstream (φ = 0–180 degrees) and +downstream (φ = 180 – 360 degrees) side down to speeds below +10 km s−1. This occurs due to interaction with the intrinsic mag- +netic field and momentum transfer with the neutral atmosphere. +The OCFB is located at roughly θ ≈ 30◦ and 150◦ in the open +MS case. Due to the perfectly anti–parallel configuration of the +stellar wind and planetary magnetic field no open planetary field +lines exist in the closed MS model. This has also been observed +in sub-Alfvénic simulations using this field topology (Ip et al. +2004; Strugarek et al. 2015). +Open Magnetosphere model (Fig. 2 left): The very nar- +row vertical extend of the downstream closed field line region +of the open MS is caused by magnetic tension due to the +magnetized stellar wind. Highest velocities are found within +the open field line region mainly at the downstream side where +plasma is accelerated downstream through magnetic tension on +open lines. +Strong Poynting fluxes occur where plasma velocities have +strong components perpendicular to the magnetic field. They +are found within the open field line region mainly at the +downstream side with outward directed Poynting fluxes. Com- +paratively strong Poynting fluxes, but directed toward the planet, +are located on the upstream side near the magnetopause. Within +the closed field line region and especially near the equator +Poynting fluxes mostly vanish. +Closed Magnetosphere model (Fig. 2 right): In the closed MS +model highest velocities can be found near the planetary poles +confined to an area below 25 degrees co–latitude and similar +in the south. This high velocities are caused by tension on high +latitude closed field lines that are strongly stretched towards +the downstream side by the stellar wind and reach up to 17 +planetary radii. +Poynting fluxes oriented away from the planet are confined to +narrow bands encircling the high latitude polar regions between +40 and 80 degrees co–latitude and similar in the south. At the +upstream side Poynting fluxes vanish near the equatorial regions +due to plasma flow being mainly aligned with planetary field +lines. Inward oriented Poynting fluxes occur near the polar axis +slightly shifted towards the downstream side. +We now study the Poynting fluxes parallel to the unper- +turbed background magnetic field because in the solar system +magnetospheres they are considered the root energy fluxes +from which a small fraction can be converted into auroral +radio emission. Poynting fluxes provide the energy from which +wave–particle interaction can draw energy to accelerate elec- +trons (e.g. for Jupiter Hill 2001; Saur et al. 2021). The resulting +energetic electrons then can be subject to the electron maser +instability (Treumann 2006; Zarka 2007). The interaction of +the stellar wind with τ Boötis b’s magnetic field perturbs the +magnetic and electric field, which causes the Poynting fluxes. To +quantitatively assess the associated Poynting flux, we therefore +use the unperturbed magnetic background field B0 = Bp,0 + Bsw +(i.e. the initial dipole and stellar wind field) to calculate the +Poynting flux on this field, S · eB0. The unit vector eB0 points +in the direction of unperturbed magnetic field lines. These +projections give insight on where electromagnetic energy is +transported either through propagating magnetic disturbances +(i.e. Alfvén waves) or convection. The bottom row of Fig. 2 +Table 2: Integrated magnetospheric Poynting fluxes for different mag- +netic field topologies +Model +Dipole tilt +Pa [W]a +Pa||B0 [W]b +Open MS +0◦ +3.46e+18 +8.73e+17 +Semi-open MS +90◦ +1.77e+18 +4.88e+17 +Closed MS +180◦ +6.91e+17 +1.09e+17 +Notes. Integrated Poynting fluxes over a sphere with radius r = 2Rp. +(a) Magnetospheric Poynting flux (Eq. 11) +(b) Magnetospheric Poynting flux parallel to B0 (Eq. 12) +shows |S · eB0| for the open MS (left) and closed MS (right). We +note that only absolute values are shown in the plots in order to +clearly identify zero or near-zero power densities. +Strongest energy transport along unperturbed field lines occurs +over narrow bands encircling the polar open field line regions at +the flanks of the planet where velocities are nearly perpendicular +to the magnetic field as seen in Fig. 2. Moreover, the spatial +structure of Poynting fluxes along B0 is strictly symmetric with +respect to the equator (at θ = 90◦). A significant amount of +energy is transported parallel to the unperturbed field within +the polar open field line regions in the open MS case. Parallel +energy fluxes reach values up to 10 Wm−2 at the flanks of +the planet just outside the closed field line regions. Poynting +fluxes up to 9 Wm−2 are found at the downstream side, above +the OCFB. For both, open and closed MS model, strongest +convected energy can be found extensively in high latitude +regions due to high velocities perpendicular to the magnetic +field. Here the planetary field lines are most mobile in a sense +that they are bent over towards the downstream side by the +stellar wind. For the closed MS parallel Poynting fluxes up to +roughly 9 Wm−2 can be found directly at the planetary poles +slightly shifted towards the upstream side. At lower latitudes +parallel Poynting fluxes up to 6 Wm−2 are confined to narrow +bands at the flanks of the planet. Auroral emission is expected to +be strong where Poynting fluxes are large, hence near the OCFB +(e.g. mostly confined to the L = 3–3.5 shell at the upstream +side) and in the polar regions for both MS models. They vanish +completely along the equator. Generally said Poynting fluxes +are significantly weaker and confined to the small polar regions +for the closed magnetosphere model compared to the open MS +case. In the open MS model strong parallel Poynting fluxes +cover the whole open field line area with their maximum at +the flanks of the planet in contrast to the closed MS where the +regions of strongest parallel Poynting fluxes are partitioned into +smaller areas around the planetary poles. +3.2.2. Energetics of the interaction +To estimate the total available Poynting flux, which serves as the +root energy flux, we assume for simplicity that the radio emis- +sion is generated in a shell 1 Rp above the surface of the exo- +planet. This particular choice is inspired by the fact that radio +emission around Jupiter and other solar system planets arises +from altitudes about 1Rp (or larger) above the planet’s surface +(e.g. Zarka 1998; Hess & Zarka 2011) where strong electron ac- +celeration takes place (e.g. for Jupiter Mauk et al. 2020). Poynt- +ing fluxes within the magnetosphere of τ Boötis b only vary little +as function of distance from the planet (see appendix A for a dis- +cussion on the choice of r). +Available electromagnetic power for possible conversion into +electron acceleration and radio emission is given by the diver- +Article number, page 7 of 19 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +gence of the Poynting flux in this shell with Volume V, +Pa = +� +V +∇ · SdV = +� +Ashell +S · ˆndAshell , +(10) +where Ashell is the surface area of the shell and ˆn the surface +normal vector. To investigate the maximal Poynting flux which +can be dissipated in the shell we assume that the Poynting flux +entering the shell from above or below is dissipated within the +shell. For mathematical simplicity we further let the thickness of +the shell grow infinitesimally small such that +Pa = +� +Asphere +|Sr| dAsphere , +(11) +with Asphere the area of the sphere located at 2 RP from the cen- +ter. In physical terms it means that the possible dissipation in the +shell can be supplied with energy fluxes from below the shell +(i.e. coming from the planet’s ionosphere) or from above the +shell (i.e. coming from the magnetosphere or stellar wind). Ul- +timately, the energy flux is coming from the stellar wind, but +the energy flux can be reflected or converted in the ionosphere +and can be redirected away from the planet again. This inte- +grated Poynting flux serves as a proxy for maximum available +electromagnetic energy dissipated within an auroral acceleration +region. +Alternatively, we integrate the components of the Poynting flux +parallel to the unperturbed magnetic field, B0 = B − δB, where +δB denotes the magnetic field perturbation generated by the in- +teraction. These Poynting fluxes take into account the energy +flux of which a fraction can directly contribute to particle ac- +celeration and powering the ECMI driven emission, +Pa,|| = +� +A +|S · eB0| dA , +(12) +where eB0 is the unit vector pointing in direction of B0. We re- +fer to this Poynting flux component as the auroral Poynting flux. +As opposed to Eq. 11, Pa,|| (Eq. 12) serves as a more realistic +estimator for calculating auroral energy dissipation since Eq. 11 +includes significant contribution of convected energy which is +likely not converted into particle acceleration. Table 2 summa- +rizes integrated Poynting fluxes according to Eq. 11 (third col- +umn) for all three intrinsic magnetic field orientations. The 4th +column shows integrated Poynting fluxes along the unperturbed +field (Eq. 12). Integrated Poynting fluxes range from 3.5 × 1018 +down to 6.9×1017 W for the open towards the closed MS model. +Poynting fluxes along the unperturbed field (Eq. 12) amount to +∼ 9 × 1017 and 1017 W for the open and closed MS respectively. +The effect of magnetic topology on convected energy within the +magnetosphere is therefore significant as the powers differ by al- +most one order of magnitude. Magnetic stress due to the stellar +wind interaction can work on the magnetospheres less strongly if +the magnetosphere is closed, thus giving rise to weaker flows and +therefore weaker convected Poynting fluxes. The trend is simi- +lar for Poynting fluxes along B0, Pa,||, but here the powers are +reduced by almost an order of magnitude below the integrated +total Poynting fluxes Pa. +3.2.3. Influence of stellar wind variability on magnetospheric +energetics +For modeling the space environment of τ Boötis b, the properties +of its surrounding stellar wind carry very large uncertainties, in +particular the stellar wind density. In Nicholson et al. (2016) and +Fig. 3: Schematic illustrating the several steps from incident stellar +wind energy flux towards auroral radio emission. The transfer function +Ta (see Sect. 4.1.1) describes the conversion from incident stellar en- +ergy to auroral Poynting fluxes. The conversion efficiency from auroral +Poynting fluxes (Eq. 12) to radio emission, ϵa, implicitly includes the +efficiency of electron acceleration and the ECMI mechanism. The steps +within the pink shaded area are not included in our model. Brown ar- +rows indicate physical processes, blue arrows denote model parameters +quantifying energy conversion and the magenta arrow the radiometric +scaling law. +Vidotto et al. (2012), the coronal base density was estimated by +choosing the electron density so that it can reproduce electron +measure (EM) observations of τ Boötis A. The energy fluxes +within the MS are powered by and limited by the maximum in- +cident power of the stellar wind flow transferring onto the mag- +netospheric obstacle. Zarka (2007) found that the observed radio +power of solar system planets is nearly a constant fraction of the +incident kinetic and magnetic energy convected through the ob- +stacle’s cross section, πR2 +mp, where Rmp is the magnetospheric +stand-off distance of the magnetized planet. This energy is uti- +lized in perturbing the topology of the planets magnetic field +which in turn results in currents induced by changes in magnetic +flux. Therefore the incident power controls the energetics within +the MS. The magnetic Poynting flux, PB, and the kinetic energy +flux, Pkin, convected through the obstacle’s cross section can be +calculated as follows +Pkin += +1 +2ρ0v2 +0 · πR2 +mpv0 +(13) +PB += +B2 +sw +µ0 +· πR2 +mpv0 . +(14) +Additionally, the thermal energy flux should be considered as +well as it cannot be neglected for close-in orbits where stellar +wind temperature, T, pressure and density are high, +Ptherm = 3 +2nswkBT · πR2 +mpv0 , +(15) +with nsw being the stellar wind particle density and v0 denoting +the incident stellar wind velocity. +The magnetopause distance Rmp can be obtained from an equi- +librium between stellar wind and planetary ram (pram = ρswv2 +0), +magnetic (pB = B2/2µ0) and thermal pressure. Both, the magne- +tospheric thermal and ram pressures are considered negligible, +thus pram,sw + pB,sw + ptherm,sw = pB,pl, where the subscript sw +stands for stellar wind and pl for planet. The magnetopause dis- +tance (or magnetospheric stand-off distance) can then be calcu- +Article number, page 8 of 19 + +Magnetospheric emission scenario +Radiometric Bode law +Stellar wind incident energy flux +Auroral radio emission +- thermal +- kinetic +- Poynting flux +Transfer function T, +Auroral Poynting fluxes +Electron Cyclotron Maser Instability +Electron accelerationF. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +Fig. 4: Top: Analytically calculated incident kinetic (red dashed line), +Poynting (red dotted line) and thermal (red dashed–dotted line) en- +ergy fluxes convected through the magnetospheric cross section πR2 +mp +as function of stellar wind density at the position of τ Boötis b. The +pressure varies accordingly through p0 ∝ ρ0 (see Table 1 for the ba- +sic stellar wind model). The powers are calculated using Eqs. 13 - 15 +and 17. The red solid line represents the sum of the three energy com- +ponents. The Magnetosphere stand-off distance Rmp (blue solid line) is +calculated with the same set of ρ0 and p0 using Eq. 16. The stand-off +distance can be obtained from the right y-axis, given in terms of plane- +tary radii Rp. The magenta arrow indicates the convected power in the +limiting case where the plasma density in the astrosphere approaches +zero and the planet is only exposed to the stellar dipole field (see Sect. +4.1.2 for discussion of this case). Red crosses show simulated Poynt- +ing fluxes along the unperturbed magnetic field over a spherical shell +with radius 2Rp for the closed MS (Eq. 12). Red stars indicate simu- +lated Poynting fluxes for the closed MS. +Bottom: Transfer function Ta = Pa/Pi as a function of ρ0 and p0 (i.e. +the conversion from incident energy to auroral Poynting fluxes). The +scaling behavior according to power laws of Ta is indicated by spectral +indices above the lines. +lated from +Rmp = RpB1/3 +p +� +2µ0 +�1 +2ρswv2 +0 + psw +� ++ B2 +sw +�−1/6 +. +(16) +All parameters can be found in Table 1. The parameter v0 refers +to the relative velocity between the stellar wind and planet and +Bp to the planetary surface magnetic field at the equator. +A certain fraction of the total incident power, +Pi = Pkin + PB + Ptherm , +(17) +intersecting with the magnetopause can eventually be converted +for the generation of radio emission within the magnetosphere. +The fraction of total incident energy, ϵ, that may result in ra- +dio emission is expected to range from 10−5 to ∼ 3 × 10−3 +(i.e. Pradio = ϵPi) in the radiometric Bode law (see Fig. 3) +(Zarka 2007). We point out that various efficiencies for con- +verting incident energy flux into electromagnetic radiation are +discussed in the literature. For example, the efficiency of con- +version from magnetospheric, auroral Poynting fluxes to radio +emission, which accounts for the efficiency of electron acceler- +ation through wave-particle interaction and the efficiency of the +electron-cyclotron maser, should be separated from the generic +efficiency factor obtained from the radiometric Bode law (Zarka +2007) (see Fig. 3). For Jupiter’s radio emission the efficiency +for conversion from magnetospheric, auroral Ponyting fluxes to +radio emission is roughly 0.3 – 3×10−4 (Saur et al. 2021). We +denote this efficiency by ϵa (Fig. 3). +As the stellar wind density is the most uncertain parameter we +performed simulations with densities ranging from 0.05 · ρ0 to +100 · ρ0 (see Table 1 for the basic model). To get an understand- +ing on how stellar wind variability affects the structure of the +magnetosphere we show xz–plane slices similar to Fig. 1 for the +two extreme cases (0.05·ρ0 and 100·ρ0) in the appendix C (Fig. +C.1). We do not solve a self consistent stellar wind model but +instead follow the solar wind solution of Parker (1958) where +the solution of the solar wind velocity v(r) is independent of +the coronal base density nc,0. In this solution, stellar mass and +base temperature control v(r) and T(r), where r is the distance +from the sun. For simplicity of the parameter study of this sub- +section, we choose an isothermal approach and change the den- +sity together with the pressure p0 (and therefore T) according to +p0 ∝ ρ0 (see equation 15). We therefore keep the temperature +constant and consequently the velocity does not change accord- +ing to Parker (1958). Given the average stellar mass loss rate of +τ Boötis A of ˙M ≈ 2.3 × 10−12 M⊙ yr−1 estimated by Nicholson +et al. (2016) (see also our basic model, Table 1) the parameter +range of stellar wind densities considered in this parameter study +translates to mass loss rates between 1.15 × 10−13 M⊙ yr−1 and +2.3 × 10−10 M⊙ yr−1 since ˙M ∝ ρ0. +We integrate the Poynting flux along the unperturbed field over +a spherical shell with radius 2Rp (e.g. Eq. 12) in order to obtain +an understanding of how much incident energy flux is eventu- +ally converted to auroral Poynting fluxes. Resulting powers are +shown as red crosses and stars (open and closed MS respectively) +in Fig. 4 as a function of ρ0 (and p0). The simulated convected +energy fluxes follow the trend of incident energy flux estimates +(red solid line in Fig. 4) but are reduced to fractions of the total +incident energy flux, Pi, between 15 and 20 % for the open MS +and between 1 and 5 % for the closed MS. Changes to the stellar +wind density ρ0 (and in the same manner p0) affect the incident +power inflicted on πR2 +mp but also influence the magnetospheric +cross section in an opposite manner, as it can be seen in Fig. +4 (blue solid line). The magnetospheric stand-off distance scales +according to Rmp ∝ (ρswv2 +0+psw)−1/6 and the incident energy flux +with Pi ∝ ρsw + psw, therefore the incident energy flux increase +dominates over the effect of a shrinking MS due to increasing +thermal and kinetic pressure. This is also validated by our sim- +ulation results (Fig. 4), implying a approximately linear scaling +of auroral Poynting fluxes with ρ0 and p0 at least in the regime +between 3 × 1011 and 3 × 1013H+ m−3. Below the point where +stellar wind magnetic energy dominates over thermal and kinetic +energy near 1011H+ m−3, auroral Poynting fluxes seem to satu- +rate near 2–3 ×1017 W (open MS) and near 1–2×1016 W (closed +MS). Above 3 × 1013H+ m−3 the increase of auroral Poynting +fluxes with ρ0 (and p0) deviates further from the course of inci- +Article number, page 9 of 19 + +Incident stellar wind powers and magnetopause stand-off distance as function of po and po +Po [μPa] +10-1 +100 +101 +102 +103 +1020 +Basic model +5 +1019 +4 +1018 +[W +[Rp] +Power +3 +1017 +2 +P with Parker magnetic fieid +Pkin +Ptherm +1016 +Pkin + PB + Ptherm +1 +integrated S ·Bo, simulations (open Ms) +integrated S·Bo, simulations (closed Ms) +PB with stellar dipole field +1015 +109 +1010 +1011 +1012 +1013 +1014 +0.25 +α p-0.06 +0.20 +0.15 +Transfer function Ta = Pa/P; (open MS) +Transferfunction Ta= Pa/P; (closed MS) +fer +0.10 +Transf +α 00.57 +0.05 +0.00 - +109 +1010 +1011 +1012 +1013 +1014 +[s-w/+ H] odA&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +dent flux, implying a saturation towards 1019 W (open MS). This, +however, has to be validated further through future simulations. +4. Discussion +In this section we discuss the importance of the stellar wind on +magnetospheric energetics (Sect. 4.1) and on possible auroral +radio emission (Sect. 4.2). +4.1. Importance of the stellar wind on magnetospheric +energetics +In the following sections we study the conversion of incident +to dissipated power within the magnetosphere (Sect. 4.1.1) as a +function of stellar wind density and pressure. We also discuss the +limiting case of an absent stellar wind (Sect. 4.1.2). +4.1.1. Stellar wind variability, its effect on magnetospheric +energetics and scaling behavior of auroral Poynting +fluxes +We separate the considered stellar wind density and pressure +range introduced in Sect. 3.2.3, Fig. 4 in two regimes: +Regime 1 ranges from a vanishing stellar wind up to a density at +roughly 1011H+ m−3 where kinetic and thermal energy fluxes fall +below the persistent magnetic energy flux which dominates the +flow (compare red curves in Fig. 4). Above roughly 1011H+ m−3 +the flow is super-Alfvénic (MA ≈ 2) and super-fast (Mf ≈ 1). +The interaction is super-Alfvénic for the whole parameter space +used in our simulations and sub-fast only for the lowest simu- +lated density (ρsw = 7 × 1010 H+ m−3, MA ≈ 1.2). The incident +energy nearly stagnates below ρ0 = 1010 H+ m−3 (red dotted +line). Below this point the incident energy flux asymptotically +approaches its minimum at 8×1017 W as we assume that only the +plasma density decrease but the incident magnetic field is kept +constant. In this regime it can be expected that the stellar wind +magnetic field solution transitions from the Parker solution (e.g. +B ∝ r−2) to a pure stellar multipole (here dipole) solution (e.g. +Bsw = Bstar ∝ r−3) with decreasing stellar wind density. Even- +tually, when the stellar wind density hypothetically approaches +zero, only the dipolar stellar magnetic field interacts with the +planetary magnetic field. This limiting case will be separately +discussed in Sect. 4.1.2. +Regime 2 ranges from roughly 1011H+ m−3 up to arbitrarily high +stellar wind densities. Here kinetic and thermal energy fluxes +dominate the flow. We will now focus on this regime. Consider- +ing the total energy flux convected through the magnetospheric +cross section πR2 +mp, Ptotal (red solid line in Fig. 4), we observe +a nearly constant efficiency of conversion from incident stellar +wind energy towards magnetospheric Poynting fluxes at auro- +ral altitudes (we assumed r ≈ 2Rp) with increasing density and +pressure. We calculate the transfer function Ta as the conversion +ratio from total incident energy flux Pi to the simulated auroral +Poynting fluxes parallel to the unperturbed field (Eq. 12), Pa (red +crosses and stars in Fig. 4), within the MS, such that Ta = Pa/Pi +(see Fig. 3 for a schematic illustrating the role of Ta). The trans- +fer function also contains information on the magnetic topology +and thus the efficiency of reconnection. The transfer function is +displayed in the lower panel of Fig. 4. For the open MS, auroral +Poynting fluxes decrease as a function of ρ0 and p0 according +to an approximately constant ratio up to ∼ 3 × 1013H+ m−3. For +higher densities and pressures, the transfer function scales with +an exponent of ∼ −0.4, indicating a decrease of efficiency for +conversion from incident to auroral energy fluxes. As the MS is +increasingly compressed due to stronger ram and thermal pres- +sures, the magnetopause eventually crosses the spherical shell +with r = 2Rp after a critical density of ∼ 3 × 1013H+ m−3 and +pressure of 3 × 10−4 Pa. +For the closed MS, Ta behaves differently. The transfer func- +tion instead increases slightly from 7 × 1010 to ∼ 3 × 1013H+ +m−3 following a power law with exponent ∼ 0.6. The opposite +behavior compared to the open MS transfer function might be +a consequence of a geometry where less reconnection occurs. +The stronger ram and thermal pressure exert stronger tension on +planetary field lines which in turn release stronger energy fluxes +during reconnection. +4.1.2. Beyond the MHD limit +The more the stellar wind density decreases, the emptier the +heliosphere of τ Boötis A becomes. In analogy with the solar +wind, the stellar wind density upstream of the magnetosphere of +τ Boötis b may change by orders of magnitude. The solar wind +density is observed to vary by more than two orders of magni- +tude (see e.g. Chané et al. 2012) such that Earth’s bow shock +can disappear and the Earth develops Alfvén wings. In the hy- +pothetical limit when the density approaches zero, τ Boötis b +will still be exposed to the stellar magnetic field Bτ(r) (which +then decreases according to Bτ(r) ∝ r−3 instead of Bsw ∝ r−2) +and will propagate through it. Therefore the incoming Poynting +flux of the star does not, in contrast to kinetic and thermal energy +flux, vanish and is maintained by the relative motion between the +stellar and planetary magnetic field. In case of an empty helio- +sphere, the interaction around τ Boötis b is not magnetohydrody- +namic any more, but turns electromagnetic. Then the movement +of τ Boötis b within the external magnetic field of the star is +a unipolar or homopolar interaction (i.e. a moving conductive +object with external field similar to a current generator in clas- +sical electromagnetism). In the case of a stellar magnetic field +rotating at the same speed as the planet orbits around its host +star (i.e. in case of total orbital and spin synchronization), non- +existent changes in magnetic flux lead to a system where no work +can be done by the magnetic fields. Therefore no currents are +induced and the magnetosphere remains energetically silent. Al- +though it is expected that close-in exoplanets are in nearly tidally +locked rotation, Murray & Dermott (2000) suggest in their text- +book that completely synchronous rotation might not be possi- +ble if the planet has no permanent magnetic quadrupole moment +and its eccentricity is not zero, which is the case for τ Boötis +b (Wang & Ford 2011). Taking the rotation period of τ Boötis +A, Pτ = 3.1 ± 0.1 d (Brown et al. 2021; Mengel et al. 2016), +and the sidereal rotation period of τ Boötis b, Porb = 3.31 d +(Butler et al. 1997; Wang & Ford 2011), we can calculate the +relative velocity between the stellar and planetary magnetic field +v0 = 2πa × (1/Pτ − 1/Porb) ≈ 10.4 km s−1, where a = 0.046 AU +is the semi-major axis of τ Boötis b. We use an average surface +magnetic flux density of τ Boötis A obtained by Marsden et al. +(2014); Mengel et al. (2016); Jeffers et al. (2018), Bτ,0 = 2.4 G, +and calculate the flux density at 0.046 AU using the dipole for- +mula, Bτ(r = a) = Bτ,0 · (a/Rτ)−3 ≈ 0.73 G, where Rτ = 1.43R⊙ +is the stellar radius (Bonfanti et al. 2016). The magnetospheric +stand-off distance in this case is defined purely by the balance of +stellar and planetary magnetic pressure, i.e. +Rmp = Rp · (Bτ(a)/Bp)−1/6 ≈ 8.6 Rp . +(18) +Article number, page 10 of 19 + +F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +Now we can calculate the stellar magnetic power convected on +the MS using +PB = Bτ(a)2/µ0 · v0πR2 +mp ≈ 8 × 1015 W . +(19) +This value is included in Fig. 4 as magenta arrow. We can con- +clude the following: +(a) The maximum possible magnetospheric stand-off distance is +reached in complete absence of a stellar wind and if the stellar +and planetary magnetic fields are anti-parallel. If the stellar and +planetary fields are parallel (i.e. open MS) the planetary field +lines are all connected to the stellar field and no magnetopause +can be determined. In the closed MS case the magnetopause lies +at roughly 8.6 planetary radii in the upstream direction. +(b) Even if there is no stellar wind, the magnetic interaction be- +tween the stellar and planetary magnetic field still has the po- +tential to drive an interaction with an available power limit of +roughly 8×1015 W due to the relative motion of τ Boötis b in the +stellar magnetic field. Radio emission would still be possible al- +though very weak; corresponding radio fluxes at Earth’s position +would be far below today’s telescope sensitivity limit. Consider- +ing Poynting flux–to–radio power efficiencies between 10−4 and +10−2, radio powers can reach values between 1011 and 1014 W. +These emitted powers exceed the strongest radio sources within +the solar system by several orders of magnitude, with Jupiter’s +aurora being the strongest radio emitter (Pradio ≈ 1010 − 1011 W +Zarka 2007), although Jupiter’s emission is, in contrast to close- +in exoplanets, powered by internal, rotationally driven mecha- +nisms. The transition from a magnetosphere interaction with a +stellar wind field (according to the Parker solution Parker 1958) +to an interaction with a pure dipolar stellar magnetic field goes +with an energetic transition followed by a decrease of maximum +emitted radio power. This might pose a possible opportunity for +constraining stellar wind densities in the future. Solving a self +consistent stellar wind model and comparing auroral Poynting +fluxes for different stellar wind base densities could reveal the +critical density range where the transition from a stellar mag- +netic field dominated electrodynamic interaction to a stellar wind +dominated magnethydrodynamic interaction takes place. Com- +paring the magnetospheric Poynting fluxes and corresponding +radio powers with possible future observations could reveal if +the stellar wind density lies below or above the critical density. +4.2. Magnetospheric Poynting fluxes and auroral radio +emission +In this section we discuss possible radio emission scenarios +(Sect. 4.2.1), study how magnetic topology of the interaction +as well as stellar wind variability affects auroral radio emission +output (Sect. 4.2.2). We also discuss the possibility of a sub– +Alfvénic emission scenario (Sect. 4.2.3) as well as a rotation +driven magnetosphere of τ Boötis b in Sect. 4.2.4, followed by +a discussion on possible source regions and radio frequencies of +auroral emission in Sect. 4.2.5. +4.2.1. On the different radio emission scenarios +There are several scenarios capable of generating observable ra- +dio emission that must be distinguished from each other. +(1) If the stellar wind is sub–Alfvénic, Alfvén waves are able to +propagate upstream towards the stellar atmosphere along Alfvén +wings and possibly drive electron acceleration and radio emis- +sion in the stellar vicinity. Local radio emission within the mag- +netosphere can also be generated in this scenario. We will refer +to this scenario as sub–Alfvénic emission scenario. This scenario +is discussed briefly in Sect. 4.2.3. +(2) If the stellar wind is super–Alfvénic, no MHD wave is able +to propagate upstream. The stellar wind–magnetosphere interac- +tion however drives Poynting fluxes within the magnetosphere +which may to some extent generate auroral radio emission (see +Fig. 3). We refer to this scenario as the magnetospheric emis- +sion scenario. Due to the stellar wind being super–Alfvénic for +all simulations we will focus on the magnetospheric emission +(Sect. 4.2.2). +(3) In a rotation dominated magnetosphere scenario the rotat- +ing planet and its magnetic field causes co–rotation of magneto- +spheric plasma that, at some point, breaks-down due to conser- +vation of angular momentum if radial mass transfer takes place. +This co–rotation breakdown exerts magnetic stresses on the field +lines that are the root cause of auroral Poynting fluxes which in +turn drive auroral radio emission. This scenario is discussed in +Sect. 4.2.4. +4.2.2. Effects of magnetic field tilt and stellar wind variability +on auroral radio emission – Magnetospheric emission +scenario +Figure 5 shows radio powers as function of magnetic axis tilt. +Radio powers are obtained from multiplying integrated auro- +ral Poynting fluxes (i.e. Eq. 12), which serve as a proxy for +maximum available electromagnetic energy that is transported +along magnetic field lines, with efficiency factors for converting +magnetospheric Poynting fluxes to radio–power, ϵ, ranging from +10−4 to 10−2. This range covers proposed (Zarka 2007) and ob- +served efficiency factors (e.g. ϵ ≈ 10−4 for Jupiter Saur et al. +2021). The modeled magnetic field tilt can also be interpreted as +stellar magnetic field orientation within this work, allowing us +to study the effect of varying stellar magnetic field polarity on +magnetospheric Poynting fluxes and limits for associated radio +emission. Radio powers within the limits inferred from observa- +tions by Turner et al. (2021) lie within the gray shaded area. It is +visible that efficiency factors in the range of ϵ ≈ (0.3 − 1) × 10−2 +deliver radio powers most consistent with observations if the MS +is open or at least semi open given the basic model (Table 1). +This indicates that the efficiency of auroral Poynting fluxes driv- +ing electron acceleration and the electron cyclotron maser emis- +sion may be higher in the magnetosphere of τ Boötis b than in +the Jovian magnetosphere (Saur et al. 2021). Electric fields gen- +erated by reconnection between stellar wind and planetary mag- +netic field lines are expected to contribute significantly to pow- +ering electron acceleration and therefore the ECMI (Jardine & +Collier Cameron 2008). In our studies we find reconnection to +indirectly play an important role (Fig. 5) because auroral Poynt- +ing fluxes and consequently radio powers drop by nearly an or- +der of magnitude from an open to a closed MS. This is due to +magnetic stress exerted by the stellar wind interaction being less +strong for closed magnetospheres. The polarity of τ Boötis A’s +magnetic field switches every approximate 360 days (Fares et al. +2013). Shorter cycles in magnetic activity levels (by means of S- +indices) were also observed (Mengel et al. 2016). A difference of +half an order of magnitude to almost an order of magnitude can +therefore be caused by a polarity reversal of τ Boötis A’s mag- +netic field. This results in radio emission whose observability is +expected to fluctuated periodically in a nearly 1-year cycle. We +note that the stellar wind magnetic field strength was kept con- +stant in our parameter study, although in reality the field strength +may vary strongly and influence produced radio emission signif- +Article number, page 11 of 19 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +Fig. 5: Expected radio powers as function of planetary magnetic axis tilt +using auroral-to-radio power conversion efficiencies between 10−4 and +10−2. The auroral Poynting fluxes S · B0 are integrated over a spherical +shell with radius 2Rp. The gray shaded area represents observational +limits given by Turner et al. (2021). +Fig. 6: Radio flux (Eq. 20) as function of stellar wind density and pres- +sure for different efficiency factors. Colored solid and dashed lines rep- +resent fluxes for the open and closed MS model. Observational limits +(Turner et al. 2021) are indicated by the gray shaded area. Horizontal +gray lines display theoretical sensitivity limits of the LOFAR telescope. +The real sensitivity for Stokes V signals obtained from Turner et al. +(2019) is plotted as yellow line. The vertical gray line marks the basic +model (Table 1). +icantly (See et al. 2015). +The emitted radio flux observed at Earth’s position can be +calculated with (Grießmeier et al. 2005, 2007b) +Φ = Pradio +Ω δν d2 , +(20) +where Ω is the solid angle of the beam and δν the emission +bandwidth that is approximately equal to the maximum gyro- +frequency (Grießmeier et al. 2007b), νg,e ≈ 24 MHz. We assume +a solid angle of Ω = 1.6 sr similar to Jupiter’s decametric radio +emission (Zarka et al. 2004). The distance to the τ Boötis sys- +tem is 15.6 pc. We calculate the radio flux for both, the open and +closed MS model, as a function of ρ0 and p0 with radio efficien- +cies ϵa = 10−4–10−2. The results are displayed in Fig. 6. Solid +and dashed colored lines represent radio fluxes originating from +open and closed MS’s, respectively. The gray shaded area again +denotes the range of observed radio fluxes form Turner et al. +(2021). Horizontal gray lines indicate theoretical sensitivity lim- +its of the LOFAR telescopes for 20MHz. As stated by Turner +et al. (2019), the realistic sensitivity might be slightly lower for +circularly polarized (Stokes V) signals. We therefore include the +expected sensitivity calculated by Turner et al. (2019) as yellow +line. The results in Fig. 6 indicate that radio efficiencies between +∼ 3 × 10−3 and ∼ 10−2 are most consistent with the tentative +observations (Turner et al. 2021). The efficiency accounts for +several steps from conversion of auroral Poynting fluxes to ra- +dio emission (e.g. wave-particle interaction, electron accelera- +tion and ECMI), therefore an efficiency of the order of 1–10% +might be unrealistic. The efficiency for Jupiter’s auroral emis- +sion is roughly ϵa = 0.3–3 × 10−4 (Saur et al. 2021), therefore +ϵa = 10−2 might be too high. Moreover high plasma densities +within the magnetosphere injected by the dense stellar wind and +due to strong irradiation which results in high ionization rates +and inflated atmospheres (e.g. for ν And b see Erkaev et al. +2022), may further decrease the ECMI efficiency or even prevent +it (Weber et al. 2017, 2018; Daley-Yates & Stevens 2018). As- +suming the radio efficiency to lie near 10−3, the radio flux from +a closed MS falls below the detection threshold (yellow line). +Therefore, in case of a polarity reversal of τ Boöts A’s magnetic +field (i.e. from aligned with the planetary field to anti-aligned), +the radio signal would not be observable anymore in case of ra- +dio efficiency equal or below ∼ 3 × 10−3. In case of ϵa ≈ 10−4 all +radio fluxes for the basic model fall below the sensitivity limit. +The observability increases, however, if stellar wind density and +pressure rises, rendering ϵa = 10−3–10−2 to possible efficiencies +to observe emission from open and closed MS’s. Additionally, +the ECMI efficiency (Treumann 2006; Weber et al. 2017) as well +as efficiency of electron acceleration through wave-particle in- +teraction decreases dramatically with increasing plasma density +(Saur et al. 2018), making the higher density and pressure regime +a less likely scenario to explain the tentative observations. As the +pressure rises, the magnetopause is getting closer to the planet, +reducing the space of magnetospheric diluted plasma regions be- +tween the magnetopause and atmosphere where radio emission +might occur. We therefore conclude that the basic model (verti- +cal gray line) and slightly different configurations represent the +most likely scenarios if the emission is indeed generated by stel- +lar wind driven auroral Poynting fluxes. In this case radio emis- +sion is only observable, if the stellar wind and planetary mag- +netic fields are aligned (i.e. the magnetosphere is open). Given +the high efficiencies (ϵ > 10−3) needed by our model in order +to generate radio emission which is consistent with the tentative +observations, the magnetospheric emission scenario might not +be energetic enough to explain the observations. +4.2.3. Sub–Alfvénic emission scenario +Although there is no sub-Alfvénic interaction within the param- +eter space we considered, the possibility of such an interaction +and its consequences on possible radio emission should not be +neglected. By choosing a stellar wind density of ρsw = 0.03 ρ0 +we find an Alfvénic Mach number of MA ≈ 0.9. In this case +Alfvén waves may propagate back to the star through Alfvén +wings connecting the planetary magnetic field with the star. The +electromagnetic energy channeled through this flux tube can be +Article number, page 12 of 19 + +Estimatedradiopowerasfunctionofmagnticfieldtiltforthebasicmode +Obs.range,Turneret al.(2o21) += 10-4 +1016 +=10-3 +=3 × 10-3 +[M] += 10-2 +Pradio +1014 +1013 +o°(OpenMS) +90°(Semi-open MS) +180*(ClosedMS) +MagneticfieldtilteB[deareesiEstimatedradiofluxasfunctionofpo(po) +104 +Obs.range,Turneret al.(2021) +103 +Radio flux Φ [mly] +102 +101 +Open MS, =10 +Closed MS, = 104 +Open MS, = 3 × 10 3 +Closed MS, = 3 × 10~3 +100 +Open MS, = 10-2 +Closed MS, = 102 +theo. sens. 6 stations +Basic model +theo. sens. 24 stations (Turner et al. (2021) +theo. sens. 40 stations +theo. sens. 48 stations +estimated sens. Stokes V @ 10 MHz (Tumer et al. 2019) +10-1 +1011 +1012 +1013 +1014 +Po [H + /m-3]F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +calculated using the model of Saur et al. (2013), +PS = 2πR2 +mp +(¯αBsw cos θ)2 +µ0 +MAv0 , +(21) +where θ = 0◦ is the angle which describes the deviation of the +flow from being perpendicular to the stellar wind magnetic field, +Rmp ≈ 5Rp the magnetospheric stand–of distance and ¯α the in- +teraction strength. Due to the planet presumably possessing an +ionosphere which favors a strong plasma interaction we choose +¯α ≈ 1. Using v0 and Bsw from our basic model (Table 1), the +energy flux channeled through Alfvén wings parallel to mag- +netic field lines amounts to PS = 1.2 × 1018 W. This energy +flux may contribute to electron acceleration and consequently to +ECMI driven radio emission in the space environment near the +star. Taking the same range of Poynting flux to radio efficiencies +(10−4–10−2) the expected radio power ranges from roughly 1014 +to 1016 W which partially overlaps with the range of radio power +deduced by Turner et al. (2021). The emission frequency would, +however, be much lower compared to the signals observed by +Turner et al. (2021) due to the low magnetic field strength of τ +Boötis A (Turner et al. 2021). Additionally Turner et al. (2021) +state that stellar emission would be less strongly circularly polar- +ized compared to planetary emission. Nevertheless the root en- +ergy flux able to possibly drive radio emission near the stellar at- +mosphere from such a sub–Alfvénic interaction exceeds the au- +roral Poynting fluxes in the magnetospheric emission scenario by +roughly half an order of magnitude. Consequently this scenario +might provide energetically a possible explanation for the ob- +served radio emission if the interaction is indeed sub–Alfvénic. +This scenario requires the stellar wind density to be reduced by +at least a factor of ∼ 30 compared to our basic model. This, how- +ever, could well be within the realm of possible densities given +the observed density variations of the solar wind even tough the +intervals of variability are short (Chané et al. 2012). +4.2.4. Is auroral radio emission from τ Boötis b rotationally +driven? +One of the possible generators for auroral emissions is radial +mass transport within the magnetosphere as it is the case for +Jupiter (e.g. Hill 2001; Zarka et al. 2018, 2021). Plasma is +moving radially outwards due to centrifugal forces within an +exoplanet’s MS. For small distances from the planet (or stel- +lar host) the plasma co-rotates with the host. After a certain +distance ˆL in units of planetary radii, ˆL × Rp, the co-rotation +breaks down due to conservation of angular momentum (i.e. the +plasma orbits its host with a smaller angular velocity than those +from the rotation of the host). The relative velocity exerts mag- +netic stresses on the frozen-in field lines, the tension accelerates +plasma along the field lines back to the host. The distance of co- +rotation breakdown can be estimated using the co-latitude of the +region, where auroral emission occurs, θa, following Hill (2001), +sin(θa) = ˆL−1/2. Saur et al. (2021) derived a so called auroral +power potential for magnetized rotation dominated hosts, +S pot = B2 +hostΩ2 +hostR2 +host , +(22) +where the subscript host refers in our case to the planet τ +Boötis b. The hosts magnetic flux density is denoted by Bhost, +Ωhost ≈ 2π/Porb = 2.2 × 10−5 rad s−1 is its angular velocity and +Rhost = Rp the planet’s radius. We note that the discussion in +this subsection is only based on theoretical considerations since +we neglected planetary rotation in our MHD model. Following +Saur et al. (2021) we can calculate the energy flux due to mass +transport, Pmag, by using +Pmag = S pot +�������π +R2 +p +ˆL2 +������� Σp , +(23) +where Σp denotes the Pedersen conductance. We assume a value +of 1 S and 10 S that are of the same order of magnitude as +observed conductances near Jupiter (e.g. Hill 2001; Strobel & +Atreya 1983; Hinson et al. 1998). For a hot Jupiter exoplanet +the ionospheric Pedersen conductivity might be larger due to the +larger ionization (Koskinen et al. 2010). However, also the height +of the resulting ionospheric layer and thus the mobility of the +electrons and ions within the atmosphere also plays an impor- +tant role for the values of its conductivity. To our knowledge no +dedicated study for the conductance of τ Boötis b is available. +We also note that in Eq. 22 the distance of co–rotation break- +down ˆL is proportional to Σ1/4 (Hill 2001). Thus Pmag in Eq. +23 depends effectively weaker on Σp (i.e. Pmag is proportional +to Σ1/2 +p ). In our simulations we integrate the ion component of +the Pedersen conductivity along the z–axis at the poles from the +surface (r = Rp) to r = 1.1 Rp where the plasma density peaks +and find the height integrated Pedersen conductance to be of the +order of 10 S. We find, with aurora occurring at field lines with +L-parameter smaller than ∼ 4 that have to lie within the MS +(e.g. ˆL ≈ 3.5), the auroral power potential to be S pot ≈ 0.5 W +m−2 S−1 and the magnetic power to lie between Pmag ≈ 1014 W +and 1015 W. The auroral power potential of Jupiter, for com- +parison, is roughly 600 W m−2 S−1. The resulting energy flux +between the magnetosphere and ionosphere lies near the Jovian +value (3.1 × 1014 W Saur et al. 2021). This result is several or- +ders of magnitude below the power of magnetospheric Poynting +fluxes powered by the stellar wind – planet interaction in our +model (e.g. 1017 – 1018 W, Table. 2). Therefore we can safely +assume that the MS of τ Boötis b is not rotationally dominated +which is in agreement with the expected slow rotation nearly +synchronized with its orbit around the star. +4.2.5. Properties and source regions of possible auroral +radio emission +Auroral radio emission is mainly characterized by signal +strength and frequency. The peak frequency is defined by the +local electron gyrofrequency, +νg,e = |e|B +2πme +, +(24) +where e is the electron charge, me its mass and B the local mag- +netic flux density (Farrell et al. 1999; Grießmeier et al. 2007a,b). +We already discussed possible radio powers from τ Boötis b in +Sects. 3.2.2 and 4.2.2, now we will focus on possible signal fre- +quencies. Turner et al. (2021) observed signals within the 21 – +30 MHz range. The magnetic flux density inferred from the max- +imal signal frequency by the authors is almost certainly under- +estimated since ECMI driven auroral radio emission typically +arises from altitudes above the polar regions of the planet’s at- +mosphere (Zarka 1998; Hess & Zarka 2011) due to three rea- +sons: (a) High ionospheric electron densities cause the electron +plasma frequency νp,e to exceed the local gyro frequency νg,e, +thus prohibiting the transmission of emitted waves; (b) radio +wave amplification through the ECMI mechanism works most +efficiently in regions where νg,e >> νp,e (Treumann 2006; Weber +Article number, page 13 of 19 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +et al. 2017) and (c) the electrons are usually accelerated in re- +gions where the plasma density along a field line is the smallest +(or B/n is maximum) (e.g. Saur et al. 2018). There wave particle +interaction to accelerate electrons is maximum efficient. +The electron plasma frequency is given by +νp,e = +� +nee2 +ϵ0me +1 +2π , +(25) +where ne = ni is the particle density in a quasi-neutral plasma +and ϵ0 the vacuum permittivity. The source regions of Jupiter’s +radio emission lie several planetary radii above the surface (e.g. +Zarka 1998) where strongest electron acceleration is expected +(e.g. Mauk et al. 2017). The maximum gyro frequency in our +model (directly above the magnetic poles) is ∼ 24 MHz. Au- +roral emission, however, typically arises from near the OCFB. +In our simulations we indeed see that strongest Poynting fluxes +are confined to narrow bands encircling the OCFB (or magne- +topause) near L=3 and L=3.5 shell field lines (Fig. 2). Since +source regions of ECMI induced emission typically lie at higher +altitudes the magnetic flux density and thus emission frequency +is even further reduced. The conditions (a–c) for efficient ECMI +driven radio emission might particularly be fulfilled for regions +with low plasma density at the night side of the planet where the +shadow exerted by the planet prevents photo-ionization. Since τ +Boötis b is likely in tidally locked rotation around its host star, +the night side might exhibit relatively low temperatures, favoring +recombination of electrons and hydrogen ions as well as further +electron capture processes. In the shadow zone (i.e. permanent +night side of the planet), ECMI induced emission might be more +likely and might as well occur in lower altitudes, where the mag- +netic field is stronger (and thus emission frequency is higher). +5. Conclusions +In this study we modeled the magnetosphere of the Hot Jupiter +exoplanet τ Boötis b and its interaction with the stellar wind. +In order to study magnetospheric energetics by means of +Poynting fluxes we performed MHD simulations of the near +space environment of τ Boötis b with the stellar wind being +injected into the simulation domain. The magnetic field of τ +Boötis b is constrained by tentative observations of auroral +radio emission (Turner et al. 2021). The stellar wind model is +based on simulations where observed magnetic surface maps of +τ Boötis A were utilized (Nicholson et al. 2016; Vidotto et al. +2012). We investigated magnetospheric energetics, available +electromagnetic power and limits of possible radio emission +originating from above the planet’s polar regions. We also +explored the possibility of a rotation dominated magnetospheric +emission scenario as well as a sub–Alfvénic (stellar) emission +scenario. The effect of stellar wind variability in terms of den- +sity, pressure and magnetic field orientation on magnetospheric +energetics was additionally investigated. +We find that the stellar wind–magnetosphere interaction is +super-fast and super-Alfvénic for the majority of stellar wind +densities and pressures considered in this study (from 1.4 × 1011 +H+ m−3 and 2.5 × 10−6 Pa to arbitrarily high stellar wind +densities and pressures). For the minimum density and pres- +sure considered in this work (ρsw = 7 × 1010 H+ m−3 and +psw = 1.3 × 10−5 Pa) the interaction is super–Alfvénic and +sub–fast. Maximum available electromagnetic energy convected +within the magnetosphere amounts to 3.5×1018 W, 1.8×1018 +W and 7×1017 W for an open, semi-open and closed magne- +tosphere configuration. Auroral Poynting fluxes at altitudes of +1Rp above the planet reach powers of 8.7×1017 W down to +1.1×1017 W for the open and closed magnetosphere, indicating +a strong decrease of stellar wind energy converted to auroral +Poynting fluxes as a function of magnetic field tilt (either stellar +or planetary). +We present Poynting flux maps of the planet’s near space +environment. Strongest Poynting fluxes are confined to nar- +row bands encircling the open-closed field line boundaries +(i.e. the magnetopause) that is displaced towards the planet’s +downstream side due to magnetic stresses exerted by the stellar +wind and its high plasma β. Electromagnetic energy transported +parallel to unperturbed field lines are as well confined to bands +near the magnetopause and the polar open-field line regions. +Considering the stellar wind density as free parameter, we +choose values between 0.05 and 100 times the basic density, +ρsw = 1.4 × 1012 H+ m−3 (Table 1), and change the stellar +wind pressure in the same manner in order to keep the wind +isothermal. Comparing the incident total energy flux convected +through the magnetospheric cross section, we find that the trans- +fer function Ta (i.e. the amount of stellar wind incident energy +flux converted to magnetospheric Poynting fluxes parallel to +unperturbed field lines) amounts to a fraction of roughly 20 % +for the open and 2 % for the closed MS. We find that Ta ∝ ρ−0.06 +0 +and Ta ∝ ρ0.57 +0 +for the open and closed MS respectively in the +density range 7×1011 – 2×1013 H+ m−3. Increasing stellar wind +ram and thermal pressure and thus incident energy flux causes +the shrinking of the magnetosphere. The reduced capability +of the MS for receiving stellar wind energy is outperformed +by the increasing pressure and density. The scaling of Ta and +thus energy output of the magnetosphere nearly stagnates from +that point on where the magnetopause turns so small that the +planetary atmosphere starts to interact with pre-bow shock and +magnetosheath flow. +We investigated different radio emission scenarios. A rotation +dominated magnetospheric emission scenario could be ruled +out due to the expected long rotation period of τ Boötis b. The +estimated magnetic power generated by the planet’s rotation +is on the order of Pmag ≈ 1014−15 W. As this would require a +very high Poynting flux to radio efficiency of 10−2, we expect +that this scenario is not powerful enough to drive the tentatively +observed radio emission. +We considered several efficiencies for conversion of auroral +Poynting fluxes to radio emission and compared the results +with observations together with theoretical limits based on the +tentative observations by Turner et al. (2021). We find that +generation of radio emission near τ Boötis b requires relatively +high efficiencies (ϵa ≈ 1–5 × 10−3) compared to estimates from +Jupiter’s rotationally driven aurora (up to∼ 3 × 10−4) (Saur et al. +2021) in order to explain the observed emission if we assume +the emission to originate from the magnetosphere of τ Boötis +b. Furthermore we find a strong dependence of radio emission +observability on magnetosphere topology. In case of a polarity +reversal of τ Boötis A’s magnetic field from an aligned to anti- +aligned (with respect to the planetary field) configuration, the +estimated radio power falls below the observational limit. We +therefore expect an on–off nature of detectable radio emission +in phase with τ Boötis A’s magnetic cycle. +We also considered the case of a sub–Alfvénic interaction +which requires the stellar wind density to be at least a factor +of 30 lower compared to our basic model. The Poynting flux +channeled towards the host star due to this interaction amounts +Article number, page 14 of 19 + +F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +to ∼ 1018 W which slightly exceeds the power converted by the +magnetospheric emission scenario. +Acknowledgements. This project has received funding from the European Re- +search Council (ERC) under the European Union’s Horizon 2020 research and +innovation programme (grant agreement No. 884711). +We furthermore thank the Regional Computing Center of the University of +Cologne (RRZK) for providing computing time on the DFG-funded (Funding +number: INST 216/512/1FUGG) High Performance Computing (HPC) system +CHEOPS as well as support. +We also thank Stefan Duling for providing several basic model implementations +for the Pluto code used in this study as well as for helpful discussions. +References +Alexander, R. D., Wynn, G. A., Mohammed, H., Nichols, J. D., & Ercolano, B. +2016, MNRAS, 456, 2766 +Atreya, S. K., Mahaffy, P. R., Niemann, H. B., Wong, M. H., & Owen, T. C. +2003, Planet. Space Sci., 51, 105 +Bastian, T., Cotton, B., & Hallinan, G. 2022, arXiv e-prints, arXiv:2206.14099 +Bisikalo, D. V., Kaigorodov, P. V., Ionov, D. E., & Shematovich, V. 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S. 2004, Journal of Geophysical Research: +Space Physics, 109 +Zarka, P., Magalhães, F. P., Marques, M. S., et al. 2021, Journal of Geophysical +Research (Space Physics), 126, e29780 +Zarka, P., Marques, M. S., Louis, C., et al. 2018, A&A, 618, A84 +Zarka, P., Treumann, R. A., Ryabov, B. P., & Ryabov, V. B. 2001, Ap&SS, 277, +293 +Zhilkin, A. G. & Bisikalo, D. V. 2019, Astronomy Reports, 63, 550 +Zhilkin, A. G. & Bisikalo, D. V. 2020, Astronomy Reports, 64, 563 +Article number, page 15 of 19 + +A&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +Appendix A: Poynting fluxes as function of radial +distance from the planet +Fig. A.1: Magnetospheric Poynting flux as function of radial distance +from the planet’s center calculated from Eq. 11 for the open (blue), +semi–open (orange) and closed (green) magnetosphere model. +Here we study the dependence of magnetospheric Poynting +fluxes as a function of radial distance from the planet. The par- +ticular choice of integrating Poynting fluxes over a spherical +shell with radius r = 2Rp is based on the fact that radio emis- +sion around Jupiter and most solar system planets with intrinsic +magnetic fields originates from altitudes of about 1 Rp or higher +above the planets (see e.g. Zarka 1998) rather than from within +the ionosphere or even further below. We do not have evidence if +this translates to extrasolar planets but simply assume so. For the +ECMI mechanism to work efficiently the ratio between electron +gyro frequency and plasma frequency, νg,e/νp,e, must be signif- +icantly larger than 1 which turns the magnetospheric region at +high altitudes to favored radio source regions where plasma den- +sities are strongly reduced compared to other regions of the mag- +netosphere. (Zarka 1998; Treumann 2006; Weber et al. 2017). +Considering the plasma and neutral particle density within the +modeled magnetosphere we also chose a location (i.e. the ra- +dius) where ion–neutral collisions are significantly reduced (e.g. +above the ionosphere). The favored radius according to this con- +straint lies between 1.3 and 3 Rp (see Sect. D and Fig. D.2). +In order to show that the choice of r = 2 Rp or any other ra- +dial distance within the magnetosphere does not influence the +results and the derived conclusions significantly (despite control- +ling the emission frequency since stronger magnetic fields cause +higher gyro frequencies) we integrated the Poynting flux (Eq. +11) for shells with radii between 1 and 2.5 Rp above the planet. +The results are displayed in Fig. A.1 for the open (blue), semi- +open (orange) and closed (green) magnetosphere model. We note +that the Poynting flux is not a conserved quantity in this system +since several possibilities for conversion from or to electromag- +netic energy exist within the magnetosphere (e.g. deceleration +due to magnetic stresses, ion–neutral collisions, conversion be- +tween electromagnetic and thermal energy). As it can be seen +in Fig. A.1 the Poynting flux variation as function of r amounts +to a factor of ∼1.5–2 for the open MS, ∼ 1 – 1.5 for the semi– +open MS and ∼2 for the closed MS. Given the uncertainties with +which our study is anyway afflicted (such as the uncertainty of +the stellar wind density, magnetic field strength, etc.), the uncer- +tainty by the choice of the shell radius upon the Poynting fluxes +is negligible. The choice of magnetic field strength, for instance, +has much larger influence on the Poynting fluxes due to its B2 +dependence. +Appendix B: On the effect of magnetic diffusion on +the results +In our MHD model we include the ionosphere and atmosphere +of τ Boötis b through collisions with the plasma. We do not +include the effects of collisions in the induction equation for +computational time reasons. In this section we will justify +this simplification by comparing simulations with different +physical diffusion rates (i.e. different ionospheric conductivities) +compared to our basic model without diffusion. +Therefore we add a diffusion term to the induction equa- +tion (Eq. 4) which then reads +∂B +∂t = ∇ × �v × B − η∇ × B� . +(B.1) +We scale the magnetic diffusivity η = η(r) with the atmospheric +scale height so that it correlates with the neutral particle density, +η(r) = η0 exp +�Rp − r +H +� +, +(B.2) +where η0 is the surface magnetic diffusivity, Rp = 72875 km the +planetary radius (Wang & Ford 2011), r the radial distance from +the planet center and H = 4373 km the scale height (see Table 1 +and Sect. 2.3). +We assume a range of height integrated Pedersen conductivities, +ΣP between 1 and 100 S. Together with the scale height we can +calculate the surface magnetic diffusivity with +η0 = Hµ−1 +0 Σ−1 +P , +(B.3) +where µ0 is the vacuum permeability. The expected range of sur- +face diffusivities is therefore η(1) +0 ≈ 3.5×1010 and η(2) +0 = 3.5×1012 +m2 s−1 for conductances of 100 S and 1 S, respectively. +In order to study the effect of magnetic diffusion on our simula- +tion results, we compare the simulated plasma variables B, p, ρ +and v within the magnetosphere of τ Boötis b with and without +magnetic diffusion near the planet at r = 2 Rp. We restrict our- +selves to simulations with open magnetospheres using the basic +stellar wind model. +The deviations of the 4 plasma variables simulated with diffu- +sion from those simulated without are displayed in Figs. B.1 +(η(1) +0 ≈ 3.5 × 1010 m2 s−1, ΣP = 100 S) and B.2 (η(2) +0 ≈ 3.5 × 1012 +m2 s−1, ΣP = 1 S) given in arbitrary normalized units. The finite +conductivity causes only very small deviations from our basic +model at r = 2Rp on the order of 10−3 for a diffusive model with +1 S Pedersen conductance. Maximum deviations at r = 2Rp for +the diffusive model with ΣP = 100 S amount to factor of 10−5 +(Fig. B.1) compared with the non–diffusive model. +We thus conclude that the effect of magnetic diffusion on our re- +sults (i.e. the magnetospheric Poynting flux) at 2 Rp above the +ionosphere/atmosphere of the planet is negligible. +Appendix C: Structure of the interactions: +Parameter study +In order to illustrate the effect of stellar wind density and pres- +sure on the magnetospheric structure, we show xz–plane slices +(Figs. C.1) similar to Fig. 1 for the extreme cases (0.05 ρsw +and 100 ρsw) of our parameter study (Sects. 3.2.3 and 4.1.1). +The plots show plasma velocity and thermal pressure in the near +space environment of τ Boötis b. We use spherical coordinates +Article number, page 16 of 19 + +Magnetospheric Poynting flux as function of radial distance from center +1019 +open Ms +semi-open Ms +closed Ms +Power [W] +1018. +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +r[Rp]F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +Fig. B.1: Deviations of resistive MHD simulations from our simulations discussed in the main text using the basic model (Table 1). The results +correspond to simulations with Pedersen conductance of 100S. The four panels display magnetic field (upper left), pressure (upper right), density +(bottom left) and velocity (bottom right) maps at r = 2Rp. +Fig. B.2: Deviations of resistive MHD simulations from our simulations discussed in the main text using the basic model (Table 1). The results +correspond to simulations with Pedersen conductance of 1S. The four panels display magnetic field (upper left), pressure (upper right), density +(bottom left) and velocity (bottom right) maps at r = 2Rp. +to numerically describe the space environment around τ Boötis +b. These coordinates have mathematical singularities along the +pole axis. For exceptionally small upstream plasma densities, +this can lead to numerical artifacts (i.e. jumps for the scalar vari- +ables along the pole axis in the PLUTO code). This is visible in +Figure C.1 (left). In the vector fields and thus the Poynting fluxes +this discontinuity is negligible. This effect occurring at the ex- +tremely low upstream conditions thus does not have an effect on +the conclusions of this work. +The effect of stellar wind pressure and density on the size of the +magnetosphere is clearly visible. The day side magnetopause lo- +cation is ∼ 5 Rp for the lowest density case (0.05 ρsw, left panels) +Article number, page 17 of 19 + +magnetic field B +1e-6 +0 +2 +25 +50 +1 +75 +[Bo] +0 +100 +B +125 +-1 +150 +-2 +175 +0 +50 +100 +150 +200 +250 +300 +350 +Φ[]thermal pressure p +1e-6 +0 +3 +25 +50 +1 +75 +[d] +0 +100 +d +-1 +125 +-2 +150 +175 +-3 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma density p +1e-5 +0 +25 +2 +50 +75 +1 +[] +100 +d +0 +125 +150 +-1 +175 +0 +50 +100 +150 +200 +250 +300 +350 +[。 Φplasma velocity v +1e-5 +0 +2.0 +25 +1.5 +50 +1.0 +75 +[] ^ +0.5 +e +100 +125 +0.0 +150 +-0.5 +175 +50 +150 +250 +200 +300 +0 +100 +350 +[。] Φmagnetic field B +0 +0.0003 +25 +0.0002 +50 +0.0001 +75 +[Bo] +0.0000 +100 +B +125 +-0.0001 +150 +-0.0002 +175 +0 +50 +100 +150 +200 +250 +300 +350 +Φ[]thermal pressure p +0 +25 +0.0004 +50 +75 +0.0002 +[d] +100 +d +0.0000 +125 +150 +-0.0002 +175 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma density p +0 +0.003 +25 +50 +0.002 +75 +[d] +0.001 +100 +Q +125 +0.000 +150 +-0.001 +175 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma velocity v +0 +0.0015 +25 +50 +0.0010 +75 +v[v。] +0.0005 +θ 100 +125 +0.0000 +150 +-0.0005 +175 +50 +0 +100 +150 +200 +250 +300 +350 +[。] ΦA&A proofs: manuscript no. Elekes_Saur_2022_TauBooB +Fig. C.1: Plasma interaction for density ρsw = 0.05 ρ0 (left panels) and density ρsw = 100 ρ0 (right panels). Displayed are velocity fields (colored +arrows, left colorbars) and plasma pressure (color contours, right colorbars) in the xz-plane for the open MS (θB = 0◦,top) and closed (θB = 180◦, +bottom) MS case. Closed and open magnetospheric field lines are colored in magenta. +and ∼ 1.2 Rp for the highest density case (100 ρsw, right panels). +Also visible is the sub–fast nature of the interaction in the low- +est density case, where no bow shock forms upstream and thus +incident plasma flow is perturbed prior to intersecting with the +magnetosphere. +Appendix D: On the neutral atmosphere model +assumptions and its interaction with the plasma +In this section we discuss some properties and assumptions on +our neutral atmosphere model presented in Sect. 2.3 as well as +how the atmosphere affects the plasma focusing on our basic +model (Table 1). +The aim of this work is not a detailed description of the iono- +sphere of the planet, but its magnetosphere and larger space en- +vironment. Our simplistic atmosphere model only acts through +collisions with the plasma and affects the photo–ionization rate +(i.e. plasma production) which both directly scale with the neu- +tral particle density (Eqs. 5, 6). Due to the exponential decrease +of neutral particle density (Eq. 5) with radial distance from the +planet the atmosphere’s effect on the plasma population drasti- +cally decreases with increasing altitude. The denser the neutral +atmosphere the more the magnetospheric plasma is decelerated +which leads to plasma pile up around the planet mimicking an +ionosphere. Figure D.2 shows plasma density profiles within the +magnetosphere as function of radial distance from the center. +The black dotted line represents the plasma density along the +polar axes. Red and magenta lines represent equatorial upstream +and downstream profiles respectively. The orange solid line de- +notes the neutral particle density according to Eq. 5, the green +dotted line shows the corresponding ion–neutral collision fre- +quency. As visible in Fig. D.2 the effect of ion–neutral collisions +and thus the amount of plasma pile up is drastically reduced +above an altitude near 1.3 Rp. There the neutral particle density +is n(r = 1.3Rp) ≈ 5 × 1010 m−3. The ion–neutral collision fre- +quency is ∼ 0.5 s−1 at the surface and drastically decreases with +altitude. In our studies we focus on the region 1Rp above the +planets surface and thus above the ionospheric shell. At r = 2 Rp +the collisions are negligible because the collision frequency has +decreased to about ∼ 10−7 s−1. +The large scale height of 4373 km increases the extent of the at- +mosphere but is needed in order to sufficiently resolve the atmo- +sphere in our model. However, the atmospheres of Hot Jupiter +exoplanets are expected to be strongly inflated due to intense +stellar irradiation (e.g. Vidal-Madjar et al. 2003) which is par- +tially mimicked by the large scale height. The surface neutral +particle density n0 corresponds to an atmosphere pressure near +10−3 nbar assuming a temperature between 1000 and 2000 K. +Article number, page 18 of 19 + +600 +16 +Open magnetosphere +8 +14 +500 +6 +12 +4 +400 +10 +2 +v [km/s] +[Ro] +[edr] +0 +300 +8 +N +p +-2 +6 +200 +-4 +4 +-6 +100 +2 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x [Ro]300 +16000 +Open magnetosphere +8 +14000 +6 +250 +12000 +4 +200 +10000 +2 +v [km/s] +[Ro] +[edr] +0 +150 +8000 +N +d +-2 +6000 +100 +-4 +4000 +-6 +50 +2000 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x[Ro]300 +16 +Closed magnetosphere +8 +14 +250 +6 +12 +4 +200 + 10 +2 +v [km/s] +[Ro] +[edr] +: C +0 +150 +8 +N +d +-2 +6 +100 +-4 +4 +-6 +50 +2 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x [Ro]300 +16000 +Closed magnetosphere +8 +14000 +250 +6 +12000 +4 +200 +10000 +2 +v [km/s] +[Ro] +[edn] +0 +150 +8000 +N +d +-2 +6000 +100 +-4 +4000 +-6 +50 +2000 +-8 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +0 +0 +x[Ro]F. Elekes and J. Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b +Fig. D.1: Plasma density maps over a sphere with r = 2 Rp for the open (left panel) and closed (right panel) MS case. +Fig. D.2: Plasma density profiles along the polar axis (black dotted +line), upstream (red) and downstream x–axis (magenta). The blue solid +line denotes the mean plasma density profile. The orange solid line +shows the neutral particle density according to our atmosphere model +(Eq. 5). The green solid line (right y–axis) denotes ion–neutral collision +frequencies. +Here the mixing ratio of plasma (hydrogen ions) and neutral +particles is roughly 50 %. The neutral atmosphere consists of +molecular hydrogen which is the most abundant molecule in all +solar system gas giants. +In order to demonstrate the minor role of the neutral atmosphere +at r = 2 Rp we show plasma density maps over a shell at this +altitude in Fig. D.1. The left and right panels display the den- +sity map of the open and closed MS case, respectively. There the +neutral particle density is n(r = 2 Rp) ≈ 4.6×105 m−3 according +to Eq. 5 and Fig. D.2 leading to a neutral–plasma mixing ratio +of about 10−6. Regions of high plasma density (i.e. at the day +side) indicate regions where the interaction between the neutral +atmosphere and plasma is strongest (at r = 2 Rp) in terms of +ion–neutral collisions and recombination. However, the effect of +the neutral atmosphere on the plasma is an order of magnitude +lower than in regions within the ionosphere below ∼ 1.3 Rp. +Article number, page 19 of 19 + +plasma density p +1el1 +0 +8 +25 +7 +50 +6 +p [H+ /m3] +75 +5 +100 +4 +125 +3 +150 +2 +175 +0 +50 +100 +150 +200 +250 +300 +350 +[。] Φplasma density p +1e12 +0 +25 +1.4 +50 +1.2 +75 +100 - +1.0 +Q +125 +0.8 +150 +175 +0.6 +0 +50 +100 +150 +200 +250 +300 +350 +[。] ΦMagnetosphericplasmaandneutraldensities +100 +nmean +npolar +1013 +10-2 +nupstream +ndownstream +nneutral +10-4 +10-6 +1012 +10-8 +col +10-10 +10-12 +1011 +10-14 +2 +3 +4 +5 +6 +r[Rp] \ No newline at end of file diff --git a/p9E4T4oBgHgl3EQfUQzO/content/tmp_files/load_file.txt b/p9E4T4oBgHgl3EQfUQzO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..313f8b6b67bd312f8cb55c675825c7cd7018eaa3 --- /dev/null +++ b/p9E4T4oBgHgl3EQfUQzO/content/tmp_files/load_file.txt @@ -0,0 +1,2063 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf,len=2062 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB ©ESO 2023 January 13, 2023 Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes1 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur1,2 1 Institute of Geophysics and Meteorology, University of Cologne, Pohligstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3, D-50969 Köln e-mail: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='elekes@uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='de 2 e-mail: jsaur@uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='de Accepted: January 8, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The first tentative detection of a magnetic field on the Hot Jupiter type exoplanet τ Boötis b was recently reported by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field was inferred from observations of circularly-polarized radio emission obtained with the LOFAR telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The observed radio emission is possibly a consequence of the interaction of the surrounding stellar wind with the planet’s magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We aim to better understand the near space environment of τ Boötis b and to shed light on the structure and energetics of its near-field interaction with the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We are particularly interested in understanding the magnetospheric energy fluxes powered by the star-planet interaction and in localizing the source region of possible auroral radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We perform magnetohydrodynamic simulations of the space environment around τ Boötis b and its interaction with the stellar wind using the PLUTO code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We investigate the magnetospheric energy fluxes and effects of different magnetic field orienta- tions in order to understand the physical processes which cause energy fluxes that may lead to the observed radio emission given the proposed magnetic field strength in Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Furthermore we study the effect of various stellar wind properties, such as density and pressure, on magnetospheric energy fluxes given the uncertainty of extrasolar stellar wind predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We find in our simulations that the interaction is most likely super–Alfvénic and energy fluxes generated by the stellar wind– planet interaction are consistent with the observed radio powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetospheric Poynting fluxes are of the order of 1–8 ×1018 W for hypothetical open, semi-open and closed magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These Poynting fluxes are energetically consistent with the radio powers in Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) for a magnetospheric Poynting flux–to–radio efficiency > 10−3 when the magnetic fields of the planet and star are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of lower efficiency factors the magnetospheric radio emission scenario is according to the parameter space modeled in this study not powerful enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A sub–Alfvénic interaction with decreased stellar wind density could channel Poynting fluxes on the order of 1018W towards the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of a magnetic polarity reversal of the host star from an aligned to anti-aligned field configuration, expected radio powers in the magnetospheric emission scenario fall below the observable threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Furthermore we constrain the possible structure of the auroral oval to a narrow band near the open-closed field line boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strongest emission is likely to originate from the night side of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' More generally, we find that stellar wind variability in terms of density and pressure does influence magnetospheric energy fluxes significantly for close-in magnetized exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetohydrodynamics (MHD) – Methods: numerical – Planet-star interactions – Planets and satellites: aurorae 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Introduction Recently, tentative measurements of auroral radio emission from the hot Jupiter exoplanet τ Boötis b were obtained with the Low Frequency Array (LOFAR) (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These obser- vations might be considered the strongest evidence so far of an intrinsic magnetic field on a planet outside the solar system if the emission indeed originates from the planets vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Then they also imply that τ Boötis b possess a magnetosphere which inter- acts with its surrounding stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The radio observations by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021), if confirmed, thus help pave the way for the field of extrasolar space physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this work we therefore use properties from the observed radio signals to derive new con- straints on the space environment around τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The massive Hot Jupiter τ Boötis b (Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1997) is a very good candidate to remotely observe a powerful interaction of a stellar wind with an exoplanet’s magnetosphere for several rea- sons: With ∼16 pc, the τ Boötis system is relatively close to the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The planet orbits its host star τ Boötis A at a short distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 astronomical units (Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Addi- tionally, its large mass (> 5MJupiter) may cause its exobase to remain close to the planet, leading to a magnetosphere (MS) not completely filled with dense plasma and thus allowing for ra- dio emission to be produced efficiently and to escape the planets vicinity (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Daley-Yates & Stevens 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The tentative radio measurements obtained with LOFAR com- prise two signals allegedly from the vicinity of τ Boötis b (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The circularly polarized signals were detected in the 21 – 30 MHz and 15 – 21 MHz frequency bands, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The emission possibly originates from gyrating, ener- getic electrons precipitating towards the planetary polar regions, emitting radio waves generated through the Electron Cyclotron Maser Instability (ECMI), which is expected to be the domi- nant mechanism for exoplanetary radio emission (Zarka 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Treumann 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' From these signals the planetary magnetic field strength can be inferred directly since the emission frequency corresponds to the local electron gyro-frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The existing observations are consistent with the expectations on the emit- ted power from the radio-magnetic Bode’s law (Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018), on the polarization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Article number, page 1 of 19 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05015v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='EP] 12 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB circular polarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2005) and on the frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' slightly above Earth’s ionospheric cutoff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2007b, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Griessmeier 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The mea- sured radio signal however needs multi-site follow up observa- tions preferably at various radio wavelengths to confirm and to further constrain the magnetic field environment of τ Boötis b (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In addition to radio emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Grießmeier 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 2007) other indirect meth- ods have been proposed to detect and constrain the magnetic fields of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These are, for example, optical signatures in the stellar chromosphere by means of Ca II H&K line excess emission induced by star-planet interactions (SPI) (Cuntz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Shkolnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2003, 2005, 2008), and asymmetries in near-ultraviolet stellar light curves together with UV absorption signatures caused by transiting planetary bow shocks (Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2010, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Llama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The SPI and transit related observations can lead to false-positives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Kislyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Preusse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Kopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Bisikalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Gurumath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Route 2019) due to sets of model assumptions involved in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Radio observations in con- trast, can directly constrain the magnetic field amplitude and are therefore less susceptible to false–positives (Grießmeier 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The success of radio observations has been demonstrated in the past in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For example, Jupiter’s magnetic field was discovered through radio observations (Franklin & Burke 1958) before spacecraft confirmed it with in-situ magnetometer measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Since τ Boötis b may be the first exoplanet with a directly observed magnetic field it provides an unique opportunity to constrain the space environment around this exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' However, various properties of τ Boötis b are unknowns such as radius, size and extend of its atmosphere above the 1 bar level as well as stellar wind parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' τ Boötis A is a solar-like F7 IV-V star (Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2001), coronal temperature and pressure might therefore be comparable to the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The coronal base density and consequently the stellar wind mass loss rate poses the most uncertain free parameter of previous studies of the stellar wind from τ Boötis A (Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Recently new constraints on stellar winds of M dwarf stars were reported utilizing astrospherical absorption signatures induced by the interaction of the stellar wind with the interstellar medium (Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The question naturally arises if stellar wind – planet interactions might as well produce observable signatures capable to provide constrains on stellar wind properties such as density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' mass loss rate) or pressure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' temperature), which will be addressed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The proximity of Hot Jupiters to their host stars can potentially cause sub–Alfvénic star – planet interactions, which is expected to produce observable signatures in the stellar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' chromo- spheric emission Ca II H & K line emission Shkolnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2003, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2019) or the planetary atmosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' au- roral radio emission Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Kavanagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Such magnetic SPI in exoplanetary systems were excessively studied by means of magnetohydrodynamic simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Preusse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2006, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zhilkin & Bisikalo 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Varela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018, 2022), partly with a focus on the far–field interaction incorporating self–consistent stellar wind models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strugarek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2014, 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidotto & Donati 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Modeling of super-Alfvénic stellar wind – magnetosphere interactions was sparsely done to our knowledge although a large number of close-in exoplanets might be exposed to, at least temporarily, super-Alfvénic wind conditions (Zhilkin & Bisikalo 2019) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' orbiting outside the Alfvén surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of super-Alfvénic or, more precisely, super-fast magnetosonic stellar wind flows a bow shock forms upstream of the planetary magnetosphere because of the flow being faster than the fastest magnetohydrodynamic wave mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this case the planet does not interact with the star since waves are not able to propagate upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This might be the case for τ Boötis b, as the planet is likely exposed to a super–fast stellar wind according to Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The generation of radio emission from exoplanets, its properties and dependence on stellar wind and planetary parameters was studied intensively using numerical simulations (Nichols & Milan 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Varela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Daley-Yates & Stevens 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Kavanagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2020) for other or generic exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' However, little to no emphasis was given to studying the detailed spatial structure and energetics of magnetospheric Poynting fluxes that ultimately deliver the available electromagnetic energy capable of driving planetary auroral emissions at various wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In order to better understand the space environment around τ Boötis b we perform magnetohydrodynamic (MHD) simulations of the near space environment of τ Boötis b and its magnetic field interacting with the surrounding stellar wind plasma using the PLUTO code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stellar wind model is based on wind sim- ulations (Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016) driven by magnetic surface maps derived from magnetic measurements of τ Boötis A (Marsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Jeffers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field estimate of the planet’s intrinsic field, based on the tentative magnetic field strengths derived by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021), is used to model the planetary magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We specifically aim to better understand the magnetospheric energy fluxes around τ Boötis b and of Hot Jupiter type exoplanets in general that are exposed to similar stellar wind conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also address the question on how stellar wind variability in the time-independent case affects magnetospheric Poynting fluxes and therefore possible radio powers generated by the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The paper is structured in the following way: An overview of the physical model to describe the plasma interaction τ Boötis b with the surrounding stellar wind is given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The numerical setup is summarized in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1, details about the stellar wind model can be found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The τ Boötis b model is described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the subsequent Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3 we show our results, starting with a general description of the interaction in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1, followed by a study of the spatial structure of Poynting fluxes in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Then we discuss the energetics of the interaction in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2, where we also compare possible radio emission output with the observations by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results are followed by a discussion about the role and importance of the stellar wind to power the energy fluxes in the magnetosphere of the exoplanet in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' At last, we discuss possible auroral radio emission and its detectability in scope of stellar wind variability (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Numerical simulation In this section we introduce our physical model and the numerics to describe the interaction of τ Boötis b and its intrinsic magnetic field with its surrounding stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The MHD model together with the numerical model and coordinate system are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We introduce the stellar wind that is included as bound- ary condition for the plasma variables in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 followed by Article number, page 2 of 19 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b the description of parametrizations of physical processes intro- duced by the planet and its atmosphere in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Method We performed single-fluid ideal, non-resistive and non-viscous MHD simulations using the open-source code PLUTO (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4) in spherical coordinates (Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The MHD equa- tions to solve are ∂ρ ∂t + ∇ · �ρv� = Pmn − Lmp (1) ∂ρv ∂t + ∇ · � ρvv + p − BB + 1 2 B2 � = −(Lmp + νnρ)v (2) ∂Et ∂t + ∇ · �(Et + pt)v − B(v · B)� = −1 2(Lmp + νnρ)v2 −3 2(Lmp + νnρ) p ρ +3 2(Pmn + νnρ)kBTn mn (3) ∂B ∂t − ∇ × [v × B] = 0 , (4) where ρv is the momentum density, v the velocity, ρ the mass density, pt the total pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' magnetic and thermal) and p the thermal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B is the magnetic flux density, −v×B in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4 is the electric field in the ideal limit with infinite electrical con- ductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Et is the total energy density, Et = ρe+ρv2/2+B2/2µ0, and e the specific internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The system is closed by the equation of state in the form p = ρe(γ − 1), where γ is the ratio of specific heats for the adiabatic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As for magnetic diffusion we do not include a diffusion term in the induction equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4) but point out that numerical dif- fusion, especially for coarse grids such as in our simulation, in- troduce numerical diffusion sufficient to allow for reconnection (see Varela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' To justify this assumption we performed test simulations incorporating magnetic diffusion and found it to not influence the results of this paper significantly (see appendix B for a detailed discussion on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We include plasma production, P, and loss terms, L, (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1-3) to account for photo-ionization, dissociative recombination to- gether with associated momentum and internal energy transfer between neutral atmospheric and magnetospheric plasma parti- cles as well as ion-neutral collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that the neutral species is not simulated and altered by the interaction with the ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Details on how plasma production and loss are mod- eled can be found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The mass of plasma particles is denoted by mp and mn describes the mass of neutral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We assume the plasma to completely consist of ionized hydro- gen atoms, mp = mH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The atmosphere only consists of neutral molecular hydrogen, mn = mH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The conservative form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 – 4 are integrated using a ap- proximate hll-Riemann solver (Harten, Lax, Van Leer) with the diffusive minmod limiter function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The ∇ · B = 0 condition was ensured by the mixed hyperbolic–parabolic divergence cleaning technique (Dedner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The spherical grid consists of 256 non–equidistant radial, 64 and 128 equally spaced angular grid cells in θ and φ dimension re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The radial grid is divided into three regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' From 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 planetary radii (Rp) the grid contains 10 uniform cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Af- ter that from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 to 12 Rp the next 150 cells increase in size with a factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='01 per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The last 96 cells from 12 Rp towards the outer boundary at 70 Rp increase gradually with a factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The positive x axis points parallel to the relative veloc- ity v0 of the stellar wind in the frame of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stellar wind magnetic field is assumed to be perpendicular to v0 and is anti-parallel to the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The y axis completes the right handed coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Co-latitude θ is measured from the positive z- axis, longitudes Φ are measured from the positive y axis within the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The origin is located at the planetary center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We run all simulations for approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 h physical time until a quasi steady-state is reached in the vicinity of the planet (r < 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Small fluctuations cannot be avoided although larger scale struc- ture and dynamics within the MS remain already almost constant after approximately 2 hours physical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Stellar wind model The derived stellar wind parameters from Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) resemble those of the sun, such as the polytropic index, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 (Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011), and the stellar coronal base tem- perature, which is not well constrained by observations, is set to 2 × 106 K as typical value for the solar coronae (Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field of τ Boötis A was studied excessively dur- ing several epochs and magnetic surface maps as well as several magnetic polarity reversals were observed (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Fares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2009, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The coronal base density remains an ed- ucated guess based on a comparison of emission measure (EM) values obtained from X-Ray spectra of τ Boötis A (Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Maggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Due to the uncertainty of the base density estimate, different stellar wind densities will be investi- gated separately in the scope of magnetospheric Poynting fluxes and possible radio powers in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stellar wind is applied through constant in–flow boundary conditions at the upstream hemisphere (Φ = 0 to 180◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field is assumed to be perpendicular to the relative ve- locity v0 of the wind (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' parallel to the negative z axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The in–flow velocity of the plasma, which we call the relative ve- locity v0, is parallel to the x–axis and is composed of the radial velocity of the wind vsw and the orbital velocity of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The adopted plasma parameters of the wind are summarized in Table 1 which were averaged over the several epochs studied by Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' τ Boötis b model We assumed a radially symmetric neutral atmosphere with a scale height of H = 4373 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Thus, the scale height extends over three radial grid cells and consequently the neutral atmo- sphere is sufficiently resolved within the numerical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We as- sume an atmosphere consisting of molecular hydrogen as it is, followed by helium, the most abundant constituent of the Jovian atmosphere (Atreya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The collisional cross-section is assumed to be σin = 2 × 10−19 m2 for H+ – H2 collisions with momentum transfer for low–eV relative velocities between the colliding particles (Tabata & Shirai 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In our simulations the collision frequency is νin ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 s−1, so that νin = ¯vσinn(r), where ¯v ≈ v0 denotes a typical velocity in the system and nn(r) is the atmosphere number density as function of radial distance from the center, nn(r) = nn,0 exp �Rp − r H � , (5) where nn,0 = 8 × 1012 m−3 is the surface number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Based on test studies, we found that for nn,0 ≈ 8 × 1012 m−3 the ion- Article number, page 3 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB Table 1: Physical simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Details of the stellar wind and planet model are discussed in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Symbol Value Source Note τ Boötis b Planet radius Rp 72875 km Wang & Ford (2011) Theoretical Orbital period Porb 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='31d Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Wang & Ford (2011) Semi-major axis a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 au Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (1997) Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' surface density nn,0 8×1012 m−3 – Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' scale height H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='06 Rp – 3 radial grid cells Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' flux density (eq) Bp 455µT Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) Average of observations 1,2 Basic Stellar wind model Therm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' pressure psw 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 × 10−5 Pa Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) Ion density nsw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 × 1012 m−3 Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) Velocity vsw 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 km s−1 Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) Stellar frame of reference Relative velocity v0 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='98 km s−1 – v0 = � v2sw + 4π2 × a2/P2 orb Magnetic flux density Bsw 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='715 µT Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) Alfvén Mach number MA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='36 Fast mode Mach number M f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='9 Plasma beta β 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='31 neutral collisions nearly completely bring the incoming plasma flow to a halt in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This results in plasma pile up in form of a shell around the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Increasing the density would thus not produce a larger interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We use a simplified description of photo-ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We neglect the shadow zone exerted by the planet’s body and parameterize plasma production through photo-ionization using only the ra- dial dependence of the neutral atmosphere density, P(r) = νionnn(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (6) The radial symmetric ionization partially mimics some night side ionization through electron impact ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the photo-ionization frequency of hydrogen exposed to a solar-like UV radiation environment at a distance of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 AU from the star we take the value from Kislyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2014), νion = 6 × 10−5 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Plasma loss is introduced through recombination of hydrogen ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The loss term therefore depends on the plasma density, L(r, t) = αn(r, t)(n(r, t) − nsw) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (7) Plasma loss is switched off if the plasma density falls below the background density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' n(r, t) ≤ nsw) as stellar wind ions and electrons recombine significantly slower due to the higher electron temperatures in the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Given an electron tem- perature of roughly Te ≈ 7500 K for a Hot Jupiter exoplanet’s ionosphere with semi-major axis of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 AU around a sun-like star derived by Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2010) and using the formula of Storey & Hummer (1995), α = 4 × 10−12 �300K Te �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='64 cm3 s−1 , (8) we find the hydrogen ion recombination rate, α, to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1×10−19 m3 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Further discussion about the underlying assumption about our atmosphere model can be found in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Recent tentative auroral radio measurements from τ Boötis b give a first observational constraint on its magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) found the polar surface magnetic flux density Bp to lie between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='7 G for two right-handed circularly polarized signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We assume a dipole field and adopt the average value of both Stokes V+ signals (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021), Bp = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 G, for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Furthermore we study the effect of dipole orientation on the stellar wind – planet interaction through simu- lating an open (0◦ tilt), semi-open (90◦ tilt) and closed MS (180◦ tilt), where the tilt is measured with respect to the negative z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The various tilts are realized by rotating the stellar back- ground magnetic field accordingly so that the planetary dipole axis is always parallel to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Given the strong magnetic variability of τ Boötis A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' several magnetic polarity reversals were observed as well as a chromospheric activity cycle in terms of S–indices of roughly 240 days Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Fares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2009, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mittag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Jeffers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018) we are also able to study the effect of the host star’s mag- netic field topology on the stellar wind–planet interaction and associated magnetospheric energy fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field is implemented using the insulating– boundary method by Duling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2014) which ensures that no radial electric currents exist within the insulating boundary, which we assume to be the planet’s neutral atmosphere below its ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Results In this section we first present results of our modeling which pro- vides an overview of the space plasma environment of τ Boötis b (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Then we study in detail the magnetospheric Poynting fluxes in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Structure of the interaction The simulated plasma velocities and pressures according to the basic model (Table 1) are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 for the open (θB = 0◦), semi-open MS (θB = 90◦) and closed MS (θB = 180◦) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field tilt θB is the angle between the external field (parallel to the z axis) and the planet’s magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that, due to the symmetries chosen in our model studies, the stellar wind and intrinsic magnetic field are not inclined with respect to the z-axis, therefore we also show projected field lines (black solid lines) in the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Color contours denote plasma pressure in µPa (right colorbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Arrows represent velocity com- ponents, their magnitudes are color coded (left color bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The length of arrows indicate the magnitudes of the shown compo- Article number, page 4 of 19 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1: Velocity fields (colored arrows, left colorbars) and plasma pres- sure (color contours, right colorbars) in the xz-plane for the open MS (θB = 0◦, top), semi-open (θB = 90◦, middle) and closed (θB = 180◦, bottom) MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Projected stellar wind magnetic field lines are indi- cated as black solid lines within the xz-plane parallel to the ambient magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Closed and open magnetospheric field lines are colored in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Spatial dimensions are given in units of planetary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The intrinsic magnetic field and its corresponding MS poses an obstacle to the stellar wind flow coming from negative x – di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The flow outside the MS is super-Alfvénic (MA = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='36) and super-fast magnetosonic (Mf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6) (see Table 1), where M f = v0/(v2 A + c2 s)1/2, with the sound speed cs = � γpsw/ρsw, polytropic index γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 (Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016) and Alfvén ve- locity vA = Bsw/ √µ0ρsw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The super-fast interaction enforces a bow shock to be formed roughly 5Rp in front of the planet fol- lowed by a fairly thick magnetosheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Since no wave is able to propagate upstream, the stellar wind plasma is unperturbed until the bow shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The structure of the MS strongly depends on the internal field orientation as visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 with an in- crease of overall MS size towards higher magnetic axis tilts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the open and semi-open MS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 top and middle plot re- spectively) two magnetic lobes form, separated by a thin plasma sheet, where open magnetic field lines connect to the stellar wind field several planetary radii downstream (not shown in the plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The day side magnetopause, defined by the location of the last closed field line, lies between 3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 Rp, while the night side magnetopause is located at roughly 5 Rp for the open and semi- open MS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The downstream side magnetopause is very narrow in the z – direction as expected due to the mag- netic field lines convected downstream together with the stel- lar wind flow and due to the magnetic stresses stretching the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The closed MS case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1, bottom plot) has a night side magnetopause lying several planetary radii (∼ 17Rp) downstream (not shown in the plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' While the day side mag- netopause is controlled by the stellar wind thermal and magnetic pressure balanced with those exerted by the planet’s surround- ings, the night side MS is influenced by reconnection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the merging of planetary with stellar wind field lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetic re- connection is most efficient for a magnetic moment parallel to the ambient field (here the z-axis), therefore the fraction of open planetary field lines connected to the star decreases significantly with an intrinsic field moment being directed anti-parallel to the stellar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As the stellar wind plasma primarily penetrates the MS along magnetic field lines, the amount of plasma and ther- mal pressure decreases as well with increasing magnetic axis tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that, as visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 (bottom), the magnetosphere is completely closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This is due to the perfect anti–parallel align- ment of the planetary and stellar wind magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Within the MS the flow velocity is strongly reduced and has weak upstream components in the negative x direction due to magnetic tension exerted on planetary field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetic re- connection takes place at the upstream and downstream side where velocities, both within and outside the MS, are strongly enhanced due to acceleration through released magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Velocities are slightly larger at the flanks of the MS compared to the upstream side and exceed the initial stellar wind velocity at the downstream side where stellar wind as well as planetary field lines merge together again and accelerate the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Thermal pressures are strongly enhanced within the magne- tosheath, where stellar wind plasma is decelerated abruptly and compressed, so that kinetic energy is converted into heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Plasma may penetrate the MS along open magnetic field lines in the polar cusps where pressure is enhanced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The cusps act as channels for plasma transport into the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' There is a trend towards lower pressures in the cusps for increasing magnetic axis tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This is directly connected to the amount of stellar wind plasma advected towards the planet as the amount of injected plasma is related to the ability of magnetic field lines to merge with the ambient field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This becomes increasingly difficult for planetary magnetic moments having components anti-parallel to the ambient field, therefore the area fraction of open magnetic field lines and thus the size of the plasma injection channel is maximal for a completely open MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Here pressures up to 160 µPa can be reached while the closed MS case shows pressures up to roughly 90 µPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Article number, page 5 of 19 400 Open magnetosphere 160 8 350 6 140 300 4 120 250 2 100 v [km/s] [Ro] [edn] 0 200 80 N d 2 150 60 4 100 40 6 50 20 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x [Ro]400 Semi-open magnetosphere 160 8 350 6 140 300 4 120 250 2 100 v [km/s] [Ro] [edr] 0 200 80 N p 2 150 60 4 100 40 6 50 20 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x [Ro]400 Closed magnetosphere 160 8 350 6 140 300 4 120 250 2 100 v [km/s] [Ro] [μPa] 0 200 80 N p 2 150 60 4 100 40 6 50 20 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x [Ro]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2: Mercator projections of the Poynting flux (upper row), plasma velocity (middle row) and absolute values of Poynting flux components parallel to the unperturbed planetary field plus small background field from stellar wind (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results are shown at an altitude of one planetary radius above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The left column displays maps for the open MS case (θB = 0◦), the right column for the closed MS case (θB = 180◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Arrows indicate normalized angular components, color contours denote radial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Red dashed lines indicate the location of the OCFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynting fluxes and aurorae We are interested in understanding the electromagnetic coupling of the stellar wind with the magnetosphere of the exoplanet, its atmosphere and ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The energy fluxes associated with these electromagnetic coupling processes provide the energet- ics for the auroral emission from the exoplanet’s magnetosphere at radio and other wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore we study the Poynt- ing flux to calculate the maximum available electromagnetic en- ergy fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We describe the spatial structure of magnetospheric Poynting fluxes in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Then we study the energetics of the interaction and effects of stellar wind variability on magne- tospheric energetics in the subsequent sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Spatial structure We first describe the spatial structure of the Poynting fluxes and plasma velocities within the MS as displayed in the top and mid- dle row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The plots show Mercator projec- tions of the angular vector components over a spherical shell with radius 2Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The angle θ denotes the co-latitude, and φ the longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Arrows indicate the orientation of vectorial quanti- ties and color contours their magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Positive values indicate radial components pointing away from the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Red dashed lines represent the open – closed planetary field line boundaries (OCFB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetic field lines with both foot points on the plan- etary surface are closed field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Each field line having only one foot point on the planet is an open field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The OCFB sep- arates areas with open from areas with closed field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Thus, the OCFB also represents the magnetopause at that specific ra- dial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The open and closed MS cases are shown left and right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The Poynting vector S can be rewritten in the ideal MHD case using the convective electrical field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2013) S = E × B µ0 = −v × B × B µ0 = B2 µ0 v⊥ , (9) which is bodily carried by the plasma flow perpendicular to magnetic field lines, denoted by the perpendicular velocity v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The Poynting flux describes the transport of magnetic enthalpy, which is a factor of two larger than the magnetic energy density B2/2µ0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the remainder of this work we mostly present Poynting fluxes, but need to consider the factor of two when we compare magnetic energy densities with Article number, page 6 of 19 Poynting flux S 0 60 25 50 40 [W/m²] 75 20 100 0 S 125 20 150 175 40 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦPoynting flux S 0 25 10 50 5 75 0 100 S 125 150 10 175 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma velocity v 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 25 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 75 [km/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 100 > 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 150 175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma velocity v 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 25 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 75 v [km/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 e 100 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 150 175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦPoynting flux iS ·: eBo 0 10 25 8 50 [W/m2 ] 75 6 eBo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' e 100 4 125 S 150 2 175 0 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦPoynting flux IS ·eBo 0 10 25 8 50 - [zw/M 75 6 S ebo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 100 - 4 125 S 2 150 175 0 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b thermal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' internal energy) or kinetic energy densities based on their flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For both, the open and closed MS case, flow velocities are strongly reduced at the upstream (φ = 0–180 degrees) and downstream (φ = 180 – 360 degrees) side down to speeds below 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This occurs due to interaction with the intrinsic mag- netic field and momentum transfer with the neutral atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The OCFB is located at roughly θ ≈ 30◦ and 150◦ in the open MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Due to the perfectly anti–parallel configuration of the stellar wind and planetary magnetic field no open planetary field lines exist in the closed MS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This has also been observed in sub-Alfvénic simulations using this field topology (Ip et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strugarek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Open Magnetosphere model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2 left): The very nar- row vertical extend of the downstream closed field line region of the open MS is caused by magnetic tension due to the magnetized stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Highest velocities are found within the open field line region mainly at the downstream side where plasma is accelerated downstream through magnetic tension on open lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strong Poynting fluxes occur where plasma velocities have strong components perpendicular to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' They are found within the open field line region mainly at the downstream side with outward directed Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Com- paratively strong Poynting fluxes, but directed toward the planet, are located on the upstream side near the magnetopause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Within the closed field line region and especially near the equator Poynting fluxes mostly vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Closed Magnetosphere model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2 right): In the closed MS model highest velocities can be found near the planetary poles confined to an area below 25 degrees co–latitude and similar in the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This high velocities are caused by tension on high latitude closed field lines that are strongly stretched towards the downstream side by the stellar wind and reach up to 17 planetary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynting fluxes oriented away from the planet are confined to narrow bands encircling the high latitude polar regions between 40 and 80 degrees co–latitude and similar in the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' At the upstream side Poynting fluxes vanish near the equatorial regions due to plasma flow being mainly aligned with planetary field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Inward oriented Poynting fluxes occur near the polar axis slightly shifted towards the downstream side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We now study the Poynting fluxes parallel to the unper- turbed background magnetic field because in the solar system magnetospheres they are considered the root energy fluxes from which a small fraction can be converted into auroral radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynting fluxes provide the energy from which wave–particle interaction can draw energy to accelerate elec- trons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' for Jupiter Hill 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The resulting energetic electrons then can be subject to the electron maser instability (Treumann 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The interaction of the stellar wind with τ Boötis b’s magnetic field perturbs the magnetic and electric field, which causes the Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' To quantitatively assess the associated Poynting flux, we therefore use the unperturbed magnetic background field B0 = Bp,0 + Bsw (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the initial dipole and stellar wind field) to calculate the Poynting flux on this field, S · eB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The unit vector eB0 points in the direction of unperturbed magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These projections give insight on where electromagnetic energy is transported either through propagating magnetic disturbances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Alfvén waves) or convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2 Table 2: Integrated magnetospheric Poynting fluxes for different mag- netic field topologies Model Dipole tilt Pa [W]a Pa||B0 [W]b Open MS 0◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='46e+18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='73e+17 Semi-open MS 90◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='77e+18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='88e+17 Closed MS 180◦ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='91e+17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='09e+17 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Integrated Poynting fluxes over a sphere with radius r = 2Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (a) Magnetospheric Poynting flux (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11) (b) Magnetospheric Poynting flux parallel to B0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12) shows |S · eB0| for the open MS (left) and closed MS (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that only absolute values are shown in the plots in order to clearly identify zero or near-zero power densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strongest energy transport along unperturbed field lines occurs over narrow bands encircling the polar open field line regions at the flanks of the planet where velocities are nearly perpendicular to the magnetic field as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Moreover, the spatial structure of Poynting fluxes along B0 is strictly symmetric with respect to the equator (at θ = 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A significant amount of energy is transported parallel to the unperturbed field within the polar open field line regions in the open MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Parallel energy fluxes reach values up to 10 Wm−2 at the flanks of the planet just outside the closed field line regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynting fluxes up to 9 Wm−2 are found at the downstream side, above the OCFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For both, open and closed MS model, strongest convected energy can be found extensively in high latitude regions due to high velocities perpendicular to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Here the planetary field lines are most mobile in a sense that they are bent over towards the downstream side by the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the closed MS parallel Poynting fluxes up to roughly 9 Wm−2 can be found directly at the planetary poles slightly shifted towards the upstream side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' At lower latitudes parallel Poynting fluxes up to 6 Wm−2 are confined to narrow bands at the flanks of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Auroral emission is expected to be strong where Poynting fluxes are large, hence near the OCFB (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' mostly confined to the L = 3–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 shell at the upstream side) and in the polar regions for both MS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' They vanish completely along the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Generally said Poynting fluxes are significantly weaker and confined to the small polar regions for the closed magnetosphere model compared to the open MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the open MS model strong parallel Poynting fluxes cover the whole open field line area with their maximum at the flanks of the planet in contrast to the closed MS where the regions of strongest parallel Poynting fluxes are partitioned into smaller areas around the planetary poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Energetics of the interaction To estimate the total available Poynting flux, which serves as the root energy flux, we assume for simplicity that the radio emis- sion is generated in a shell 1 Rp above the surface of the exo- planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This particular choice is inspired by the fact that radio emission around Jupiter and other solar system planets arises from altitudes about 1Rp (or larger) above the planet’s surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Hess & Zarka 2011) where strong electron ac- celeration takes place (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' for Jupiter Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynt- ing fluxes within the magnetosphere of τ Boötis b only vary little as function of distance from the planet (see appendix A for a dis- cussion on the choice of r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Available electromagnetic power for possible conversion into electron acceleration and radio emission is given by the diver- Article number, page 7 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB gence of the Poynting flux in this shell with Volume V, Pa = � V ∇ · SdV = � Ashell S · ˆndAshell , (10) where Ashell is the surface area of the shell and ˆn the surface normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' To investigate the maximal Poynting flux which can be dissipated in the shell we assume that the Poynting flux entering the shell from above or below is dissipated within the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For mathematical simplicity we further let the thickness of the shell grow infinitesimally small such that Pa = � Asphere |Sr| dAsphere , (11) with Asphere the area of the sphere located at 2 RP from the cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In physical terms it means that the possible dissipation in the shell can be supplied with energy fluxes from below the shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' coming from the planet’s ionosphere) or from above the shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' coming from the magnetosphere or stellar wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Ul- timately, the energy flux is coming from the stellar wind, but the energy flux can be reflected or converted in the ionosphere and can be redirected away from the planet again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This inte- grated Poynting flux serves as a proxy for maximum available electromagnetic energy dissipated within an auroral acceleration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Alternatively, we integrate the components of the Poynting flux parallel to the unperturbed magnetic field, B0 = B − δB, where δB denotes the magnetic field perturbation generated by the in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These Poynting fluxes take into account the energy flux of which a fraction can directly contribute to particle ac- celeration and powering the ECMI driven emission, Pa,|| = � A |S · eB0| dA , (12) where eB0 is the unit vector pointing in direction of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We re- fer to this Poynting flux component as the auroral Poynting flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As opposed to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11, Pa,|| (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12) serves as a more realistic estimator for calculating auroral energy dissipation since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11 includes significant contribution of convected energy which is likely not converted into particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Table 2 summa- rizes integrated Poynting fluxes according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11 (third col- umn) for all three intrinsic magnetic field orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The 4th column shows integrated Poynting fluxes along the unperturbed field (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Integrated Poynting fluxes range from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 × 1018 down to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='9×1017 W for the open towards the closed MS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Poynting fluxes along the unperturbed field (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12) amount to ∼ 9 × 1017 and 1017 W for the open and closed MS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The effect of magnetic topology on convected energy within the magnetosphere is therefore significant as the powers differ by al- most one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetic stress due to the stellar wind interaction can work on the magnetospheres less strongly if the magnetosphere is closed, thus giving rise to weaker flows and therefore weaker convected Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The trend is simi- lar for Poynting fluxes along B0, Pa,||, but here the powers are reduced by almost an order of magnitude below the integrated total Poynting fluxes Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Influence of stellar wind variability on magnetospheric energetics For modeling the space environment of τ Boötis b, the properties of its surrounding stellar wind carry very large uncertainties, in particular the stellar wind density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3: Schematic illustrating the several steps from incident stellar wind energy flux towards auroral radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The transfer function Ta (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) describes the conversion from incident stellar en- ergy to auroral Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The conversion efficiency from auroral Poynting fluxes (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12) to radio emission, ϵa, implicitly includes the efficiency of electron acceleration and the ECMI mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The steps within the pink shaded area are not included in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Brown ar- rows indicate physical processes, blue arrows denote model parameters quantifying energy conversion and the magenta arrow the radiometric scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2012), the coronal base density was estimated by choosing the electron density so that it can reproduce electron measure (EM) observations of τ Boötis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The energy fluxes within the MS are powered by and limited by the maximum in- cident power of the stellar wind flow transferring onto the mag- netospheric obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka (2007) found that the observed radio power of solar system planets is nearly a constant fraction of the incident kinetic and magnetic energy convected through the ob- stacle’s cross section, πR2 mp, where Rmp is the magnetospheric stand-off distance of the magnetized planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This energy is uti- lized in perturbing the topology of the planets magnetic field which in turn results in currents induced by changes in magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore the incident power controls the energetics within the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic Poynting flux, PB, and the kinetic energy flux, Pkin, convected through the obstacle’s cross section can be calculated as follows Pkin = 1 2ρ0v2 0 · πR2 mpv0 (13) PB = B2 sw µ0 πR2 mpv0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (14) Additionally, the thermal energy flux should be considered as well as it cannot be neglected for close-in orbits where stellar wind temperature, T, pressure and density are high, Ptherm = 3 2nswkBT · πR2 mpv0 , (15) with nsw being the stellar wind particle density and v0 denoting the incident stellar wind velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetopause distance Rmp can be obtained from an equi- librium between stellar wind and planetary ram (pram = ρswv2 0), magnetic (pB = B2/2µ0) and thermal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Both, the magne- tospheric thermal and ram pressures are considered negligible, thus pram,sw + pB,sw + ptherm,sw = pB,pl, where the subscript sw stands for stellar wind and pl for planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetopause dis- tance (or magnetospheric stand-off distance) can then be calcu- Article number, page 8 of 19 Magnetospheric emission scenario Radiometric Bode law Stellar wind incident energy flux Auroral radio emission thermal kinetic Poynting flux Transfer function T, Auroral Poynting fluxes Electron Cyclotron Maser Instability Electron accelerationF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4: Top: Analytically calculated incident kinetic (red dashed line), Poynting (red dotted line) and thermal (red dashed–dotted line) en- ergy fluxes convected through the magnetospheric cross section πR2 mp as function of stellar wind density at the position of τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The pressure varies accordingly through p0 ∝ ρ0 (see Table 1 for the ba- sic stellar wind model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The powers are calculated using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 13 - 15 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The red solid line represents the sum of the three energy com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The Magnetosphere stand-off distance Rmp (blue solid line) is calculated with the same set of ρ0 and p0 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stand-off distance can be obtained from the right y-axis, given in terms of plane- tary radii Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magenta arrow indicates the convected power in the limiting case where the plasma density in the astrosphere approaches zero and the planet is only exposed to the stellar dipole field (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 for discussion of this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Red crosses show simulated Poynt- ing fluxes along the unperturbed magnetic field over a spherical shell with radius 2Rp for the closed MS (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Red stars indicate simu- lated Poynting fluxes for the closed MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Bottom: Transfer function Ta = Pa/Pi as a function of ρ0 and p0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the conversion from incident energy to auroral Poynting fluxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The scaling behavior according to power laws of Ta is indicated by spectral indices above the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' lated from Rmp = RpB1/3 p � 2µ0 �1 2ρswv2 0 + psw � + B2 sw �−1/6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (16) All parameters can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The parameter v0 refers to the relative velocity between the stellar wind and planet and Bp to the planetary surface magnetic field at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A certain fraction of the total incident power, Pi = Pkin + PB + Ptherm , (17) intersecting with the magnetopause can eventually be converted for the generation of radio emission within the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The fraction of total incident energy, ϵ, that may result in ra- dio emission is expected to range from 10−5 to ∼ 3 × 10−3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Pradio = ϵPi) in the radiometric Bode law (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3) (Zarka 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We point out that various efficiencies for con- verting incident energy flux into electromagnetic radiation are discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For example, the efficiency of con- version from magnetospheric, auroral Poynting fluxes to radio emission, which accounts for the efficiency of electron acceler- ation through wave-particle interaction and the efficiency of the electron-cyclotron maser, should be separated from the generic efficiency factor obtained from the radiometric Bode law (Zarka 2007) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For Jupiter’s radio emission the efficiency for conversion from magnetospheric, auroral Ponyting fluxes to radio emission is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 – 3×10−4 (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We denote this efficiency by ϵa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As the stellar wind density is the most uncertain parameter we performed simulations with densities ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 · ρ0 to 100 · ρ0 (see Table 1 for the basic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' To get an understand- ing on how stellar wind variability affects the structure of the magnetosphere we show xz–plane slices similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 for the two extreme cases (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05·ρ0 and 100·ρ0) in the appendix C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We do not solve a self consistent stellar wind model but instead follow the solar wind solution of Parker (1958) where the solution of the solar wind velocity v(r) is independent of the coronal base density nc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this solution, stellar mass and base temperature control v(r) and T(r), where r is the distance from the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For simplicity of the parameter study of this sub- section, we choose an isothermal approach and change the den- sity together with the pressure p0 (and therefore T) according to p0 ∝ ρ0 (see equation 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We therefore keep the temperature constant and consequently the velocity does not change accord- ing to Parker (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Given the average stellar mass loss rate of τ Boötis A of ˙M ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 × 10−12 M⊙ yr−1 estimated by Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016) (see also our basic model, Table 1) the parameter range of stellar wind densities considered in this parameter study translates to mass loss rates between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='15 × 10−13 M⊙ yr−1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 × 10−10 M⊙ yr−1 since ˙M ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We integrate the Poynting flux along the unperturbed field over a spherical shell with radius 2Rp (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12) in order to obtain an understanding of how much incident energy flux is eventu- ally converted to auroral Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Resulting powers are shown as red crosses and stars (open and closed MS respectively) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4 as a function of ρ0 (and p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The simulated convected energy fluxes follow the trend of incident energy flux estimates (red solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4) but are reduced to fractions of the total incident energy flux, Pi, between 15 and 20 % for the open MS and between 1 and 5 % for the closed MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Changes to the stellar wind density ρ0 (and in the same manner p0) affect the incident power inflicted on πR2 mp but also influence the magnetospheric cross section in an opposite manner, as it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4 (blue solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetospheric stand-off distance scales according to Rmp ∝ (ρswv2 0+psw)−1/6 and the incident energy flux with Pi ∝ ρsw + psw, therefore the incident energy flux increase dominates over the effect of a shrinking MS due to increasing thermal and kinetic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This is also validated by our sim- ulation results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4), implying a approximately linear scaling of auroral Poynting fluxes with ρ0 and p0 at least in the regime between 3 × 1011 and 3 × 1013H+ m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Below the point where stellar wind magnetic energy dominates over thermal and kinetic energy near 1011H+ m−3, auroral Poynting fluxes seem to satu- rate near 2–3 ×1017 W (open MS) and near 1–2×1016 W (closed MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Above 3 × 1013H+ m−3 the increase of auroral Poynting fluxes with ρ0 (and p0) deviates further from the course of inci- Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' page 9 of 19 Incident stellar wind powers and magnetopause stand-off distance as function of po and po Po [μPa] 10-1 100 101 102 103 1020 Basic model 5 1019 4 1018 [W [Rp] Power 3 1017 2 P with Parker magnetic fieid Pkin Ptherm 1016 Pkin + PB + Ptherm 1 integrated S ·Bo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' simulations (open Ms) integrated S·Bo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' simulations (closed Ms) PB with stellar dipole field 1015 109 1010 1011 1012 1013 1014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='25 α p-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='15 Transfer function Ta = Pa/P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (open MS) Transferfunction Ta= Pa/P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (closed MS) fer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='10 Transf α 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='00 - 109 1010 1011 1012 1013 1014 [s-w/+ H] odA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB dent flux, implying a saturation towards 1019 W (open MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This, however, has to be validated further through future simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Discussion In this section we discuss the importance of the stellar wind on magnetospheric energetics (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) and on possible auroral radio emission (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Importance of the stellar wind on magnetospheric energetics In the following sections we study the conversion of incident to dissipated power within the magnetosphere (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) as a function of stellar wind density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also discuss the limiting case of an absent stellar wind (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Stellar wind variability, its effect on magnetospheric energetics and scaling behavior of auroral Poynting fluxes We separate the considered stellar wind density and pressure range introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4 in two regimes: Regime 1 ranges from a vanishing stellar wind up to a density at roughly 1011H+ m−3 where kinetic and thermal energy fluxes fall below the persistent magnetic energy flux which dominates the flow (compare red curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Above roughly 1011H+ m−3 the flow is super-Alfvénic (MA ≈ 2) and super-fast (Mf ≈ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The interaction is super-Alfvénic for the whole parameter space used in our simulations and sub-fast only for the lowest simu- lated density (ρsw = 7 × 1010 H+ m−3, MA ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The incident energy nearly stagnates below ρ0 = 1010 H+ m−3 (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Below this point the incident energy flux asymptotically approaches its minimum at 8×1017 W as we assume that only the plasma density decrease but the incident magnetic field is kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this regime it can be expected that the stellar wind magnetic field solution transitions from the Parker solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B ∝ r−2) to a pure stellar multipole (here dipole) solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Bsw = Bstar ∝ r−3) with decreasing stellar wind density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Even- tually, when the stellar wind density hypothetically approaches zero, only the dipolar stellar magnetic field interacts with the planetary magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This limiting case will be separately discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Regime 2 ranges from roughly 1011H+ m−3 up to arbitrarily high stellar wind densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Here kinetic and thermal energy fluxes dominate the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We will now focus on this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Consider- ing the total energy flux convected through the magnetospheric cross section πR2 mp, Ptotal (red solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4), we observe a nearly constant efficiency of conversion from incident stellar wind energy towards magnetospheric Poynting fluxes at auro- ral altitudes (we assumed r ≈ 2Rp) with increasing density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We calculate the transfer function Ta as the conversion ratio from total incident energy flux Pi to the simulated auroral Poynting fluxes parallel to the unperturbed field (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12), Pa (red crosses and stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4), within the MS, such that Ta = Pa/Pi (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3 for a schematic illustrating the role of Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The trans- fer function also contains information on the magnetic topology and thus the efficiency of reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The transfer function is displayed in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the open MS, auroral Poynting fluxes decrease as a function of ρ0 and p0 according to an approximately constant ratio up to ∼ 3 × 1013H+ m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For higher densities and pressures, the transfer function scales with an exponent of ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4, indicating a decrease of efficiency for conversion from incident to auroral energy fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As the MS is increasingly compressed due to stronger ram and thermal pres- sures, the magnetopause eventually crosses the spherical shell with r = 2Rp after a critical density of ∼ 3 × 1013H+ m−3 and pressure of 3 × 10−4 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the closed MS, Ta behaves differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The transfer func- tion instead increases slightly from 7 × 1010 to ∼ 3 × 1013H+ m−3 following a power law with exponent ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The opposite behavior compared to the open MS transfer function might be a consequence of a geometry where less reconnection occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stronger ram and thermal pressure exert stronger tension on planetary field lines which in turn release stronger energy fluxes during reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Beyond the MHD limit The more the stellar wind density decreases, the emptier the heliosphere of τ Boötis A becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In analogy with the solar wind, the stellar wind density upstream of the magnetosphere of τ Boötis b may change by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The solar wind density is observed to vary by more than two orders of magni- tude (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Chané et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012) such that Earth’s bow shock can disappear and the Earth develops Alfvén wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the hy- pothetical limit when the density approaches zero, τ Boötis b will still be exposed to the stellar magnetic field Bτ(r) (which then decreases according to Bτ(r) ∝ r−3 instead of Bsw ∝ r−2) and will propagate through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore the incoming Poynting flux of the star does not, in contrast to kinetic and thermal energy flux, vanish and is maintained by the relative motion between the stellar and planetary magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of an empty helio- sphere, the interaction around τ Boötis b is not magnetohydrody- namic any more, but turns electromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Then the movement of τ Boötis b within the external magnetic field of the star is a unipolar or homopolar interaction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' a moving conductive object with external field similar to a current generator in clas- sical electromagnetism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the case of a stellar magnetic field rotating at the same speed as the planet orbits around its host star (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' in case of total orbital and spin synchronization), non- existent changes in magnetic flux lead to a system where no work can be done by the magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore no currents are induced and the magnetosphere remains energetically silent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Al- though it is expected that close-in exoplanets are in nearly tidally locked rotation, Murray & Dermott (2000) suggest in their text- book that completely synchronous rotation might not be possi- ble if the planet has no permanent magnetic quadrupole moment and its eccentricity is not zero, which is the case for τ Boötis b (Wang & Ford 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Taking the rotation period of τ Boötis A, Pτ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 d (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016), and the sidereal rotation period of τ Boötis b, Porb = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='31 d (Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Wang & Ford 2011), we can calculate the relative velocity between the stellar and planetary magnetic field v0 = 2πa × (1/Pτ − 1/Porb) ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 km s−1, where a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 AU is the semi-major axis of τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We use an average surface magnetic flux density of τ Boötis A obtained by Marsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Jeffers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2018), Bτ,0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 G, and calculate the flux density at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='046 AU using the dipole for- mula, Bτ(r = a) = Bτ,0 · (a/Rτ)−3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='73 G, where Rτ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='43R⊙ is the stellar radius (Bonfanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetospheric stand-off distance in this case is defined purely by the balance of stellar and planetary magnetic pressure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Rmp = Rp · (Bτ(a)/Bp)−1/6 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 Rp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (18) Article number, page 10 of 19 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b Now we can calculate the stellar magnetic power convected on the MS using PB = Bτ(a)2/µ0 · v0πR2 mp ≈ 8 × 1015 W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (19) This value is included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4 as magenta arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We can con- clude the following: (a) The maximum possible magnetospheric stand-off distance is reached in complete absence of a stellar wind and if the stellar and planetary magnetic fields are anti-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' If the stellar and planetary fields are parallel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' open MS) the planetary field lines are all connected to the stellar field and no magnetopause can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the closed MS case the magnetopause lies at roughly 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 planetary radii in the upstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (b) Even if there is no stellar wind, the magnetic interaction be- tween the stellar and planetary magnetic field still has the po- tential to drive an interaction with an available power limit of roughly 8×1015 W due to the relative motion of τ Boötis b in the stellar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Radio emission would still be possible al- though very weak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' corresponding radio fluxes at Earth’s position would be far below today’s telescope sensitivity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Consider- ing Poynting flux–to–radio power efficiencies between 10−4 and 10−2, radio powers can reach values between 1011 and 1014 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These emitted powers exceed the strongest radio sources within the solar system by several orders of magnitude, with Jupiter’s aurora being the strongest radio emitter (Pradio ≈ 1010 − 1011 W Zarka 2007), although Jupiter’s emission is, in contrast to close- in exoplanets, powered by internal, rotationally driven mecha- nisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The transition from a magnetosphere interaction with a stellar wind field (according to the Parker solution Parker 1958) to an interaction with a pure dipolar stellar magnetic field goes with an energetic transition followed by a decrease of maximum emitted radio power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This might pose a possible opportunity for constraining stellar wind densities in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Solving a self consistent stellar wind model and comparing auroral Poynting fluxes for different stellar wind base densities could reveal the critical density range where the transition from a stellar mag- netic field dominated electrodynamic interaction to a stellar wind dominated magnethydrodynamic interaction takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Com- paring the magnetospheric Poynting fluxes and corresponding radio powers with possible future observations could reveal if the stellar wind density lies below or above the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Magnetospheric Poynting fluxes and auroral radio emission In this section we discuss possible radio emission scenarios (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1), study how magnetic topology of the interaction as well as stellar wind variability affects auroral radio emission output (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also discuss the possibility of a sub– Alfvénic emission scenario (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3) as well as a rotation driven magnetosphere of τ Boötis b in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4, followed by a discussion on possible source regions and radio frequencies of auroral emission in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' On the different radio emission scenarios There are several scenarios capable of generating observable ra- dio emission that must be distinguished from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (1) If the stellar wind is sub–Alfvénic, Alfvén waves are able to propagate upstream towards the stellar atmosphere along Alfvén wings and possibly drive electron acceleration and radio emis- sion in the stellar vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Local radio emission within the mag- netosphere can also be generated in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We will refer to this scenario as sub–Alfvénic emission scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This scenario is discussed briefly in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2) If the stellar wind is super–Alfvénic, no MHD wave is able to propagate upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stellar wind–magnetosphere interac- tion however drives Poynting fluxes within the magnetosphere which may to some extent generate auroral radio emission (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We refer to this scenario as the magnetospheric emis- sion scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Due to the stellar wind being super–Alfvénic for all simulations we will focus on the magnetospheric emission (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (3) In a rotation dominated magnetosphere scenario the rotat- ing planet and its magnetic field causes co–rotation of magneto- spheric plasma that, at some point, breaks-down due to conser- vation of angular momentum if radial mass transfer takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This co–rotation breakdown exerts magnetic stresses on the field lines that are the root cause of auroral Poynting fluxes which in turn drive auroral radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This scenario is discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Effects of magnetic field tilt and stellar wind variability on auroral radio emission – Magnetospheric emission scenario Figure 5 shows radio powers as function of magnetic axis tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Radio powers are obtained from multiplying integrated auro- ral Poynting fluxes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 12), which serve as a proxy for maximum available electromagnetic energy that is transported along magnetic field lines, with efficiency factors for converting magnetospheric Poynting fluxes to radio–power, ϵ, ranging from 10−4 to 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This range covers proposed (Zarka 2007) and ob- served efficiency factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' ϵ ≈ 10−4 for Jupiter Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The modeled magnetic field tilt can also be interpreted as stellar magnetic field orientation within this work, allowing us to study the effect of varying stellar magnetic field polarity on magnetospheric Poynting fluxes and limits for associated radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Radio powers within the limits inferred from observa- tions by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) lie within the gray shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' It is visible that efficiency factors in the range of ϵ ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 − 1) × 10−2 deliver radio powers most consistent with observations if the MS is open or at least semi open given the basic model (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This indicates that the efficiency of auroral Poynting fluxes driv- ing electron acceleration and the electron cyclotron maser emis- sion may be higher in the magnetosphere of τ Boötis b than in the Jovian magnetosphere (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Electric fields gen- erated by reconnection between stellar wind and planetary mag- netic field lines are expected to contribute significantly to pow- ering electron acceleration and therefore the ECMI (Jardine & Collier Cameron 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In our studies we find reconnection to indirectly play an important role (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5) because auroral Poynt- ing fluxes and consequently radio powers drop by nearly an or- der of magnitude from an open to a closed MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This is due to magnetic stress exerted by the stellar wind interaction being less strong for closed magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The polarity of τ Boötis A’s magnetic field switches every approximate 360 days (Fares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Shorter cycles in magnetic activity levels (by means of S- indices) were also observed (Mengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A difference of half an order of magnitude to almost an order of magnitude can therefore be caused by a polarity reversal of τ Boötis A’s mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This results in radio emission whose observability is expected to fluctuated periodically in a nearly 1-year cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that the stellar wind magnetic field strength was kept con- stant in our parameter study, although in reality the field strength may vary strongly and influence produced radio emission signif- Article number, page 11 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5: Expected radio powers as function of planetary magnetic axis tilt using auroral-to-radio power conversion efficiencies between 10−4 and 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The auroral Poynting fluxes S · B0 are integrated over a spherical shell with radius 2Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The gray shaded area represents observational limits given by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 6: Radio flux (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 20) as function of stellar wind density and pres- sure for different efficiency factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Colored solid and dashed lines rep- resent fluxes for the open and closed MS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Observational limits (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021) are indicated by the gray shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Horizontal gray lines display theoretical sensitivity limits of the LOFAR telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The real sensitivity for Stokes V signals obtained from Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2019) is plotted as yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The vertical gray line marks the basic model (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' icantly (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The emitted radio flux observed at Earth’s position can be calculated with (Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2005, 2007b) Φ = Pradio Ω δν d2 , (20) where Ω is the solid angle of the beam and δν the emission bandwidth that is approximately equal to the maximum gyro- frequency (Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2007b), νg,e ≈ 24 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We assume a solid angle of Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 sr similar to Jupiter’s decametric radio emission (Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The distance to the τ Boötis sys- tem is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We calculate the radio flux for both, the open and closed MS model, as a function of ρ0 and p0 with radio efficien- cies ϵa = 10−4–10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Solid and dashed colored lines represent radio fluxes originating from open and closed MS’s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The gray shaded area again denotes the range of observed radio fluxes form Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Horizontal gray lines indicate theoretical sensitivity lim- its of the LOFAR telescopes for 20MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As stated by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2019), the realistic sensitivity might be slightly lower for circularly polarized (Stokes V) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We therefore include the expected sensitivity calculated by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2019) as yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 6 indicate that radio efficiencies between ∼ 3 × 10−3 and ∼ 10−2 are most consistent with the tentative observations (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The efficiency accounts for several steps from conversion of auroral Poynting fluxes to ra- dio emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' wave-particle interaction, electron accelera- tion and ECMI), therefore an efficiency of the order of 1–10% might be unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The efficiency for Jupiter’s auroral emis- sion is roughly ϵa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3–3 × 10−4 (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021), therefore ϵa = 10−2 might be too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Moreover high plasma densities within the magnetosphere injected by the dense stellar wind and due to strong irradiation which results in high ionization rates and inflated atmospheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' for ν And b see Erkaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2022), may further decrease the ECMI efficiency or even prevent it (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Daley-Yates & Stevens 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As- suming the radio efficiency to lie near 10−3, the radio flux from a closed MS falls below the detection threshold (yellow line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore, in case of a polarity reversal of τ Boöts A’s magnetic field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' from aligned with the planetary field to anti-aligned), the radio signal would not be observable anymore in case of ra- dio efficiency equal or below ∼ 3 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of ϵa ≈ 10−4 all radio fluxes for the basic model fall below the sensitivity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The observability increases, however, if stellar wind density and pressure rises, rendering ϵa = 10−3–10−2 to possible efficiencies to observe emission from open and closed MS’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Additionally, the ECMI efficiency (Treumann 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017) as well as efficiency of electron acceleration through wave-particle in- teraction decreases dramatically with increasing plasma density (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018), making the higher density and pressure regime a less likely scenario to explain the tentative observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As the pressure rises, the magnetopause is getting closer to the planet, reducing the space of magnetospheric diluted plasma regions be- tween the magnetopause and atmosphere where radio emission might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We therefore conclude that the basic model (verti- cal gray line) and slightly different configurations represent the most likely scenarios if the emission is indeed generated by stel- lar wind driven auroral Poynting fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this case radio emis- sion is only observable, if the stellar wind and planetary mag- netic fields are aligned (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the magnetosphere is open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Given the high efficiencies (ϵ > 10−3) needed by our model in order to generate radio emission which is consistent with the tentative observations, the magnetospheric emission scenario might not be energetic enough to explain the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Sub–Alfvénic emission scenario Although there is no sub-Alfvénic interaction within the param- eter space we considered, the possibility of such an interaction and its consequences on possible radio emission should not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' By choosing a stellar wind density of ρsw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='03 ρ0 we find an Alfvénic Mach number of MA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this case Alfvén waves may propagate back to the star through Alfvén wings connecting the planetary magnetic field with the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The electromagnetic energy channeled through this flux tube can be Article number, page 12 of 19 Estimatedradiopowerasfunctionofmagnticfieldtiltforthebasicmode Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='range,Turneret al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2o21) = 10-4 1016 =10-3 =3 × 10-3 [M] = 10-2 Pradio 1014 1013 o°(OpenMS) 90°(Semi-open MS) 180*(ClosedMS) MagneticfieldtilteB[deareesiEstimatedradiofluxasfunctionofpo(po) 104 Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='range,Turneret al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) 103 Radio flux Φ [mly] 102 101 Open MS, =10 Closed MS, = 104 Open MS, = 3 × 10 3 Closed MS, = 3 × 10~3 100 Open MS, = 10-2 Closed MS, = 102 theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 6 stations Basic model theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 24 stations (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 40 stations theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 48 stations estimated sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Stokes V @ 10 MHz (Tumer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2019) 10-1 1011 1012 1013 1014 Po [H + /m-3]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b calculated using the model of Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2013), PS = 2πR2 mp (¯αBsw cos θ)2 µ0 MAv0 , (21) where θ = 0◦ is the angle which describes the deviation of the flow from being perpendicular to the stellar wind magnetic field, Rmp ≈ 5Rp the magnetospheric stand–of distance and ¯α the in- teraction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Due to the planet presumably possessing an ionosphere which favors a strong plasma interaction we choose ¯α ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Using v0 and Bsw from our basic model (Table 1), the energy flux channeled through Alfvén wings parallel to mag- netic field lines amounts to PS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 × 1018 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This energy flux may contribute to electron acceleration and consequently to ECMI driven radio emission in the space environment near the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Taking the same range of Poynting flux to radio efficiencies (10−4–10−2) the expected radio power ranges from roughly 1014 to 1016 W which partially overlaps with the range of radio power deduced by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The emission frequency would, however, be much lower compared to the signals observed by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) due to the low magnetic field strength of τ Boötis A (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Additionally Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) state that stellar emission would be less strongly circularly polar- ized compared to planetary emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Nevertheless the root en- ergy flux able to possibly drive radio emission near the stellar at- mosphere from such a sub–Alfvénic interaction exceeds the au- roral Poynting fluxes in the magnetospheric emission scenario by roughly half an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Consequently this scenario might provide energetically a possible explanation for the ob- served radio emission if the interaction is indeed sub–Alfvénic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This scenario requires the stellar wind density to be reduced by at least a factor of ∼ 30 compared to our basic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This, how- ever, could well be within the realm of possible densities given the observed density variations of the solar wind even tough the intervals of variability are short (Chané et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Is auroral radio emission from τ Boötis b rotationally driven?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' One of the possible generators for auroral emissions is radial mass transport within the magnetosphere as it is the case for Jupiter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Hill 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Plasma is moving radially outwards due to centrifugal forces within an exoplanet’s MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For small distances from the planet (or stel- lar host) the plasma co-rotates with the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' After a certain distance ˆL in units of planetary radii, ˆL × Rp, the co-rotation breaks down due to conservation of angular momentum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the plasma orbits its host with a smaller angular velocity than those from the rotation of the host).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The relative velocity exerts mag- netic stresses on the frozen-in field lines, the tension accelerates plasma along the field lines back to the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The distance of co- rotation breakdown can be estimated using the co-latitude of the region, where auroral emission occurs, θa, following Hill (2001), sin(θa) = ˆL−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) derived a so called auroral power potential for magnetized rotation dominated hosts, S pot = B2 hostΩ2 hostR2 host , (22) where the subscript host refers in our case to the planet τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The hosts magnetic flux density is denoted by Bhost, Ωhost ≈ 2π/Porb = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 × 10−5 rad s−1 is its angular velocity and Rhost = Rp the planet’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that the discussion in this subsection is only based on theoretical considerations since we neglected planetary rotation in our MHD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Following Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) we can calculate the energy flux due to mass transport, Pmag, by using Pmag = S pot �������π R2 p ˆL2 ������� Σp , (23) where Σp denotes the Pedersen conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We assume a value of 1 S and 10 S that are of the same order of magnitude as observed conductances near Jupiter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Hill 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strobel & Atreya 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Hinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For a hot Jupiter exoplanet the ionospheric Pedersen conductivity might be larger due to the larger ionization (Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' However, also the height of the resulting ionospheric layer and thus the mobility of the electrons and ions within the atmosphere also plays an impor- tant role for the values of its conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' To our knowledge no dedicated study for the conductance of τ Boötis b is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 22 the distance of co–rotation break- down ˆL is proportional to Σ1/4 (Hill 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Thus Pmag in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 23 depends effectively weaker on Σp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Pmag is proportional to Σ1/2 p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In our simulations we integrate the ion component of the Pedersen conductivity along the z–axis at the poles from the surface (r = Rp) to r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 Rp where the plasma density peaks and find the height integrated Pedersen conductance to be of the order of 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We find, with aurora occurring at field lines with L-parameter smaller than ∼ 4 that have to lie within the MS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' ˆL ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5), the auroral power potential to be S pot ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 W m−2 S−1 and the magnetic power to lie between Pmag ≈ 1014 W and 1015 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The auroral power potential of Jupiter, for com- parison, is roughly 600 W m−2 S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The resulting energy flux between the magnetosphere and ionosphere lies near the Jovian value (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 × 1014 W Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This result is several or- ders of magnitude below the power of magnetospheric Poynting fluxes powered by the stellar wind – planet interaction in our model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1017 – 1018 W, Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore we can safely assume that the MS of τ Boötis b is not rotationally dominated which is in agreement with the expected slow rotation nearly synchronized with its orbit around the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Properties and source regions of possible auroral radio emission Auroral radio emission is mainly characterized by signal strength and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The peak frequency is defined by the local electron gyrofrequency, νg,e = |e|B 2πme , (24) where e is the electron charge, me its mass and B the local mag- netic flux density (Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2007a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We already discussed possible radio powers from τ Boötis b in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2, now we will focus on possible signal fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021) observed signals within the 21 – 30 MHz range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic flux density inferred from the max- imal signal frequency by the authors is almost certainly under- estimated since ECMI driven auroral radio emission typically arises from altitudes above the polar regions of the planet’s at- mosphere (Zarka 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Hess & Zarka 2011) due to three rea- sons: (a) High ionospheric electron densities cause the electron plasma frequency νp,e to exceed the local gyro frequency νg,e, thus prohibiting the transmission of emitted waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (b) radio wave amplification through the ECMI mechanism works most efficiently in regions where νg,e >> νp,e (Treumann 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Weber Article number, page 13 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017) and (c) the electrons are usually accelerated in re- gions where the plasma density along a field line is the smallest (or B/n is maximum) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' There wave particle interaction to accelerate electrons is maximum efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The electron plasma frequency is given by νp,e = � nee2 ϵ0me 1 2π , (25) where ne = ni is the particle density in a quasi-neutral plasma and ϵ0 the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The source regions of Jupiter’s radio emission lie several planetary radii above the surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 1998) where strongest electron acceleration is expected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The maximum gyro frequency in our model (directly above the magnetic poles) is ∼ 24 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Au- roral emission, however, typically arises from near the OCFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In our simulations we indeed see that strongest Poynting fluxes are confined to narrow bands encircling the OCFB (or magne- topause) near L=3 and L=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 shell field lines (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Since source regions of ECMI induced emission typically lie at higher altitudes the magnetic flux density and thus emission frequency is even further reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The conditions (a–c) for efficient ECMI driven radio emission might particularly be fulfilled for regions with low plasma density at the night side of the planet where the shadow exerted by the planet prevents photo-ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Since τ Boötis b is likely in tidally locked rotation around its host star, the night side might exhibit relatively low temperatures, favoring recombination of electrons and hydrogen ions as well as further electron capture processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the shadow zone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' permanent night side of the planet), ECMI induced emission might be more likely and might as well occur in lower altitudes, where the mag- netic field is stronger (and thus emission frequency is higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Conclusions In this study we modeled the magnetosphere of the Hot Jupiter exoplanet τ Boötis b and its interaction with the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In order to study magnetospheric energetics by means of Poynting fluxes we performed MHD simulations of the near space environment of τ Boötis b with the stellar wind being injected into the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The magnetic field of τ Boötis b is constrained by tentative observations of auroral radio emission (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The stellar wind model is based on simulations where observed magnetic surface maps of τ Boötis A were utilized (Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We investigated magnetospheric energetics, available electromagnetic power and limits of possible radio emission originating from above the planet’s polar regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also explored the possibility of a rotation dominated magnetospheric emission scenario as well as a sub–Alfvénic (stellar) emission scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The effect of stellar wind variability in terms of den- sity, pressure and magnetic field orientation on magnetospheric energetics was additionally investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We find that the stellar wind–magnetosphere interaction is super-fast and super-Alfvénic for the majority of stellar wind densities and pressures considered in this study (from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 × 1011 H+ m−3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 × 10−6 Pa to arbitrarily high stellar wind densities and pressures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the minimum density and pres- sure considered in this work (ρsw = 7 × 1010 H+ m−3 and psw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 × 10−5 Pa) the interaction is super–Alfvénic and sub–fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Maximum available electromagnetic energy convected within the magnetosphere amounts to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5×1018 W, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='8×1018 W and 7×1017 W for an open, semi-open and closed magne- tosphere configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Auroral Poynting fluxes at altitudes of 1Rp above the planet reach powers of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='7×1017 W down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1×1017 W for the open and closed magnetosphere, indicating a strong decrease of stellar wind energy converted to auroral Poynting fluxes as a function of magnetic field tilt (either stellar or planetary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We present Poynting flux maps of the planet’s near space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Strongest Poynting fluxes are confined to nar- row bands encircling the open-closed field line boundaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the magnetopause) that is displaced towards the planet’s downstream side due to magnetic stresses exerted by the stellar wind and its high plasma β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Electromagnetic energy transported parallel to unperturbed field lines are as well confined to bands near the magnetopause and the polar open-field line regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Considering the stellar wind density as free parameter, we choose values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 and 100 times the basic density, ρsw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 × 1012 H+ m−3 (Table 1), and change the stellar wind pressure in the same manner in order to keep the wind isothermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Comparing the incident total energy flux convected through the magnetospheric cross section, we find that the trans- fer function Ta (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the amount of stellar wind incident energy flux converted to magnetospheric Poynting fluxes parallel to unperturbed field lines) amounts to a fraction of roughly 20 % for the open and 2 % for the closed MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We find that Ta ∝ ρ−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='06 0 and Ta ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='57 0 for the open and closed MS respectively in the density range 7×1011 – 2×1013 H+ m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Increasing stellar wind ram and thermal pressure and thus incident energy flux causes the shrinking of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The reduced capability of the MS for receiving stellar wind energy is outperformed by the increasing pressure and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The scaling of Ta and thus energy output of the magnetosphere nearly stagnates from that point on where the magnetopause turns so small that the planetary atmosphere starts to interact with pre-bow shock and magnetosheath flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We investigated different radio emission scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A rotation dominated magnetospheric emission scenario could be ruled out due to the expected long rotation period of τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The estimated magnetic power generated by the planet’s rotation is on the order of Pmag ≈ 1014−15 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As this would require a very high Poynting flux to radio efficiency of 10−2, we expect that this scenario is not powerful enough to drive the tentatively observed radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We considered several efficiencies for conversion of auroral Poynting fluxes to radio emission and compared the results with observations together with theoretical limits based on the tentative observations by Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We find that generation of radio emission near τ Boötis b requires relatively high efficiencies (ϵa ≈ 1–5 × 10−3) compared to estimates from Jupiter’s rotationally driven aurora (up to∼ 3 × 10−4) (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2021) in order to explain the observed emission if we assume the emission to originate from the magnetosphere of τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Furthermore we find a strong dependence of radio emission observability on magnetosphere topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In case of a polarity reversal of τ Boötis A’s magnetic field from an aligned to anti- aligned (with respect to the planetary field) configuration, the estimated radio power falls below the observational limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We therefore expect an on–off nature of detectable radio emission in phase with τ Boötis A’s magnetic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also considered the case of a sub–Alfvénic interaction which requires the stellar wind density to be at least a factor of 30 lower compared to our basic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The Poynting flux channeled towards the host star due to this interaction amounts Article number, page 14 of 19 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b to ∼ 1018 W which slightly exceeds the power converted by the magnetospheric emission scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This project has received funding from the European Re- search Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 884711).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We furthermore thank the Regional Computing Center of the University of Cologne (RRZK) for providing computing time on the DFG-funded (Funding number: INST 216/512/1FUGG) High Performance Computing (HPC) system CHEOPS as well as support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We also thank Stefan Duling for providing 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V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2001, Ap&SS, 277, 293 Zhilkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' & Bisikalo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2019, Astronomy Reports, 63, 550 Zhilkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' & Bisikalo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2020, Astronomy Reports, 64, 563 Article number, page 15 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB Appendix A: Poynting fluxes as function of radial distance from the planet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1: Magnetospheric Poynting flux as function of radial distance from the planet’s center calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11 for the open (blue), semi–open (orange) and closed (green) magnetosphere model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Here we study the dependence of magnetospheric Poynting fluxes as a function of radial distance from the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The par- ticular choice of integrating Poynting fluxes over a spherical shell with radius r = 2Rp is based on the fact that radio emis- sion around Jupiter and most solar system planets with intrinsic magnetic fields originates from altitudes of about 1 Rp or higher above the planets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Zarka 1998) rather than from within the ionosphere or even further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We do not have evidence if this translates to extrasolar planets but simply assume so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For the ECMI mechanism to work efficiently the ratio between electron gyro frequency and plasma frequency, νg,e/νp,e, must be signif- icantly larger than 1 which turns the magnetospheric region at high altitudes to favored radio source regions where plasma den- sities are strongly reduced compared to other regions of the mag- netosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (Zarka 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Treumann 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Considering the plasma and neutral particle density within the modeled magnetosphere we also chose a location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the ra- dius) where ion–neutral collisions are significantly reduced (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' above the ionosphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The favored radius according to this con- straint lies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 and 3 Rp (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In order to show that the choice of r = 2 Rp or any other ra- dial distance within the magnetosphere does not influence the results and the derived conclusions significantly (despite control- ling the emission frequency since stronger magnetic fields cause higher gyro frequencies) we integrated the Poynting flux (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 11) for shells with radii between 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 Rp above the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 for the open (blue), semi- open (orange) and closed (green) magnetosphere model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We note that the Poynting flux is not a conserved quantity in this system since several possibilities for conversion from or to electromag- netic energy exist within the magnetosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' deceleration due to magnetic stresses, ion–neutral collisions, conversion be- tween electromagnetic and thermal energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 the Poynting flux variation as function of r amounts to a factor of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5–2 for the open MS, ∼ 1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 for the semi– open MS and ∼2 for the closed MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Given the uncertainties with which our study is anyway afflicted (such as the uncertainty of the stellar wind density, magnetic field strength, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' ), the uncer- tainty by the choice of the shell radius upon the Poynting fluxes is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The choice of magnetic field strength, for instance, has much larger influence on the Poynting fluxes due to its B2 dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Appendix B: On the effect of magnetic diffusion on the results In our MHD model we include the ionosphere and atmosphere of τ Boötis b through collisions with the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We do not include the effects of collisions in the induction equation for computational time reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In this section we will justify this simplification by comparing simulations with different physical diffusion rates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' different ionospheric conductivities) compared to our basic model without diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Therefore we add a diffusion term to the induction equa- tion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 4) which then reads ∂B ∂t = ∇ × �v × B − η∇ × B� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) We scale the magnetic diffusivity η = η(r) with the atmospheric scale height so that it correlates with the neutral particle density, η(r) = η0 exp �Rp − r H � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2) where η0 is the surface magnetic diffusivity, Rp = 72875 km the planetary radius (Wang & Ford 2011), r the radial distance from the planet center and H = 4373 km the scale height (see Table 1 and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We assume a range of height integrated Pedersen conductivities, ΣP between 1 and 100 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Together with the scale height we can calculate the surface magnetic diffusivity with η0 = Hµ−1 0 Σ−1 P , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3) where µ0 is the vacuum permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The expected range of sur- face diffusivities is therefore η(1) 0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5×1010 and η(2) 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5×1012 m2 s−1 for conductances of 100 S and 1 S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In order to study the effect of magnetic diffusion on our simula- tion results, we compare the simulated plasma variables B, p, ρ and v within the magnetosphere of τ Boötis b with and without magnetic diffusion near the planet at r = 2 Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We restrict our- selves to simulations with open magnetospheres using the basic stellar wind model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The deviations of the 4 plasma variables simulated with diffu- sion from those simulated without are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 (η(1) 0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 × 1010 m2 s−1, ΣP = 100 S) and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 (η(2) 0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 × 1012 m2 s−1, ΣP = 1 S) given in arbitrary normalized units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The finite conductivity causes only very small deviations from our basic model at r = 2Rp on the order of 10−3 for a diffusive model with 1 S Pedersen conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Maximum deviations at r = 2Rp for the diffusive model with ΣP = 100 S amount to factor of 10−5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) compared with the non–diffusive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We thus conclude that the effect of magnetic diffusion on our re- sults (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' the magnetospheric Poynting flux) at 2 Rp above the ionosphere/atmosphere of the planet is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Appendix C: Structure of the interactions: Parameter study In order to illustrate the effect of stellar wind density and pres- sure on the magnetospheric structure, we show xz–plane slices (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1) similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1 for the extreme cases (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 ρsw and 100 ρsw) of our parameter study (Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The plots show plasma velocity and thermal pressure in the near space environment of τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' We use spherical coordinates Article number, page 16 of 19 Magnetospheric Poynting flux as function of radial distance from center 1019 open Ms semi-open Ms closed Ms Power [W] 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 r[Rp]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1: Deviations of resistive MHD simulations from our simulations discussed in the main text using the basic model (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results correspond to simulations with Pedersen conductance of 100S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The four panels display magnetic field (upper left), pressure (upper right), density (bottom left) and velocity (bottom right) maps at r = 2Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2: Deviations of resistive MHD simulations from our simulations discussed in the main text using the basic model (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The results correspond to simulations with Pedersen conductance of 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The four panels display magnetic field (upper left), pressure (upper right), density (bottom left) and velocity (bottom right) maps at r = 2Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' to numerically describe the space environment around τ Boötis b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' These coordinates have mathematical singularities along the pole axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' For exceptionally small upstream plasma densities, this can lead to numerical artifacts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' jumps for the scalar vari- ables along the pole axis in the PLUTO code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This is visible in Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In the vector fields and thus the Poynting fluxes this discontinuity is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' This effect occurring at the ex- tremely low upstream conditions thus does not have an effect on the conclusions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The effect of stellar wind pressure and density on the size of the magnetosphere is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The day side magnetopause lo- cation is ∼ 5 Rp for the lowest density case (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 ρsw, left panels) Article number, page 17 of 19 magnetic field B 1e-6 0 2 25 50 1 75 [Bo] 0 100 B 125 1 150 2 175 0 50 100 150 200 250 300 350 Φ[]thermal pressure p 1e-6 0 3 25 50 1 75 [d] 0 100 d 1 125 2 150 175 3 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma density p 1e-5 0 25 2 50 75 1 [] 100 d 0 125 150 1 175 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Φplasma velocity v 1e-5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 75 [] ^ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 e 100 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 175 50 150 250 200 300 0 100 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φmagnetic field B 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0003 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0002 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0001 75 [Bo] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0000 100 B 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0001 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0002 175 0 50 100 150 200 250 300 350 Φ[]thermal pressure p 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0004 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0002 [d] 100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0000 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0002 175 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma density p 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='003 25 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='002 75 [d] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='001 100 Q 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='000 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='001 175 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma velocity v 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0015 25 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0010 75 v[v。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0005 θ 100 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0000 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0005 175 50 0 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes_Saur_2022_TauBooB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1: Plasma interaction for density ρsw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='05 ρ0 (left panels) and density ρsw = 100 ρ0 (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Displayed are velocity fields (colored arrows, left colorbars) and plasma pressure (color contours, right colorbars) in the xz-plane for the open MS (θB = 0◦,top) and closed (θB = 180◦, bottom) MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Closed and open magnetospheric field lines are colored in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 Rp for the highest density case (100 ρsw, right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Also visible is the sub–fast nature of the interaction in the low- est density case, where no bow shock forms upstream and thus incident plasma flow is perturbed prior to intersecting with the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Appendix D: On the neutral atmosphere model assumptions and its interaction with the plasma In this section we discuss some properties and assumptions on our neutral atmosphere model presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 as well as how the atmosphere affects the plasma focusing on our basic model (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The aim of this work is not a detailed description of the iono- sphere of the planet, but its magnetosphere and larger space en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Our simplistic atmosphere model only acts through collisions with the plasma and affects the photo–ionization rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' plasma production) which both directly scale with the neu- tral particle density (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Due to the exponential decrease of neutral particle density (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5) with radial distance from the planet the atmosphere’s effect on the plasma population drasti- cally decreases with increasing altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The denser the neutral atmosphere the more the magnetospheric plasma is decelerated which leads to plasma pile up around the planet mimicking an ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 shows plasma density profiles within the magnetosphere as function of radial distance from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The black dotted line represents the plasma density along the polar axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Red and magenta lines represent equatorial upstream and downstream profiles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The orange solid line de- notes the neutral particle density according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5, the green dotted line shows the corresponding ion–neutral collision fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' As visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 the effect of ion–neutral collisions and thus the amount of plasma pile up is drastically reduced above an altitude near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' There the neutral particle density is n(r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3Rp) ≈ 5 × 1010 m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The ion–neutral collision fre- quency is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 s−1 at the surface and drastically decreases with altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In our studies we focus on the region 1Rp above the planets surface and thus above the ionospheric shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' At r = 2 Rp the collisions are negligible because the collision frequency has decreased to about ∼ 10−7 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The large scale height of 4373 km increases the extent of the at- mosphere but is needed in order to sufficiently resolve the atmo- sphere in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' However, the atmospheres of Hot Jupiter exoplanets are expected to be strongly inflated due to intense stellar irradiation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Vidal-Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 2003) which is par- tially mimicked by the large scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The surface neutral particle density n0 corresponds to an atmosphere pressure near 10−3 nbar assuming a temperature between 1000 and 2000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Article number, page 18 of 19 600 16 Open magnetosphere 8 14 500 6 12 4 400 10 2 v [km/s] [Ro] [edr] 0 300 8 N p 2 6 200 4 4 6 100 2 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x [Ro]300 16000 Open magnetosphere 8 14000 6 250 12000 4 200 10000 2 v [km/s] [Ro] [edr] 0 150 8000 N d 2 6000 100 4 4000 6 50 2000 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x[Ro]300 16 Closed magnetosphere 8 14 250 6 12 4 200 10 2 v [km/s] [Ro] [edr] : C 0 150 8 N d 2 6 100 4 4 6 50 2 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x [Ro]300 16000 Closed magnetosphere 8 14000 250 6 12000 4 200 10000 2 v [km/s] [Ro] [edn] 0 150 8000 N d 2 6000 100 4 4000 6 50 2000 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='5 0 0 x[Ro]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Elekes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Saur: Space environment and magnetospheric Poynting fluxes of the exoplanet τ Boötis b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1: Plasma density maps over a sphere with r = 2 Rp for the open (left panel) and closed (right panel) MS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2: Plasma density profiles along the polar axis (black dotted line), upstream (red) and downstream x–axis (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The blue solid line denotes the mean plasma density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The orange solid line shows the neutral particle density according to our atmosphere model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The green solid line (right y–axis) denotes ion–neutral collision frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Here the mixing ratio of plasma (hydrogen ions) and neutral particles is roughly 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The neutral atmosphere consists of molecular hydrogen which is the most abundant molecule in all solar system gas giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' In order to demonstrate the minor role of the neutral atmosphere at r = 2 Rp we show plasma density maps over a shell at this altitude in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' The left and right panels display the den- sity map of the open and closed MS case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' There the neutral particle density is n(r = 2 Rp) ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6×105 m−3 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 leading to a neutral–plasma mixing ratio of about 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Regions of high plasma density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' at the day side) indicate regions where the interaction between the neutral atmosphere and plasma is strongest (at r = 2 Rp) in terms of ion–neutral collisions and recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' However, the effect of the neutral atmosphere on the plasma is an order of magnitude lower than in regions within the ionosphere below ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='3 Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content=' Article number, page 19 of 19 plasma density p 1el1 0 8 25 7 50 6 p [H+ /m3] 75 5 100 4 125 3 150 2 175 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] Φplasma density p 1e12 0 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='4 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='2 75 100 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='0 Q 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='8 150 175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='6 0 50 100 150 200 250 300 350 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} +page_content='] ΦMagnetosphericplasmaandneutraldensities 100 nmean npolar 1013 10-2 nupstream ndownstream nneutral 10-4 10-6 1012 10-8 col 10-10 10-12 1011 10-14 2 3 4 5 6 r[Rp]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E4T4oBgHgl3EQfUQzO/content/2301.05015v1.pdf'} diff --git a/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/2301.13204v1.pdf.txt b/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/2301.13204v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a944811d0c01973d7e56fddb2f6b32493fa1b8cf --- /dev/null +++ b/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/2301.13204v1.pdf.txt @@ -0,0 +1,316 @@ +arXiv:2301.13204v1 [q-fin.CP] 29 Jan 2023 +Forex Trading Strategy That Might Be Executed Due to the Popularity of Gotobi Anomaly +Hiroki Bessho1, Takanari Sugimoto2, Tomoya Suzuki1 +1 Major in Mechanical Systems Engineering, Graduate School of Science and Engineering, Ibaraki University +4-12-1 Nakanarusawa-cho, Hitachi-shi, Ibaraki, 316-8511 Japan. Phone: +81-294-38-5195 +E-mail: {21nm480s, tomoya.suzuki.lab}@vc.ibaraki.ac.jp +2 Derivatives Business Department, Gaika ex byGMO, Inc. E-mail: takanari sugimoto@gaikaex.jp +Abstract +Our previous research has confirmed that the USD/JPY rate +tends to rise toward 9:55 every morning in the Gotobi days, +which are divisible by five. This is called the Gotobi anomaly. +In the present study, we verify the possible trading strategy +and its validity under the condition that investors recognize +the existence of the anomaly. Moreover, we illustrate the pos- +sibility that the wealth of Japanese companies might leak to +FX traders due to the arbitrage opportunity if Japanese com- +panies blindly keep making payments in the Gotobi days as a +business custom. +1. Introduction +According to the efficient market hypothesis[1] and the +random walk theory, it is impossible to predict future fluc- +tuations of financial markets, but in real financial markets, +several statistical properties such as the weekend effect[2], +the month-end effect[3], and the holiday effect[4] have been +discovered, which contradicts the efficient market hypothesis. +Therefore, these properties are called “anomaly.” Basically, +the reason why anomalies occur cannot be explained, but the +Gotobi anomaly reported by Ref.[5] has a possible reason to +some extent. +Gotobi is a Japanese word and means the days divisible by +five or ten, that is, 5, 10, 15, 20, 25, and 30 of each month. +As one of Japanese business customs, there has been “Goto- +barai” since before the Edo period, and lots of Japanese com- +panies still keep this business custom. In particular, since +domestic import companies mainly pay in US dollars, the de- +mand for selling the yen and buying the dollar by TTM (Tele- +graphic Transfer Middle Rate) that is the intraday exchange +rate tends to be increased at 9:55 a.m. when TTM is decided. +Moreover, since banks can know the demand in advance, they +try to purchase the dollars earlier and cheeper enough to deal +with the demand, that is, the cover transaction, which puts up- +ward pressure on the USD/JPY rate toward 9:55. On the other +hand, since domestic export companies are waiting for pay- +ment from their counterparties that do not have the business +custom of Goto-barai, the demand for buying the yen and sell- +ing the dollar does not occur in the Gotobi days. Therefore, +by the asymmetry of demand, there might be the mechanism +that the yen becomes relatively cheaper than the dollar. This +phenomenon has been rumored in FX traders as an empirical +rule of the USD/JPY market, but Ref.[5] had confirmed it by +statistical analyses with real data. +In the present study, we discuss what kind of trading strate- +gies FX traders including speculators come up with ideas on +after the anomaly became widely known. The anomaly will +disappear if Japanese companies stop the business custom of +Goto-barai, the purpose of this research is to warn about the +economic losses caused by possible trading strategies per- +formed by rational market participants if the anomaly is left +unattended. +In Section 2, we consider what kind of hypotheses might +be derived by FX traders who recognized the existence of the +Gotobi anomaly by analyzing the recent historical data of the +USD/JPY rate during 2018 to 2020. In Section 3, we con- +sider the rational trading strategy that FX traders might come +up with by combining the hypotheses derived in Sect. 2 and +some basic concepts of the technical analysis to detect trad- +ing timings. By confirming the effectiveness of this trading +strategy, we recognize that the Gotobi anomaly is a kind of +arbitrage opportunity and the possibility that the wealth of +Japanese companies is being leaked to FX traders owing to +leaving the anomaly without attention. +For the analysis of our study, we use the order book infor- +mation of the Electronic Broking System (EBS), which en- +ables us to analyze not only contract prices but also trading +volumes in short time scale. Since the previous study[5] an- +alyzed in hourly time scale by approximating 9:55 as 10:00, +the detailed analysis around 9:55 when TTM is decided has +not yet been conducted. +2. Statistical Properties Before and After 9:55 +We analyze what kind of hypotheses might be derived +by FX traders who recognized the existence of the Go- +tobi anomaly by analyzing the recent historical data of the +USD/JPY rate during 2018 to 2020 in order to make trad- + +ing strategies. For this analysis, we distinguish between Go- +tobi and non-Gotobi days, and try to extract some statisti- +cal properties hidden in Gotobi days. However, the number +of Gotobi days is small, and therefore the previous business +days were included if Gotobi days were holiday as in our pre- +vious study[5]. On the other hand, because the number of +non-Gotobi days is large, non-Gotobi days were randomly +selected for the same number of Gotobi days. Moreover, be- +cause the world’s FX markets are closed early in the morning +on Mondays in Japan, we used daily data from Tuesday to +Friday on business days for the sample of Gotobi and non- +Gotobi days. +As shown in Figure 1, the movement of the USD/JPY +rate is clearly different between the Gotobi days and the +non-Gotobi days. In particular, we can confirm the Gotobi +anomaly that the USD/JPY rate keeps increasing until 9:55 +a.m. in the Gotobi days. Next, Figure 2 shows the probability +that the Gotobi anomaly gets started, and the anomaly might +be started around 3:00 a.m. (JST). These results are consis- +tent with the previous study[5], and therefore we derive the +first hypothesis as follows: +Hypothesis 1 Entering the USD/JPY market by selling the +yen and buying the dollar around 3 a.m. (JST) is likely +to benefit from the Gotobi anomaly. +Next, according to Figures 1 and 2, the USD/JPY rate is +suddenly decreased just after 9:55 a.m. that is the decision +and announcement time of TTM. If the Gotobi anomaly is +caused by the mispricing due to market overreaction, the mis- +pricing must be corrected by the market efficiency after 9:55 +at which time the settlement demand of Japanese import com- +panies is satisfied and TTM is fixed. As its evidence, Figure +3 shows the movement of the USD/JPY rates for a minute +just after 9:55. We can confirm the fact that the exchange rate +is clearly decreased only in Gotobi days, which means that +the increase of exchange rates due to the Gotobi anomaly is +caused by the mispricing, and consequently its modification +is stated just after 9:55. From this viewpoint, we derive the +second hypothesis as follows: +Hypothesis 2 The mispricing caused by the Gotobi anomaly +is immediately corrected, and therefore it is effective to +reverse trading positions of buy and sell just after 9:55 +when the occurrence of mispricing was comfirmed. +3. Verification of Hypotheses by Technical Analysis +We verify the validity of two hypotheses and consider the +rational trading strategy that ordinary FX traders might come +up with, assuming that FX traders use the moving average, +which is the most popular tool of technical analysis, to de- +tect trading timings. In order to calculate the profit obtained +by trading strategy, we subtracted the bid-ask spread when +entering into the market and exiting from the market. Then, +Japan Standard Time (JST) +Gotobi +non-Gotobi +average +Figure 1: Movement of the dollar-yen rate every minute dur- +ing 1/1/2018 ∼ 12/31/2020: the vertical axis at 9:55 (i.e., +the decision and announcement time of TTM) is set to 0.00. +The blue line is the average movement of Gotobi days, each +green line is that of non-Gotobi days randomly sampled for +the same number of Gotobi days, and the red line is the aver- +age of all green lines. +Japan Standard Time (JST) +Gotobi +non-Gotobi +average +Figure 2: Probability that the dollar-yen rate every minute is +more than that of 9:55 on the same day. +Number of Gotobi days +Gotobi +non-Gotobi +average +Figure 3: Cumulative movement of the dollar-yen rates dur- +ing 9:55 to 9:56 (i.e., one minute just after the announcement +of TTM at 9:55). Two vertical dotted lines represent the fist +days of 2019 and 2020, respectively. + +the profit factor (PF ) was calculated by dividing the total of +positive profits by the total of negative profits, the payoff ratio +(PR) was calculated by dividing the average of positive profits +by the average of negative profits, and the winning percent- +age (W) was calculated by dividing the number of getting +positive profits by the total number of all trades N. +3.1 Verification of Hypothesis 1 +Figure 4 shows the profitability of entering the market at +3:00 and exiting it at 9:55 in the Gotobi days, and its earn- +ings are stably increasing, which can be an evidence of the +first hypothesis. On the other hand, Figure 5 shows the same +simulation but in non-Gotobi days, and its earnings are stably +decreased by the bid-ask spreads and are completely different +in the Gotobi days. +If FX traders recognized the existence of the anomaly, +they might try to optimize the timing of entering the mar- +ket by using technical analysis as a time-series filter. For +this reason, we apply the golden cross (GC) based on the +25-minute (short-term) moving average (SMA) and the 100- +minute (long-term) moving average (LMA). and enter the +market if the GC occurs during 2:30 to 3:00 and then exit +it at 9:55. As shown in Figures 4 and 5, using the GC can im- +prove the total earnings and trading efficiency of the Gotobi +days even though the number of trades N is reduced. +Figure 6 shows the case of changing the time to apply the +above GC strategy. As a result, the performance of n = 3 is +the best, which also shows the validity of the first Hypothesis. +3.2 Verification of Hypothesis 2 +If the Gotobi anomaly, that is, the irregular increase of ex- +change rate is caused by the mispricing of market, rational +traders might attempt to reverse their buy positions into sell +positions for following its modification process. Therefore, +we verify the profitability of taking a sell position just after +9:55 and closing its position at 12:00. Here, because the oc- +currence of the Gotobi anomaly is a necessary condition for +the mispricing, this strategy is performed only when the profit +obtained until 9:00 is more than 0. +Figure 7 shows the profits of taking a sell position during +9:55 to 12:00. If the Gotobi anomaly occurs, this strategy +works well. However, even in Gotobi days, this strategy is +meaningless if the Gotobi anomaly does not occur. Therefore, +the usefulness of taking sell positions is caused by the Gotobi +anomaly, which also concludes that the anomaly is caused by +the mispricing of market and supports the second hypothesis. +Figure 8 shows the same simulation but in non-Gotobi days. +Even if the profit obtained until 9:00 is more than 0, which +is shown as “Anomaly occurred”, this strategy does not work +because it has nothing to do with the mispricing caused by +the Gotobi anomaly. +3:00 +2:30 ~ 3:00 (GC) +Number of Gotobi days +Earnings [pips] +Figure 4: Cumulative earnings in Gotobi days by two cases +of using the GC strategy during 2:30 to 3:00 and entering +the market at 3:00 without the GC strategy. In using the GC +strategy, N = 65, PF = 2.62, PR = 1.11, and W = 0.68. In +not using the GC strategy, N = 185, PF = 1.46, PR = 0.94, +and W = 0.60. +3:00 +2:30 ~ 3:00 (GC) +Number of Gotobi days +Earnings [pips] +Figure 5: +Same as Figure 4, but in non-Gotobi days. In +using the GC strategy, N = 69, PF = 0.52, PR = 0.60, +and W = 0.46. In not using the GC strategy, N = 185, +PF = 0.51, PR = 0.69, and W = 0.41. +n = 1 +n = 2 +n = 4 +n = 5 +n = 3 +Number of Gotobi days +Earnings [pips] +Figure 6: Same as Figure 4, but by the case of changing the +time to apply the GC strategy during 30 minutes past n− 1 to +n. If n = 1, it is during 0:30 to 1:00. When n = 1, N = 55, +PF = 1.24, PR = 1.11, and W = 0.52. When n = 2, N = +72, PF = 0.94, PR = 0.79, and W = 0.50. When n = 4, +N = 79, PF = 1.47, PR = 1.02, and W = 0.56. When +n = 5, N = 69, PF = 1.27, PR = 0.79, and W = 0.60. + +Number of Gotobi days +Earnings [pips] +Anomaly occurred +Anomaly did not occur +Figure 7: Cumulative earnings in Gotobi days by taking a sell +position during 9:55 to 12:00. In the days when the Gotobi +anomaly occurred, N = 113, PF = 2.09, PR = 1.51, and +W = 0.57. In the days when the Gotobi anomaly did not +occur, N = 72, PF = 1.18, PR = 1.15, and W = 0.48. +Number of Gotobi days +Earnings [pips] +Anomaly occurred +Anomaly did not occur +Figure 8: Same as Figure 7, but in non-Gotobi days. In the +days when the Gotobi anomaly occurred, N = 88, PF = +1.20, PR = 1.07, and W = 0.51. In the days when the +Gotobi anomaly did not occur, N = 72, PF = 1.08, PR = +1.11, and W = 0.47. +H1 and H2 +H1 +H2 +Number of Gotobi days +Earnings [pips] +Figure 9: +Cumulative earnings in Gotobi days by the com- +bination of two hypotheses: H1 and H2 where N = 167, +PF = 2.60, PR = 1.48, and W = 0.63. Here, H1 cor- +responds to the blue-colored line in Figure 4, and H2 corre- +sponds to the red-colored line in Figure 7. +3.3 Combination of Two Hypotheses +Finally, by combining the trading strategies based on two +hypotheses, Figure 9 shows the final earnings that ordinary +but rational FX traders can obtain. As a result, the combi- +nation can get the most stable and largest profits by making +the most of the Gotobi anomaly, which means the possibility +that the wealth of Japanese companies leaks to FX traders if +they blindly keep making payments in the Gotobi days as a +business custom. +4. Conclusion +In this study, we presented two basic hypotheses that can +be derived by FX traders who recognized the Gotobi anomaly, +and considered trading strategies that combines the hypothe- +ses with the golden cross of popular technical analysis. By +confirming the usefulness of trading strategies through in- +vestment simulations, it can be concluded that the Gotobi +anomaly corresponds to a kind of arbitrage opportunity. This +is caused by the traditional business custom of Japanese im- +port companies that make payments by TTM in the Gotobi +days, and therefore maintaining this business custom contin- +ues to drain their financial wealth to ordinary FX traders. The +scale of its loss is not necessarily huge, but it should be noted +that it leaks stably. +This research was partially supported by a JSPS Grant-in- +Aid for Scientific Research (20K11969). The contents of this +article are the personal views of its authors and not the official +views of the institutions with which they are affiliated. +References +[1] B. G. Malkiel and E. F. Fama: “Efficient capital markets: +A review of theory and empirical work,” Journal of Fi- +nance, vol.25, no.2, pp.383–417, 1970. +[2] K. R. French: “Stock returns and the weekend effect,” +Journal of Financial Economics, vol.8, no.1, pp.55–69, +1980. +[3] R. A. Ariel: “A monthly effect in stock returns,” Journal +of Financial Economics, vol.18, no.1, pp.161–174, 1987. +[4] J. Lakonishok and S. Smidt: “Are seasonal anomalies +real? A ninety-year perspective,” Review of Financial +Studies, vol.1, no.4, pp.403–425, 1988. +[5] T. Akiyama, T. Sugimoto, R. Sakemoto, T. Suzuki: “Re- +lationship between the Gotobi anomaly and share of in- +voice currency in Japanese imports,” JAFEE Journal, +vol.19, pp.57–78, 2021 (in Japanese). + diff --git a/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/load_file.txt b/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19a81ca68f2daa6b11ded833036db3e6e1f1394b --- /dev/null +++ b/pNFPT4oBgHgl3EQf7TWt/content/tmp_files/load_file.txt @@ -0,0 +1,196 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf,len=195 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='13204v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='CP] 29 Jan 2023 Forex Trading Strategy That Might Be Executed Due to the Popularity of Gotobi Anomaly Hiroki Bessho1, Takanari Sugimoto2, Tomoya Suzuki1 1 Major in Mechanical Systems Engineering, Graduate School of Science and Engineering, Ibaraki University 4-12-1 Nakanarusawa-cho, Hitachi-shi, Ibaraki, 316-8511 Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Phone: +81-294-38-5195 E-mail: {21nm480s, tomoya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='lab}@vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='ibaraki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='jp 2 Derivatives Business Department, Gaika ex byGMO, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' E-mail: takanari sugimoto@gaikaex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='jp Abstract Our previous research has confirmed that the USD/JPY rate tends to rise toward 9:55 every morning in the Gotobi days, which are divisible by five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' This is called the Gotobi anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the present study, we verify the possible trading strategy and its validity under the condition that investors recognize the existence of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Moreover, we illustrate the pos- sibility that the wealth of Japanese companies might leak to FX traders due to the arbitrage opportunity if Japanese com- panies blindly keep making payments in the Gotobi days as a business custom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Introduction According to the efficient market hypothesis[1] and the random walk theory, it is impossible to predict future fluc- tuations of financial markets, but in real financial markets, several statistical properties such as the weekend effect[2], the month-end effect[3], and the holiday effect[4] have been discovered, which contradicts the efficient market hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Therefore, these properties are called “anomaly.” Basically, the reason why anomalies occur cannot be explained, but the Gotobi anomaly reported by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' [5] has a possible reason to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Gotobi is a Japanese word and means the days divisible by five or ten, that is, 5, 10, 15, 20, 25, and 30 of each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As one of Japanese business customs, there has been “Goto- barai” since before the Edo period, and lots of Japanese com- panies still keep this business custom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In particular, since domestic import companies mainly pay in US dollars, the de- mand for selling the yen and buying the dollar by TTM (Tele- graphic Transfer Middle Rate) that is the intraday exchange rate tends to be increased at 9:55 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' when TTM is decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Moreover, since banks can know the demand in advance, they try to purchase the dollars earlier and cheeper enough to deal with the demand, that is, the cover transaction, which puts up- ward pressure on the USD/JPY rate toward 9:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' On the other hand, since domestic export companies are waiting for pay- ment from their counterparties that do not have the business custom of Goto-barai, the demand for buying the yen and sell- ing the dollar does not occur in the Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Therefore, by the asymmetry of demand, there might be the mechanism that the yen becomes relatively cheaper than the dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' This phenomenon has been rumored in FX traders as an empirical rule of the USD/JPY market, but Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' [5] had confirmed it by statistical analyses with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the present study, we discuss what kind of trading strate- gies FX traders including speculators come up with ideas on after the anomaly became widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' The anomaly will disappear if Japanese companies stop the business custom of Goto-barai, the purpose of this research is to warn about the economic losses caused by possible trading strategies per- formed by rational market participants if the anomaly is left unattended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In Section 2, we consider what kind of hypotheses might be derived by FX traders who recognized the existence of the Gotobi anomaly by analyzing the recent historical data of the USD/JPY rate during 2018 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In Section 3, we con- sider the rational trading strategy that FX traders might come up with by combining the hypotheses derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 2 and some basic concepts of the technical analysis to detect trad- ing timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' By confirming the effectiveness of this trading strategy, we recognize that the Gotobi anomaly is a kind of arbitrage opportunity and the possibility that the wealth of Japanese companies is being leaked to FX traders owing to leaving the anomaly without attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' For the analysis of our study, we use the order book infor- mation of the Electronic Broking System (EBS), which en- ables us to analyze not only contract prices but also trading volumes in short time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Since the previous study[5] an- alyzed in hourly time scale by approximating 9:55 as 10:00, the detailed analysis around 9:55 when TTM is decided has not yet been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Statistical Properties Before and After 9:55 We analyze what kind of hypotheses might be derived by FX traders who recognized the existence of the Go- tobi anomaly by analyzing the recent historical data of the USD/JPY rate during 2018 to 2020 in order to make trad- ing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' For this analysis, we distinguish between Go- tobi and non-Gotobi days, and try to extract some statisti- cal properties hidden in Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' However, the number of Gotobi days is small, and therefore the previous business days were included if Gotobi days were holiday as in our pre- vious study[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' On the other hand, because the number of non-Gotobi days is large, non-Gotobi days were randomly selected for the same number of Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Moreover, be- cause the world’s FX markets are closed early in the morning on Mondays in Japan, we used daily data from Tuesday to Friday on business days for the sample of Gotobi and non- Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As shown in Figure 1, the movement of the USD/JPY rate is clearly different between the Gotobi days and the non-Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In particular, we can confirm the Gotobi anomaly that the USD/JPY rate keeps increasing until 9:55 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' in the Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Next, Figure 2 shows the probability that the Gotobi anomaly gets started, and the anomaly might be started around 3:00 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' (JST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' These results are consis- tent with the previous study[5], and therefore we derive the first hypothesis as follows: Hypothesis 1 Entering the USD/JPY market by selling the yen and buying the dollar around 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' (JST) is likely to benefit from the Gotobi anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Next, according to Figures 1 and 2, the USD/JPY rate is suddenly decreased just after 9:55 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' that is the decision and announcement time of TTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' If the Gotobi anomaly is caused by the mispricing due to market overreaction, the mis- pricing must be corrected by the market efficiency after 9:55 at which time the settlement demand of Japanese import com- panies is satisfied and TTM is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As its evidence, Figure 3 shows the movement of the USD/JPY rates for a minute just after 9:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' We can confirm the fact that the exchange rate is clearly decreased only in Gotobi days, which means that the increase of exchange rates due to the Gotobi anomaly is caused by the mispricing, and consequently its modification is stated just after 9:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' From this viewpoint, we derive the second hypothesis as follows: Hypothesis 2 The mispricing caused by the Gotobi anomaly is immediately corrected, and therefore it is effective to reverse trading positions of buy and sell just after 9:55 when the occurrence of mispricing was comfirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Verification of Hypotheses by Technical Analysis We verify the validity of two hypotheses and consider the rational trading strategy that ordinary FX traders might come up with, assuming that FX traders use the moving average, which is the most popular tool of technical analysis, to de- tect trading timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In order to calculate the profit obtained by trading strategy, we subtracted the bid-ask spread when entering into the market and exiting from the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Then, Japan Standard Time (JST) Gotobi non-Gotobi average Figure 1: Movement of the dollar-yen rate every minute dur- ing 1/1/2018 ∼ 12/31/2020: the vertical axis at 9:55 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=', the decision and announcement time of TTM) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' The blue line is the average movement of Gotobi days, each green line is that of non-Gotobi days randomly sampled for the same number of Gotobi days, and the red line is the aver- age of all green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Japan Standard Time (JST) Gotobi non-Gotobi average Figure 2: Probability that the dollar-yen rate every minute is more than that of 9:55 on the same day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Number of Gotobi days Gotobi non-Gotobi average Figure 3: Cumulative movement of the dollar-yen rates dur- ing 9:55 to 9:56 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=', one minute just after the announcement of TTM at 9:55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Two vertical dotted lines represent the fist days of 2019 and 2020, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' the profit factor (PF ) was calculated by dividing the total of positive profits by the total of negative profits, the payoff ratio (PR) was calculated by dividing the average of positive profits by the average of negative profits, and the winning percent- age (W) was calculated by dividing the number of getting positive profits by the total number of all trades N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='1 Verification of Hypothesis 1 Figure 4 shows the profitability of entering the market at 3:00 and exiting it at 9:55 in the Gotobi days, and its earn- ings are stably increasing, which can be an evidence of the first hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' On the other hand, Figure 5 shows the same simulation but in non-Gotobi days, and its earnings are stably decreased by the bid-ask spreads and are completely different in the Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' If FX traders recognized the existence of the anomaly, they might try to optimize the timing of entering the mar- ket by using technical analysis as a time-series filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' For this reason, we apply the golden cross (GC) based on the 25-minute (short-term) moving average (SMA) and the 100- minute (long-term) moving average (LMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' and enter the market if the GC occurs during 2:30 to 3:00 and then exit it at 9:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As shown in Figures 4 and 5, using the GC can im- prove the total earnings and trading efficiency of the Gotobi days even though the number of trades N is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Figure 6 shows the case of changing the time to apply the above GC strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As a result, the performance of n = 3 is the best, which also shows the validity of the first Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='2 Verification of Hypothesis 2 If the Gotobi anomaly, that is, the irregular increase of ex- change rate is caused by the mispricing of market, rational traders might attempt to reverse their buy positions into sell positions for following its modification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Therefore, we verify the profitability of taking a sell position just after 9:55 and closing its position at 12:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Here, because the oc- currence of the Gotobi anomaly is a necessary condition for the mispricing, this strategy is performed only when the profit obtained until 9:00 is more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Figure 7 shows the profits of taking a sell position during 9:55 to 12:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' If the Gotobi anomaly occurs, this strategy works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' However, even in Gotobi days, this strategy is meaningless if the Gotobi anomaly does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Therefore, the usefulness of taking sell positions is caused by the Gotobi anomaly, which also concludes that the anomaly is caused by the mispricing of market and supports the second hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Figure 8 shows the same simulation but in non-Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Even if the profit obtained until 9:00 is more than 0, which is shown as “Anomaly occurred”, this strategy does not work because it has nothing to do with the mispricing caused by the Gotobi anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3:00 2:30 ~ 3:00 (GC) Number of Gotobi days Earnings [pips] Figure 4: Cumulative earnings in Gotobi days by two cases of using the GC strategy during 2:30 to 3:00 and entering the market at 3:00 without the GC strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In using the GC strategy, N = 65, PF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='62, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='11, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In not using the GC strategy, N = 185, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='46, PR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='94, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3:00 2:30 ~ 3:00 (GC) Number of Gotobi days Earnings [pips] Figure 5: Same as Figure 4, but in non-Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In using the GC strategy, N = 69, PF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='52, PR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='60, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In not using the GC strategy, N = 185, PF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='51, PR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='69, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' n = 1 n = 2 n = 4 n = 5 n = 3 Number of Gotobi days Earnings [pips] Figure 6: Same as Figure 4, but by the case of changing the time to apply the GC strategy during 30 minutes past n− 1 to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' If n = 1, it is during 0:30 to 1:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' When n = 1, N = 55, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='24, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='11, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' When n = 2, N = 72, PF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='94, PR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='79, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' When n = 4, N = 79, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='47, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='02, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' When n = 5, N = 69, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='27, PR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='79, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Number of Gotobi days Earnings [pips] Anomaly occurred Anomaly did not occur Figure 7: Cumulative earnings in Gotobi days by taking a sell position during 9:55 to 12:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the days when the Gotobi anomaly occurred, N = 113, PF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='09, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='51, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the days when the Gotobi anomaly did not occur, N = 72, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='18, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='15, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Number of Gotobi days Earnings [pips] Anomaly occurred Anomaly did not occur Figure 8: Same as Figure 7, but in non-Gotobi days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the days when the Gotobi anomaly occurred, N = 88, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='20, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='07, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' In the days when the Gotobi anomaly did not occur, N = 72, PF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='08, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='11, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' H1 and H2 H1 H2 Number of Gotobi days Earnings [pips] Figure 9: Cumulative earnings in Gotobi days by the com- bination of two hypotheses: H1 and H2 where N = 167, PF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='60, PR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='48, and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Here, H1 cor- responds to the blue-colored line in Figure 4, and H2 corre- sponds to the red-colored line in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='3 Combination of Two Hypotheses Finally, by combining the trading strategies based on two hypotheses, Figure 9 shows the final earnings that ordinary but rational FX traders can obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' As a result, the combi- nation can get the most stable and largest profits by making the most of the Gotobi anomaly, which means the possibility that the wealth of Japanese companies leaks to FX traders if they blindly keep making payments in the Gotobi days as a business custom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Conclusion In this study, we presented two basic hypotheses that can be derived by FX traders who recognized the Gotobi anomaly, and considered trading strategies that combines the hypothe- ses with the golden cross of popular technical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' By confirming the usefulness of trading strategies through in- vestment simulations, it can be concluded that the Gotobi anomaly corresponds to a kind of arbitrage opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' This is caused by the traditional business custom of Japanese im- port companies that make payments by TTM in the Gotobi days, and therefore maintaining this business custom contin- ues to drain their financial wealth to ordinary FX traders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' The scale of its loss is not necessarily huge, but it should be noted that it leaks stably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' This research was partially supported by a JSPS Grant-in- Aid for Scientific Research (20K11969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' The contents of this article are the personal views of its authors and not the official views of the institutions with which they are affiliated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content=' Malkiel and E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} +page_content='57–78, 2021 (in Japanese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFPT4oBgHgl3EQf7TWt/content/2301.13204v1.pdf'} diff --git a/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf b/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b20914d7bb43d3e7b93a603dbc8605fd00f8ba27 --- /dev/null +++ b/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0b1c8ef937d48c9ff3198c745311222dee47b590d33a74255b7df55868f95f6f +size 8355928 diff --git a/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/2301.05246v1.pdf.txt b/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/2301.05246v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bedca9aeeca40d180628def2aff95e8d1b6bc5af --- /dev/null +++ b/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/2301.05246v1.pdf.txt @@ -0,0 +1,727 @@ +ONLINE CLASS-INCREMENTAL LEARNING FOR +REAL-WORLD FOOD CLASSIFICATION +Siddeshwar Raghavan, Jiangpeng He, Fengqing Zhu +School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana USA +ABSTRACT +Online Class-Incremental Learning (OCIL) aims to continu- +ously learn new information from single-pass data streams to +update the model and mitigate catastrophic forgetting. How- +ever, most existing OCIL methods make several assumptions, +including non-overlapped classes across phases and an equal +number of classes in each learning phase. This is a highly +simplified view of typical real-world scenarios. In this pa- +per, we extend OCIL to the real-world food image classifi- +cation task by removing these assumptions and significantly +improving the performance of existing OCIL methods. We +first introduce a novel probabilistic framework to simulate re- +alistic food data sequences in different scenarios, including +strict, moderate, and open diets, as a new benchmark experi- +ment protocol. Next, we propose a novel plug-and-play mod- +ule to dynamically select relevant images during training for +the model update to improve learning and forgetting perfor- +mance. Our proposed module can be incorporated into exist- +ing Experience Replay (ER) methods, which store representa- +tive samples from each class into an episodic memory buffer +for knowledge rehearsal. We evaluate our method on the chal- +lenging Food-101 dataset and show substantial improvements +over the current OCIL methods, demonstrating great potential +for lifelong learning of real-world food image classification. +1. INTRODUCTION +Food image classification [1, 2] is an integral part of health +status indicators for chronic disease detection, and image- +based dietary assessment [3, 4]. These systems aim to pro- +vide valuable insights into a person’s health status based on +the consumed food [5], which is the primary source of energy +and macro-nutrients[6, 7, 8, 9]. However, food image patterns +evolve continuously in the real world, requiring the system +to learn and update new knowledge sequentially. Methods +that continuously learn from sequentially arriving data with- +out catastrophic forgetting of previously learned knowledge +are termed Class-Incremental Learning (CIL) [10, 11], which +recently received increased attention for the image classifica- +tion task. In this paper, we focus on Online Class-Incremental +Learning (OCIL) [12, 13, 14, 15] where each data is used only +once during training; this is more challenging but more prac- +tical, especially for real-life applications such as the detection +and monitoring food intake [16] +Experience Replay (ER) based OCIL methods [17, 18, 19] +have shown remarkable performance, which stores a small +number of representative samples from each learned class for +knowledge replay to mitigate catastrophic forgetting. How- +ever, existing methods work in a simplified scenario where +(1) the number of new classes in each learning phase is fixed, +and (2) the classes are disjoint across phases, i.e. no over- +lapped classes appear across different learning phases, which +rarely happens in reality. A few existing OCIL work target the +food image data [13], but these assumptions make it less ef- +fective due to high intra-class variance (same food ingredients +but cooked differently) and unpredictable future class distri- +butions observed in real-world food classification scenarios. +In this paper, we remove the above assumptions on Online +Class-Incremental Learning for food image classification and +additionally propose a novel method to further improve the +accuracy of learning. First, we introduce a Realistic Data Dis- +tribution Module (RDDM) to simulate real-world food pat- +terns and set up more realistic experimental benchmarks by +removing the aforementioned assumptions. We allow images +of classes to appear more realistically across multiple phases1 +through a probabilistic framework as shown in Figure 1. In +addition, the RDDM allows a varied number of classes and +samples to appear across phases to mimic the high intra-class +variance and unpredictable future class distributions predom- +inantly present in real-world food image sequences. +Sec- +ond, we propose a simple yet efficient plug-and-play module +named Dynamic Model Update (DMU) to dynamically select +the most beneficial new data used for training, which can be +incorporated into the existing ER methods [18, 17, 19] in the +online setting [20, 21] to attain performance improvement. +Our method is evaluated on the challenging Food-101 [22] +dataset by categorizing the food data distribution into three +scenarios, including strict diet, moderate diet, and open diet, +using our proposed RDDM framework, which is illustrated +in section 2.1.1. We observe the performance degradation by +analyzing the results of the existing ER models in our newly +defined benchmarks. Thus, showing the unsuitability of the +existing methods to directly apply in real-world applications. +1A phase denotes a stream of data input to the OCIL system, which can +contain new and old classes of data (images). +arXiv:2301.05246v1 [cs.CV] 12 Jan 2023 + +Fig. 1: Realistic Data Distribution Module (RDDM) takes +probabilistic distributions, the dataset, and tunable parameters +α and β to produce realistic data sequences that function as +the input to the OCIL model. +Our proposed DMU module improves the classification accu- +racy by a large margin of 15-18% on average and can better +handle the uncertainty in data distribution. +The contributions of the paper are summarized as follows. +(1) To the best of our knowledge, we are the first to general- +ize Online Class Incremental Learning (OCIL) towards food +image classification through a probabilistic data formulation +framework named RDDM, which creates more realistic data +sequences. (2) We propose a novel dynamic plug-and-play +module, termed DMU, which can be incorporated into ex- +isting Experience Replay (ER) methods to improve learning +accuracy and mitigate catastrophic forgetting in realistic sce- +narios. (3) We conduct extensive experiments on the chal- +lenging Food-101 dataset and show that the performance of +our Dynamic Model Update (DMU) module greatly surpasses +existing OCIL methods. +2. METHOD +Our proposed method contains two main components. We +first introduce a novel Realistic Data Distribution Module +(RDDM) framework to simulate realistic sequences for food +image classification by removing the assumptions in existing +OCIL formulations [23, 14, 20]. Secondly, we propose a new +Dynamic Model Update (DMU) module, which monitors the +model’s current performance to dynamically select only the +new image data (from the input) or the entirety of the input +image sequence in any given phase to improve overall learn- +ing and reduce forgetting. +2.1. Framework for Realistic Data Distribution Module +(RDDM) +The objective is to create realistic data sequences and address +the current issues of simplified assumptions. As shown in +Figure 1, the input to the framework is the dataset, parame- +ters (α and β), and probabilistic distributions. The outputs +from the framework are realistic data sequences occurring in +each phase k, along with the sample size of the classes. An +unbalanced sample occurs due to possible class repetitions, +i.e. if a class repeats m times across k phases, the total sam- +ples are divided by m in each occurrence. Thus there are no +repeated mini-batches of data encountered by the online class +incremental learner. +2.1.1. Formulae For Our Distribution +We consider different values for the hyperparameters used +in the RDDM framework described above. The parameter +α controls the degree of class overlap across phases, and β +controls the extent of occurrence of new classes (Nn). These +hyperparameters can be configured to (i) ensure at least one +new class is added in each subsequent phase and (ii) restrict +the occurrence of all classes in a single phase which is unreal- +istic in the real-world food classification task. Without these +constraints, we would shift the focus from OCIL to a domain +adaptation problem [24]. The number of new classes Nn to +appear in a phase k depends on a phase-dependent distribution +DN. Different scenarios can be simulated by varying the dis- +tribution DN. In our experiments, we set DN as a truncated +exponential distribution to include the influence of previously +occurred classes (Np), set the upper bound as the total number +of unseen classes, and still be able to maintain the properties +of the exponential distribution in the bounded space. This can +be formally defined as: +Nn ∼ exp +� +µ = β Nt +Np +� +(1) +where Nt denotes the total number of classes (dataset depen- +dent quantity). The total number of classes to appear in a +phase Nk depends on another phase-dependent distribution +DT . By varying the distribution DT , various scenarios can be +simulated. In our experiments, we fix DT as a uniform distri- +bution as it is reasonable to assume that each class is equally +likely to appear in any phase k. +Nk ∼ Uniform +� +Nn, min +� +(Np + Nn) , Nt +α +�� +(2) +Table 1: Classification of diets based on parameters α and β +Diet Type +Alpha +Beta +Classes +Strict Diet +[9,10,11,12] +[1, 2, 3] +30 - 50 +Moderate Diet +[5,6,7,8] +[4,5,6] +50 - 80 +Open Diet +[1,2,3,4] +[7,8,9] +80 - 101 +2.1.2. Formulation Of Diet Categories +We encounter various dietary patterns in the real-world due to +region, culture, dietary constraints, and personal preferences. +Therefore, it is essential to consider the patterns while de- +signing a data distribution framework for food classification + +RDDM(Realistic +Data DistributionFig. 2: Overview of proposed method: The center part +shows the overview of our Dynamic model update module. +The accuracy of the two virtual models is compared with the +threshold set. The input image sequence, along with the cor- +responding exemplars of the best-performing virtual branch, +is propagated through the main OCIL model for training and +finally to the BufferUpdate module. +in the OCIL setting. In this work, we incorporate this ideol- +ogy to segregate the simulated distributions into three cate- +gories corresponding to dietary restrictions: strict, moderate, +and open diets [25, 26], as shown in Table 1. It is also possi- +ble that we may only encounter some of the classes from the +dataset during the simulation of food image sequences due to +the probabilistic nature of the formulation. Hence, catego- +rizing is critical to cover a broad range of diet patterns and +analyze the performance of the proposed methods. +We categorize the total number of classes across all phases +using the most representative parameters. +The number of +classes in a phase (indicates the flexibility of a diet), the de- +gree of overlap (visualized as an unchanged preference for ex- +isting foods), and the degree of occurrence of new classes in a +phase (pictured as adding newer foods to break the monotony +of repetitive diets). Using the two defined hyperparameters +we describe the different diet categories. Hyperparameter α +is inversely proportional to the degree of overlap of classes +across phases and, β is directly proportional to the degree of +new class occurrence in a phase. +• Strict diet - Larger values of α set an upper bound on +the total number of classes in a phase, and smaller val- +ues of β reduce the overlap as seen in Equations 1 and +2. This setting leads to fewer classes and a low degree +of overlap, simulating a strict diet regime. +• Moderate diet - Decreasing the values of α and increas- +ing the values of β, we obtain moderate classes in +each phase and a moderate degree of overlap com- +pared to the strict diet, and we call this category the +moderate diet. +• Open diet (An unrestricted case) - Low values of α and +high values of β generate a large number of classes +and overlap, simulating an open/ unrestricted diet. +Even though α, β can take a wide range of values, we +have defined the above three combinations of values as best +representing the most common dietary practices as shown in +Table 1. +2.2. Dynamic Model Update +In this section, we propose a simple yet effective model +update strategy to learn new and overlapping classes effi- +ciently and address catastrophic forgetting on realistic se- +quences created by our RDDM framework as illustrated in +Section 2.1.1. The overview of our proposed Dynamic Model +Update (DMU) module is shown in Figure 2. Our motivation +is to avoid blindly training the model without being aware of +the input image data from the sequence, which leads to (1) +overfitting if there are images belonging to a particular class +appear repeatedly, and (2) longer training time for the low +efficiency of model update in the online scenario. The nov- +elty of our method lies in the plug-and-play nature of the dy- +namic update module into existing ER replay methods and +having dynamic decision capacity regarding model update +during training to improve performance by utilizing image se- +quences simulated by RDDM. +Specifically, in the training phase, the input image se- +quences are fed from the RDDM framework. Two virtual +branches, V M1 and V M2 are created by reproducing the +OCIL model to infer performance without gradient calcula- +tion. The first branch is trained with the entire input batch +[X, Y ] and corresponding exemplars [XE, YE]. The second +branch is trained only with new class image [XH, YH] (non- +overlapped) samples and their corresponding exemplars ob- +tained by comparing the prior distribution of labels (PD) and +the current input [X, Y ], where PD(k) is the cumulative +stored value of all the unique set of labels occurred until phase +k. We obtain the non-overlapped labels and, subsequently, the +images using the following equation: +[YH] = [Y ] ∩ PD(k) +(3) +The exemplars selected for the two virtual branches are +different as the input image data is varied for the two +branches. Additionally, the exemplar retrieval methodology +varies with different ER methods. +The better-performing +branch’s data samples (input and buffer images and their cor- +responding labels) are propagated to the main model. The +accuracy of the initial phase is stored as threshold accuracy +as this value is relative to the input image sequence and diet +categorization as shown in Table 1. View supplementary ma- +terial for detailed training algorithm. Irrespective of the vir- +tual branches, the entire training batch is used to update the +episodic memory buffer. Each ER method has a unique ap- +proach to updating the buffer (BufferUpdate) and retrieving +exemplars (BufferRetrieval). + +8 +Max(Acc(VM1, VM2 )) +Main +Virtual model - 1 +BufferUpdate +model +Threshold +OOOOO +Virtual model - 2 +OOOOOTable 2: Average Accuracy (%) and Average Forgetting (%) rates with phase sizes of 5, 10, and 20 on the Food-101 +dataset. Results include existing ER methods and our method (DMU) leveraging the RDDM framework for realistic food +image sequences. The best accuracy results are highlighted in boldface. +Phase size +5 +10 +20 +Distribution categories +Strict +Moderate +Open +Strict +Moderate +Open +Strict +Moderate +Open +Accuracy (%) / Forgetting(%) +Finetune Accuracy (%) +54.52 / NA +45.26 / NA +44.08 / NA +36.89 / NA +43.35 / NA +41.45 / NA +26.36 / NA +25.43 / NA +29.01 / NA +MIR [18] +74.71 / 29.24 +69.79 / 11.19 +57.55 / 22.73 +64.0 /16.7 +57.0 / 16.7 +48.9 / 25.5 +54.15 / 10.38 +48.39 / 16.32 +43.85 / 20.81 +GSS [19] +72.17 / 28.32 +69.77 / 11.06 +58.54 / 21.54 +51.5 / 12.5 +50.5 /16.9 +47.8 / 23.9 +40.33 / 7.9 +42.8 / 10.64 +38.51 / 18.07 +iCaRL [17, 20] +75.02 / 29.71 +69.34 / 10.51 +57.75 / 21.94 +67.0 / 16.11 +56.6 / 18.9 +49.65 / 25.6 +57.24 /12.01 +51.69 / 16.79 +47.13 / 22.35 +MIR + DMU +82.14 / 30.54 +75.69 / 11.08 +67.97 / 24.63 +72.73 / 15.33 +71.17 / 19.01 +67.72 / 30.89 +65.50 / 11.08 +63.03 / 18.2 +62.04 / 26.59 +GSS + DMU +76.29 / 29.89 +74.01 / 11.33 +68.19 / 24.56 +55.07 / 13.37 +66.17 / 16.6 +66.06/ 29.9 +47.43 / 7.09 +49.86 / 10.7 +50.13 / 20.45 +iCaRL + DMU +76.04 / 29.79 +70.68 / 10.72 +59.22 / 21.76 +67.14 / 14.95 +62.29 / 17.39 +52.19 / 26.27 +59.72 / 12.21 +53.65 / 16.54 +50.37 / 23.61 +Fig. 3: Average accuracy of the proposed DMU module with +existing ER methods, Finetune, and existing ER methods as +the baseline in realistic sequences generated by RDDM for 10 +phases in the moderate diet categorization. +3. EXPERIMENTS +This section compares the existing ER methods, including +iCaRL[17], MIR[18], and GSS[19] in the online setting [20] +with our proposed Dynamic Model Update (DMU) described +in Section 2.2. These experiments are performed under the +proposed benchmarks created by our RDDM framework in +all three diet categories as illustrated in Section 2.1. In ad- +dition, we include Finetune for comparison, which considers +only the new image data, does not depend on the previous +phases, does not use exemplars for knowledge rehearsal, and +can be considered the lower bound. +3.1. Evaluation Metrics +It is critical to understand how the model performs in each +phase, how well it learns and how much knowledge it has +forgotten, along with the overall performance averages of the +OCIL model. We define ai,j as the accuracy evaluated on a +held-out phase j after training the network from phases 1 to +i, and assume there are k phases in total [20]. +The Average Accuracy is defined in Equation 4. When +i = k, Ak represents the average accuracy by the end of train- +ing using the entire image data sequence (from phase 1 to k) +[20]. Here Ai denotes the Average Accuracy at any phase i +such that i ̸= k [20]. +Average Accuracy(Ai) = 1 +i +i +� +j=1 +a(i,j) +(4) +After the model finishes learning at phase k, we test the +performance on all k phases by constructing a k × k matrix +A where A ∈ IRk×k. Each element of the matrix represented +as a(i,j) is the test classification accuracy of the model on +phase j after observing the last sample from phase i [27]. The +average accuracy is calculated at the end of training with all +the encountered classes. +The Average Forgetting at phase i is defined in Equa- +tion5, where f(i,j) represents how much the model has forgot- +ten about phase j after being trained on phase i. From the k× +k matrix, the best test accuracy of the model on phase j before +learning phase m can be obtained with maxl∈[1,...,m−1] (al,j) +and a(m,j) is the test accuracy on phase j after learning phase +m [20]. +Average Forgetting(Fi) = +1 +i − 1 +i−1 +� +j=1 +fi,j, +wherefm,j = +max +l∈[1,...,m−1] (al,j) − am,j ∀ j < m +(5) +We require the models to have high average accuracy and +low average forgetting to indicate superior performance. +3.2. Experimental Setup +Dataset: We use Food-101 [22] to evaluate our proposed +method. Food-101 is a challenging dataset consisting of 101 +food classes, and each class has 1,000 images. The dataset is +split into 600 training images, 150 validation images, and 250 +test images. We apply our proposed RDDM data distribution +protocol to split the 101 classes into phases of 5, 10, and 20 +for the experiment. +Implementation details: We use ResNet-18 [28] pre-trained +on ImageNet [29] as the backbone structure for all experi- +ments in comparison, and our implementation is based on Py- +Torch. Pre-trained networks are employed to reduce difficulty + +100.00% +90.00% +80.00% +70.00% +Accuracy(%) +60.00% +50.00% +40.00% +30.00% +20.00% +10.00% +0.00% +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Phases +-MIR+DMU +MIR +iCaRL+DMU +iCaRL +GSS+DMU +GSS +-Finetuneand time consumption during training, especially while us- +ing a complex dataset like Food-101 [22]. The input image +size is cropped to 224 × 224. We use stochastic gradient de- +scent (SGD) optimizer with a fixed learning rate of 0.001 and +weight decay of 0.00001. The batch size is set as 32 (16 phase +input image data with 16 selected exemplars from the buffer). +We follow the existing experimental protocol [20] to use a +5K episodic memory buffer to store exemplars for all exper- +iments except finetune (does not use exemplars). The OCIL +model encounters each image data sample only once in all ex- +periments. The degree of overlap and the total classes learned +depends on the RDDM formulation and diet categories, as ex- +plained in Section 2.1.1. We repeat each experiment 3 times +using random seeds for individual diet category settings (α +and β) based on Table 1 and report the average classification +accuracy rate and average forgetting rate. +3.3. Experimental Results and Discussion +Results on the Food-101 dataset are summarized in Table 2, +where higher accuracy and lower forgetting represent better +performance. +Strict Diet has the highest accuracy since it has the least +number of total classes and the least overlap of classes when +the number of phases is low, as seen in Table 1. +When +the number of phases increases, the overlap across phases +increases as the total number of classes encountered are fixed +with a range. The DMU module helps increase accuracy and +mitigate catastrophic forgetting by training the OCIL model +with relevant input images. +Moderate diet has the best accuracy forgetting trade-off. +Due to the increased overlap of classes across phases, the +model encounters different samples from the classes learned +throughout the training. +The increased overlap improves +the model’s accuracy in training. The Figure. 3 shows the +improved average accuracy of ER methods with our proposed +module (DMU) compared with the existing ER methods +(baseline); see supplementary material for accuracy graphs +across all categories. In the subsequent phases, the DMU +module leads training toward new classes (non-overlapped) +to improve accuracy and reduce overall forgetting. +Open diet has the lowest accuracy combined with a high +forgetting rate. Maximum overlap and a maximum number +of classes are encountered in each phase. The ability of the +OCIL model to learn a large number of new classes in a +single phase and not forget them is challenging, leading to +lower accuracy and higher forgetting. The model’s accuracy +predominantly does not cross the threshold. +Thus, DMU +struggles to steer the training with only new class samples. +Our proposed DMU module focuses on the training pro- +cess, utilizing the accuracy rate set as a threshold and driving +the model toward higher accuracy. We do not modify the ex- +isting ER method’s exemplar update and retrieval strategies; +this may be the reason for our module’s slightly higher for- +getting rate than the existing methods. However, the improve- +ment in the average accuracy of our proposed (DMU) module +is achieved without introducing a significant trade-off with +forgetting by selecting the appropriate part of the input image +sequence for training. We notice iCaRL to have the lowest +forgetting compared to MIR and GSS, but it also has the low- +est accuracy. This could be because the Nearest Class Mean +(NCM) Classifier is used in iCaRL instead of the fully con- +nected layers. In the OCIL setting, NCM approximates buffer +data to calculate the prototype vector and assigns class labels +with the most similar one [20], which may not be the best +representation due to repeated classes. The low forgetting +can happen due to the model’s inability to learn new classes +because of the strong regularization imposed by Knowledge +Distillation in iCaRL. +3.3.1. Influence of Plug and Play Dynamic Modules +The addition of our Dynamic Model Update module signif- +icantly improves the accuracy rate. The dynamic selection +of new image data from overlapping classes during training +boosts learning and hence accuracy rate by a large margin. +The increased learning results in a small degradation in for- +getting rate, as seen in Table 2. The overall performance of +ER methods with the DMU module largely exceeds the ex- +isting ER methods. Our module attempts to efficiently steer +the learning by training the model with input image samples +and exemplar images that lead to potential performance in- +crease while reducing overfitting when the model repeatedly +encounters images from well-trained classes. +Overall, the +DMU module has the best accuracy with MIR, with an im- +provement of 15-18% from the baselines. The fundamental +structure of MIR focuses on BufferRetrieval [18] and, com- +bined with the dynamic updates, leads to better accuracy. On +the other hand, GSS inherently focuses on BufferUpdate [19], +and the dynamic update leads to the least forgetting. +4. CONCLUSION +In this paper, we focus on Online Class-Incremental Learn- +ing for food image classification. We introduce a novel Re- +alistic Data Distribution Module (RDDM) to simulate real- +world food consumption patterns in different scenarios along +with a plug-and-play Dynamic Model Update (DMU) mod- +ule, which is compatible with existing ER methods by inde- +pendently targeting the training pipeline for improving perfor- +mance. We show that our data distribution considers more re- +alistic scenarios than existing OCIL systems, including a cor- +responding experimental benchmark. Furthermore, experi- +mental results demonstrate our proposed method outperforms +existing methods in more realistic food image classification +settings on the challenging Food-101 dataset. Our future work +will focus on improving the exemplar storage procedure by + +exploiting the most representative samples simulated by the +RDDM framework. We also plan to evaluate our method on +large supervised open-world datasets in the future. +5. 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Brytek-Matera, Kamila Czepczor-Bernat, Helena Ju- +rzak, Monika Kornacka, and Natalia Kołodziejczyk, +“Strict +health-oriented eating patterns (orthorexic eating behaviours) +and their connection with a vegetarian and vegan diet,” Eat. +Weight Disord., vol. 24, no. 3, pp. 441–452, June 2019. +[27] David Lopez-Paz and Marc’Aurelio Ranzato, +“Gradient +episodic memory for continual learning,” Red Hook, NY, USA, +2017, p. 6470–6479, Curran Associates Inc. +[28] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, +“Deep residual learning for image recognition,” 2016, pp. 770– +778. +[29] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and +Li Fei-Fei, +“Imagenet: +A large-scale hierarchical image +database,” Ieee, 2009, pp. 248–255. + diff --git a/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/load_file.txt b/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fad8840b693499457b30b93e899674c77b391c48 --- /dev/null +++ b/q9E4T4oBgHgl3EQfwA0l/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf,len=424 +page_content='ONLINE CLASS-INCREMENTAL LEARNING FOR REAL-WORLD FOOD CLASSIFICATION Siddeshwar Raghavan, Jiangpeng He, Fengqing Zhu School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana USA ABSTRACT Online Class-Incremental Learning (OCIL) aims to continu- ously learn new information from single-pass data streams to update the model and mitigate catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' How- ever, most existing OCIL methods make several assumptions, including non-overlapped classes across phases and an equal number of classes in each learning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' This is a highly simplified view of typical real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In this pa- per, we extend OCIL to the real-world food image classifi- cation task by removing these assumptions and significantly improving the performance of existing OCIL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We first introduce a novel probabilistic framework to simulate re- alistic food data sequences in different scenarios, including strict, moderate, and open diets, as a new benchmark experi- ment protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Next, we propose a novel plug-and-play mod- ule to dynamically select relevant images during training for the model update to improve learning and forgetting perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our proposed module can be incorporated into exist- ing Experience Replay (ER) methods, which store representa- tive samples from each class into an episodic memory buffer for knowledge rehearsal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We evaluate our method on the chal- lenging Food-101 dataset and show substantial improvements over the current OCIL methods, demonstrating great potential for lifelong learning of real-world food image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' INTRODUCTION Food image classification [1, 2] is an integral part of health status indicators for chronic disease detection, and image- based dietary assessment [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' These systems aim to pro- vide valuable insights into a person’s health status based on the consumed food [5], which is the primary source of energy and macro-nutrients[6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' However, food image patterns evolve continuously in the real world, requiring the system to learn and update new knowledge sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Methods that continuously learn from sequentially arriving data with- out catastrophic forgetting of previously learned knowledge are termed Class-Incremental Learning (CIL) [10, 11], which recently received increased attention for the image classifica- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In this paper, we focus on Online Class-Incremental Learning (OCIL) [12, 13, 14, 15] where each data is used only once during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' this is more challenging but more prac- tical, especially for real-life applications such as the detection and monitoring food intake [16] Experience Replay (ER) based OCIL methods [17, 18, 19] have shown remarkable performance, which stores a small number of representative samples from each learned class for knowledge replay to mitigate catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' How- ever, existing methods work in a simplified scenario where (1) the number of new classes in each learning phase is fixed, and (2) the classes are disjoint across phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' no over- lapped classes appear across different learning phases, which rarely happens in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' A few existing OCIL work target the food image data [13], but these assumptions make it less ef- fective due to high intra-class variance (same food ingredients but cooked differently) and unpredictable future class distri- butions observed in real-world food classification scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In this paper, we remove the above assumptions on Online Class-Incremental Learning for food image classification and additionally propose a novel method to further improve the accuracy of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' First, we introduce a Realistic Data Dis- tribution Module (RDDM) to simulate real-world food pat- terns and set up more realistic experimental benchmarks by removing the aforementioned assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We allow images of classes to appear more realistically across multiple phases1 through a probabilistic framework as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In addition, the RDDM allows a varied number of classes and samples to appear across phases to mimic the high intra-class variance and unpredictable future class distributions predom- inantly present in real-world food image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Sec- ond, we propose a simple yet efficient plug-and-play module named Dynamic Model Update (DMU) to dynamically select the most beneficial new data used for training, which can be incorporated into the existing ER methods [18, 17, 19] in the online setting [20, 21] to attain performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our method is evaluated on the challenging Food-101 [22] dataset by categorizing the food data distribution into three scenarios, including strict diet, moderate diet, and open diet, using our proposed RDDM framework, which is illustrated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We observe the performance degradation by analyzing the results of the existing ER models in our newly defined benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Thus, showing the unsuitability of the existing methods to directly apply in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 1A phase denotes a stream of data input to the OCIL system, which can contain new and old classes of data (images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='05246v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='CV] 12 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 1: Realistic Data Distribution Module (RDDM) takes probabilistic distributions, the dataset, and tunable parameters α and β to produce realistic data sequences that function as the input to the OCIL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our proposed DMU module improves the classification accu- racy by a large margin of 15-18% on average and can better handle the uncertainty in data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The contributions of the paper are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' (1) To the best of our knowledge, we are the first to general- ize Online Class Incremental Learning (OCIL) towards food image classification through a probabilistic data formulation framework named RDDM, which creates more realistic data sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' (2) We propose a novel dynamic plug-and-play module, termed DMU, which can be incorporated into ex- isting Experience Replay (ER) methods to improve learning accuracy and mitigate catastrophic forgetting in realistic sce- narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' (3) We conduct extensive experiments on the chal- lenging Food-101 dataset and show that the performance of our Dynamic Model Update (DMU) module greatly surpasses existing OCIL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' METHOD Our proposed method contains two main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We first introduce a novel Realistic Data Distribution Module (RDDM) framework to simulate realistic sequences for food image classification by removing the assumptions in existing OCIL formulations [23, 14, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Secondly, we propose a new Dynamic Model Update (DMU) module, which monitors the model’s current performance to dynamically select only the new image data (from the input) or the entirety of the input image sequence in any given phase to improve overall learn- ing and reduce forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Framework for Realistic Data Distribution Module (RDDM) The objective is to create realistic data sequences and address the current issues of simplified assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' As shown in Figure 1, the input to the framework is the dataset, parame- ters (α and β), and probabilistic distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The outputs from the framework are realistic data sequences occurring in each phase k, along with the sample size of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' An unbalanced sample occurs due to possible class repetitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' if a class repeats m times across k phases, the total sam- ples are divided by m in each occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Thus there are no repeated mini-batches of data encountered by the online class incremental learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Formulae For Our Distribution We consider different values for the hyperparameters used in the RDDM framework described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The parameter α controls the degree of class overlap across phases, and β controls the extent of occurrence of new classes (Nn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' These hyperparameters can be configured to (i) ensure at least one new class is added in each subsequent phase and (ii) restrict the occurrence of all classes in a single phase which is unreal- istic in the real-world food classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Without these constraints, we would shift the focus from OCIL to a domain adaptation problem [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The number of new classes Nn to appear in a phase k depends on a phase-dependent distribution DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Different scenarios can be simulated by varying the dis- tribution DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In our experiments, we set DN as a truncated exponential distribution to include the influence of previously occurred classes (Np), set the upper bound as the total number of unseen classes, and still be able to maintain the properties of the exponential distribution in the bounded space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' This can be formally defined as: Nn ∼ exp � µ = β Nt Np � (1) where Nt denotes the total number of classes (dataset depen- dent quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The total number of classes to appear in a phase Nk depends on another phase-dependent distribution DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' By varying the distribution DT , various scenarios can be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In our experiments, we fix DT as a uniform distri- bution as it is reasonable to assume that each class is equally likely to appear in any phase k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Nk ∼ Uniform � Nn, min � (Np + Nn) , Nt α �� (2) Table 1: Classification of diets based on parameters α and β Diet Type Alpha Beta Classes Strict Diet [9,10,11,12] [1, 2, 3] 30 - 50 Moderate Diet [5,6,7,8] [4,5,6] 50 - 80 Open Diet [1,2,3,4] [7,8,9] 80 - 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Formulation Of Diet Categories We encounter various dietary patterns in the real-world due to region, culture, dietary constraints, and personal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Therefore, it is essential to consider the patterns while de- signing a data distribution framework for food classification RDDM(Realistic Data DistributionFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 2: Overview of proposed method: The center part shows the overview of our Dynamic model update module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The accuracy of the two virtual models is compared with the threshold set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The input image sequence, along with the cor- responding exemplars of the best-performing virtual branch, is propagated through the main OCIL model for training and finally to the BufferUpdate module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' in the OCIL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In this work, we incorporate this ideol- ogy to segregate the simulated distributions into three cate- gories corresponding to dietary restrictions: strict, moderate, and open diets [25, 26], as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' It is also possi- ble that we may only encounter some of the classes from the dataset during the simulation of food image sequences due to the probabilistic nature of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Hence, catego- rizing is critical to cover a broad range of diet patterns and analyze the performance of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We categorize the total number of classes across all phases using the most representative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The number of classes in a phase (indicates the flexibility of a diet), the de- gree of overlap (visualized as an unchanged preference for ex- isting foods), and the degree of occurrence of new classes in a phase (pictured as adding newer foods to break the monotony of repetitive diets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Using the two defined hyperparameters we describe the different diet categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Hyperparameter α is inversely proportional to the degree of overlap of classes across phases and, β is directly proportional to the degree of new class occurrence in a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Strict diet - Larger values of α set an upper bound on the total number of classes in a phase, and smaller val- ues of β reduce the overlap as seen in Equations 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' This setting leads to fewer classes and a low degree of overlap, simulating a strict diet regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Moderate diet - Decreasing the values of α and increas- ing the values of β, we obtain moderate classes in each phase and a moderate degree of overlap com- pared to the strict diet, and we call this category the moderate diet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Open diet (An unrestricted case) - Low values of α and high values of β generate a large number of classes and overlap, simulating an open/ unrestricted diet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Even though α, β can take a wide range of values, we have defined the above three combinations of values as best representing the most common dietary practices as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Dynamic Model Update In this section, we propose a simple yet effective model update strategy to learn new and overlapping classes effi- ciently and address catastrophic forgetting on realistic se- quences created by our RDDM framework as illustrated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The overview of our proposed Dynamic Model Update (DMU) module is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our motivation is to avoid blindly training the model without being aware of the input image data from the sequence, which leads to (1) overfitting if there are images belonging to a particular class appear repeatedly, and (2) longer training time for the low efficiency of model update in the online scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The nov- elty of our method lies in the plug-and-play nature of the dy- namic update module into existing ER replay methods and having dynamic decision capacity regarding model update during training to improve performance by utilizing image se- quences simulated by RDDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Specifically, in the training phase, the input image se- quences are fed from the RDDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Two virtual branches, V M1 and V M2 are created by reproducing the OCIL model to infer performance without gradient calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The first branch is trained with the entire input batch [X, Y ] and corresponding exemplars [XE, YE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The second branch is trained only with new class image [XH, YH] (non- overlapped) samples and their corresponding exemplars ob- tained by comparing the prior distribution of labels (PD) and the current input [X, Y ], where PD(k) is the cumulative stored value of all the unique set of labels occurred until phase k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We obtain the non-overlapped labels and, subsequently, the images using the following equation: [YH] = [Y ] ∩ PD(k) (3) The exemplars selected for the two virtual branches are different as the input image data is varied for the two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Additionally, the exemplar retrieval methodology varies with different ER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The better-performing branch’s data samples (input and buffer images and their cor- responding labels) are propagated to the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The accuracy of the initial phase is stored as threshold accuracy as this value is relative to the input image sequence and diet categorization as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' View supplementary ma- terial for detailed training algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Irrespective of the vir- tual branches, the entire training batch is used to update the episodic memory buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Each ER method has a unique ap- proach to updating the buffer (BufferUpdate) and retrieving exemplars (BufferRetrieval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 8 Max(Acc(VM1, VM2 )) Main Virtual model - 1 BufferUpdate model Threshold OOOOO Virtual model - 2 OOOOOTable 2: Average Accuracy (%) and Average Forgetting (%) rates with phase sizes of 5, 10, and 20 on the Food-101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Results include existing ER methods and our method (DMU) leveraging the RDDM framework for realistic food image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The best accuracy results are highlighted in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Phase size 5 10 20 Distribution categories Strict Moderate Open Strict Moderate Open Strict Moderate Open Accuracy (%) / Forgetting(%) Finetune Accuracy (%) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='52 / NA 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='26 / NA 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='08 / NA 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='89 / NA 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='35 / NA 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='45 / NA 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='36 / NA 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='43 / NA 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='01 / NA MIR [18] 74.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='01 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='69 / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='79 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='13 / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='35 MIR + DMU 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='14 / 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='54 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='43 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='09 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='86 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='13 / 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='45 iCaRL + DMU 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='04 / 29.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='29 / 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='39 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='19 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='27 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='72 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='21 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='65 / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='54 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='37 / 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='61 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3: Average accuracy of the proposed DMU module with existing ER methods, Finetune, and existing ER methods as the baseline in realistic sequences generated by RDDM for 10 phases in the moderate diet categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' EXPERIMENTS This section compares the existing ER methods, including iCaRL[17], MIR[18], and GSS[19] in the online setting [20] with our proposed Dynamic Model Update (DMU) described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' These experiments are performed under the proposed benchmarks created by our RDDM framework in all three diet categories as illustrated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In ad- dition, we include Finetune for comparison, which considers only the new image data, does not depend on the previous phases, does not use exemplars for knowledge rehearsal, and can be considered the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Evaluation Metrics It is critical to understand how the model performs in each phase, how well it learns and how much knowledge it has forgotten, along with the overall performance averages of the OCIL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We define ai,j as the accuracy evaluated on a held-out phase j after training the network from phases 1 to i, and assume there are k phases in total [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The Average Accuracy is defined in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' When i = k, Ak represents the average accuracy by the end of train- ing using the entire image data sequence (from phase 1 to k) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Here Ai denotes the Average Accuracy at any phase i such that i ̸= k [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Average Accuracy(Ai) = 1 i i � j=1 a(i,j) (4) After the model finishes learning at phase k, we test the performance on all k phases by constructing a k × k matrix A where A ∈ IRk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Each element of the matrix represented as a(i,j) is the test classification accuracy of the model on phase j after observing the last sample from phase i [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The average accuracy is calculated at the end of training with all the encountered classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The Average Forgetting at phase i is defined in Equa- tion5, where f(i,j) represents how much the model has forgot- ten about phase j after being trained on phase i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' From the k× k matrix, the best test accuracy of the model on phase j before learning phase m can be obtained with maxl∈[1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=',m−1] (al,j) and a(m,j) is the test accuracy on phase j after learning phase m [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Average Forgetting(Fi) = 1 i − 1 i−1 � j=1 fi,j, wherefm,j = max l∈[1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=',m−1] (al,j) − am,j ∀ j < m (5) We require the models to have high average accuracy and low average forgetting to indicate superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Experimental Setup Dataset: We use Food-101 [22] to evaluate our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Food-101 is a challenging dataset consisting of 101 food classes, and each class has 1,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The dataset is split into 600 training images, 150 validation images, and 250 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We apply our proposed RDDM data distribution protocol to split the 101 classes into phases of 5, 10, and 20 for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Implementation details: We use ResNet-18 [28] pre-trained on ImageNet [29] as the backbone structure for all experi- ments in comparison, and our implementation is based on Py- Torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Pre-trained networks are employed to reduce difficulty 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% Accuracy(%) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00% 1 2 3 4 5 6 7 8 9 10 Phases MIR+DMU MIR iCaRL+DMU iCaRL GSS+DMU GSS Finetuneand time consumption during training, especially while us- ing a complex dataset like Food-101 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The input image size is cropped to 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We use stochastic gradient de- scent (SGD) optimizer with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='001 and weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='00001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The batch size is set as 32 (16 phase input image data with 16 selected exemplars from the buffer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We follow the existing experimental protocol [20] to use a 5K episodic memory buffer to store exemplars for all exper- iments except finetune (does not use exemplars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The OCIL model encounters each image data sample only once in all ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The degree of overlap and the total classes learned depends on the RDDM formulation and diet categories, as ex- plained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We repeat each experiment 3 times using random seeds for individual diet category settings (α and β) based on Table 1 and report the average classification accuracy rate and average forgetting rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Experimental Results and Discussion Results on the Food-101 dataset are summarized in Table 2, where higher accuracy and lower forgetting represent better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Strict Diet has the highest accuracy since it has the least number of total classes and the least overlap of classes when the number of phases is low, as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' When the number of phases increases, the overlap across phases increases as the total number of classes encountered are fixed with a range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The DMU module helps increase accuracy and mitigate catastrophic forgetting by training the OCIL model with relevant input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Moderate diet has the best accuracy forgetting trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Due to the increased overlap of classes across phases, the model encounters different samples from the classes learned throughout the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The increased overlap improves the model’s accuracy in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3 shows the improved average accuracy of ER methods with our proposed module (DMU) compared with the existing ER methods (baseline);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' see supplementary material for accuracy graphs across all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In the subsequent phases, the DMU module leads training toward new classes (non-overlapped) to improve accuracy and reduce overall forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Open diet has the lowest accuracy combined with a high forgetting rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Maximum overlap and a maximum number of classes are encountered in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The ability of the OCIL model to learn a large number of new classes in a single phase and not forget them is challenging, leading to lower accuracy and higher forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The model’s accuracy predominantly does not cross the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Thus, DMU struggles to steer the training with only new class samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our proposed DMU module focuses on the training pro- cess, utilizing the accuracy rate set as a threshold and driving the model toward higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We do not modify the ex- isting ER method’s exemplar update and retrieval strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' this may be the reason for our module’s slightly higher for- getting rate than the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' However, the improve- ment in the average accuracy of our proposed (DMU) module is achieved without introducing a significant trade-off with forgetting by selecting the appropriate part of the input image sequence for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We notice iCaRL to have the lowest forgetting compared to MIR and GSS, but it also has the low- est accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' This could be because the Nearest Class Mean (NCM) Classifier is used in iCaRL instead of the fully con- nected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' In the OCIL setting, NCM approximates buffer data to calculate the prototype vector and assigns class labels with the most similar one [20], which may not be the best representation due to repeated classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The low forgetting can happen due to the model’s inability to learn new classes because of the strong regularization imposed by Knowledge Distillation in iCaRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Influence of Plug and Play Dynamic Modules The addition of our Dynamic Model Update module signif- icantly improves the accuracy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The dynamic selection of new image data from overlapping classes during training boosts learning and hence accuracy rate by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The increased learning results in a small degradation in for- getting rate, as seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The overall performance of ER methods with the DMU module largely exceeds the ex- isting ER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our module attempts to efficiently steer the learning by training the model with input image samples and exemplar images that lead to potential performance in- crease while reducing overfitting when the model repeatedly encounters images from well-trained classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Overall, the DMU module has the best accuracy with MIR, with an im- provement of 15-18% from the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' The fundamental structure of MIR focuses on BufferRetrieval [18] and, com- bined with the dynamic updates, leads to better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' On the other hand, GSS inherently focuses on BufferUpdate [19], and the dynamic update leads to the least forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' CONCLUSION In this paper, we focus on Online Class-Incremental Learn- ing for food image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We introduce a novel Re- alistic Data Distribution Module (RDDM) to simulate real- world food consumption patterns in different scenarios along with a plug-and-play Dynamic Model Update (DMU) mod- ule, which is compatible with existing ER methods by inde- pendently targeting the training pipeline for improving perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We show that our data distribution considers more re- alistic scenarios than existing OCIL systems, including a cor- responding experimental benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Furthermore, experi- mental results demonstrate our proposed method outperforms existing methods in more realistic food image classification settings on the challenging Food-101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' Our future work will focus on improving the exemplar storage procedure by exploiting the most representative samples simulated by the RDDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' We also plan to evaluate our method on large supervised open-world datasets in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' REFERENCES [1] Runyu Mao, Jiangpeng He, Zeman Shao, Sri Kalyan Yarla- gadda, and Fengqing Zhu, “Visual aware hierarchy based food recognition,” arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content='03368, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E4T4oBgHgl3EQfwA0l/content/2301.05246v1.pdf'} +page_content=' [2] Runyu Mao, Jiangpeng He, Luotao Lin, Zeman Shao, Heather A.' 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sha256:21f5aaf643b49b584df9cf6bdc938f83f109540cf9b7ebb3bb845855d77d2dd2 +size 4849709 diff --git a/udAzT4oBgHgl3EQfsP1l/content/tmp_files/2301.01656v1.pdf.txt b/udAzT4oBgHgl3EQfsP1l/content/tmp_files/2301.01656v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad6f1c0052daa1ef03c27e03e377265a7222acc2 --- /dev/null +++ b/udAzT4oBgHgl3EQfsP1l/content/tmp_files/2301.01656v1.pdf.txt @@ -0,0 +1,712 @@ +arXiv:2301.01656v1 [math.CO] 4 Jan 2023 +On the maximum number of edges in k-critical graphs +Cong Luo∗ +Jie Ma† +Tianchi Yang‡ +Abstract +A graph is called k-critical if its chromatic number is k but any proper subgraph has +chromatic number less than k. An old and important problem in graph theory asks to +determine the maximum number of edges in an n-vertex k-critical graph. This is widely +open for any integer k ≥ 4. Using a structural characterization of Greenwell and Lov´asz +and an extremal result of Simonovits, Stiebitz proved in 1987 that for k ≥ 4 and sufficiently +large n, this maximum number is less than the number of edges in the n-vertex balanced +complete (k −2)-partite graph. In this paper we obtain the first improvement on the above +result in the past 35 years. Our proofs combine arguments from extremal graph theory as +well as some structural analysis. A key lemma we use indicates a partial structure in dense +k-critical graphs, which may be of independent interest. +1 +Introduction +All graphs we consider are finite and simple. A graph G is k-colorable if we can assign k +colors to its vertices such that no adjacent vertices receive the same color. We say a graph G +is k-chromatic if it is k-colorable but not (k − 1)-colorable. A graph G is called k-critical if +G is k-chromatic but every its proper subgraph is (k − 1)-colorable. For k ∈ {1, 2} the only +k-critical graph is Kk, and the family of 3-critical graphs is precisely the family of odd cycles. +In this paper, we consider k-critical graphs for k ≥ 4. +A central problem in graph theory asks to determine the maximum number of edges fk(n) +in an n-vertex k-critical graph (see [6]). Before we discuss the literature on fk(n), we would +like to point out a relevant yet easy fact that the Tur´an graph Tk(n) (that is, the n-vertex +balanced complete k-partite graph) has the maximum number of edges among all n-vertex +∗School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, +China. Email: luoc@mail.ustc.edu.cn. +†School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, +China. Research supported in part by the National Key R and D Program of China 2020YFA0713100, National +Natural Science Foundation of China grant 12125106, and Anhui Initiative in Quantum Information Technologies +grant AHY150200. Email: jiema@ustc.edu.cn. +‡Department of Mathematics, National University of Singapore, 119076, Singapore. Research supported in +part by Professor Hao Huang’s start-up grant at NUS and an MOE Academic Research Fund (AcRF) Tier 1 +grant. Email: tcyang@nus.edu.sg. +1 + +k-chromatic graphs. Dirac [2] gave f6(n) ≥ 1 +4n2 + n by considering the graphs obtained by +joining two vertex-disjoint odd cycles with the same number of vertices. Toft [12] proved that +for every k ≥ 4, there exists a positive constant ck such that fk(n) ≥ ckn2 holds for all integers +n ≥ k (except n = k + 1). In the most basic and interesting cases k = 4, 5, the constants are +given by +c4 ≥ 1 +16 = 0.0625 +and +c5 ≥ 4 +31 ≥ 0.129. +In the general case when k ≥ 6, explicit constructions in [12] show that there exist infinitely +many values of n such that +fk(n) ≥ +�1 +2 − +3 +2k − δk +� +n2, +where δk = 0 if k ≡ 0 (mod 3), δk = 8/7 if k ≡ 1 (mod 3), and δk = 44/23 if k ≡ 2 (mod 3). +To our best knowledge, no construction for giving better constants fk(n)/n2 have been found +since. +It is also an open question if limn→∞ +fk(n) +n2 +exits for any k ≥ 4. +In 2013, Pegden +[8] considered dense triangle-free k-critical graphs. He constructed infinitely many n-vertex +triangle-free 4-critical graphs with at least +� 1 +16 − o(1) +� +n2 edges, triangle-free 5-critical graphs +with at least +� 4 +31 − o(1) +� +n2 edges, and triangle-free k-critical graphs with at least +�1 +4 − o(1) +� +n2 +edges for every k ≥ 6. The last bound is asymptotically best possible by Tur´an’s theorem. He +also showed the existence of dense k-critical graphs without containing any odd cycle of length +at most ℓ for any ℓ, which is again asymptotically tight for k ≥ 6. +Turning to the upper bound of fk(n), since any n-vertex k-critical graph with n > k does not +contain Kk as a subgraph, by Tur´an’s theorem one can easily obtain that fk(n) < e(Tk−1(n)) +for any n > k ≥ 4. Using a characterization of Greenwell and Lov´asz [5] for subgraphs of +k-critical graphs and a classical theorem of Simonovits [10], Stiebitz [11] improved this trivial +bound in 1987 by showing that +fk(n) < e(Tk−2(n)) for sufficiently large integer n. +(1) +It has been 35 years since then and as far as we are aware, this remains the best upper bound. +There is a natural relation between fk(n) and the problem of determining the maximum +number of copies of Kk−1 in k-critical graphs. Abbott and Zhou [1] generalized an earlier result +of Stiebitz [11] on 4-critical graphs and showed that for any k ≥ 4 every k-critical graph on n +vertices contains at most n copies of Kk−1. The bound was further improved in [7]. Recently, +Gao and Ma [4] proved a sharp result that for any n > k ≥ 4, any k-critical graph on n vertices +contains at most n − k + 3 copies of Kk−1. If we delete one edge for every Kk−1 in a k-critical +graph on n vertices, then this can result in a graph without containing Kk−1. Using Tur´an’s +theorem and the above result of [4], we can derive that +fk(n) ≤ e(Tk−2(n)) + n − k + 3 for any n > k ≥ 4. +In this paper, we focus on the upper bound of fk(n). Our first result improves the long- +standing upper bound (1) of Stiebitz [11]. +2 + +Theorem 1.1. For any integer k ≥ 4 and sufficiently large integers n, there exists a constant +ck ≥ +1 +36(k−1)2 such that fk(n) ≤ e(Tk−2(n)) − ckn2. +Our second result considers 4-critical graphs. A better upper bound for f4(n) than Theo- +rem 1.1 is obtained in the following. +Theorem 1.2. For sufficiently large integers n, it holds that f4(n) < 0.164n2. +The proofs of both theorems rely on arguments from extremal graph theory (such as the +stability of F¨uredi [3]) and a structural lemma (Lemma 2.1) given in the coming section. +Lemma 2.1 indicates a partial structure in dense critical graphs (under certain constraints), +which can be witnessed in many classical constructions of dense critical graphs (see the discus- +sion at the beginning of Section 2). For that, we would like to give a full construction for the +well-known Toft graph (see [12]). The vertex set of the Toft graph is formed by 4 disjoint sets +A, B, C, D with the same odd size, where A and D are odd cycles, B and C are independent +sets, the edges between B and C form a complete bipartite graph, and both of the edges in +(A, B) and in (C, D) form perfect matchings. It is easy to check that the n-vertex Toft graph is +4-critical and has 1 +16n2+n edges. We remark that the Toft graph remains the best construction +for dense 4-critical graphs. +We use standard notation in graph theory. Let G denote the complement of the graph +G. +For a vertex v in a graph G, let NG(v) denote the neighborhood of v in G, and let +dG(v) := |NG(v)| denote the degree of v in G. When G is clear from the context, we often drop +the subscript. Let d(G) denote the average degree of the graph G. Also, for any S ⊆ V (G), let +G[S] denote the induced subgraph of G on the vertex set S. For any disjoint sets A, B ⊆ V (G), +let G[A, B] denote the induced bipartite subgraph of G with bipartition (A, B). +The rest of the paper is organized as follows. In Section 2, we prove a lemma which is key +for the coming proofs. Then we prove Theorem 1.1 in Section 3 and Theorem 1.2 in Section 4. +2 +Key lemma +In this section we prove our key lemma, which roughly says that if a k-critical graph G contains +certain t copies of Kk−2 sharing k−3 common vertices, then there exists an “induced” matching +of size t in G which are connected to these cliques. This indicates a substructure similar to +the Toft graph (and many other examples of k-critical graphs). In particular, it reveals that +the structure of k-critical graphs cannot be close to the Tur´an graph Tk−2(n) and thus the +inequality (1) should not be tight . +Lemma 2.1. Let k ≥ 4 and let G be a k-critical graph. Suppose that G [{x1, x2, . . . , xk−3}] +forms a copy of Kk−3 and there exists a set W ⊆ N(x1) ∩ · · · ∩ N(xk−3) ∩ N(u) for some +vertex u /∈ {x1, x2, . . . , xk−3}. Then there exist a set W ′ and a bijection ϕ : W → W ′ such that +3 + +N(ϕ(w)) ∩ W = {w} and N(w) ∩ W ′ = {ϕ(w)} hold for each w ∈ W. Moreover, if |W| ≥ 3, +then W is an independent set in G, and W ′ ∩ W = ∅. +Proof. For each vertex w ∈ W, by deleting the edge uw from the k-critical graph G, we can +get a (k − 1)-chromatic graph G′. We denote the color classes of G′ by C1, C2, . . . , Ck−1. It is +easy to see the vertices u and w are in the same color class. Since G[{x1, x2, . . . , xk−3, w}] is +a (k − 2)-clique, we can assume x1 ∈ C1, x2 ∈ C2,. . . , xk−3 ∈ Ck−3, and u, w ∈ Ck−2. The fact +W ⊆ N(x1)∩· · ·∩N(xk−3)∩N(u) tells us that the set W\{w} (if not empty) must be contained +in Ck−1, and thus W\{w} is an independent set in G. We claim N(w) ∩ Ck−1 must contain a +vertex, say ϕ(w). Since otherwise C1, . . . , Ck−3, Ck−2 −{w}, Ck−1 ∪{w} can be a (k−1)-coloring +of G, which contradicts the fact that G is k-critical. Besides, {ϕ(w)} ∪ (W\{w}) ⊆ Ck−1 tells +us that N(ϕ(w)) ∩ W = {w}. Now we define W ′ := {ϕ(w) : w ∈ W}. As we have shown that +N(ϕ(w)) ∩ W = {w} holds for each w ∈ W, it is easy to see |W ′| = |W|, ϕ : W → W ′ is a +bijection, and N(w) ∩ W ′ = {ϕ(w)} holds for each w ∈ W. +Moreover, if |W| ≥ 3, then W is an independent set in G (since W\{v} is an independent +set in G for each vertex v ∈ W). By the fact that the edges between W ′ and W precisely form +a matching, we can see W ′ ∩ W = ∅ in this case. +It would be very interesting to see if this lemma (or its proof) can be extended further. +3 +The general case: k-critical +Providing a simple and new proof of the stability for the Tur´an number ex(n, Kr+1), F¨uredi +[3] showed that if an n-vertex graph G is Kr+1-free and has at least e(Tr(n)) − t edges +where 0 ≤ t < e(Tr(n)) < n2, then there exists a partition V1, . . . , Vr of V (G) such that +�r +i=1 e(G[Vi]) ≤ t. The proof of [3] (see Corollary 3) also indicates that if the complete r- +chromatic graph with color classes V1, . . . , Vr is denoted by K, then |E(K)\E(G)| ≤ 2t and +moreover, �r +i=1 (|Vi| − n/r)2 < 4t + o(n2). We summarize in the following lemma. +Lemma 3.1 (F¨uredi [3]). Suppose that G is an n-vertex Kr+1-free graph with e(G) ≥ e(Tr(n))− +t where 0 ≤ t < e(Tr(n)) < n2. +Then there exists a complete r-chromatic graph K := +K(V1, . . . , Vr) with V (K) = V (G) such that +|E(K)\E(G)| ≤ 2t, +and +r +� +i=1 +� +|Vi| − n +r +�2 +< 4t + o(n2). +We are ready to use Lemmas 2.1 and 3.1 to prove Theorem 1.1. +4 + +Proof of Theorem 1.1. Fix k ≥ 4 and let C = +1 +36(k−1)2 . Let G be a k-critical graph on n +vertices with e(G) > e(Tk−2(n)) − Cn2. In the rest of the proof, we will always assume that +n is large enough, and we denote V (G) by V for convenience. The result in [1] tells us the +number of copies of Kk−1 in G is at most n. So by deleting at most n edges in G, we obtain a +spanning subgraph G′ which is Kk−1-free. Obviously we have e(G′) > e(Tk−2(n)) − (Cn2 + n). +With the application of Lemma 3.1, we get a partition V1, . . . , Vk−2 of V and a complete +(k − 2)-chromatic graph K := K(V1, . . . , Vk−2) such that |E(K)\E(G′)| ≤ 2(Cn2 + n) and +���|Vi| − +n +k − 2 +��� < +� +4Cn2 + o(n2) < +n +3(k − 1) + o(n) for each 1 ≤ i ≤ k − 2. +Without loss of generality, we assume |V1| ≤ · · · ≤ |Vk−2|. Thus |Vk−2| ≥ n/(k−2). We call the +edges in E(K)\E(G′) as missing edges. And the number of missing edges incident to the vertex +v in K is called the missing degree of v. For each 1 ≤ i ≤ k − 2, we define Bi to be the set of +� +n +3(k−1) +� +vertices in Vi satisfying that there exists some mi such that the missing degree of any +vertex in Bi is at least mi, and the missing degree of any vertex in Ui := Vi −Bi is at most mi. +Since there are at most 2(Cn2 + n) missing edges in total, we have �k−2 +i=1 mi|Bi| < 4(Cn2 + n), +and thus we can get +k−2 +� +i=1 +mi < 4(Cn2 + n) +� � +n +3(k − 1) +� +≤ +n +3(k − 1) + 12(k − 1). +And we can check that for each 1 ≤ i ≤ k − 2, we have +|Ui| = |Vi| − |Bi| > n/(k − 2) − +n +3(k − 1) − +n +3(k − 1) − o(n) > +n +3(k − 2) ≥ +k−2 +� +i=1 +mi + Θ(n). (2) +Fix an arbitrary vertex x0 ∈ Uk−2 and let Y := NG′(x0)\Vk−2. It is clear that +|Y | ≥ n − |Vk−2| − mk−2. +We can find a copy of Kk−3 in G′ on vertices x1, x2, . . . , xk−3 with xi ∈ Ui ∩ Y = Ui ∩ NG′(x0) +by greedily choosing the vertex xi ∈ Ui ∩NG′(x0)∩· · ·∩NG′(xi−1) for 1 ≤ i ≤ k −3 one by one +since (2) holds for each 1 ≤ i ≤ k − 3. Then, since |Ui| − mk−2 ≥ |Ui| − �k−2 +i=1 mj > k − 2 holds +for each 1 ≤ i ≤ k − 3 by (2), we can find a vertex u ∈ Ui0 ∩ Y distinct from x1, x2, . . . , xk−3, +where we choose i0 such that mi0 = min{m1, . . . , mk−3}. Let W := NG′(x1)∩· · ·∩NG′(xk−3)∩ +NG′(u) ∩ Vk−2. We can see W ∋ x0, W ∩ Y = ∅, and +|W| ≥ |Vk−2| − +k−3 +� +i=1 +mj − mi0 ≥ |Vk−2| − +� +1 + +1 +k − 3 +� k−3 +� +i=1 +mj. +Then by using Lemma 2.1, we get a set W ′ with |W ′| = |W| such that |NG(w) ∩ W ′| = 1 for +each w ∈ W ′, and |W ′ ∩W| ≤ 2. Note that all vertices in Y are adjacent to the vertex x0 ∈ W +in G′ ⊆ G, so we can see |W ′ ∩ Y | ≤ 1. +5 + +As W ∩ Y = ∅, |W ′ ∩ W| ≤ 2, |W ′ ∩ Y | ≤ 1 and |W ′| = |W|, we get n ≥ |W ∪ Y ∪ W ′| ≥ +2|W| + |Y | − 3. Thus +2|W| + |Y | ≤ n + 3. +But on the other hand, we can check that +2|W| + |Y | ≥ 2 + +|Vk−2| − +� +1 + +1 +k − 3 +� k−3 +� +j=1 +mj + + + (n − |Vk−2| − mk−2) +≥ n + |Vk−2| − 2 +� +1 + +1 +k − 3 +� k−2 +� +j=1 +mj +≥ n + +n +k − 2 − 2 +� +1 + +1 +k − 3 +� � +n +3(k − 1) + 12(k − 1) +� +> n + 3. +This derives a contradiction. So we have fk(n) ≤ e(Tk−2(n)) − Cn2 for n sufficiently large. +We would like to remark that the above proof relies on the existence of Kk−2. (Recall that +in Lemma 2.1, G[{w, x1, x2, . . . , xk−3}] forms a copy of Kk−2 for each vertex w ∈ W.) So using +this approach, we will not be able to improve the upper bound to the following +e(G) ≤ ex(n, Kk−2) = e(Tk−3(n)) ≤ e(Tk−2(n)) − +n2 +2(k − 2)(k − 3); +that says, we are not able to obtain a constant ck better than the order of magnitude k−2. +4 +The 4-critical case +In this section we consider 4-critical graphs and prove Theorem 1.2. +Before presenting the proof of Theorem 1.2, we like to give a short proof of a slightly +weaker bound (see Theorem 4.1) than Theorem 1.2 to illustrate the proof ideas. In doing this, +we study certain local structure based on 2-paths (i.e., a path of length two) in the proof of +Theorem 4.1, while we consider 4-cycles (i.e., a cycle of length four) in replace of 2-paths in +the proof of Theorem 1.2. +4.1 +A weaker upper bound +We first show the following result. +Theorem 4.1. For any integer n ≥ 4, it holds that f4(n) < 1 +6n2 + 10n ≤ 0.167n2 + 10n. +We also need two lemmas as follows. For a graph G, we denote t(G) to be the number of +triangles in G. For a vertex v, let tG(v) be the number of triangles containing the vertex v in +G. When G is clear, we often drop the subscript. +6 + +Lemma 4.2. Suppose G has at most n triangles and minimum degree at least 3. Then G +contains a 2-path xyz such that +d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≥ 6e(G) +n +− 9n2 +e(G). +Proof. For some vertex v ∈ V (G), write N(v) = {v1, v2, . . . , vt} for some t ≥ 3. Let +Pv := {v1vv2, . . . , vt−1vvt, vtvv1} +be a family of 2-paths with center v. We have |Pv| = d(v), and +� +xyz∈Pv +(d(x) + d(y) + d(z)) = d(v)2 + 2 +� +u∈N(v) +d(u), +� +xyz∈Pv +(t(x) + t(z)) = 2 +� +u∈N(v) +t(u). +Then let P := � +v∈V (G) Pv. We have +|P| = +� +v∈V (G) +d(v) = 2e(G). +Using Jensen’s inequality, we get +� +xyz∈P +(d(x) + d(y) + d(z)) = +� +v∈V (G) +d(v)2 + 2 +� +v∈V (G),u∈N(v) +d(u) = +� +v∈V (G) +d(v)2 + 2 +� +u∈V (G),v∈N(u) +d(u) += +� +v∈V (G) +d(v)2 + 2 +� +u∈V (G) +d(u)2 = 3 +� +v∈V (G) +d(v)2 ≥ 12e(G)2/n. +Since every vertex in G has degree at most n − 1 and � +u∈V (G) t(u) = 3t(G) ≤ 3n, we get +� +xyz∈P +(t(x) + t(z)) = 2 +� +v∈V (G) +� +u∈N(v) +t(u) = 2 +� +u∈V (G) +d(u)t(u) ≤ 2n +� +u∈V (G) +t(u) ≤ 6n2. +So by picking a 2-path xyz in P uniformly and randomly, we see +E[d(x) + d(y) + d(z) − 3t(x) − 3t(z)] ≥ 12e(G)2/n − 18n2 +|P| += 6e(G) +n +− 9n2 +e(G). +Thus we can find a 2-path xyz as desired. +Lemma 4.3. For any 2-path xyz in a 4-critical graph G, we have +d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≤ n + 1. +7 + +Proof. Let X := N(x), Y := N(y), Z := N(z), and W := X ∩ Z. If u ∈ X ∩ Y , uxy is a +triangle. So |X ∩ Y | ≤ t(x). Similarly, |Z ∩ Y | ≤ t(z). Then we have +|X ∪Y ∪Z| ≥ |X|+|Y |+|Z|−|X ∩Y |−|Z ∩Y |−|X ∩Z| ≥ d(x)+d(y)+d(z)−t(x)−t(z)−|W|. +By Lemma 2.1, we can find a set W ′ ⊆ V (G) and a bijection ϕ : W → W ′ such that +W ′ = {ϕ(w) : w ∈ W ′}, and for each w ∈ W, we have both N(ϕ(w)) ∩ W = {w} and +N(w) ∩ W ′ = {ϕ(w)}. +We consider the size of W ′∩(X ∪Y ∪Z). Since both N(ϕ(w))∩W = {w} and N(w)∩W ′ = +{ϕ(w)} hold for each w ∈ W, and we know y ∈ W, we can see |W ′ ∩ Y | ≤ |W ′ ∩ N(y)| ≤ 1. +Suppose v′ ∈ W ′ ∩ X. There is a vertex v ∈ W such that vv′ is an edge. Then we see xvv′ is +a triangle. So |W ′ ∩ X| ≤ 2t(x). Similarly, |W ′ ∩ Z| ≤ 2t(z). Totally, we have +|W ′ ∩ (X ∪ Y ∪ Z)| ≤ |W ′ ∩ X| + |W ′ ∩ Y | + |W ′ ∩ Z| ≤ 2t(x) + 2t(z) + 1. +Finally, we get +n ≥ |X ∪ Y ∪ Z ∪ W ′| = |X ∪ Y ∪ Z ∪ W ′| + |W ′| − |W ′ ∩ (X ∪ Y ∪ Z)| += (d(x) + d(y) + d(z) − t(x) − t(z) − |W|) + |W| − (2t(x) + 2t(z) + 1) += d(x) + d(y) + d(z) − 3t(x) − 3t(z) − 1, +completing the proof of this lemma. +Now we can finish the proof of this subsection. +Proof of Theorem 4.1. Let G be an n-vertex 4-critical graph. It is easy to see that the +minimum degree of G is at least 3. +By the result in [11], G contains at most n copies of +triangles. Applying Lemma 4.2, we can get a 2-path xyz with +d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≥ 6e(G) +n +− 9n2 +e(G). +Together with Lemma 4.3, we have +6e(G) +n +− 9n2 +e(G) ≤ n + 1. +This implies that e(G) < n2/6 + 10n. +4.2 +The proof of Theorem 1.2 +To show Theorem 1.2, we need some new lemmas. The coming lemma can be easily obtained +by averaging, which says that every graph contains an edge such that the sum of the degrees +of its two endpoints is at least twice the average degree of the graph. +8 + +Lemma 4.4. Any graph G contains an edge xy such that +d(x) + d(y) ≥ 2d(G). +Proof. By Jensen’s inequality, we can get +� +xy∈E +(d(x) + d(y)) = +� +v∈V +d(v)2 ≥ nd(G)2. +Note that |E| = (nd(G)) /2. Thus there exists an edge xy ∈ E such that +d(x) + d(y) ≥ +nd(G)2 +(nd(G)) /2 = 2d(G), +proving the lemma. +We now give the following lemma about 4-cycles, which can be viewed as a generalization +of the previous lemma. Recall the well-known result of Reiman [9] that any n-vertex graph +without containing 4-cycles has at most n +4(1 + √4n − 3) < n +3 +2 edges. +Lemma 4.5. Any n-vertex graph G with e(G) > n +4 (1 + √4n − 3) contains a 4-cycle v1v2v3v4 +satisfying that +d(v1) + d(v2) + d(v3) + d(v4) ≥ 4d(G) − O(n +3 +4 ). +Proof. Fix ǫ := 9n− 1 +4 . Note that G must contain 4-cycles by the result of Reiman [9]. Suppose +to the contrary that any 4-cycle v1v2v3v4 in G satisfies d(v1)+d(v2)+d(v3)+d(v4) < 4d(G)−4ǫn. +Let A := {v ∈ V : d(v) < d(G)} and B := {v ∈ V : d(v) ≥ d(G)}. Then A∪B forms a partition +of V (G) such that G[B] does not contain any 4-cycle. +For each 1 ≤ i ≤ d(G)/ǫn, let Ai := {v ∈ V : d(G) − iǫn ≤ d(v) < d(G) − (i − 1)ǫn}. +Then these Ai’s form a partition of A. For each 1 ≤ i ≤ (n − d(G)) /ǫn, let Bi := {v ∈ V : +d(G) + (i − 1)ǫn ≤ d(v) < d(G) + iǫn}. Then these Bi’s form a partition of B. It is not +hard to check that G[A1] does not contain any 4-cycle, and for each 1 ≤ i ≤ (n − d(G)) /ǫn, +G +��i+1 +j=1 Aj, Bi +� +does not contain any 4-cycle. +We delete all edges in G[B], G[A1] and G +��i+1 +j=1 Aj, Bi +� +for each 1 ≤ i ≤ (n − d(G)) /ǫn to +get a spanning subgraph G′ of G. By the result of Reiman [9], we can obtain +e(G′) ≥ e(G) − (2 + (n − d(G)) /ǫn) n +3 +2 ≥ e(G) − 2n +3 +2 − 1 +9n +7 +4 ≥ e(G) − 19 +9 n +7 +4 . +Thus we have +d(G′) ≥ d(G) − 38 +9 n +3 +4 . +Note that any edge of G′ is either contained in A, or between Aj and Bi for some j ≥ i + 2; +moreover, we have e(G′[A1]) = 0. Thus, as n is large enough, it is easy to check that for any +edge xy in G′, +dG′(x) + dG′(y) < 2d(G) − ǫn = 2d(G) − 9n +3 +4 < 2d(G′). +This contradicts Lemma 4.4, thus proving Lemma 4.5. +9 + +The following lemma is derived from Lemma 2.1, which provides an essential structure to +the proof of Theorem 1.2. +Lemma 4.6. Let G be a 4-critical graph. Suppose v1v2v3v4 is a 4-cycle in G, and V1, V2, V3, V4 +are four sets such that {v2, v4} ⊆ V1 ⊆ N(v1), {v1, v3} ⊆ V2 ⊆ N(v2), {v2, v4} ⊆ V3 ⊆ N(v3), +and {v1, v3} ⊆ V4 ⊆ N(v4). Let X = V1 ∩ V3 and Y = V2 ∩ V4. Then there exist sets X′′ and +Y ′′ such that +• X′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ = Y ′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4), +• e(G[X′′, X]) ≤ |X| and e(G[Y ′′, Y ]) ≤ |Y |, and +• |X′′| ≥ |X| − 2tG(v1) − 2tG(v3) − 2 and |Y ′′| ≥ |Y | − 2tG(v2) − 2tG(v4) − 2. +Proof. As X ⊆ N(v1) ∩ N(v3), by Lemma 2.1 for k = 4, there exists a set X′ ⊆ V (G) and +a bijection ϕ : X → X′ such that X′ = {ϕ(x) : x ∈ X}, and for each x ∈ X, we have both +N(ϕ(x)) ∩ X = {x} and N(x) ∩ X′ = {ϕ(x)}. We define X′′ := X′\ (V1 ∪ V2 ∪ V3 ∪ V4), then +obviously X′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ and e(G[X′′, X]) ≤ |X|. +As Y ⊆ N(v2) ∩ N(v4), by Lemma 2.1 for k = 4, there exists a set Y ′ ⊆ V (G) and a +bijection φ : Y → Y ′ such that Y ′ = {φ(y) : y ∈ Y }, and for each y ∈ Y , we have both +N(φ(y)) ∩ Y = {y} and N(y) ∩ Y ′ = {φ(y)}. We define Y ′′ := Y ′\ (V1 ∪ V2 ∪ V3 ∪ V4), then +obviously Y ′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ and e(G[Y ′′, Y ]) ≤ |Y |. +Then we want to show the last property. +All vertices in V2 are adjacent to the vertex v2 ∈ X. Then we have |X′ ∩ V2| ≤ 1 since +|N(x)∩X′| = 1 for each x ∈ X. Similarly, we have |X′∩V4| ≤ 1, |Y ′∩V1| ≤ 1, and |Y ′∩V3| ≤ 1. +All vertices in V1 are adjacent to the vertex v1. Since each vertex in X′ has a neighbor +in X ⊆ N(v1), we can check that |X′ ∩ V1| ≤ 2t(v1). Similarly, we have |X′ ∩ V3| ≤ 2t(v3), +|Y ′ ∩ V2| ≤ 2t(v2), |Y ′ ∩ V4| ≤ 2t(v4). Therefore, +|X′′| = |X′| − |X′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) | ≥ |X| − 2t(v1) − 2t(v3) − 2, +and +|Y ′′| = |Y ′| − |Y ′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) | ≥ |Y | − 2t(v2) − 2t(v4) − 2, +completing the proof. +Now we are ready to prove Theorem 1.2. +Proof of Theorem 1.2. Throughout this proof, we assume that n is sufficiently large, and +the subscripts of the notations such as vi’s and Vi’s are under module 4. Suppose by contra- +diction that there exists an n-vertex 4-critical graph G with e(G) ≥ 0.164n2. By the result +in [11], G contains at most n copies of triangles. +Let V0 := {v ∈ V (G) : tG(v) ≥ √n}. +Then clearly we have |V0| < 3√n. Let G′ := G[V (G) − V0]. It is not hard to see e(G′) ≥ +10 + +e(G) − n|V0| > e(G) − 3n +3 +2 ≥ 0.164n2 − o(n2). +Note that t(G′) ≤ t(G) ≤ n. +Therefore, +by deleting at most n edges from G′, we can get a subgraph G′′ ⊆ G′ such that t(G′′) = 0, +e(G′′) ≥ e(G′) − n ≥ 0.164n2 − o(n2), and tG(v) < √n for each v ∈ V (G′′) = V (G) − V0. By +applying Lemma 4.5 to G′′, we can get a 4-cycle v1v2v3v4 in G′′ such that +|V1| + |V2| + |V3| + |V4| ≥ 8e(G′′)/n − o(n) ≥ 1.312n − o(n), +(3) +where Vi := NG′′(vi) for each 1 ≤ i ≤ 4. Note that for each 1 ≤ i ≤ 4, every vertex in Vi ∩ Vi+1 +must form a triangle with the vertices vi, vi+1 in G′′, which contradicts the fact t(G′′) = 0. So +it is clear that +Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4. +Also it is easy to check that {vi−1, vi+1} ⊆ Vi ⊆ NG(vi) for each 1 ≤ i ≤ 4. Define X = V1 ∩ V3 +and Y = V2 ∩ V4. +Applying Lemma 4.6, we can get two sets X′′, Y ′′ satisfying the three +properties of Lemma 4.6. Note that X′′ and Y ′′ are disjoint from V1 ∪V2 ∪V3 ∪V4, V1 ∩V3 = X, +V2 ∩ V4 = Y , and Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4. So we can see that +|V1| + |V2| + |V3| + |V4| − |X| − |Y | + |X′′ ∪ Y ′′| ≤ n. +Besides, by using the last property in Lemma 4.6, we have +|X′′ ∪ Y ′′| ≥ max{|X′′|, |Y ′′|} ≥ |X′′| + |Y ′′| +2 +≥ |X| + |Y | +2 +− O(√n). +(4) +The above two inequalities tell us that +|X| + |Y | +2 +≥ |V1| + |V2| + |V3| + |V4| − n − O(√n) ≥ 0.312n − o(n). +(5) +Then we consider the non-edges of the graph G, i.e., the edges of the graph G. First, since +Vi = NG′′(vi) ⊆ NG(vi) and vi ∈ V (G′′), we can see e(G[Vi]) ≤ tG(vi) ≤ √n for each 1 ≤ i ≤ 4. +So +e(G[Vi]) ≥ +�|Vi| +2 +� +− o(n2) = 1 +2|Vi|2 − o(n2) for each 1 ≤ i ≤ 4. +Thus by noting V1 ∩ V3 = X, V2 ∩ V4 = Y , and Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4, we can get +����� +4� +i=1 +E(G[Vi]) +����� ≥ +4 +� +i=1 +e(G[Vi]) − +�|X| +2 +� +− +�|Y | +2 +� +≥ 1 +2 +� 4 +� +i=1 +|Vi|2 − |X|2 − |Y |2 +� +− o(n2). +Next, since G can be made triangle-free by deleting at most n edges and any n-vertex triangle- +free graph has at most 1 +4n2 edges, we can see e(G[X′′ ∪ Y ′′]) ≤ 1 +4|X′′ ∪ Y ′′|2 + n, and thus +e(G[X′′ ∪ Y ′′]) ≥ 1 +4|X′′ ∪ Y ′′|2 − o(n2). +By the properties on X′′, Y ′′ we derive from Lemma 4.6, we can obtain +e(G[X′′, X]) ≥ |X′′||X| − |X| ≥ |X|2 − o(n2), +11 + +e(G[Y ′′, Y ]) ≥ |Y ′′||Y | − |Y | ≥ |Y |2 − o(n2). +By the above three inequalities, we can deduce that +e(G) = +�n +2 +� +− e(G) ≤ +�n +2 +� +− 1 +2 +� 4 +� +i=1 +|Vi|2 − |X|2 − |Y |2 +� +− 1 +4|X′′ ∪ Y ′′|2 − |X|2 − |Y |2 + o(n2) +≤ 1 +2n2 − 1 +8 (|V1| + |V2| + |V3| + |V4|)2 − 1 +4 +�|X| + |Y | +2 +�2 +− +�|X| + |Y | +2 +�2 ++ o(n2) +≤ 1 +2n2 − 1 +8(1.312n)2 − 5 +4(0.312n)2 + o(n2) < 0.1632n2 + o(n2), +where the second inequality comes from the inequality (4), and the third inequality comes from +the inequalities (3) and (5). This contradicts the assumption that e(G) ≥ 0.164n2, completing +the proof of Theorem 1.2. +Our understanding for the functions fk(n) is generally poor, and it is not even known if +f4(n) < f5(n) holds for sufficiently large integers n. +(6) +So it seems to be a natural next step to pursue the question that if f4(n) ≤ cn2 holds for some +constant c < +4 +31 and sufficiently large n. Note that if this is true, then it would imply (6). +Acknowledgement. The authors would like to thank Prof. Alexandr Kostochka for many +valuable comments on a preliminary version of the manuscript. +References +[1] H. L. Abbott and B. Zhou, On a conjecture of Gallai concerning complete subgraphs of +k-critical graphs, Discrete Math. 100 (1992), 223–228. +[2] G. A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. +London Math. Soc. 27 (1952), 85–92. +[3] Z. F¨uredi, A proof of the stability of extremal graphs, Simonovits’ stability from Sze- +mer´edi’s regularity, J. Combin. Theory Ser. B 115 (2015), 66–71. +[4] J. Gao and J. Ma, Tight bounds towards a conjecture of Gallai, Combinatorica, to appear. +[5] D. Greenwell and L. Lov´asz, Applications of product colouring, Aeta Math. Acad. Sci. +Hungar. 25 (1974), 335–340. +[6] T. R. Jensen and B. Toft, Graph coloring problems, John Wiley & Sons, 2011. +[7] A. E. K´ezdy and H. S. Snevily, On extensions of a conjecture of Gallai, J. Combin. Theory +Ser. B 70 (1997), 317–324. +12 + +[8] W. Pegden, Critical graphs without triangles: an optimum density construction, Combi- +natorica 33 (2013), 495–512. +[9] I. Reiman, +¨Uber ein Problem von K. Zarankiewicz, Acta Math. Acad. Sci. Hungar. 9 +(1958), 269–273. +[10] M. Simonovits, A method for solving extremal problems in graph theory, stability prob- +lems, in: +Theory of Graphs, Proc. Colloq. Tihany, 1966, Academic Press, New York, +1968. +[11] M. Stiebitz, Subgraphs of colour-critical graphs, Combinatorica 7 (1987), 303–312. +[12] B. Toft, On the maximal number of edges of critical k-chromatic graphs, Studia Sci. +Math. Hungar. 5 (1970), 461–470. +13 + diff --git a/udAzT4oBgHgl3EQfsP1l/content/tmp_files/load_file.txt b/udAzT4oBgHgl3EQfsP1l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6d4ef9a306282f41ae95336bb67aedc29d5ab43 --- /dev/null +++ b/udAzT4oBgHgl3EQfsP1l/content/tmp_files/load_file.txt @@ -0,0 +1,449 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf,len=448 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='01656v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='CO] 4 Jan 2023 On the maximum number of edges in k-critical graphs Cong Luo∗ Jie Ma† Tianchi Yang‡ Abstract A graph is called k-critical if its chromatic number is k but any proper subgraph has chromatic number less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' An old and important problem in graph theory asks to determine the maximum number of edges in an n-vertex k-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This is widely open for any integer k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Using a structural characterization of Greenwell and Lov´asz and an extremal result of Simonovits, Stiebitz proved in 1987 that for k ≥ 4 and sufficiently large n, this maximum number is less than the number of edges in the n-vertex balanced complete (k −2)-partite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In this paper we obtain the first improvement on the above result in the past 35 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Our proofs combine arguments from extremal graph theory as well as some structural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' A key lemma we use indicates a partial structure in dense k-critical graphs, which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 1 Introduction All graphs we consider are finite and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' A graph G is k-colorable if we can assign k colors to its vertices such that no adjacent vertices receive the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We say a graph G is k-chromatic if it is k-colorable but not (k − 1)-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' A graph G is called k-critical if G is k-chromatic but every its proper subgraph is (k − 1)-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For k ∈ {1, 2} the only k-critical graph is Kk, and the family of 3-critical graphs is precisely the family of odd cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In this paper, we consider k-critical graphs for k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' A central problem in graph theory asks to determine the maximum number of edges fk(n) in an n-vertex k-critical graph (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Before we discuss the literature on fk(n), we would like to point out a relevant yet easy fact that the Tur´an graph Tk(n) (that is, the n-vertex balanced complete k-partite graph) has the maximum number of edges among all n-vertex ∗School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Email: luoc@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' †School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Research supported in part by the National Key R and D Program of China 2020YFA0713100, National Natural Science Foundation of China grant 12125106, and Anhui Initiative in Quantum Information Technologies grant AHY150200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Email: jiema@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' ‡Department of Mathematics, National University of Singapore, 119076, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Research supported in part by Professor Hao Huang’s start-up grant at NUS and an MOE Academic Research Fund (AcRF) Tier 1 grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Email: tcyang@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 1 k-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Dirac [2] gave f6(n) ≥ 1 4n2 + n by considering the graphs obtained by joining two vertex-disjoint odd cycles with the same number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Toft [12] proved that for every k ≥ 4, there exists a positive constant ck such that fk(n) ≥ ckn2 holds for all integers n ≥ k (except n = k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In the most basic and interesting cases k = 4, 5, the constants are given by c4 ≥ 1 16 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='0625 and c5 ≥ 4 31 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In the general case when k ≥ 6, explicit constructions in [12] show that there exist infinitely many values of n such that fk(n) ≥ �1 2 − 3 2k − δk � n2, where δk = 0 if k ≡ 0 (mod 3), δk = 8/7 if k ≡ 1 (mod 3), and δk = 44/23 if k ≡ 2 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' To our best knowledge, no construction for giving better constants fk(n)/n2 have been found since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is also an open question if limn→∞ fk(n) n2 exits for any k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In 2013, Pegden [8] considered dense triangle-free k-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' He constructed infinitely many n-vertex triangle-free 4-critical graphs with at least � 1 16 − o(1) � n2 edges, triangle-free 5-critical graphs with at least � 4 31 − o(1) � n2 edges, and triangle-free k-critical graphs with at least �1 4 − o(1) � n2 edges for every k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The last bound is asymptotically best possible by Tur´an’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' He also showed the existence of dense k-critical graphs without containing any odd cycle of length at most ℓ for any ℓ, which is again asymptotically tight for k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Turning to the upper bound of fk(n), since any n-vertex k-critical graph with n > k does not contain Kk as a subgraph, by Tur´an’s theorem one can easily obtain that fk(n) < e(Tk−1(n)) for any n > k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Using a characterization of Greenwell and Lov´asz [5] for subgraphs of k-critical graphs and a classical theorem of Simonovits [10], Stiebitz [11] improved this trivial bound in 1987 by showing that fk(n) < e(Tk−2(n)) for sufficiently large integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (1) It has been 35 years since then and as far as we are aware, this remains the best upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' There is a natural relation between fk(n) and the problem of determining the maximum number of copies of Kk−1 in k-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Abbott and Zhou [1] generalized an earlier result of Stiebitz [11] on 4-critical graphs and showed that for any k ≥ 4 every k-critical graph on n vertices contains at most n copies of Kk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The bound was further improved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Recently, Gao and Ma [4] proved a sharp result that for any n > k ≥ 4, any k-critical graph on n vertices contains at most n − k + 3 copies of Kk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' If we delete one edge for every Kk−1 in a k-critical graph on n vertices, then this can result in a graph without containing Kk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Using Tur´an’s theorem and the above result of [4], we can derive that fk(n) ≤ e(Tk−2(n)) + n − k + 3 for any n > k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In this paper, we focus on the upper bound of fk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Our first result improves the long- standing upper bound (1) of Stiebitz [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For any integer k ≥ 4 and sufficiently large integers n, there exists a constant ck ≥ 1 36(k−1)2 such that fk(n) ≤ e(Tk−2(n)) − ckn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Our second result considers 4-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' A better upper bound for f4(n) than Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 is obtained in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For sufficiently large integers n, it holds that f4(n) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='164n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The proofs of both theorems rely on arguments from extremal graph theory (such as the stability of F¨uredi [3]) and a structural lemma (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1) given in the coming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 indicates a partial structure in dense critical graphs (under certain constraints), which can be witnessed in many classical constructions of dense critical graphs (see the discus- sion at the beginning of Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For that, we would like to give a full construction for the well-known Toft graph (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The vertex set of the Toft graph is formed by 4 disjoint sets A, B, C, D with the same odd size, where A and D are odd cycles, B and C are independent sets, the edges between B and C form a complete bipartite graph, and both of the edges in (A, B) and in (C, D) form perfect matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is easy to check that the n-vertex Toft graph is 4-critical and has 1 16n2+n edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We remark that the Toft graph remains the best construction for dense 4-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We use standard notation in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let G denote the complement of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For a vertex v in a graph G, let NG(v) denote the neighborhood of v in G, and let dG(v) := |NG(v)| denote the degree of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' When G is clear from the context, we often drop the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let d(G) denote the average degree of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Also, for any S ⊆ V (G), let G[S] denote the induced subgraph of G on the vertex set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For any disjoint sets A, B ⊆ V (G), let G[A, B] denote the induced bipartite subgraph of G with bipartition (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In Section 2, we prove a lemma which is key for the coming proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 in Section 3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 2 Key lemma In this section we prove our key lemma, which roughly says that if a k-critical graph G contains certain t copies of Kk−2 sharing k−3 common vertices, then there exists an “induced” matching of size t in G which are connected to these cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This indicates a substructure similar to the Toft graph (and many other examples of k-critical graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In particular, it reveals that the structure of k-critical graphs cannot be close to the Tur´an graph Tk−2(n) and thus the inequality (1) should not be tight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let k ≥ 4 and let G be a k-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose that G [{x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3}] forms a copy of Kk−3 and there exists a set W ⊆ N(x1) ∩ · · · ∩ N(xk−3) ∩ N(u) for some vertex u /∈ {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then there exist a set W ′ and a bijection ϕ : W → W ′ such that 3 N(ϕ(w)) ∩ W = {w} and N(w) ∩ W ′ = {ϕ(w)} hold for each w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Moreover, if |W| ≥ 3, then W is an independent set in G, and W ′ ∩ W = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For each vertex w ∈ W, by deleting the edge uw from the k-critical graph G, we can get a (k − 1)-chromatic graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We denote the color classes of G′ by C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is easy to see the vertices u and w are in the same color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since G[{x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3, w}] is a (k − 2)-clique, we can assume x1 ∈ C1, x2 ∈ C2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3 ∈ Ck−3, and u, w ∈ Ck−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The fact W ⊆ N(x1)∩· · ·∩N(xk−3)∩N(u) tells us that the set W\\{w} (if not empty) must be contained in Ck−1, and thus W\\{w} is an independent set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We claim N(w) ∩ Ck−1 must contain a vertex, say ϕ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since otherwise C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Ck−3, Ck−2 −{w}, Ck−1 ∪{w} can be a (k−1)-coloring of G, which contradicts the fact that G is k-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Besides, {ϕ(w)} ∪ (W\\{w}) ⊆ Ck−1 tells us that N(ϕ(w)) ∩ W = {w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Now we define W ′ := {ϕ(w) : w ∈ W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' As we have shown that N(ϕ(w)) ∩ W = {w} holds for each w ∈ W, it is easy to see |W ′| = |W|, ϕ : W → W ′ is a bijection, and N(w) ∩ W ′ = {ϕ(w)} holds for each w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Moreover, if |W| ≥ 3, then W is an independent set in G (since W\\{v} is an independent set in G for each vertex v ∈ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the fact that the edges between W ′ and W precisely form a matching, we can see W ′ ∩ W = ∅ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It would be very interesting to see if this lemma (or its proof) can be extended further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 3 The general case: k-critical Providing a simple and new proof of the stability for the Tur´an number ex(n, Kr+1), F¨uredi [3] showed that if an n-vertex graph G is Kr+1-free and has at least e(Tr(n)) − t edges where 0 ≤ t < e(Tr(n)) < n2, then there exists a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Vr of V (G) such that �r i=1 e(G[Vi]) ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The proof of [3] (see Corollary 3) also indicates that if the complete r- chromatic graph with color classes V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Vr is denoted by K, then |E(K)\\E(G)| ≤ 2t and moreover, �r i=1 (|Vi| − n/r)2 < 4t + o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We summarize in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 (F¨uredi [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose that G is an n-vertex Kr+1-free graph with e(G) ≥ e(Tr(n))− t where 0 ≤ t < e(Tr(n)) < n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then there exists a complete r-chromatic graph K := K(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Vr) with V (K) = V (G) such that |E(K)\\E(G)| ≤ 2t, and r � i=1 � |Vi| − n r �2 < 4t + o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We are ready to use Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Fix k ≥ 4 and let C = 1 36(k−1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let G be a k-critical graph on n vertices with e(G) > e(Tk−2(n)) − Cn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In the rest of the proof, we will always assume that n is large enough, and we denote V (G) by V for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The result in [1] tells us the number of copies of Kk−1 in G is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So by deleting at most n edges in G, we obtain a spanning subgraph G′ which is Kk−1-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Obviously we have e(G′) > e(Tk−2(n)) − (Cn2 + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' With the application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, we get a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Vk−2 of V and a complete (k − 2)-chromatic graph K := K(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , Vk−2) such that |E(K)\\E(G′)| ≤ 2(Cn2 + n) and ���|Vi| − n k − 2 ��� < � 4Cn2 + o(n2) < n 3(k − 1) + o(n) for each 1 ≤ i ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Without loss of generality, we assume |V1| ≤ · · · ≤ |Vk−2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus |Vk−2| ≥ n/(k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We call the edges in E(K)\\E(G′) as missing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' And the number of missing edges incident to the vertex v in K is called the missing degree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For each 1 ≤ i ≤ k − 2, we define Bi to be the set of � n 3(k−1) � vertices in Vi satisfying that there exists some mi such that the missing degree of any vertex in Bi is at least mi, and the missing degree of any vertex in Ui := Vi −Bi is at most mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since there are at most 2(Cn2 + n) missing edges in total, we have �k−2 i=1 mi|Bi| < 4(Cn2 + n), and thus we can get k−2 � i=1 mi < 4(Cn2 + n) � � n 3(k − 1) � ≤ n 3(k − 1) + 12(k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' And we can check that for each 1 ≤ i ≤ k − 2, we have |Ui| = |Vi| − |Bi| > n/(k − 2) − n 3(k − 1) − n 3(k − 1) − o(n) > n 3(k − 2) ≥ k−2 � i=1 mi + Θ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (2) Fix an arbitrary vertex x0 ∈ Uk−2 and let Y := NG′(x0)\\Vk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is clear that |Y | ≥ n − |Vk−2| − mk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We can find a copy of Kk−3 in G′ on vertices x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3 with xi ∈ Ui ∩ Y = Ui ∩ NG′(x0) by greedily choosing the vertex xi ∈ Ui ∩NG′(x0)∩· · ·∩NG′(xi−1) for 1 ≤ i ≤ k −3 one by one since (2) holds for each 1 ≤ i ≤ k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then, since |Ui| − mk−2 ≥ |Ui| − �k−2 i=1 mj > k − 2 holds for each 1 ≤ i ≤ k − 3 by (2), we can find a vertex u ∈ Ui0 ∩ Y distinct from x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3, where we choose i0 such that mi0 = min{m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , mk−3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let W := NG′(x1)∩· · ·∩NG′(xk−3)∩ NG′(u) ∩ Vk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We can see W ∋ x0, W ∩ Y = ∅, and |W| ≥ |Vk−2| − k−3 � i=1 mj − mi0 ≥ |Vk−2| − � 1 + 1 k − 3 � k−3 � i=1 mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, we get a set W ′ with |W ′| = |W| such that |NG(w) ∩ W ′| = 1 for each w ∈ W ′, and |W ′ ∩W| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that all vertices in Y are adjacent to the vertex x0 ∈ W in G′ ⊆ G, so we can see |W ′ ∩ Y | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 5 As W ∩ Y = ∅, |W ′ ∩ W| ≤ 2, |W ′ ∩ Y | ≤ 1 and |W ′| = |W|, we get n ≥ |W ∪ Y ∪ W ′| ≥ 2|W| + |Y | − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus 2|W| + |Y | ≤ n + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' But on the other hand, we can check that 2|W| + |Y | ≥ 2 \uf8eb \uf8ed|Vk−2| − � 1 + 1 k − 3 � k−3 � j=1 mj \uf8f6 \uf8f8 + (n − |Vk−2| − mk−2) ≥ n + |Vk−2| − 2 � 1 + 1 k − 3 � k−2 � j=1 mj ≥ n + n k − 2 − 2 � 1 + 1 k − 3 � � n 3(k − 1) + 12(k − 1) � > n + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This derives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So we have fk(n) ≤ e(Tk−2(n)) − Cn2 for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We would like to remark that the above proof relies on the existence of Kk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (Recall that in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, G[{w, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , xk−3}] forms a copy of Kk−2 for each vertex w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=') So using this approach, we will not be able to improve the upper bound to the following e(G) ≤ ex(n, Kk−2) = e(Tk−3(n)) ≤ e(Tk−2(n)) − n2 2(k − 2)(k − 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' that says, we are not able to obtain a constant ck better than the order of magnitude k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 4 The 4-critical case In this section we consider 4-critical graphs and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Before presenting the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2, we like to give a short proof of a slightly weaker bound (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1) than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2 to illustrate the proof ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' In doing this, we study certain local structure based on 2-paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=', a path of length two) in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, while we consider 4-cycles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=', a cycle of length four) in replace of 2-paths in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 A weaker upper bound We first show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For any integer n ≥ 4, it holds that f4(n) < 1 6n2 + 10n ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='167n2 + 10n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We also need two lemmas as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For a graph G, we denote t(G) to be the number of triangles in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For a vertex v, let tG(v) be the number of triangles containing the vertex v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' When G is clear, we often drop the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 6 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose G has at most n triangles and minimum degree at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then G contains a 2-path xyz such that d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≥ 6e(G) n − 9n2 e(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For some vertex v ∈ V (G), write N(v) = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , vt} for some t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let Pv := {v1vv2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' , vt−1vvt, vtvv1} be a family of 2-paths with center v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We have |Pv| = d(v), and � xyz∈Pv (d(x) + d(y) + d(z)) = d(v)2 + 2 � u∈N(v) d(u), � xyz∈Pv (t(x) + t(z)) = 2 � u∈N(v) t(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then let P := � v∈V (G) Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We have |P| = � v∈V (G) d(v) = 2e(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Using Jensen’s inequality, we get � xyz∈P (d(x) + d(y) + d(z)) = � v∈V (G) d(v)2 + 2 � v∈V (G),u∈N(v) d(u) = � v∈V (G) d(v)2 + 2 � u∈V (G),v∈N(u) d(u) = � v∈V (G) d(v)2 + 2 � u∈V (G) d(u)2 = 3 � v∈V (G) d(v)2 ≥ 12e(G)2/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since every vertex in G has degree at most n − 1 and � u∈V (G) t(u) = 3t(G) ≤ 3n, we get � xyz∈P (t(x) + t(z)) = 2 � v∈V (G) � u∈N(v) t(u) = 2 � u∈V (G) d(u)t(u) ≤ 2n � u∈V (G) t(u) ≤ 6n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So by picking a 2-path xyz in P uniformly and randomly, we see E[d(x) + d(y) + d(z) − 3t(x) − 3t(z)] ≥ 12e(G)2/n − 18n2 |P| = 6e(G) n − 9n2 e(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus we can find a 2-path xyz as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For any 2-path xyz in a 4-critical graph G, we have d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let X := N(x), Y := N(y), Z := N(z), and W := X ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' If u ∈ X ∩ Y , uxy is a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So |X ∩ Y | ≤ t(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Similarly, |Z ∩ Y | ≤ t(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then we have |X ∪Y ∪Z| ≥ |X|+|Y |+|Z|−|X ∩Y |−|Z ∩Y |−|X ∩Z| ≥ d(x)+d(y)+d(z)−t(x)−t(z)−|W|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, we can find a set W ′ ⊆ V (G) and a bijection ϕ : W → W ′ such that W ′ = {ϕ(w) : w ∈ W ′}, and for each w ∈ W, we have both N(ϕ(w)) ∩ W = {w} and N(w) ∩ W ′ = {ϕ(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We consider the size of W ′∩(X ∪Y ∪Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since both N(ϕ(w))∩W = {w} and N(w)∩W ′ = {ϕ(w)} hold for each w ∈ W, and we know y ∈ W, we can see |W ′ ∩ Y | ≤ |W ′ ∩ N(y)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose v′ ∈ W ′ ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' There is a vertex v ∈ W such that vv′ is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then we see xvv′ is a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So |W ′ ∩ X| ≤ 2t(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Similarly, |W ′ ∩ Z| ≤ 2t(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Totally, we have |W ′ ∩ (X ∪ Y ∪ Z)| ≤ |W ′ ∩ X| + |W ′ ∩ Y | + |W ′ ∩ Z| ≤ 2t(x) + 2t(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Finally, we get n ≥ |X ∪ Y ∪ Z ∪ W ′| = |X ∪ Y ∪ Z ∪ W ′| + |W ′| − |W ′ ∩ (X ∪ Y ∪ Z)| = (d(x) + d(y) + d(z) − t(x) − t(z) − |W|) + |W| − (2t(x) + 2t(z) + 1) = d(x) + d(y) + d(z) − 3t(x) − 3t(z) − 1, completing the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Now we can finish the proof of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let G be an n-vertex 4-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is easy to see that the minimum degree of G is at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the result in [11], G contains at most n copies of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2, we can get a 2-path xyz with d(x) + d(y) + d(z) − 3t(x) − 3t(z) ≥ 6e(G) n − 9n2 e(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='3, we have 6e(G) n − 9n2 e(G) ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This implies that e(G) < n2/6 + 10n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2 The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2 To show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2, we need some new lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' The coming lemma can be easily obtained by averaging, which says that every graph contains an edge such that the sum of the degrees of its two endpoints is at least twice the average degree of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 8 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Any graph G contains an edge xy such that d(x) + d(y) ≥ 2d(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By Jensen’s inequality, we can get � xy∈E (d(x) + d(y)) = � v∈V d(v)2 ≥ nd(G)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that |E| = (nd(G)) /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus there exists an edge xy ∈ E such that d(x) + d(y) ≥ nd(G)2 (nd(G)) /2 = 2d(G), proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We now give the following lemma about 4-cycles, which can be viewed as a generalization of the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Recall the well-known result of Reiman [9] that any n-vertex graph without containing 4-cycles has at most n 4(1 + √4n − 3) < n 3 2 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Any n-vertex graph G with e(G) > n 4 (1 + √4n − 3) contains a 4-cycle v1v2v3v4 satisfying that d(v1) + d(v2) + d(v3) + d(v4) ≥ 4d(G) − O(n 3 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Fix ǫ := 9n− 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that G must contain 4-cycles by the result of Reiman [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose to the contrary that any 4-cycle v1v2v3v4 in G satisfies d(v1)+d(v2)+d(v3)+d(v4) < 4d(G)−4ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let A := {v ∈ V : d(v) < d(G)} and B := {v ∈ V : d(v) ≥ d(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then A∪B forms a partition of V (G) such that G[B] does not contain any 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For each 1 ≤ i ≤ d(G)/ǫn, let Ai := {v ∈ V : d(G) − iǫn ≤ d(v) < d(G) − (i − 1)ǫn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then these Ai’s form a partition of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' For each 1 ≤ i ≤ (n − d(G)) /ǫn, let Bi := {v ∈ V : d(G) + (i − 1)ǫn ≤ d(v) < d(G) + iǫn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then these Bi’s form a partition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is not hard to check that G[A1] does not contain any 4-cycle, and for each 1 ≤ i ≤ (n − d(G)) /ǫn, G ��i+1 j=1 Aj, Bi � does not contain any 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We delete all edges in G[B], G[A1] and G ��i+1 j=1 Aj, Bi � for each 1 ≤ i ≤ (n − d(G)) /ǫn to get a spanning subgraph G′ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the result of Reiman [9], we can obtain e(G′) ≥ e(G) − (2 + (n − d(G)) /ǫn) n 3 2 ≥ e(G) − 2n 3 2 − 1 9n 7 4 ≥ e(G) − 19 9 n 7 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus we have d(G′) ≥ d(G) − 38 9 n 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that any edge of G′ is either contained in A, or between Aj and Bi for some j ≥ i + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' moreover, we have e(G′[A1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus, as n is large enough, it is easy to check that for any edge xy in G′, dG′(x) + dG′(y) < 2d(G) − ǫn = 2d(G) − 9n 3 4 < 2d(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='4, thus proving Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' 9 The following lemma is derived from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1, which provides an essential structure to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let G be a 4-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose v1v2v3v4 is a 4-cycle in G, and V1, V2, V3, V4 are four sets such that {v2, v4} ⊆ V1 ⊆ N(v1), {v1, v3} ⊆ V2 ⊆ N(v2), {v2, v4} ⊆ V3 ⊆ N(v3), and {v1, v3} ⊆ V4 ⊆ N(v4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let X = V1 ∩ V3 and Y = V2 ∩ V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then there exist sets X′′ and Y ′′ such that X′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ = Y ′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4), e(G[X′′, X]) ≤ |X| and e(G[Y ′′, Y ]) ≤ |Y |, and |X′′| ≥ |X| − 2tG(v1) − 2tG(v3) − 2 and |Y ′′| ≥ |Y | − 2tG(v2) − 2tG(v4) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' As X ⊆ N(v1) ∩ N(v3), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 for k = 4, there exists a set X′ ⊆ V (G) and a bijection ϕ : X → X′ such that X′ = {ϕ(x) : x ∈ X}, and for each x ∈ X, we have both N(ϕ(x)) ∩ X = {x} and N(x) ∩ X′ = {ϕ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We define X′′ := X′\\ (V1 ∪ V2 ∪ V3 ∪ V4), then obviously X′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ and e(G[X′′, X]) ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' As Y ⊆ N(v2) ∩ N(v4), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1 for k = 4, there exists a set Y ′ ⊆ V (G) and a bijection φ : Y → Y ′ such that Y ′ = {φ(y) : y ∈ Y }, and for each y ∈ Y , we have both N(φ(y)) ∩ Y = {y} and N(y) ∩ Y ′ = {φ(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' We define Y ′′ := Y ′\\ (V1 ∪ V2 ∪ V3 ∪ V4), then obviously Y ′′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) = ∅ and e(G[Y ′′, Y ]) ≤ |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then we want to show the last property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' All vertices in V2 are adjacent to the vertex v2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then we have |X′ ∩ V2| ≤ 1 since |N(x)∩X′| = 1 for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Similarly, we have |X′∩V4| ≤ 1, |Y ′∩V1| ≤ 1, and |Y ′∩V3| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' All vertices in V1 are adjacent to the vertex v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Since each vertex in X′ has a neighbor in X ⊆ N(v1), we can check that |X′ ∩ V1| ≤ 2t(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Similarly, we have |X′ ∩ V3| ≤ 2t(v3), |Y ′ ∩ V2| ≤ 2t(v2), |Y ′ ∩ V4| ≤ 2t(v4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Therefore, |X′′| = |X′| − |X′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) | ≥ |X| − 2t(v1) − 2t(v3) − 2, and |Y ′′| = |Y ′| − |Y ′ ∩ (V1 ∪ V2 ∪ V3 ∪ V4) | ≥ |Y | − 2t(v2) − 2t(v4) − 2, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Now we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Throughout this proof, we assume that n is sufficiently large, and the subscripts of the notations such as vi’s and Vi’s are under module 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Suppose by contra- diction that there exists an n-vertex 4-critical graph G with e(G) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='164n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the result in [11], G contains at most n copies of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let V0 := {v ∈ V (G) : tG(v) ≥ √n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Then clearly we have |V0| < 3√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Let G′ := G[V (G) − V0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' It is not hard to see e(G′) ≥ 10 e(G) − n|V0| > e(G) − 3n 3 2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='164n2 − o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that t(G′) ≤ t(G) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Therefore, by deleting at most n edges from G′, we can get a subgraph G′′ ⊆ G′ such that t(G′′) = 0, e(G′′) ≥ e(G′) − n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='164n2 − o(n2), and tG(v) < √n for each v ∈ V (G′′) = V (G) − V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='5 to G′′, we can get a 4-cycle v1v2v3v4 in G′′ such that |V1| + |V2| + |V3| + |V4| ≥ 8e(G′′)/n − o(n) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='312n − o(n), (3) where Vi := NG′′(vi) for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that for each 1 ≤ i ≤ 4, every vertex in Vi ∩ Vi+1 must form a triangle with the vertices vi, vi+1 in G′′, which contradicts the fact t(G′′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So it is clear that Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Also it is easy to check that {vi−1, vi+1} ⊆ Vi ⊆ NG(vi) for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Define X = V1 ∩ V3 and Y = V2 ∩ V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='6, we can get two sets X′′, Y ′′ satisfying the three properties of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that X′′ and Y ′′ are disjoint from V1 ∪V2 ∪V3 ∪V4, V1 ∩V3 = X, V2 ∩ V4 = Y , and Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So we can see that |V1| + |V2| + |V3| + |V4| − |X| − |Y | + |X′′ ∪ Y ′′| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Besides, by using the last property in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='6, we have |X′′ ∪ Y ′′| ≥ max{|X′′|, |Y ′′|} ≥ |X′′| + |Y ′′| 2 ≥ |X| + |Y | 2 − O(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (4) The above two inequalities tell us that |X| + |Y | 2 ≥ |V1| + |V2| + |V3| + |V4| − n − O(√n) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='312n − o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (5) Then we consider the non-edges of the graph G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=', the edges of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' First, since Vi = NG′′(vi) ⊆ NG(vi) and vi ∈ V (G′′), we can see e(G[Vi]) ≤ tG(vi) ≤ √n for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' So e(G[Vi]) ≥ �|Vi| 2 � − o(n2) = 1 2|Vi|2 − o(n2) for each 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Thus by noting V1 ∩ V3 = X, V2 ∩ V4 = Y , and Vi ∩ Vi+1 = ∅ for each 1 ≤ i ≤ 4, we can get ����� 4� i=1 E(G[Vi]) ����� ≥ 4 � i=1 e(G[Vi]) − �|X| 2 � − �|Y | 2 � ≥ 1 2 � 4 � i=1 |Vi|2 − |X|2 − |Y |2 � − o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Next, since G can be made triangle-free by deleting at most n edges and any n-vertex triangle- free graph has at most 1 4n2 edges, we can see e(G[X′′ ∪ Y ′′]) ≤ 1 4|X′′ ∪ Y ′′|2 + n, and thus e(G[X′′ ∪ Y ′′]) ≥ 1 4|X′′ ∪ Y ′′|2 − o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the properties on X′′, Y ′′ we derive from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='6, we can obtain e(G[X′′, X]) ≥ |X′′||X| − |X| ≥ |X|2 − o(n2), 11 e(G[Y ′′, Y ]) ≥ |Y ′′||Y | − |Y | ≥ |Y |2 − o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' By the above three inequalities, we can deduce that e(G) = �n 2 � − e(G) ≤ �n 2 � − 1 2 � 4 � i=1 |Vi|2 − |X|2 − |Y |2 � − 1 4|X′′ ∪ Y ′′|2 − |X|2 − |Y |2 + o(n2) ≤ 1 2n2 − 1 8 (|V1| + |V2| + |V3| + |V4|)2 − 1 4 �|X| + |Y | 2 �2 − �|X| + |Y | 2 �2 + o(n2) ≤ 1 2n2 − 1 8(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='312n)2 − 5 4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='312n)2 + o(n2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='1632n2 + o(n2), where the second inequality comes from the inequality (4), and the third inequality comes from the inequalities (3) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' This contradicts the assumption that e(G) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='164n2, completing the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Our understanding for the functions fk(n) is generally poor, and it is not even known if f4(n) < f5(n) holds for sufficiently large integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' (6) So it seems to be a natural next step to pursue the question that if f4(n) ≤ cn2 holds for some constant c < 4 31 and sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Note that if this is true, then it would imply (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfsP1l/content/2301.01656v1.pdf'} 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